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NOAA Technical Report EDS 1 9
Separation of Mixed
Data Sets into
Homogeneous Sets
Washington D.C.
January 1977
U.S. DEPARTMENT OF COMMERCE
National Oceanic and Atmospheric Administration
Environmental Data Service
NOAA TECHNICAL REPORTS
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Reports in the series are available from the National Technical Information Service, U.S. Department
of Commerce, Sills Bldg., 5285 Port Royal Road, Springfield, Va. 22151. Price: $3.00 paper copy;
$1.45 microfiche. When available, order by accession number shown in parentheses.
ESSA Technical Reports
EDS 1 Upper Wind Statistics of the Northern Western Hemisphere. Harold L. Crutcher and Don K. Halli-
gan, April 1967. (PB-174-921)
Direct and Inverse Tables of the Gamma Distribution. H. C. S. Thom, April 1968. (PB-178-32b)
Standard Deviation of Monthly Averane Temperature. H. C. S. Thom, April 1968. (PB-178-309)
Prediction of Movement and Intensity of Tropical Storms Over the Indian Seas During the October
to December Season. P. Jagannathan and H. L. Crutcher, May 1968. (PB-178-497)
An Application of the Ramma Distribution Function to Indian Rainfall. D. A. Mooley and H. L.
Crutcher, August 1968. (PB-180-056)
Quantiles of Monthly Precioitation for Selected Stations in the Contiguous United States. H. C.
S. Thom and Ida B. Vestal, August 1968. (PB-180-057)
A Comparison of Radiosonde Temperatures at the 100- , 80-, 50-, and 30-mb Levels. Harold L.
Crutcher and Frank T. Quinlan, August 1968. (PB-180-058)
EDS 8 Characteristics and Probabilities of Precioitation in China. Augustine Y. M. Yao, September
1969. (PB-188-420)
EDS 9 Markov Chain Models for Probabilities of Hot and Cool Days Seguences and Hot Spells in Nevada.
Clarence M. Sakamoto, March 1970. (PB-193-221)
NOAA Technical Renorts
EDS 10 BOMEX Temporary Archive Description of Available Data. Terry de la Moriniere, January 1972.
(COM- 72-50289)
EDS 11 A Note on a Gamma Distribution Computer Proaram and Graph Paper. Harold L. Crutcher, Gerald L.
Barger, and Grady F. McKay, April 1973. (COM-73-11401)
EDS 12 BOMEX Permanent Archive: Description of Data. Center for Experiment Design and Data Analysis,
May 1975.
EDS 13 Precipitation Analysis for BOMEX Period III. M. D. Hudlow and W. D. Scherer, September 1975.
(PB-246-870)
EDS 14 IFYGL Rawinsonde System: Description of Archived Data. Sandra M. Hoexter, May 1976.
(PB-258-057)
EDS 15 IFYGL Physical Data Collection System: Descriotion of Archived Data. Jack Foreman, September
1976.
(Continued on inside back cover)
EDS
2
EDS
3
EDS
4
EDS
5
EDS
6
EDS
7
■'WfMT Of *
"^'^^^J^^co^
I
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NOAA Technical Report EDS 1 9
Separation of Mixed
Data Sets into
■Homogeneous Sets
Harold L. Crutcher and Raymond L. Joiner
National Climatic Center
Asheville N.C.
January 1977
>. U.S. DEPARTMENT OF COMMERCE
^ Elliot L. Richardson, Secretary
o
^^ National Oceanic and Atmospheric Administration
Robert M. White, Administrator
Environmental Data Service
Thomas S. Austin, Director
Stock Number 003-019-00036-5 Price $2.45
ACKNOWLEDGMENTS
Appreciation is expressed to the many who have helped us in the prepara-
tion of this paper. Among these are the personnel of the National Climatic
Center's Science Advisory Staff, ADP Services Division, Audio-Visual Services
Section, and the Library Group. Specific acknowledgment is made to Miss Lisa
Green for preparation of the original typescript and to Mrs. Margaret Larabee
for the careful preparation of the final typescript.
Acknowledgment is made to Prof. E. S. Pearson and the Trustees of Biometrika
to use the data and format displayed in figures 2a and 2b. Acknowledgment is
made to University Microfilms, Ann Arbor, Michigan, for permission to repro-
duce or to modify figures as these appear on figures 4, 5, and 6.
Appreciation is expressed to the U. S. Navy and to Dr. John H. Wolfe of the
U. S. Navy Personnel Research and Development Center, San Diego, California,
to adapt and use his NORMIX and NORMAP computer programs and for his ever
ready response to written, telephonic, and personal visit requests.
Acknowledgment is made to the National Oceanic and Atmospheric Administra-
tion for permission to quote material from the Monthly Weather Review.
Acknowledgment is made also to Prof. A, Clifford Cohen, Jr. and to Mr. Lee
Falls for permission to use their computer program which they furnished and
which was adapted to separate two mixed univariate normal distributions.
Appreciation also is expressed to the staff at EDS's Environmental Science
Information Center, in particular to Mr. Patrick McHugh, for the care given
to the editing of this paper.
Mention of a commercial company or product does not constitute an endorse-
ment by the NOAA Environmental Data Service. Use for publicity or advertising
purposes of information from this publication concerning proprietary products
or the tests of such products is not authorized.
n
CONTENTS
Acknowledgments ii
Symbols used in this report xiii
Abstract 1
1. Introduction 1
2. Regression techniques 6
3. Discriminant techniques 10
3.1 Discriminant functions 10
3.2 Factor analysis 15
3.3 Principal component analysis 16
3.4 Multivariate statistical methods 17
3.5 Multivariate logit 18
4. Clustering 19
5. Transformations 21
6. Separation of mixtures 23
6.1 Univariate mixtures 23
6.2 Bivariate mixtures 23
6.3 Multivariate mixtures 24
7. Wolfe - NORMIX 360 computer program 26
7.1 Maximum likelihood estimation 26
7.2 Initial estimation 27
7.3 Significance tests for number of clusters 28
7 . 4 Strategy of use 28
7.5 Usage 28
7.5.1 Storage requirements 28
7.5.2 Restrictions 28
7.5.3 Error messages 28
7.5.4 Input deck 29
7.5.5 Validation examples 29
8. Examples 31
8.1 Introduction 31
8.2 Brief descriptions of data and locations 31
8.2.1 Land-sea breeze data 31
8.2.2 Tropical stratospheric wind data 32
8.2.3 Mid-latitude troDospheric wind data 33
8.2.4 Mountain pass wind data 34
8.2.5 Marine surface data 34
8.2.6 Radiosonde and rawinsonde data 35
m
8.3 Selected data 35
8.3.1 Land-sea breeze data set 35
8.3.1.1 Input information 35
8.3.1.2 Tables - output information 36
8.3.1.3 Figures and discussion '. 36
8.3.2 Tropical stratospheric wind data set 41
8.3.2.1 Input information 41
8.3.2.2 Tables for wind configurations (1954-1964) ... 42
8.3.2.3 Figures and discussion for wind configurations
(1954-1964) 57
8.3.2.4 Tables and discussions for height, temperature
and wind configuration (1957-1967) 76
8.3.3 Mid-latitude tropospheric wind data set 82
8.3.3.1 Input information 82
8.3.3.2 Tables 82
8.3.3.3 Figures and discussion 104
8.3.4 Mountain pass wind data set Ill
8.3.4.1 Input information Ill
8.3.4.2 Tables Ill
8.3.4.3 Figures and discussion 116
8.3.5 Marine surface data set 121
8.3.5.1 Input information 121
8.3.5.2 Tables - output information 121
8.3.5.3 Figures and discussion 129
8.3.6 Radiosonde and rawinsonde data set 134
8.3.6.1 Input information 134
8.3.6.2 Tables and discussion 134
9. Multivariate quality assurance and control 137
10. Prediction 155
Summary 1 56
References 1 57
Author i ndex 1 66
IV
FIGURES
Figure 1. A mixed set of trivariate distributions showing clusters
or modes of varying sizes and shapes 4
Figure 2a. Regression of sons' statures on the fathers' stature 7
Figure 2b. Regression of sister's span for given forearm of brother.. 7
Figure 3. General schematic of points and clusters with implied
correlations, r 8
Figure 4. Schematic illustrations of discrimination for two
cl usters 11
Figure 5. Schematic illustration of discrimination in two-
dimensional form with projection onto plane and onto
one axis for linear discrimination 12
Figure 6. Schematic representation of the discriminate function
when the two bivariate populations, Pi and P2> have
unequal means but equal variances and covariances 13
Figure 7. Schematic "D" illustration of constellations of annual
temperature climates for North American stations 14
Figure 8. San Juan, Puerto Rico, surface wind distributions; period
of record, October 1-31, 1955, October 1-3, 1956, hours
0600 and 0800 and 1200-1400 local standard time; n = 200,
100 from each period; separation shown for two and three
types with covariances assumed equal and then unequal 40
Figure 9. Canton Island, U.S.A. and U.K., upper wind distribution
plots; period of record, the months of July 1954-1964;
pressure levels (a) 50-, (b) 30-, (c) 20-, and (d) 10-mb.. 59
Figure 10. Canton Island, U.S.A. and U.K., upper wind distributions;
period of record, July 1954-1964; pressure levels, 50-,
30-, and 20-mb; wind plot shown in figure 9; separation
shown for two types, assumption of unequal covariances
matrices 61
Figure 11. Canton Island, U.S.A. and U.K., upper wind distributions;
period of record, July 1954-1964; pressure levels, 50-,
30-, and 20-mb; wind plot shown in figure 9; separation
shown for three types, assumption of equal covariance
matrices 62
Figure 12. Canton Island, U.S.A. and U.K., upper wind distributions;
period of record, July 1954-1964; pressure level, 30-mb;
n = 244; wind plot shown in figure 9; separation shown
for four types, assumption of equal covariance matrices... 63
V
Figure 13. Plot of Discriminant Functions 1 versus 2 for Canton
Island, U.S.A. and U.K.; July winds shown in figure 9
for winds shown in figure 11, based on assumption of
equal covariance matrices; period of record 1954-1964;
functions 1 versus 2 are plotted for (a) three types
at 50-mb, (b) three types at 30-mb, (c) four types at
30-mb , and (d ) three types at 20-mb 64
Figure 14. Canton Island, U.S.A. and U.K., upper wind distributions;
period of record, July 1954-1964; pressure levels, 50-,
30-, and 20-mb; n = 263, 244, and 162, respectively;
assumption of unequal covariance matrices: distributions
are shown for (a) total for the three levels, (b) type 1
for the three levels, and (c) type 2 for the three levels. 68
Figure 15. Canton Island, U,S.A. and U.K., upper wind distributions;
period of record, July 1954-1964; pressure levels, 50-,
30-, and 20-mb; n = 263, 244, and 162, respectively,
separation shows two types, assumption of unequal co-
variance matrices; distributions are shown for (a) total
and two 2 types at 50-mb, (b) total and 2 types at 30-mb,
and (c) total and 2 types at 20-mb 69
Figure 16. Canton Island, U.S.A. and U.K., upper wind distributions;
period of record, July 1954-1964; pressure levels, 50-,
30-, and 20-mb; n = 263, 244, and 162, respectively;
separation shows 3 types, assumption of unequal covari-
ance matrices; distributions are shown for (a) three types
at 50-mb, (b) three types at 30-mb, and (c) three types
at 20-mb 70
Figure 17. Canton Island, U.S.A. and U.K., upper wind distributions;
period of record, July 1954-1964; pressure levels, 50-
and 30-mb; n = 263 and 244, respectively; separation
shows 4 types, assumption of unequal covariance matrices;
distributions are shown for (a) four types at 50-mb and
(b) four types at 30-mb 71
Figure 18. Canton Island, U.S.A. and U.K., upper wind distribution
plots; period of record, January 1954-1964; pressure
levels (a) 50-, (b) 30-, (c) 20-, and (d) 10-mb 72
Figure 19. Canton Island, U.S.A. and U.K., upper wind distributions;
period of record, January 1954-1964; pressure level,
50-mb; wind plot shown in figure 18; separation shows for
two and three types with assumption of equal then unequal
covariance matrices • 74
Figure 20. Bivariate distributions of winds at Rantoul , Illinois,
October 1950-1955 at the 700- , 500- , and 300-mb levels.
Two cluster types (1 and 2) are assumed in the total mixed
observed distribution (0.171 + 0.829 = 1.000) 105
vi
Figure 21. Bivariate distributions of winds at Rantoul , Illinois,
October 1950-1955 at the 700-, 500-, and 300-mb levels.
Three cluster types (1, 2, and 3) are assumed in the
total mixed observed distribution (0.182 + 0.183 +
0.635 = 1 .000) 107
Figure 22. Bivariate distributions of winds at Rantoul, Illinois,
October 1950-1955 at the 700- , 500- , and 300-mb levels.
Four cluster types (1, 2, 3, and 4) are assumed in the
total mixed observed distribution (0.094 + 0.193 +
0.293 + 0.421-1.000) 109
Figure 23. Stampede Pass, Easton, Washington, U.S.A.; winds and
temperatures, December 1966-1970, showing breakdown of
winds only into groups 2 and 3 from group 1 (a) and
breakdown of wind- temperature combination into 2, 3,
and 4 groups (b, c, and d) 119
Figure 24. Selected area of North American chart, 1200Z, Monday,
December 3, 1968, NMC analysis. The star represents
the approximate position of Stampede Pass, Easton,
Washington 120
Figure 25. OSV "C" surface distribution of pressure, temperature,
dew point, and wind components, February, 1200 G.C.T.,
1964 through 1972; n = 251 . Covariances are assumed
to be unequal. The 0.25 probability ellipses are
shown for the wind distribution. The total distribu-
tion and the breakout into two clusters are shown 131
Figure 26. OSV "C" surface distribution of pressure, temperature,
dew point, and wind components, February, 1200 G.C.T.,
1964 through 1972; n = 251 . Covariances are assumed
to be unequal. The 0.25 probability ellipses are shown
for the wind distribution. The total distribution and
the breakout into three clusters are shown 132
Figure 27. OSV "C" surface distribution of pressure, temperature,
dew point, and wind components, February, 1200 G.C.T.,
1964 through 1972; n = 251 . Covariances are assumed
to be unequal. The 0.25 probability ellipses are shown
for the wind distribution. The total distribution and
the breakout into four clusters are shown 133
Figure 28. Example of a two-tailed Gaussian filter operating on a
set of heterogeneous data to isolate, set aside, and
eliminate outlying data 141
Figure 29. Distribution of wind standardized components along the
two principal axes of the Canton Island, U.S.A. and U.K.,
July, 30 mb 143
vn
Figure 30a. Schematic illustration of a sample drawn from a homo-
geneous bivariate distribution contaminated by a lone
outlier and two groups of data. The result is a hetero-
geneous distribution. The ellipse shown is a theoretical
0.95 probability ellipse. The lone outlier will be
rejected as not being part of the homogeneous distri-
bution 144
Figure 30b. Schematic illustration of a sample drawn from a homo-
geneous bivariate distribution contaminated by two small
sets. The result is a heterogeneous sample. The lone
outlier of figure 30a has been eliminated as it did not
appear within the 0.95 probability ellipse. Here, the
two contaminating sets exist outside the 0.95 probability
ellipse of this figure and will be eliminated in figure
30c '. 145
Figure 30c. Schematic illustration of a sample drawn from a homo-
geneous bivariate distribution. It exists as a result
of the filtering action of the 0.95 probability ellipses
illustrated in figures 30a and 30b. Here, the 0.95
probability ellipse contains all sample data points of
the remaining group 146
Figure 31. Distribution of wind and temperature standardized com-
ponents along the three principal axis of the Stampede
Pass, Easton, Washington, December 1968-1970 data 147
Figure 32. Distribution of wind, temperature, height, and dew
point standardized components along the 20 principal
axes of the Balboa, C.Z., July data. Four levels are
involved: surface, 850-, 700-, and 500-mb 149
VI n
TABLES
Table 1. Surface wind statistics for San Juan, Puerto Rico,
October 1-31, 1955, and October 1-3, 1956, 0600-0800
and 1200-1400 l.s.t. The assumption is that the
covariance matrices are the same in any breakdown 37
Table 2. Surface wind statistics for San Juan, Puerto Rico,
October 1-31, 1955, and October 1-3, 1956, 0600-0800
and 1200-1400 l.s.t. The assumption is that the
covariance matrices are not equal 38
Table 3. Upper wind statistics for Canton Island, U.S.A.
and U.K. The period of record is the months of
July during 1954-1964. The pressure level is 50-mb.
Sample size is 263. The assumption is that the
covariance matrices are the same 43
Table 4. Upper wind statistics for Canton Island, U.S.A.
and U.K. The period of record is the months of
July during 1954-1964. The pressure level is 30-mb.
Sample size is 244. The assumption is that the
covariance matrices are the same 44
Table 5. Upper wind statistics for Canton Island, U.S.A.
and U.K. The period of record is the months of
July during 1954-1964. The pressure level is 20-mb.
Sample size is 162. The assumption is that the
covariance matrices are the same 45
Table 6. Upper wind statistics for Canton Island, U.S.A.
and U.K. The period of record is the months of
July during 1954-1964. The pressure level is 50-mb.
Sample size is 263. The assumption is that the
covariance matrices are not the same.... 46
Table 7. Upper wind statistics for Canton Island, U.S.A.
and U.K. The period of record is the months of
July during 1954-1964. The pressure level is 30-mb.
Sample size is 244. The assumption is that the
covariance matrices are not the same 49
Table 8. Upper wind statistics for Canton Island, U.S.A.
and U.K. The period of record is the months of
July during 1954-1964. The pressure level is 20-mb.
Sample size is 162. The assumption is that the
covariance matrices are not the same. 52
Table 9. Upper wind statistics for Canton Island, U.S.A.
and U.K. The period of record is the months of
January 1954-1964. The pressure level is 50-mb.
Sample size is 168. The assumption is that the
covariance matrices are the same 54
ix
Table 10. Upper wind statistics for Canton Island, U.S.A.
and U.K. The period of record is the months of
January 1954-1964. The pressure level is 50-mb.
Sample size is 168. The assumption is that the
covariance matrices are unequal , 55
Table 11. January upper air statistics for Canton Island,
U.S.A. and U.K. The period of record is the months
of January during 1957-1967. The pressure level is
30-mb. The sample size is 244. The assumption is
that the covariance matrices are not the same 78
Table 12. January correlation coefficients for data shown in
tabl e 11 79
Table 13. July upper air statistics for Canton Island, U.S.A.
and U.K. The period of record is the months of
July during 1957-1967. The pressure level is 30-mb.
Sample size is 244. The assumption is that the
covariance matrices are not the same 80
Table 14. July correlation coefficients for data shown in
table 13 81
Table 15. A multivariate (6) set of Rantoul , Illinois,
October 1950-55, upper wind components, zonal and
meridional, at the 700-, 500-, and 300-mb levels 84
Table 16. Separation of a multivariate (6) set of Rantoul,
Illinois, October 1950-55, upper wind components,
zonal and meridional mixed distribution, at the
700- , 500- , and 300-mb levels into two separate
distributions 90
Table 17. Separation of a multivariate (6) set of Rantoul,
Illinois, October 1950-55, upper wind components,
zonal and meridional mixed distribution, at the
700- , 500- , and 300-mb levels into three distinct
distributions 94
Table 18. Separation of a multivariate (6) set of Rantoul,
Illinois, October 1950-55, upper wind components,
zonal and meridional mixed distribution, at the
700-, 500-, and 300-mb levels into four distributions 98
Table 19. Surface wind statistics for Stampede Pass, Easton, WA,
U.S.A. The period of record is the month of December
1966-1970. The sample size is 310 taken 155 from each
of the local standard time hours 0700 and 1300. The
assumption is that the covariance matrices are not the
same 112
Table 20.
Table 21
Table 22,
Table 23,
Table 24.
Surface temperature and wind statistics for Stampede
Pass, Easton, WA, U.S.A. The period of record is
the month of December 1966-1970. The sample size is
310 taken 155 from each of the local standard time
hours 0700 and 1300. The assumption is that the
covariance matrices are not the same 113
Marine observations from Ocean Station C. Februaries
12Z 1964 through 1972. Sample size 1s 251. Number
of variables is 5. Number of types is 2 122
Marine observations from Ocean Station C. Februaries
12Z 1964 through 1972. Sample size is 251. Number
of variables is 5. Number of types is 3 124
Marine observations from Ocean Station C. Februaries
12Z 1964 through 1972. Sample size is 251. Number
of variables is 5. Number of types is 4 126
Means and standard deviations for the total set and
clusters 1 and 2 of the data for Balboa, C.Z. These
data are the pressure (or height), temperature, dew
point, and the u and v components of the wind at the
surface, 850-, 700-, and 500-mb levels. The dimensions
are 20. Equal covariance matrices are assumed for types
1 and 2 136
Table 25,
Separation of standardized transformed components along
the major axis of the distribution of the Canton Island:
U.S.A. and U.K., winds at the 30-mb level during the
Julys 1954-1964. The sample size is 244. There are no
dimensions in terms of units. The assumption is that
the variances are not the same ,
153
Table 26.
Separation of standardized transformed components along
the major and minor axis of the distribution of the
Canton Island, U.S.A. and U.K., winds at the 30-mb level
during the Julys 1954-1964, The sample size is 244.
There are no dimensions in terms of units. The assumption
is that the variances are not the same.. 154
XI
Symbols used in this report
d difference; deviation
d.f. degrees of freedom
f function
ft. feet
gdkm geodynamic kilometer
i subscript or superscript
j subscript or superscript
k kth point in a sample; number of clusters; kth cluster
km kilometer
m mth item; meter
mb pressure in millibars
mi. miles
n nth item; number in a sample
r sample correlation coefficient
s sample standard deviation; second; cluster
s^ sample variance
t Student's "t"
X variate
X' variate transpose
y variate
C covariance matrix; Celsius
F function
G.C.T. Greenwich Civil Time
P probability
P(S|X. ) probability of membership of X. in the cluster s
R correlation matrix
|R| determinant of the correlation matrix R
S type; cluster type
X observed value of variate
X mean of variate X
X|^ vector of observations for the kth point in the sample
Y observed value of variate
Y mean of variate Y
xm
a alpha; proportionality factor (^5); probability level of rejection
for the null hypothesis
A lambda hat; mixing proportion for type cluster s
A lambda; the diagonal matrix of eigenvectors
y mu; population mean
y mu hat; mean vector for cluster s
IT pi
p rho; population correlation
a sigma; population standard deviation
a^ population variance
a sigma hat; covariance matrix for cluster s
z sigma; summation
ij; psi
hat (caret)
equal to
^ approximately
overbar; averaging process
transpose
XIV
SEPARATION OF MIXED DATA SETS
INTO HOMOGENEOUS SETS
Harold L. Crutcher and Raymond L. Joiner
National Climatic Center
Environmental Data Service, NOAA
Asheville, N.C.
ABSTRACT. In any study, the collection, processing, and
storage of data are important. Whether the data are clean,
biased or contaminated is also important. Pollution or
adulteration of data confuse the investigator.
Data do not necessarily fall into neatly packaged boxes or
groups. Usually the data sets are mixtures of several
types of phenomena. Some of these are basically determin-
istic in nature while others are not.
This paper illustrates the use of a clustering technique
to separate mixed data sets into subsets which exhibit
group characteristics. The investigator then assesses the
relative importance of the subsets, the nature of the sub-
sets, and perhaps makes an assumption as to whether a
particular subset is biased, contaminated, or adulterated.
That is, an assessment of the quality of the data may be
made.
The techniques are applicable to any data set which is
multivariate normal. Here, they are applied to weather
data subsets, (1) land-sea breeze, (2) tropical strato-
spheric winds, (3) mid-latitude tropospheric winds, (4)
mountain pass winds and temperatures, (5) surface marine
weather temperatures, dew points and winds, and (6) radio-
sonde observation of heights, winds, temperatures, and dew
points.
1. INTRODUCTION
Prehistoric man differentiated between the good and the harmful, between
winter and summer, between drought and floods, and among many variable
factors affecting him.
In the real world, the experienced hunter knew the differences between the
bear, boar, deer, and turkey. His senses of sight, smell, and hearing aided
him when he could not see the animal, but he could sense its presence. When
he had some models in mind, he drew symbols or wrote words for the benefit of
the inexperienced. He could even provide measurements of a sort. At this
stage of the game, he began to deal more with the abstract.
Later, man attempted to record experiences, his thoughts, and his aspira-
tions in pictographs and monuments. (Even today, artists present forms in
1
symbolic representation or as configurative concepts.) Undoubtedly, many
numbering systems had also been developed and were then lost in antiquity,
for there are some which remain indecipherable today.
The above examples may be generalized to any field. Clear-cut numerical
descriptions and configurations remain important whether the field be sociol-
ogy, psychology, geophysics or medicine, four of an infinite number of fields.
In this paper we illustrate some of the techniques used to differentiate
groups within sets of given measurements. Hopefully, the measurements are
both accurate and precise. Also, hopefully, the measurements obtained and
used are those which will provide good differentiation bases.
As Friedman and Rubin (1967) quote from Bose and Roy (1938): "The problems
of discrimination and classification are insistent in sciences."
Once we enter the realm of measurements and numbers in multidimensional
space, assumptions and decisions are made as to the better or best character-
istics. The metrics that can be used are many. Those chosen ought to provide
the most useful differentiation possible.
There are many techniques and procedures used to differentiate groups.
These usually involve some measures of central tendencies within the groups,
some measures of differences of these central tendencies, variability within
and among the groups, group shapes, and scales, etc. As more is known about
the normal distribution than other distributions, it is wise, wherever possi-
ble, to transform non-normally distributed data sets to data sets which may
be approximated by the normal distribution. Transformations have been of
interest for many years. In fact, these are represented in the change of
base in many counting systems. For example, the logarithm transformation
changes a zero bounded positively skewed distribution to an unbounded distri-
bution at both ends where the values more distant from one are scaled down-
ward faster than those nearer one.
If the various features of a data set are each transformed to normal or
near normal distributions, then the combined features may be multivariate
normal. A multivariate normal distribution has normal marginal distributions.
However, the fact that the marginal distributions are normal does not assure
multivariate normality. If multivariate normality is assumed, then proba-
bilistic statements can be made. Transformation will be discussed in greater
detail later in the paper.
If multivariate normality is assumed and the distribution has one centroid,
i.e., it is unimodal , then multivariate regression techniques can be used to
produce forecast equations. If the distributions are normally distributed
in the multivariate sense but the total set is multimodal, then ordinary
linear regression techniques will not serve. Regression equations for clus-
ters must be developed and the better sets of regression equations clustered
and examined. The unique or best equation or set of equations can be used.
If there is a best equation, it is unique.
Figure 1 illustrates a multimodal trivariate distribution. The various
clusters take various ellipsoidal forms such as spheres, ellipsoids, and
disks. Each one of these is trivariate normal. Each one represents a
centroid or grouping of characteristics.
Some may be equal in all directions, such as in the spheres. Some may be
equal in two directions but not the third, such as in the football- and disk-
shaped ellipsoids. In others the distribution may be unequal in all direc-
tions. In figure 1, no scales are indicated as the illustration is for
concept only. The illustration also could be considered as a higher dimen-
sional ensemble projected onto three dimensions. When multimodal features
are evident, and if one wishes to study the modes or clusters, then tech-
niques other than regression are required to separate the data into appro-
priate subsets.
There are many techniques to separate distributions into parts which can be
studied individually. Some of these are:
(a) discriminant function analysis
(b) factor analysis
(c) principal cluster analysis
(d) dendritic (tree) analysis
(e) cluster analysis
(f) clumping
(g) numerical taxonomy
(h) unsupervised pattern recognition
(i) typology
Some of these are essentially the same.
Mixtures always present problems when they must be separated. Characteris-
tics may or may not be so noticeable that classification and discrimination
can be made. A mixed herd of cattle, sheep, goats, and horses may be easily
separated though the sheep and goats may sometimes present a few problems.
The herdsman, the separator, or the investigator must have a clear picture in
his mind, i.e., a model which includes the necessary characterization(s) of
the populations in which he is interested. If he doesn't have his senses of
sight, smell, hearing and touch to guide him (for these provide him explicit
models) and he has only some measurements of the mixture provided to him, he
is in a quandry. He no longer has an explicit model. He has to start some-
where. Good (1965) discusses the philosophical problem of deciding what can
be the best beginning in the problem of classification and discrimination.
First of all, t>'e investigator decides to accept the characterization of each
object by a set of measurements. He believes that there should be some cate-
gories or sub-categories which will be helpful in distinguishing group char-
acteristics. Explicitness is lost and there is no external criterion with
which to define the categories. An internal criterion (Good, 1965) would be
acceptable. That is, the data themselves may suggest "natural categories."
The word "suggest" is necessary, for a different beginning in the treatment
of the data may lead to slightly different categories. This item will be
treated in more detail later.
Figure 1 A mixed set of trivariate distributions showing clusters or modes
of varying sizes and shapes. These are viewed from four different
points in 3-space. In each view, the three axes and a suggested
line of best fit are indicated.
The next few sections take the reader through a short discussion of regres-
sion techniques and through some of the various techniques used to provide
classification and discrimination within heterogeneous mixtures.
2. REGRESSION TECHNIQUES
Inevitably, due to inherent laziness or an inherent desire to get to a goal
with the least expenditure of energy, physical or mental, one attempts to read
relationships into sets of experiences or of data. This. is a canon of science.
For example, if one knows that a certain thing will happen provided that some-
thing else specific is done, then there is a clear-cut and obvious relation-
ship between the reaction and the prior action. Though the converse may not
be true, it is disregarded here.
Galton (1889) and Snedecor and Cochran (1967) noted the height of sons as
related to the heights of the fathers. With the heights of the fathers as
one set of data and the heights of the sons as a second set of data, Galton
plotted the heights of the fathers against the heights of the sons, respec-
tively. He noted that tall fathers did not always produce as tall or taller
sons but that the height of a son seemed to be oetv/een the height of the
father and the mean height of the group. That is, the height of a son re-
gressed towards the group norm. The line of best fit for the data set was
then and since then labeled the line of regression for any line relationship
between data sets. Pearson and Lee (1903), Galton's associates, collected a
set of data (more than a thousand) of stature, cubit, and span in family
groups.
Figure 2a shows the regression of sons' statures on the fathers' statures.
Please note that the heights of the sons of short fathers also tend (or re-
gress) toward the mean. This figure is taken from Pearson's and Lee's data
as illustrated, but is also shown by Snedecor and Cochran (1967). The data
are scaled in the metric system here. Also, note that the data scatter uni-
formly along the line and do not cluster.
Figure 2b illustrates the relationship between a sister's span for a given
forearm length of her brother. Both figures are adapted from Pearson and Lee
with the kind permission of the Trustees of Biometrika.
Figure 3a illustrates schematically the correlation at one point. The
correlation may be said to be perfect on the one hand, but also it can be
said to be indeterminant.
Figure 3b illustrates the correlation between two points. The correlation
may be said to be perfect on the one hand but for the space in between the
points, on the line connecting the points, it may be said to be indeterminant
or even zero.
Figure 3c illustrates a correlation of one with three col linear data.
Figure 3d illustrates the case of two clusters each with zero correlation,
one superposed on the other.
Figure 3e illustrates the case of two clusters shown in figure 3d where the
clusters are slightly separated with cluster 2 moving away on a line of 45
degrees. The cc'^relation coefficient is something greater than zero but much
less than one; .e., 0<r<<l .
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Figure 3f illustrates the case of two clusters shown in figure 3d where the
clusters are separated still further with cluster 2 moving farther away on
the 45 degree line.
Figure 3g illustrates the case of two superposed clusters. One with a
correlation of plus one, the other with a correlation of minus one. The
total cluster has a correlation of zero.
Figure 3h illustrates the case of two clusters one with a correlation of
plus one, the other with a correlation of minus one. The second cluster is
moving on a 45-degree line from the first. The distance between cluster
centroids is the same as in figure 3c. The group correlation is greater than
zero but less than that shown in figure 3e.
Figure 3i illustrates the case of the two clusters shown in figure 3g
moving along the 45-degree line farther than shown in figure 3h. The corre-
lation is increasing and the coefficient approaches one but not as rapidly as
in figure 3f as the distance between the centroids increases.
The above discussion implies that there may be two or more clusters in any
data ensemble. The scatter may be more in one cluster than in another. Also,
the internal correlation (or dispersion) within each cluster may be the same
or may be different from the other clusters. The total correlation attained
thus may be more of the relationship among the clusters than points. As the
distance between the centroids increases, the clusters are more like singleton
points in the regression analyses. If there is a clustering, i.e., effec-
tively a dearth of observations between groups either real or simply unob-
served, then regression analysis whether linear or non-linear will fail.
Please refer again to figure 1 which represents clusters in 3-space or clus-
ters in n-space projected into 3-space. The representation here is multi-
normal. This may not be the case sometimes and some clusters or all clusters
may be eccentric in shape such as eggs, starch grains, or bivalve shells.
If the variates are not all normally distributed, it is assumed that the
user will transform the variates to normal variates. It is helpful to know
by means of the Central Limit Theorem that, though individual data character-
istics may not be normally distributed, linear functions of these tend to be
normally distributed. Also, it is assumed that the user will extract the
deterministic part of the variables wherever possible. Therefore, the prob-
lems of nonlinear regression are not discussed here.
The point of the entire discussion above is simply that linear regression
analysis ought to be used only with a unimodal univariate or multivariate
distribution. Linear predictor equations developed in linear regression
models then become more useful and accurate.
Gupta and Sobel -(1962) consider this problem. Aversen and McCabe (1975)
present some subset selection problems for variances associated with applica-
tions to regression analysis.
3. DISCRIMINANT TECHNIQUES
3.1 Discriminant Functions
Many workers realized the problems induced by heterogeneous or mixed dis-
tributions. Among the first to attack the problem in a systematic mathemati-
cal treatment was Pearson (1894, 1901) in work on univariate distributions.
Many others also have worked on this problem.
Barnard (1935) and Fisher (1936) may be considered to have first attacked
the problem of classification with discrimination techniques, though the
problems of classification had involved many other workers up to that time.
Let us look at a two cluster mixture from the viewpoint of separation or
discrimination with subsequent rules for classification. Figure 4 shows an
assemblage composed of two clusters. The clusters are shown first within a
dashed circle with no axes chosen (a). Suppose that the measurements are
made in terms of the x-axis drawn horizontally (b).
Projections of the cluster points are on the x-axis and in (d) on the y-
axis. Visually there is separation in the (x, y) space or two space. This
separation can also be shown in one space, i.e., linearly. In one space or
one dimension chosen first on the x-axis, there is mixture of the projections.
In (c) where the x-axis is rotated through an angle to x", there is some
separation but two points indicate some mixing. In (d) where the rotation
has been carried through 90 degrees so that the x' axis is equivalent to the
former y-axis, separation is complete though perhaps not the optimum. There
is some angle of rotation which will produce a major separation between the
clusters and a minimum variance within the clusters. The computed linear
function which describes the above line after rotation is called the linear
discriminant function. Brown (1947) applies these techniques to establish
the discriminating procedures for azotobacter, the nitrogen-fixing bacteria.
Smith (1947) provides some discrimination examples. Crutcher (1960) applies
the techniques developed by Rao (1950) to the annual march of temperature and
rainfall climates in the United States. Figure 5 (adapted from Crutcher,
1960) shows a two cluster (bimodal) bivariate distribution with the two di-
mensions shown in three-dimensional form. These three-dimensional forms are
projected into two dimensions on the xz plane. Linear discrimination is
effected along the single axis pointing to the lower right (xy plane). Figure
6 illustrates the same idea with all projections being made onto the xy plane.
Tatsuoka (1971) presents similar ideas to illustrate geometrically how the
discriminant function operates.
Figure 7 (Crutcher, 1960) shows three constellations for monthly average
temperature galaxies where the basic variability of the constellations is
different. The covariance within a galaxy is the same where the circles pro-
vide a measure of individual cluster variances and the distances between
clusters is a measure of cluster variance. A point indicates one station
only. Miller (1962) applies the technique to weather prediction. Applica-
tions as indicated by Fix and Hodges (1952) ran into the hundreds.
10
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projections on an arbitrary x-axis, (c) with projections on an angle
from the x-axis, and (d) with projections on an axis, y-axis, tt/2
radians from the x-axis.
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climates for North American stations. The constellations and
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Dispersion matrices within constellations are not significantly
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14
The works of Hotel ling (1931) and Mahalanobis (1936) are related closely
to Fisher's (1936) discriminant criterion. Study of these papers provides
excellent background. Hotelling describes these relationships in 1954.
Mahalanobis' work in a sense deals with a set of multi-dimensional data for
which the ensemble variance is reduced to one, i.e., the total ensemble is
standardized. Euclidean measurements then are used.
There are other techniques to categorize and classify data. All of these
require some transformation of data, the selection of those measurements
which provide the greatest amount of information, and the elimination of
those that provide the least if these may be termed unimportant. Usually,
their importance is assessed in how much they contribute to the overall
variability.
3.2 Factor Analysis
The basic mean structure and variance-covariance structure, i.e., the
matrices of means and covariances, are the bases for factor analysis and
principal component analysis. It is in the details and interpretation, some
complete and some incomplete, of the internal structures that the techniques
differ.
The term "factor analysis" comes from factoring procedures and techniques.
There are many variants each of which is a special case of the general method
of independent dimensional analysis (Tryon, 1959, Tryon and Bailey, 1970).
Briefly, as with the discriminant function analysis described previously, the
correlation (or covariance) matrix is the initial starting point. The methods
used to extract information from this matrix are many and varied. Some are
more complex than others. Some use weighting schemes either based on "a
priori" knowledge and experience or physical constraints and bases. Some
simply attempt to let the matrix itself determine the factors. All this leads
to some confusion and some competition among the proponents of the various
systems.
Multiple factor analysis with rotation to simple structure is the usual
procedure in factor analysis.
The purpose of factor analysis is to explain the matrix of covariances of a
multidimensional set by the least number of hypothetical factors. The corre-
lation matrix is used. First of all, the matrix is examined to determine
whether it is significantly different from zero (the identity matrix). If so,
the technique then identifies, extracts, and weights proportional amounts of
the correlation until the residual matrix is not significantly different from
zero. Factor analysis stems mainly from the initial work of Spearman (1904,
1926). Essentially, the technique serves to study the similarities in a set
of data.
Other pertinent references on factor analysis techniques are Thurstone
(1947), Kendall and Smith (1950), Cattell (1952), Bartlett (1953), Fruchter
(1954), Harman (1967), Sokal and Sneath (1963), Lawley and Maxwell (1963),
Mulaik (1972), and Anderson and Rubin (1956).
15
3.3 Principal Component Analysis
Karl Pearson (1901) first proposed an empirical method for the reduction
of a large body of data so that a maximum of variance could be extracted.
Hotel ling (1933) developed this method fully as the principle component
method. This under some conditions is identical to the discriminant function
[Kullback (1968), Anderson (1958), and Girshick (1936)].
For example, principal components according to Anderson (1958) are linear
combinations of random or statistical variables which have specified proper-
ties in terms of variances. The first principal component is the normalized
linear combination with maximum variance. In a swarm of points, each point
representing an n-component vector, the distribution may be ellipsoidal.
Unless the distribution is spherical, there will be an axis which is as long
or longer than any other axis. This axis is called the major or principal
axis. A plot of these points will reveal the ellipsoid. It is difficult to
represent such an ellipsoid in more than three dimensions. Measures of the
components may not directly reveal the ellipsoidal nature of the swarm. How-
ever, the variance-covariance or correlation matrix may be rotated in space
such that new axes are obtained where the components, projections, or linear
functions along these new axes are not correlated. The directions of these
axes are known variously as characteristic vectors, latent vectors, direction
cosines, or eigenvectors. If there are observations in n-dimensions, each
observation as an n-dimensional vector may be projected onto each of the n-
orthogonal (mutually uncorrelated) axes obtained above. The variability of
these components along each axis may be determined. These variances are known
respectively as characteristic roots, latent roots or eigenvalues. The sum
of these is the total variance or trace of the matrix. Their product is the
determinant. Each one divided by the total provides the proportional amount
of variance contributed by the components along each axis. The largest axis
is the major axis. The principal axes are those axes whose sum, from the
largest in sequence to the next largest, accounts for all or some preselected
or specified proportion, say 95 percent. It is the components along each axis
that are used. Thus, it is possible to reduce a large dimensional problem to
a small dimensional problem. It is also possible to select from the same data
a dimensional subset (n-1 or less) to eliminate that (those) dimension(s)
which contribute(s) little to the total overall variability.
There are tests to determine which principal components may be considered
to be significantly different from the others if indeed they are. Hotel ling
(1933) and Bartlett (1950, 1951) provide subtests. Mulaik (1972) presents a
good discussion of this problem. However, with all this, one must heed the
advice of Hotelling (1957) that there may be difficulties involved in neglect-
ing one principal axis even though it is the least in variance. Such an
omission may change radically the multiple correlation used in regression.
Only adequate testing will reveal whether an axis may be neglected.
In addition, the components along any one axis may be checked for evidence
of heterogeneity or mixtures. If multimodal ity is present on any principal
axis, then separation or clustering can be done for these data.
The principal component analysis is distinguished from the factor analysis
16
in that the principal component analysis studies the data structure from the
viewpoint of differences rather than similarities.
Any degeneracy which exists in a multivariate distribution is revealed in a
principal component analysis [Tatsuoka (1971)]. Therefore, principal compo-
nent analysis perhaps ought to be considered to be a first stage in factor
analysis, though this does not meet the views of the proponents and the oppo-
nents of the two techniques.
3.4 Multivariate Statistical Methods
As mentioned previously, the ways in which factor analyses are developed
and used lead to newer systems with their respective proponents and opponents.
This is so because the data sets must be discussed on the basis of each set.
From set to set the bases may be different. Hotel ling (1936a & b, 1957)
discusses the relations between two sets of variates, simplified calculation
of principal components, and then the newer multivariate statistical methods
to factor analysis.
Up to the time of Hotelling's (1957) paper, factor analyses of the usual
kinds were often inferior to other procedures. Unless the research worker
determines and uses (an) invariant statistic(s), the results always will be
difficult to assess. However, as in all investigative work of this nature,
these analyses may have only heuristic or suggestive value. Hypotheses may
be exposed which may be better tested by other methods.
In examination of the use of statistics in the various scientific fields,
it is readily apparent that each field develops its own names and "jargon"
for statistical terms within their specialized fields. Also, workers trying
to reach a common goal will independently develop similar techniques. Most
such techniques are beset with "nuisance" parameters induced by an arbitrarily
chosen statistic. As Hotelling (1957) points out, it is the invariance of the
multiple correlation coefficient as deduced by Fisher (1928) that illustrates
the possibility of eliminating the "nuisance" parameters which can only be-
cloud the issue. This puts a premium on the use of invariant statistics.
Student's (1925) "t" distribution is well known for its use to test the sig-
nificance of means and difference of means and to establish confidence inter-
vals with known probabilities. Its wery usefulness in the one dimensional case
led to much work towards generalizing this procedure to the multivariate case.
Hotelling (1933) in his work on principal components led to such a generali-
zation in the "T^" test. The value of "T^" is invariant under all non-
singular linear transformations among the variates. The trace or the sum of
the diagonal variances of a covariance matrix is invariant. The "T^" distri-
bution is a beta distribution or a variance ratio distribution.
The "D2" stability of Mahalanobis (1936) is closely allied to the "T^"
stability of Hotelling (1931). This also was an attempt to arrive at the use
of invariant statistics. Wilks (1932) considered generalization in multi-
variate analysis. Roy (1939, 1942a & b), Hsu (1938), Bartlett (1950), and
Bose and Roy (1938) did considerable work in the field of multivariate analysis
17
In the application of principal components in factor analysis, all compo-
nents must be used. The exclusion of even the smallest component may lead to
a complete change in the obtained functions to such an extent that interpreta-
tions or decisions may be quite erroneous.
Ridge regression is mentioned as an evolving technique which should receive
the attention of some readers [Hoerl (1962), Hoerl and Kennard (1970a, b), and
Bannerjee and Carr (1971)]. From the viewpoint of response surfaces, Davies
(1956), Draper (1963), and Myers (1971) may be consulted. Though these have
their impact on the problems of clustering and classification, they are not
discussed further.
3.5 Multivariate Logit
Distribution of quantal responses to drugs or poisons may be better de-
scribed by distributions other than the normal. The logistic curve is one of
these. As Kendall and Stuart (1968) state, "The Probit and Logit transforma-
tions of percentages, respectively to normal and logistic distribution devi-
ates, arise mainly in biological contexts and are discussed by Finney (1952).'
It has not found wide application in the geophysical field. Anderson (1958)
and Dempster (1973) discuss the Logit model. This would be most appropriate
if the mixture is composed of a set of variables one or more of which would
be considered fixed while the others are normal. This is beyond the scope of
the present paper as we here utilize only those distributions which from
experience are known to be approximated by the normal distribution. The use
of the multivariate Logit distribution mixture separation must be deferred to
later research. Hopefully, this will be within the next five years.
18
4. CLUSTERING
Grouping of individuals with similarities and separation of individuals with
dissimilarities is a continual process. This process is evident from the
smallest to the largest features in any ensemble whether it be in the universe
in the development and dissolution of galaxies, in the earth in its geologic
processes, or in life on earth in whatever form it may be. It is evident in
the abstract world, too. Readers of Aristotle and the followers of Hippo-
crates and Linnaeus will recognize attempts to order chaos in the sense of
clustering and differentiation, whether it be the assignment of some attribute
or the recognition of some measurable quality or quantity. Much has been done
and much has been written. The techniques of clustering and differentiation
are as varied as those who attempt to perform those functions. The logic and
the metrics may vary but the insistent theme is to group those that are alike
and separate those that are unlike. Each individual is, of course, an entity
in itself. A set of measurements on that individual undoubtedly constitutes
a group in itself. This much is acknowledged by any worker. The job is to
group those individuals together whose sets of measurements are not too far
different and to put aside those whose measurements are different at some
level of perception and thinking to another group or groups. Much sometimes
depends on the individual whose measurements are used as the starting point.
In a sense, those measurements of the first individual are in an "a priori"
sense weighed most heavily in the clustering process. Coalescence begins on
this individual.
Clustering is simply another term among the many in the general field of
taxonomy. Taxonomy is the scientific ordering and classification of informa-
tion. Taxonomic systems must be simplified representations of group char-
acteristics and all their interrelationships. Because these are most general
and therefore quite usable, perfection cannot be attained except in the case
of an individual. The main function of any such system is to reduce the
complex problem to a simpler problem, thus reducing the requirements of memory.
References, some of which have been given previously, are Sokal and Sneath
(1963), Tryon and Bailey (1970), Duda and Hart (1973), Fisher (1936), and
Anderson (1958). Hartigan (1975) offers a considerable number of clustering
algorithms.
Many terms are used in the grouping concept. Some are more suggestive of
the process than others. These terms depend on the field of endeavor and on
the basic background of the investigators. Some of the techniques are the
same though the words are different. Let us look at a few of them: clusters,
clumps, coalescence, condensations, aggregations, agglomerations, divisions,
similarities, cohesions, linkages, hierarchies, communal i ties, types, and
affinities. The classes obtained may be given names in the specialized fields.
In the biological sciences these are the families, genera, species, subspecies,
etc. In the field of the natural sciences, though not restricted to these
fields, these may be clusters, universes, constellations, galaxies, etc. In
the latter example, the order may be interchanged depending on the viewpoint
of the problem.
In any taxonomic problem dealing with measurements, there must be some way
to collect the like individuals together and to isolate, reject, and form new
19
groups with those which are most alike among the unlike. Some measures of
likeness or similarity must be developed and accepted. As individuals are
collected into small groups and as the smaller groups are collected into
larger and larger ensembles, a tracing of the procedures may be kept. As
diagrams or the concepts of the trace resemble some feature of the known
world, these may be called tree-diagrams (branch diagrams), root diagrams
(dendrodiagrams or dendritic diagrams), link diagrams, or hierarchical dia-
grams. These may be shown in two or three dimensions or left in abstract form,
Baker and Hubert (1975) consider procedures to measure the power of hier-
archical cluster analysis. This important feature in studying the various
techniques and their alternatives is not examined here in further detail.
The clustering procedures are allied closely to the previously discussed
techniques of factor analysis, principal component analysis, and discriminant
function analysis. Canonical analysis also may be used. The last is simply
the techniques which maximize the correlations among linear functions of the
data.
The specific clustering techniques used here are discussed in more detail
in Wolfe's (1971b) NORMIX program, section 7.
20
5. TRANSFORMATIONS
The multivariate normal distribution is only one of an infinite number of
useful multivariate distributions. The univariate normal distribution has
been exploited more than any other because the necessary statistical tools
have been developed. It is nearly so with the multivariate normal. If a
distribution is multivariate normal, then any of the subspace marginal dis-
tributions are normal. Though the normality of all marginal distributions
does not guarantee multivariate normality, such normality is a necessary
condition for multivariate normality.
If prior experience indicates that a certain measure is usually normally
distributed and tests imply non-normality, then one inference that can be
made is that the data set is mixed. It is precisely the thrust of this re-
port to separate such mixed distributions into their homogeneous parts. If
any marginal distribution is mixed, then certainly any higher order is mixed
or any lower order variate distribution is or may be mixed. If the set of
projections onto any principal axis is mixed (multimodal) and the set of
projections is a linear function of the multivariate set, then this set can
be used to establish estimates of the parameters of the various homogeneous
collectives.
With normality of distribution, the usual tests of significance or tests of
hypothesis may be made. Otherwise, decisions based on such tests may be
invalid or questionable. The robustness encountered, though, usually is such
that a lack of normality does not impair the decisions yery much. Far more
important perhaps is the practical significance of the decision.
If it is known that an unmixed marginal distribution is not normal, then
some transformation towards normality of that distribution should be sought.
A marginal distribution is a subset of the higher order distribution. If the
marginal distribution itself is not normal in the multivariate sense, then
the still lower order marginals or smaller subsets should be checked. It may
be that only one of the lowest order marginals (a one dimensional distribu-
tion) is the non-normal. If this is the case, then transformation of this
distribution should be sought.
The above does not imply that it is impossible to use the non-normal dis-
tributions. It does imply that the distributions must be normally distributed
or mixed normal in their distribution if the techniques discussed in this
paper can be used to provide valid results.
There may be some cases where appropriate normality cannot be achieved
through any transformation (Graybill, 1961, pg. 318).
For those distributions that can be normalized, the transformation process
is often carried forward in graphical procedures. Boehm (1974) uses this
procedure and has an electronic computer program to provide such trans-
normalization.
The methods of factor analysis, principal components, and discriminant
functions are techniques to linearize a multidimensional situation to a
21
single univariate situation. Hopefully the final distribution or distribu-
tions in the components, factors, or clusters will allow linearization within
these to the univariate problem.
There are many tests for univariate normality. Some of these (Graybill,
1961) are (1) likelihood ratio tests associated with transformation toward
normality, (2) skewness and kurtosis tests, (3) omnibus tests, and (4) normal
probability plots.
A useful transformation to improve normality is that proposed by Box and
Tidwell (1962) for estimating a shifted power transformation (x + ?)^ of a
single variable. This is a generalization of many tests developed through
the years [Tukey (1957) and Moore (1957)]. Simultaneous transformation of a
multiple set of variables has been the subject of much research. Andrews et
al. (1971) discuss the problem in more detail for the bivariate distribution.
Extensions to the multivariate case are indicated.
Transformations are not discussed further here as the climatological
examples used in this report do not require transformation to normality. The
clustering technique used here will be applied to other climatological and
geophysical problems. Many of the data distributions of the future study
will require transformation to normality. The two techniques described above
then will be treated in detail.
22
6. SEPARATION OF MIXTURES
6.1 Univariate Mixtures
Karl Pearson (1894), in his paper on "Contributions to the Mathematical
Theory of Evolution," gives the procedure to mathematically dissect a mixture
of two univariate normal distributions. In the general case the five param-
eters to be estimated are two means, two variances, and a proportionality
factor. It is necessary to find a particular solution of a ninth degree
polynomial equation, a nonic.
The following list is neither exhaustive nor the most important but it does
provide sufficient references for the reader to pursue the subject further:
Charlier (1906), Charlier and Wicksell (1924), Burrau (1934), Stromgren (1934),
Essenwanger (1954), Schneider-Carius and Essenwanger (1955), Cohen (1965),
and Cohen and Falls (1967). Essenwanger's work is applied to meteorology
while Cohen's and Fall's work provides electronic computer routines for dis-
section of heterogeneous mixtures.
Hald (1952) discusses the subject of heterogeneity in the univariate case.
Graphical as well as analytical techniques are discussed. The general case
may be written as
m
f(x) = E a^. f^. (x); la. = 1 ,
i = i
where
fi(x) = (2710^.)"'-^ exp{-[(X-vi^.)/a^.]^2}
Even for a mixture of two distributions the solution of the nonic is a big
task. Cohen and Falls (1967), using procedures developed by Charlier and
Wicksell (1924) and a modern electronic computer, provide procedures to
effect the requisite dissection and to obtain estimates of the parameters.
Discriminant criteria can then be developed to enable classification of a new
measure to one of the two groups or to classify each datum from the original
data set.
The computer program, kindly lent to the authors by Cohen and Falls, has
been used to dissect a mixed distribution in Section 9.
6.2 Bivariate Mixtures
Hartley (1959) provides techniques to separate two mixed bivariate normal
wind distributions. Crutcher and Clutter (1962) apply these to wind distri-
butions but encounter difficulty when the covariance matrices are singular or
near singular or when the variances or covariance matrices are assumed to be
different.
In the bivariate case of a unimodal distribution there are five parameters
to be estimated. These are the two means, the two variances, and the corre-
lation. When there is a mixture of two bivariate normal distributions, there
23
are eleven parameters to be estimated. These are the two means in each group,
the two variances in each group, the correlation in each group, and the mix-
ture parameter. If two assumptions are made, (1) the variances are all equal
one to each of the other three and (2) the correlation is. zero, then the
simplest mixture of two circular normal distributions is obtained. If the
component variances within each are equal, but unequal from one group to the
other, the more complicated mixture of two circular normal distributions with
different variances is obtained.
If there are "a priori" reasons to suspect that there are two groups and if
the assumption of circularity provides meaningful and useful results, then it
is suggested that Hartley's procedures be used. Computing time will be much
less than in the clustering program discussed here.
The authors know of no other computer program available specifically to take
a set of mixed bivariate normal data and dissect it even for the simplest case
of two subsets with unequal means but equal variances, component and vector,
and zero correlation. The cluster program used here will do the above as well
as the general case. But this clustering program is far from an optimum one
in terms of least time and cost. For the one or two time case, the clustering
program could be used. However, for repetitive processing of many many sets
of data it would be advisable to develop an optimized specific program based
on Hartley's procedures. This, the authors plan to do.
The mathematics may be written for the simplest bimodal bivariate circular
case as
2 2
f(x,y) = I a.f.(x,y); E a. = 1 ,
i=i i=i
where
fi(x,y) = {Zttc!)-' exp{-[(X-y^.)'/a^^ + (Y-yy.)2/ay^]2"'}
\
and the a. are the proportionality factors. In the slightly more general case,
fi(x,y) = [27Ta^a^,(l-p^p]~^^ exp{-[((X-y^.)Va^?)
6.3 Multivariate Mixtures
The general multivariate-multimodal case may be written as
n
f(x,y,...) = z a.f .(x,y,...); Za. = 1 ,
24
where
fi(x,y,...) = [(2TTa^.,o^.,...)|R|]"^' exp-(i|; ID ,
a is a proportionality factor, R is the correlation matrix, |R| is the
determinant of the correlation matrix, R. . is the respective cofactor, and
Sample estimates such as X, Y,..., s ., s .,..., r , ..., replace the popula-
tion parameters y ., y .,..., a . , a .,..., p ,..., where the bar represents
XI y I XI y I xy
an averaging process.
25
7. WOLFE - NORMIX 360 COMPUTER PROGRAM
Culminating over a decade of work, Wolfe (1971b) published his NORMIX com-
puter program. Earlier versions and other pertinent papers by Wolfe appeared
in 1965, 1967 a and b, 1968, 1970, and 1971a. He provides background dis-
cussion, the necessary programs, and an example. The example chosen is the
old standby example of Fisher (1936) which so many investigators use.
7.1 Maximum Likelihood Estimation
Unless "a priori" conditions indicate otherwise, the first hypothesis made
is that there is a specified number of types and the probability of a datum
being assigned to any one of the types is the same as to any other type. With
two groups, the probability of being assigned to one of the two types is 0.50.
With four groups, the probability of being assigned to one of the four types
is 0.250. With no "a priori" indication, a default option of the program then
provides equal mixing proportions as a first guess.
As shown by Wolfe (1970), the maximum likelihood estimates of the mixing
proportion (a ), mean (y^)^ and covariance (a ) of the cluster (type s) in a
mixture satisfy the following conditions:
n .
L = (1/n) z P(SIX. )
^ k=i ^
n
yg = (V(nx^)) E X,^ P(S|X^)
K~" 1
^3 = 0/{nX^)) I (Xj^-y^) (X,^-;^)^ P(S|X^)
where the prime indicates a transpose, X. is the vector of observations for
the kth point in the sample, and P(S|X. ) is the probability of membership of
X. in clusters and is equal to x times the ratio of the normal density of
type s at X. to the density of the mixture. In the special case where the
clusters have a common covariance matrix, a is a and may be written as
o = {^/{nX^)) E X^ X^ - y^ y^.
K~ 1
As the unknown parameters appear on both sides of the equation, it is
necessary to use an iterative process to solve the equations. For this
reason, if "a priori" estimates are used the iterative processing is
diminished. Computing costs therefore are much lower. Several sets of
values may satisfy the equations, and the results may depend on the starting
values for the iteration process. As indicated previously, a change in the
order of the input data will usually change the initial estimates obtained.
26
Local or relative maxima or saddle points may be obtained. The procedure
does not guarantee to find an absolute maximum. The initial estimates are
simply modified and improved upon. Bad initial estimates or diverging itera-
tions can cause strange estimates in the parameters such as negative variances
or singular correlation matrices. When this happens, re-initialization or
change of initial estimates may resolve the problem. The program also attempts
to resolve the problem of singular matrices by adding a small normal random
number to the diagonal terms.
7.2 Initial Estimation
It is assumed that the problem involves clustering within a multimodal
multivariate data set. All measurements are standardized. That is, estimates
of the individual component means and variances are computed. The squares of
the deviations from the means divided by the variances produce the square of
a standardized deviate. Thus, the set is reduced to a multimodal multivariate
set with a zero mean and a variance of one. This is the basic Mahalanobis
distance technique.
Within the above framework, if initial estimates of the cluster means and
covariances are not provided, the program generates initial estimates in a
KMEAN subroutine which was adapted from Ward, Hall, and Buchhorn's (1967)
hierarchical grouping for minimum variance. The variance which is minimized
is the sum of Mahalanobis distances between points within a cluster given by
'ij' = ('<r^j'' c;' (^i-^j) '
where X. and X. are points in the same cluster and Ci is the covariance
matrix. The prime indicates a transpose and the superscript indicates an
inverse.
The subroutine first transforms the data to principal axis factor scores,
■ ' , Y = a"^ F'X ,
where F is the eigenvector matrix of C^ and A is the diagonal matrix of
eigenvectors. (See discussion on principal component analysis.) Then
d./ = (Y.-Yj)- (Y,-Yj)
which is easier to compute.
After hierarchical grouping, the within-group covariance matrix C2 for ten
clusters is determined. The number ten is arbitrary and could be changed.
With this modified matrix, hierarchical grouping is performed again. A third
covariance matrix C3 within the ten clusters is obtained. New distances are
obtained which are used for the third and final hierarchical grouping.
The program permits the assignment of an arbitrary number of clusters or
means known as KMEANS. If the number setup is 150 or less, the procedure
uses the number established. If the number of input data is greater than 150,
27
a leading subroutine for the KMEANS (MacQueen, 1967) precedes the hierarchical
grouping. The first 150 input data points form the centroids of 150 clusters.
The two closest points (clusters) are merged into a two-point cluster with a
cluster mean in the location of the first of the two points read. The 151st
datum is moved into the place vacated by the second of the two points. The
next two closest cluster centroids are merged and the process continues until
all data points have been collected into 150 clusters which then are grouped
hierarchically.
7.3 Significance Tests for Number of Clusters
The user has some idea that clusters of data exist in the data set or he
would not be concerned with this report or its uses in his field of endeavor.
Suppositions or hypotheses concerning the number of types in the sample are
listed and control is placed on each of these in the program. For example,
he suspects two groups at least or perhaps four at most. Just to make sure,
as far as this technique is concerned, he might place an upper limit of six.
The program reaches a solution in each case which provides a relative (local)
maximum to the likelihood function. The ratio of the likelihoods for any two
of the hypotheses above 2, 4, or 6 provides a basis for a significance test
for rejecting the null hypothesis that there is no difference between the two
hypotheses, i.e., the smaller number of types against the alternative of a
larger number of types.
7.4 Strategy of Use
As computing time increases with number of variates, number of input data,
and null hypotheses to be checked, care should be exercised to keep time,
which is transformable to cost, to a minimum. There are certain strategies
for use which Wolfe (1971b) discusses.
7.5 Usage
It is appropriate that, since the program can be obtained from Wolfe (1971b),
a few usage comments from his program be placed here (with his permission).
7.5.1 Storage Requirements
Storage requirements are variable, depending on the data. The arrays are
dimensional at execution time. A minimal problem would require 120,000 bytes
of storage (IBM 360). The first two lines of the printout give the storage
required for a particular problem.
7.5.2 Restrictions
The number of types or clusters must not exceed 20. There are no fixed
limits on the number of variables or sample size. However, all data and
arrays must fit into core.
7.5.3 Error Messages
Bad initial estimates or diverging iterations can cause strange estimates
28
I
of the parameters, such as negative variances or singular correlation matrices
which result in system diagnostics. In such a case, the user may have to
specify his own initial estimates on the input form or rescale the data to
eliminate dichotomous variables.
7.5.4 Input Deck
Input Deck set up cards are:
Input Form, Cards 01-11
Data Cards (omit if data are on tape)
Input Form, Card 12
Initial estimates, if any, for one hypothesis
Input Form, Card 12
Initial estimates, if any, for a greater number of types
Blank Card
For example:
Card 1 provides the user and the data used.
Card 2 provides the title.
Card 3 provides room for comments.
Card 4 provides room for comments.
Card 5 provides for the number of variables, the sample size, and input
tape.
Card 6 provides hypotheses for number of clusters.
Card 7 provides space for assumption of same or different covariance
matrices and the minimum number to be accepted into a cluster.
Card 8 provides for the significance level for rejection of the null
hypothesis.
Card 9 provides for the entry of the number of KMEANS to be first
examined and whether the iterations are to be printed.
Card 10 provides for the establishment of a time limit and a limit on
the number of iterations.
Card 11 provides for the data format.
Card 12 provides for the initial estimates, if any, for the number of
types J the means, the standard deviations, and the correlations.
7.5.5 Validation Examples
Wolfe (1967, 1971) uses two examples to test his program. The first is the
Fisher (1936) treatment of Anderson's (1935) measurements on Irises. Nearly
all discrimination papers or clustering papers refer to this classic paper.
The second example is an artificial set. These and another set of data on
azotobacter, the nitrogen fixing bacteria [Cox and Martin (1937)], were used
to validate the Wolfe program adapted for use at the National Climatic Center.
The NORMIX program has been adapted for use at the National Climatic Center.
Considerable difficulty was experienced in adapting the program for use on the
Univac Spectra 70/45 due to language configurations. One adaptation is an
29
output routine to furnish a sequential leading copy of the input data. This
permits ready referencing as to a datum and its assignment to a cluster. Some
modification of the output sequences were also made.
Though this point is made by Wolfe, it is given here again for emphasis.
The techniques apply to only normally or near normally distributed variates.
Any significant departure from normality will invalidate the probability de-
cision levels. Therefore, if data distributions cannot be approximated by
the normal distribution, transformations to normality or near normality of
distribution must be made before input to the program.
A brief summary by Wolfe (1971b) is quoted below:
"The problem was to develop a computer program for cluster analysis
and unsupervised pattern recognition of types with different cluster
shape
"The program will cluster a sample of thousands of objects measured
on many continuous variables....
"The approach seeks maximum likelihood estimates of the parameters
of multivariate distributions. The likelihood equations are solved
iteratively by continually re-estimating the probability membership
of each sample point in each cluster until the likelihood reaches a
relative maximum. The initial estimates are derived from a minimum
variance hierarchical grouping subroutine, which itself is itera-
tive in seeking an appropriate distance function. The program prints
out the means, standard deviations, and intercorrelations of the
variables within clusters and the proportions of the population for
each cluster. The probabilities of membership for each cluster are
also printed."
Briefly some points mentioned in the foregoing section are iterated. In
clustering procedures, unless there is "a priori" knowledge as to the char-
acteristics of the clusters in the data set, the data themselves determine
the clustering. This is called the unlearned procedure or learning process.
The use of "a priori" estimates is termed the learned procedure, there is,
then, the necessary decision as to just where to start. This is true of any
discriminating procedure whether it be any of the allied techniques of analysis
such as discriminant function, factor, principal component, or any other
analysis.
Any change in the order of the input data will produce a difference in the
output cluster characteristics (MacQueen, 1967, p. 290). Therefore, one run
only provides an estimate of the clusters. A number of runs with a different
ordering of the data will produce different results. Hopefully, these will
not be too different and usually are not too different even though their co-
variance matrices may be unequal. The more distinctly different the clusters
are, the morti the various runs will provide the same estimates. This is not
unexpected. There will be some data which will not be assigned to the same
cluster each time.
30
8. EXAMPLES
8.1 Introduction
The techniques used here are applicable in any field as they reduce all
problems to non-dimensional ones in the sense of units. Here, climatological
(weather) observations are used. These were obtained from the archives of
the National Climatic Center (NCC) in Asheville, N.C.
We present six data sets from among the many that could have been used.
The first four present situations which are recognized as producing mixed
distributions. Obviously, these are selected to demonstrate the usefulness
of this procedure.
Output of this program for the six data sets used for this paper is not
presented in total. The data sets are:
1 . Land-sea breeze data
2. Tropical stratospheric wind data
3. Mid-latitude tropospheric wind data
4. Mountain pass wind data
5. Marine surface data
6. Radiosonde and rawinsonde data
In the first and second data sets, only selected tabular output data and
computer-drawn, 0.50 probability ellipses are shown. In the third data set,
a total input-output of the Wolfe (1971b) program adapted for use at the NCC
is shown. Except for the input data and the modified eigenvalue-eigenvector
output, the format is essentially that of the Wolfe program. Also, a con-
siderable number of 0.50 probability ellipses are shown. The computer plot
routines for these exist in programs written for the CALCOMP drum plotter and
Computer Output Microfilm and the Hewlett-Packard desk-top plotter at the
National Climatic Center. The computer print plot of the discriminant function
for the common covariance assumption is part of the Wolfe program. The last
three data sets simply illustrate selected and re-arranged output as well as
the 0.25 probability ellipses.
Winds are used in all six data sets as upper winds usually are distributed
in the bivariate normal sense. In the second, fifth, and sixth data sets,
additional weather elements are used. Where mixtures are evident, the usual
unimodal distribution cannot be used. These mixtures occur in the planetary
boundary layer, in the trade-wind inversion region, in the tropopause, or in
the monsoon change region. It is precisely the advisability of separating
these and other mixtures that prompted the investigation of the problem and
the publication of these results.
8.2 Brief Descriptions of Data and Locations
8.2.1 Land-Sea Breeze Data
San Juan, Puerto Rico, U.S.A., is a seaport located on the northern shores
of the Island in its eastern portion. It is a sub-tropical station at 18°26'
31
north latitude and 66°00' west longitude. The data are taken from the obser-
vations made at the airport at Isle Verde. Its elevation is 20 m. This
location was selected because it is a logical one for a land-sea breeze effect.
The hours selected, 0600-0800 and 1200-1400, are in local standard time. The
month of October was selected though any other period could have been used.
One hundred observations for each hour group are used for October 1955 and
the first portion of the first 3 days of October 1956. The early morning hours
should show the land breeze or a balanced condition. The 1200-1400 observation
hour should begin to show the effects of the sea breeze though perhaps not as
strongly as later hours. The hours of 0900-1100 were not used so as to elimi-
nate some of the overlapping of the land-sea breeze effect if any existed.
This would provide a clearer separation for the technique demonstration.
8.2.2 Tropical Stratospheric Wind Data
Canton Island is in the Southern Hemisphere at latitude 2°46' and longitude
171 °43' west, elevation 4 m. This location was selected because of its known,
distinct quasi-biennial wind oscillation in the stratosphere. During the
preparation of Technical Paper 34 (Crutcher, 1958) and U.S. NAVAER 50-1C-535
(Crutcher, 1959), the apparent biennial oscillation of the tropical tropo-
spheric winds had been noted. U.S. NAVAER 50-1C-535 is apparently the first
atlas in chart form to use the elliptical bivariate normal distribution to
describe the distribution of upper winds, particularly those of the strato-
sphere. This followed the very important atlases of the British groups
[Brooks and Carruthers (1950)] using the circular bivariate distribution and
preceded their update, [Heastie and Stephenson (1960) and Tucker (I960)].
The British groups assume the circular distribution to adequately describe an
upper wind distribution. This appears to be a good assumption as a first
approximation if the winds are not mixed. The extension (Crutcher, 1957) of
this assumption to that of the elliptical normal as a second approximation
increased the representativeness of that bivariate normal from about 70 per-
cent of the cases to about 90 percent in the troposphere. It was noted during
the preparation of NAVAER 50-1C-535 that, in the tropics at the higher alti-
tudes (lower pressure levels), the ratio of the major axis to the minor axis
was seemingly extraordinarily large, near four ranging up to ten at times.
An examination of the distributions generally revealed two clusters with the
easterly winds being stronger and with a higher constancy than those from the
west. In the meantime other investigators also were studying the problems.
McCreary (1959) reported that in the stratosphere during October 1956 - July
1957 at Christmas Island, westerly winds overlay the stratospheric easterlies.
This was a reversal of what was considered to be the usual pictures of east-
erlies over westerlies. Graystone (1959) then reported on a year to year
reversal of the equatorial stratospheric winds. In the following year Reed
(1960), Ebdon (I960), and Veryard (1960) provided the impetus for a great
amount of research into the phenomenon now known as the quasi-biennial oscil-
lation (QBO) of the winds in the tropical stratosphere. These same authors
soon produced further work in the field [Reed et al . » (1961) and Reed and
Rogers (1962) in the U.S. and Veryard and Ebdon (1961e. b, and 1963) in the
U.K.]. Later, an oscillation in the temperatures associated with the winds
was soon reported as were oscillations in the tropopause heights, ozone, and
other phenomena. Newell et al., (1974) discuss the development of research
32
in this field in the second volume of an important work on the general circu-
lation of the tropical atmosphere and interactions with extratropical
latitudes.
In these wery important studies, the oscillation was shown to start at the
higher altitude of the 10-mb level and propagate downward through the strato-
sphere to the 100-mb level.
Clayton (1885), Berlage (1956), and Landsberg et al . , (1963) had noted this
feature in surface phenomena. This feature now occupies a rather solid part
in the literature of the atmospheric circulation.
Upper winds during the months of July and January for the years 1954-1964
at Canton Island at the 50-, 30-, 20-, and 10-mb pressure levels were used.
For the period 1957-1967, heights and temperatures were added. Data from this
station have been used many times in studies. The quasi-biennial oscillation
(QBO) is barely evident at 100 mb and becomes strongest at about 30 to 25 mb.
Apparently it weakens at lower pressure levels (higher altitudes), such as
10 mb, though this effect may be due to loss of observations. There is a time
dependence as noted by Reed et al . , (1961), Reed and Rogers (1962), Veryard
and Ebdon (1961a, b), Angell and Korshover (1962), and later works.
The purpose of this paper is to provide the techniques to dissect such dis-
tributions. Dynamic or synoptic considerations are not made here except in
the sense that these do effect the QBO. Newell et al., (1974) discuss the
dynamic properties of the phenomenon. Dynamic considerations do create the
separate distributions insofar as the calendar year dating system is involved.
The results may be used to assess the extent of these effects. Only the
horizontal winds and temperatures are used at the individual levels. An ana-
lytic solution or more detailed examination of the atmospheric circulation and
weather is not attempted here. For example, the problem could have been made
much more multidimensional by including all stratospheric levels and concomi-
tant surface weather. The results of this section clearly demonstrate the
usefulness of this separation technique.
8.2.3 Mid-Latitude Tropospheric Wind Data
A continental U.S. station was needed. Rantoul , Illinois, was selected
since, in an earlier unpublished work, Crutcher and Clutter (1962) experienced
some difficulty in applying the techniques of Hartley (1959). The difficulty
involved singularity problems in the data when assumptions of unequal variance
in the groups were made. Its latitude is 40°18' north, longitude is 88°09'
west, and elevation is 227 m. There are some data sets which produce such
near singular matrices that the techniques cannot be used in their present
form. This is occasionally true of the present technique here. Ordinarily,
the singularity problem is resolved in the technique by the addition of a
random small amount to the matrix diagonal elements. Even then, negative
variances may be encountered. Re-initialization or change in order of data
entry may resolve the problem.
The period of record of data used is the month of October for the years
1950-1955. The upper wind data used are those at the 700- , 500- , and 300-mb
33
levels. Both input and output data are shown to better illustrate the pro-
cedures. The first two data sets described were two-dimensional. This data
set example illustrates a distribution in six dimensions, two for each of
three levels. For this purpose there had to be a simultaneous wind observa-
tion at each of the three levels. Therefore, each input data vector given
was composed of the vector formed by the zonal and meridional component of the
wind at each level, i.e., a vector composed of six components. The two-
dimensional illustration for a given level then is comparable to that of
another level for each datum at the given level having a matching datum at
the other level .
As more than three space is difficult to illustrate except for linear
reduction to three space or less, output for the six-dimensional forms is
given in eigenvector and covariance (correlation matrix) form.
No attempt is made here to correlate the wind data with current or later
weather. This can be done by simply extending the six-dimensional vectors to
a greater number of dimensions by including the desired surface weather data.
8.2.4 Mountain Pass Wind Data
Stampede Pass, Easton, Washington, U.S.A., was selected to demonstrate the
existence of two or more distinct clusters of wind, and of wind with tempera-
ture regimes, in a constricted geographic location. Its latitude is 47°17'
north, its longitude is 121°20' west, and its elevation is 1206 m. It is
almost in the lowest part of a saddle-back with a ridge running north-south
upwards from the station location to a height of about 200 m above the station
at a distance of about 3 km. North of the saddle, the ridges rise in an east-
west fashion to heights near 3,000 m at distances ranging from 20 to 80 km.
8.2.5 Marine Surface Data
Weather observations over oceanic areas constitute an important part of the
archives at the National Climatic Center. These have been the basis for the
Marine Climatic Atlases program of the U.S. Navy since 1950. In some regions
the weather is composed of quite distinct weather regimes over the year.
Within shorter time periods there may or may not be such distinctness. There
may or may not be distinct separation of weather factors between the traveling
cyclones and anti -cyclones.
Observations from transient ships offer no real opportunity for time studies
For a particular spot, over time, this may or may not be important. For
reasons of time continuity and a measure of pressure gradients, however, an
ocean station vessel location was selected. The ocean station vessel is OSV
"C" (Charlie) at 52''45' north latitude and 35°30' west longitude. The month
of February was selected for the years of 1965-1970. The 1200Z observations
are used. The elements selected are the surface winds in m-s" , the tempera-
tures (°C), the sea level atmospheric pressures (mb), and the dew-point
temperatures (°C). Other elements such as wave height, wave frequency, visi-
bility, cloud, cloud height, precipitation, and other obstructions to vision
can be used. However, some of these have a lower bound of zero or are
dichotomous, i.e., a yes or no condition. Transformation to near normality
- 34
of the distributions of these elements should be made before they are used
with this technique.
The input data are not shown. Only the pertinent output data are shown,
i.e., the means, variances or standard deviations, the correlation matrices,
and the eigenvalue-eigenvector and mixture proportions.
8.2.6 Radiosonde and Rawinsonde Data
The foregoing examples clearly illustrate the usefulness of Wolfe's NORMIX
and NORMAP techniques to separate mixed multivariate normal distributions.
Two- or higher-dimensional vectors have been used. A radiosonde observation
is composed of observations at levels in the atmosphere where significant
changes are made from level to level. Also included are observations at
mandatory levels so that all stations transmit data for the same levels.
These can be used to produce synoptic charts of the data. Ordinarily, these
charts can be studied by element and level by level or by changes within and
between levels. Also, a complete observation can be studied as one vector in
multidimensional space. Only the time and cost limitations of the computers
restrict the problem. For example, if the winds, temperatures, dew points,
and heights of selected pressure levels are used for 30 levels, an observation
may be characterized by a vector of 150 components, i.e., a 150-dimension
problem. Extension of this to other levels and features as well as differences
level to level such as lapse rates, shears, etc., would extend the vector to
still higher dimensions.
As an illustration, radiosonde and rawinsonde observations at Balboa, C.Z.,
Panama, are used. The 1500Z observations in the month of July for the years
1961-70 were selected. The levels selected were the surface, 950-, 850-, and
700-mb. The data used are winds in m-s"^, dry bulb and dew-point temperatures
in °C, and heights in gdkm. Each vector then is a 4 x 5 or 20-dimensional
vector. Assumption of both equal and unequal covariance matrices is made.
No input data are shown. Only the output data in terms of means, variance
(standard deviations), covariance (correlation) matrices, eigenvalue and
eigenvector, and mixtures are shown.
8.3 Selected Data
8.3.1 Land-Sea Breeze Data Set
8.3.1.1 Input Information.
a. San Juan. Puerto Rico
b. The period of record is October 1-31, 1955, and
October 1-3, 1956.
c. The data are surface winds.
d. The number of variables is two; these are zonal and
meridional wind components.
e. The number in the sample is 200; the first 100 are from
the 0600-1800 local standard time (l.s.t.) while the
second 100 are from the 1200-1400 l.s.t.
f. The minimum number to be accepted into a cluster is three.
35
g. The null hypotheses are made that (k + 1) clusters are not
significantly different from the k clusters. The decision
probability level selected is 0.01. Rejection of the hypothesis
then permits the assumption of (k + 1) clusters.
h. The first 40 two-dimensional vector entries are .set up as
the 40 means of 40 separate and individual clusters. These
are 40 points in two dimensions.
i. Two assumptions are made. The first is the equality of co-
variances while the second is the non-equality of covariances.
8.3.1.2 Tables - Output Information. The program computes the necessary
statistics for the three-cluster versus two-cluster including the discriminant
functions and classification of entries into the clusters.
The output statistics are now shown in tables 1 and 2 for the two-cluster
and the three-cluster even though the hypothesis that three types and two
types were not significantly different was not rejected.
8.3.1.3 Figures and Discussion. San Juan, Puerto Rico, was selected because
this is a known land-sea breeze effect location. The hour groups of 0500-0800
and 1200-1400 were selected when the land-sea breeze effect would most likely
be operating. The periods of October 1-31, 1955, and October 1-3, 1956, were
selected so as to provide 100 observations of the wind at each of the hour
groups. The wind direction and speed are used to provide the zonal and merid-
ional components of the wind. The first variable is the x-component or zonal
(west-east) component of the wind while the second variable is the y-component
or meridional (south-north) of the wind. Minus signs indicate components from
the east or north.
The procedural techniques of the Wolfe NORMIX-NORMAP electronic computer
routines (1971) worked well in this case.
Figure 8a shows a two-dimensional representation for the entire group of
200 observations. This may or may not be an adequate representation. No plot
is made of data to visually assess the fit of the mathematical elliptical form.
This form is obtained under the assumption that there is one single group.
Here it is realized that such an assumption is not valid, yet it is shown for
illustrative purposes. Under the assumption of two groups, the land and the
sea breezes, and the equality of covariances, the breakout into two groups is
shown. Elliptical error probable (e.e.p.) ellipses are shown. These are the
0.50 probability ellipses. When the axes of the ellipses are equal, the
ellipses are circles and the 0.50 probability circle is called the circular
error probable (c.e.p.). Figure 8b then shows a breakout into three clusters.
The assumption of equality of covariances is exemplified by the fact that the
ellipses all have the same shape, size, and orientation. This is not the case
as shown in figures 8c and 8d where the covariances are assumed to be different.
In these figures, the shape, size, and orientation may be different from ellipse
to ellipse. The authors believe that assumption of unequal covariances is
valid. The component means of the two-cluster configuration under the assump-
tion of equal versus unequal covariance matrices are in the first type -0.4620
versus -0.5764 and 1.4574 versus 1.3866 and in the second type -4.7149 versus
-4.7913 and -2.3420 versus -2.4596 m-s"^. The proportions assigned to the
36
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OCT SFC 2 TYPES
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6flN JUfIN DIFF COVRR.
10
L 10
Figure 8 San Juan, Puerto Rico, surface wind distributions; period of record
October l-Sl, 1955, and October 1-3, 1956, hours 0600 and 0800 and
1200-1400 local standard time; n = 200, 100 from each period; units,
m-s"^; separation shown for two and three types with covariances
assumed equal and then unequal .
(a) Total and two types, equal covariance matrices.
(b) Three types, equal covariance matrices.
c) Total and two types, unequal covariance matrices.
d) Three types, unequal covariance matrices.
An
first type are 0.5887 versus 0.612 and the second type are 0.4113 versus 0.388.
The comparisons indicate similarity. Under the assumption of unequal co-
variance matrices, the second type shown above (though losing a part to the
first) then breaks down into two clusters which are proportionally 0.255 and
0.110.
Quite clearly the unequal covariance ensemble breaks down into at least
three clusters, a south-southeast wind and an east-northeast wind, the early
morning wind versus the noon wind, and the land breeze versus the sea breeze.
The mean wind is from the east by southeast with components -2.2114 and -0.1054
m-s"^. This is the easterly trade though it is somewhat less than the average
of 3.5 m-s"i for that region [Crutcher, Wagner, and Arnett (1966)].
The land-sea breeze situation illustrates the output of the technique in
tabular form and in two-dimensional illustrations with ellipses. Two assump-
tions are shown. First, there is the assumption that the underlying statistics
have the same covariance matrix; i.e., the underlying physical bases are
operating the same. Second, there is the assumption that the underlying sta-
tistics have differing covariance matrices; i.e., there is some reason to
believe that the underlying physical bases operate differently. Whether the
assumptions are correct in any particular case is not known for these are not
tested here. Both assumptions are made. The results are presented. The
reader can make his own assessment and choose the assumption he pleases. How-
ever, statistical tests are used to reach decisions as to whether there are
two groups or less, three groups or two, etc. As with all other examples, the
decision level chosen to work with is the probability level of 0.01. The null
hypothesis is that the distribution of (k + 1) groups is not different from
(k) groups; that is, rejection of the null hypothesis that there are (k + 1)
groups rather than (k) groups is sought.
Output or input data in computer format and the intermediate steps are not
shown in this example. For these the reader is referred to Wolfe (1971b)
whose electronic computer program is adapted for use here. The mid-latitude
tropospheric wind data set (paragraph 8.3.3) does contain some of the inter-
mediate steps.
8.3.2 Tropical Stratospheric Wind Data Set
8.3.2.1 Input Information.
a. Canton Island, South Pacific, (U.S.A. and Great Britain)
b. The periods of record are the months of July and January,
1954-1964, and for 1957-1967.
c. The data are stratospheric winds, heights of pressure
surfaces, and temperatures. The pressure levels are 50-,
30-, 20-, and 10-mb.
d. The number of variables is two for the first period and four
for the second. These are for the first period, zonal and
meridional components of the wind, positive from the west and
south. For the second period, these are the winds, heights,
and temperatures. The units are m-s"^, m, and °C.
e. The number in the samples vary from level to level because at
41
times the balloons failed to reach the higher altitudes.
The numbers are 263, 244, and 162, respectively, for the
first three levels above. Though the 10-mb data were
processed, only one cluster was determined. The results
are not shown.
f. The minimum number to be accepted into a cluster is three
for the first period and five for the second.
g. The null hypotheses are made that (k + 1) clusters are not
significantly different from the k clusters. The decision
probability level selected is 0.01. Rejection of the hypothesis
then permits the assumption of (k + 1) clusters.
h. The first 40 two-dimensional vector entries in each level are
set up as the 40 means of 40 separate and individual clusters.
These are 40 points in two dimensions,
i. Two assumptions are made. The first assumption is the equality
of covariances. The second assumption is the non-equality of
covariances.
8.3.2.2 Tables for Wind Configurations (1954-1964). Tables 3 through 10
provide the output data in tabular form provided by the Wolfe (1971b) NORMIX-
NORMAP computer routine for the first period. An asterisk indicates the re-
jection of the null hypothesis that (k + 1) types are not significantly
different from the (k) types. Thus, under the following assumption of equal
then unequal covariances and at the 0.01 probability level, the greatest
number of clusters (types) for the wind distributions 1954-1964 is indicated
below:
July
Equal Unequal (Covariances)
50-mb
2
3
30-mb
at least 4
3
20-mb
3
2
10-mb
1
1
Janua
ry
Equal
Unequal
2
1
1
1
2
1
1
1
The statistics for the single clusters are not shown.
42
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56
8.3.2,3 Figures and Discussion for Wind Configurations (1954-1964). Figures
9 through 19 show various combinations of the 0.50 probability ellipses as
well as the wind plot diagrams to which these pertain. For example, figure
10a gives the 0.50 probability ellipse for the total group and a breakout into
two groups. This 0.50 probability ellipse is the ellipse estimated to contain
one-half of the winds from the cluster (type) to which it pertains. Figures
10b and 10c are similar illustrations for the 30- and 20~mb levels. Comparison
between the results of the assumptions of equal and unequal covariance matrices
are possible. Under the assumption of equal covariance matrices, the size,
shape, and orientation of the cluster breakout are the same but not necessarily
the same as the complete (total) distribution. That is, the flux indicated by
the total may be quite different from that within the individual clusters
though the individual clusters imply that the flux across the pertinent air-
stream in each cluster is the same as in another. Under the assumption of
unequal covariance matrices, the various clusters may show different size,
scale, shape, and orientation.
In addition, other figures in the above group permit comparison from level
to level of the totals and of the various types. These tabular and graphical
representations quite clearly indicate that elongated elliptical distributions
computed from ordinary bivariate normal statistical routines should be backed
up by plot or scatter diagrams. If a plot is not available, then a wind rose
of some type should be available for examination. The wind rose usually re-
ferred to is the WBAN-120 (revised) available from the National Climatic Cen-
ter (1958). This is based on the work by Crutcher (1957). A decision can
then be made as to whether the computed bivariate statistics are valid. These
assume a unimodal bivariate distribution. In the output statistics of WBAN-
120 revised format, for example, a ratio of the major to minor axes in excess
of four should indicate a need for study to see whether clustering is evident.
See table 9 for such a comparison. Both plots and analytic procedures such
as are available in discriminant function analysis, factor analysis, or prin-
cipal component analysis or other clustering routines can be used.
"A priori" considerations may also indicate that clustering techniques
should be used. That is, previous research may indicate that the use of the
total distribution as a unimodal bivariate (or multivariate) distribution is
unwarranted and that probabilistic statements from such a model may be erro-
neous.
The previous example of the land-sea breeze effect at San Juan, Puerto Rico,
and the present example of the stratospheric winds at Canton Island are "a
priori" types.
The procedures also are helpful in establishing the quality assurance of the
data. Clusters composed of only one or a few observations and far distant from
the main cluster or clusters may be examined for validity. This feature will
be discussed in a later section. In particular, in tables 6 and 7 and in
figures 17a and 17b for the four-cluster breakout, the wery low proportions in
one of the clusters and the wery large ratio of the major axis to minor axis
indicate that these groups may be suspect for one or more reasons. The non-
rejection of the null hypothesis, four clusters versus the three clusters,
implies that the four-cluster breakout was not significantly different from
57
the three-cluster breakout. Thus, the investigator can simply stay with the
three-cluster configuration while examining the isolated questionable groups
indicated in the four-cluster configuration.
The equatorial stratospheric winds situation illustrates the output of the
technique in tabular form and in two-dimensional depiction with 0.50 ellipses.
Two assumptions are used. First, there is the assumption that the underlying
distributions are the same, i.e., that the statistics have the same covariance
matrices (that the underlying bases are the same). Second, there is the alter-
native assumption that the statistics represent different physical and dynamic
situations, i.e., the covariance matrices are not the same. There is some
reason to believe that the underlying physical bases operate differently. The
reader can select the assumption that best fits his knowledge and experience.
Under the assumption of equal covariance matrices, one of the outputs is a
plot of discriminant function assignments, such as one versus two or one versus
three, etc. This is a computer tabulation type of two-dimensional plotting
with each individual data point carrying the number of the cluster type to
which it is assigned by the classification (discrimination) procedure. Figures
13, 14, and 15 illustrate the output for Canton Island during July at the 50-mb
and 30-mb levels. Figure 13 for the July 50-mb level shows the print plot of
discriminant function 1 and 2 where three types are computed. Figures 14 and
15 for the July 30-mb level show the print plots of discriminant functions 1
and 2 for the first three-type dissection and then four-type dissection. It
appears here that type 2 of 3 becomes type 3 of 4 while type 3 of 2 breaks down
into types 2 and 4 of 4.
58
6O703 7 50
« 36 32 2S M 20 It 12
■12 16 -20 -24 28 -32
— [
1
1 1 ; :
■OJ
X
[ l^Ll .
i
i
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X
X
X
X
f .
X
X
X
i
X
X
^
X
1
(b)
—
X
1
Figure 9 Canton Island, U.S.A. and U.K.; upper wind distribution plots;
period of record, the months of July 1954-1964; pressure levels
(a) 50-, (b) 30-, (c) 20-, and (d) 10-mb; units, m-s"i.
59
60703 7 20
40 36 32 28 24 20 16 12
•24 28 32
1 !
j
i
1
1
1
1
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i j
i 1
1
1 — -^
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i
1 X
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^ 1 ix >
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60703 7 10
40 36 32 2a 24 20 16 12 a 4 -4 8 -12 16 -20 -24 -28 32 '36 40
: ! i
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1
i
X
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1
j
i
1
(d) : '■
\
1
■
1
Figure 9 (continued)
60
Figure 10 Canton Island, U.S.A. and U.K.; upper wind distributions; period
of record July 1954-1964; pressure levels, 50-, 30-, amd 20-mb;
units, m-s"^; wind plot shown in figure 9; separation shown for
two types, assumption of unequal covariance matrices; elliptical
error probable, (e.e.p.), 0.50 probability ellipses.
(a) Total distribution and two types at 50-mb, n = 263
(b) Total distribution and two types at 30-mb, n = 244
(c) Total distribution and two types at 20-mb, n = 162
61
(a) Three types at 50-mb, n=263.
(b) Three types at 30-mb, n=244.
(c) Three types at 20-mb, n=162.
(0
Figure 11 Canton Island, U.S.A. and U.K.; upper wind distributions; period
of record July 1954-1964; pressure levels, 50-, 30-, and 20-mb;
units, m-s"^; wind plot shown in figure 9; separation shown for
three types, assumption of equal covariance matrices; elliptical
error probable, (e.e.p.), 0.50 probability ellipses.
(a) Three types at 50-mb, n = 263.
(b) Three types at 30-mb, n = 244.
(c) Three types at 20-mb, n = 162.
62
SOHB 4 TYPES
CANTON IS- SRHE COV
1 L
T r
"1 r
-J i_
Figure 12 Canton Island, U.S.A. and U.K.; upper wind distributions; period
of record July 1954-1964; pressure level, 30-mb; units, nrs"^;
n = 244; wind plot shown in figure 9; separation shown for four
types, assumption of equal covariance matrices; elliptical error
probable, (e.e.p.), 0.50 probability ellipses.
63
SAMPLE p CANTON ISLAND
JULY
~T0 — PE
X AND Y WIND COMPONENTS
"DISCRIMINANT PlINCTIDM 2
NUMBER DF TV|>PS» 3
r
1 I
1 rr
1 1
"1 r~T
3
"T rr
U 11 11
3 3~?3 Tm. 11 lU I 1 11
3 33333 33 33 3 3 1 HI 111111 111 1 1
TT
3
77T
3 3
333*1
33 33
iiiiniui ill
111 1 111 1
llUllll
"5~3"
T3T
3 3
TT"
lllliUl
■iT
1 11
T3 33 i 1 I Hi ill
3 2 22 1 11 lUU
Tl i 1 11
-2
T5
-4
■^
• ••«•••••
■51 ^n, ^r, ^T, ^^, ^Ti 07^ u rr~
"31 Wt Wt 6^ 7^ 37
DISCRIMINANT FUNCTION 1
Figure 13 Plot of Discriminant Functions 1 versus 2 for Canton Island, U.S.A.
and U.K.; July winds shown in figure 9 for winds shown in figure 11,
based on assumption of equal covariance matrices; period of record
1954-1964; elliptical error probable, (e.e.p.), 0.50 probability
ellipses.
(a) Three types at 50-mb.
(b) Three types at 30-mb.
■ (c) Four types at 30-mb.
(d) Three types at 20-mb.
64
SAMPLE • CANTON ISLAND
JULY 50" MB
X AND Y WIND CnMRONENTS
DISCRIMINANT FUNCTIUN 2
NUMBER PP TYPES" 3
3 3
3 3 3
I 1
T I — 1
Hill
1 I
3 3
3
33
~T5 333 33 3 3
33 335333 3 33
Ti — TTTT
1 11
1 1
1 HI
1
I 11 I 1 1 3 353 33333353
II 1 1 333 333 3 33 3 3 3
nr
11 111
I r
HI
"~33 ^ 333333 T
3 33 3 33 33 3
TTTTT
3 3
33
-1
13
T
-2
^ 3
3 3 33 3
-3
-4
■^
-6
^^
(b)
"^f: ^3i =^4^ ^y; ^r: ;t; c"; ri i; t,
DI$CRI^'.I^ANT FUNCTION 1.
"t: 5"; 6^ T, 57
Figure 13 (continued)
65
SAMPLE
CANTON ISLAND
JULY W MB
X AND Y WIND COMPONENTS
DISCRIMINANT FUNCTION 2
NUMBER OF TYPES* 4
T
6
3
5
! 1
4
3
3
3
3
1
2
1 2
I 22 444
11 1 2 A4 4
1 2 2 2 4 <» 4A<»4
1,
,111111 22 44<>4*
lllUlll 1 nil 2 22 4 4 4* 4<f 4 4 4
0.
11 \l mill 1 1 1 2i 2 2 A 44 44444 44 44
1 lllllll 1 2 2 22 22 2 4 444 44 4 44 4
nil U 1 22 2 4 4 44 44 4
1 1 11 11 11 1 2 2 24 4 444 44 4
-1.
1 11 1 1 2 2 44 4 4
1 2 2 2 4 4
-2!
1 2 4<^ 4
11 4 4 4 4
2 4 4
1
-3.
-4!
-5.
-6.
(c)
-7.
'T: ^rr, =^4; =t; ^n. ^r; o^ r; r* ^» ^, rr
DISCRIMINANT FUNCTION I
Ti t; 87
Figure 13 (continued)
66
SAMPLE ■ CANTON ISLAND
JULY 20 MB
X AND Y WIND CHMPONEVTS
DIStKIMINANT FUNCTIUN 2 NUMBER OF TYPES- 3
7.
^1
5
4
3
1
3
', 1 2
2
3 1
3
3 33 3 1 2 2 22
3 3 3 1
I
3 3 33 I 1 11 2 2
3 3 33333 1 11 2 2
3 3 3333 3 3 3 1 1 2Ji 2^222 22
3333 33333333 1 22 2 2 2
33 3 333 333 I 1 2 2 22 22 2
33 3 33 3 1 2 2 2 2
-1
3 3 33 I 2 2
3 3 1 2 2 2
^2
2 2
,3 3 3 3 2
2
3
-3
04,
-5.
-6,
(d)
-7,
■6, wS ,
■<f. -3, -2. -1. 0, 1, 2. 3.
DISCRIMINANT FUNCTION 1
4. 5,
6.
TT 87
Figure 13 (continued)
67
CfiNTON rS, OIFF COVIH.
T
JULY TYrC 1
— 1 1 1 r-
"T 1 1 r-
-1 1 J ] I u
(b)
10 1-
-1 1-
JULT TfPE 2
CRNTQN IS. UIrr COVIN.
Figure 14 Canton Island, U.S.A. and U.K.; upper wind distributions; period
of record July 1954-1964; pressure levels, 50-, 30-, and 20-mb;
units, m-s~^; n = 263, 244, and 162, respectively, assumption of
unequal covariance matrices; elliptical error probable, (e.e.p.),
0.50 probability ellipses.
(a) Total for the three levels.
(b) Type 1 for the three levels.
(c) Type 2 for the three levels.
68
JUL SOPie 2 TTPeS chnton is. srme cqvrr.
JX aOfIB 2 TTPES CSNTON IS. SfinE COVflR
1 1 r r —
(b)
^^^ ..^^A^QO...
JUL 20ne 2 TTPES CfiNTON 18. 3fine CQVflR.
n r
T r
li .mn
(c)
Figure 15 Canton Island, U.S.A. and U.K.; upper wind distributions; period
of record July 1954-1964; pressure levels, 50-, 30-, and 20-mb;
units, m-s"^; n = 263, 244, and 162, respectively; separation shows
two types; assumption of unequal covariance matrices; elliptical
err-or probable, (e.e.p.), 0.50 probability ellipses.
(a) Total and 2 types at 50-mb.
(b) Total and 2 types at 30-mb.
(c) Total and 2 types at 20-mb.
69
~I 1 1 1 1 —
JUl 60t«. 3 TYPES CflMTON [S. DIFF CQVIM.
- - -10 -[■
Figure 16 Canton Island, U.S.A. and U.K.; upper wind distributions; period
of record July 1954-1964; pressure levels, 50-, 30-, and 20-mb;
units, m-s~^; n = 263, 244, and 162, respectively; separation shows
three types, assumption of unequal covariance matrices; elliptical
error probable, (e.e.p.), 0.50 probability ellipses.
(a) Three types at 50-mb.
(b) Three types at 30-mb.
(c) "hree types at 20-mb.
70
JUL SOHB. « TYPES CflllTON IS. OIFF COVIN.
(a)
-I , 1 , 1 ^
-1— 1 ] L.
-1 L.
_l L_
_i J -L- L.
JUL 30H8. < TYPES CANTON IS- OIFF COVIU.
Figure 17 Canton Island, U.S.A. and U.K.; upper wind distributions; period
of record July 1954-1964; pressure levels, 50- and 30-mb; units,
m-s~i; n = 263 and 244, respectively; separation shows four types,
assumption of unequal covariance matrices; elliptical error probable,
(e.e.p.), 0.50 probability ellipses.
(a) Four types at 50-mb.
(b) Four types at 30-mb.
71
60703 1 60
24 20 16 12 e
-16 -20 -24 -28 -32 -36
1
-
■
X X
X
X
X
X *
X
X
>
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X
x^
X X
X
X
<
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X
X
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* V X V
X
X
X
X X
P""^ >^ X ,, x""
^ X
>: >:
X ''x
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X
K X
V
fxM '• X
^ * X
X X **
^ X it
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X Xx
X X
X
X
X /
>< 1^ ^$x
X X
X X
X
Xv
^x X
X ^
X X
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!
X X
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X xX
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(a)
60703 1 30
24 20 16 12
-16 20 24 -28
-36 -40
1
j
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! ■
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! 1
1 i
! 1
i '
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<
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i
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j
( < X ! >; x
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^ X >,
X ^,
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>
X 1 1 '
x>^x >^x ix
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1 X X T X "
f i M^ f xx
i ! M^ X : XX
K "
X
<
y
: X
X
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X i
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X ^n
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X
X
X
X
X
X , ; '-x
i 1 x
X
X X
i
i
X
X
1 : 1 X
;x j
i
1
(b)
1 1
: i
i 1 ix j
1
> ; 1
i
1
-12
Figure 18 Canton Island, U.S.A. and U.K.; upper wind distribution plots;
period of record January 1954-1964; pressure levels, (a) 50-,
(b) 30-, (c) 20-, and (d) 10-mb; units, m-s'i.
72
60703 1 20
24 2D 16 12
-12 -16 -20 -24 -28 -32 -36 -40
X
X
X
X
X
X
X X
X
X
X
^x X
^ X
>
X
.. X
>
X X
X
>
1
X
X
X X
X
x'
x X
Xn/
X
X
>
X
>
X
>^^ X
X
X
>
X
i^ "
>
X '" '
XX ! ^
1 X
Xx
X
X
X
X
X
(c)
1
-16
12
60703 1 10
20 16 12 8 4 0-4-8 12 -16 -20 -24 -28 -32 36 -40
[
u -^
X
X
X
<
>
X
X
X
X
X
X
X
X
■
X
X
X
X
X
X
X
X
1
X
X
1 ^
1
X
X
X
X
X
X
X
L..
—^
16
Figure 18 (continued)
73
JfiN 50MB 2 TYPES
r
CANTON IS. SOME COVRR.
JfiW 50f1B. 2 TYPES
MNTON IS. nrFF COVIN.
-1 1 r -10
Figure 19 Canton Island, U.S.A. and U.K.; upper wind distributions; period
of record January 1954-1964; pressure level, 50-mb; wind plot
shown in figure 18; units, m-s"^; separation shows for two and
three types with assumption of equal then unequal covariance
matrices; elliptical error probable, (e.e.p.), 0.50 probability
ellipses.
(a) Total and two types with equal covariances.
(b) Total and two types with unequal covariances.
(c) Three types with equal covariances.
(d) Three types with unequal covariances.
74
JAH 50MB. B TYPES CANTON IS. aiFF COVIN,
r 1 1- 1 r -^° r
Figure 19 (continued)
75
8.3.2.4 Tables and Discussions for Height, Temperature and Wind Configura-
tion (1957-1967). The foregoing sections discussed wind configurations only
for the 50-, 30-, 20-, and 10-mb levels for January and July 1954-1964. The
height, temperature, and wind configurations now are discussed for the 30-mb
level during the months of January and July 1957-1967.
Veryard and Ebdon (1961b) in their study of the quasi-biennial oscillation
with tropical stratospheres reported that the westerly regimes were warmer
than the easterlies. There was a slight lag in the temperatures behind the
wind changes. Most of the data examined were for the 80- to 50-mb levels.
Substantiating this feature for July but not for January are the statistics
presented in tables 11 and 13 for the 30-mb level. Here, the temperature for
the westerlies and easterlies was -56.8 and -56.7 during January and -51.0
and -55.0 during July. In a further breakdown, not illustrated here, one
cluster in January has a temperature of -59.1 though the westerly component
for the cluster is only +4.2 m-s"^.
Tables 11 and 13 have been assembled differently from previous material.
This permits a better reference for this discussion. For example, there are
essentially only three clusters for the January clustering and only four
clusters for the July clustering. In both months, except for a fourth cluster
in July, the total sample breaks down into westerly regimes with slightly
greater heights than the easterly regimes; but while there are no temperature
differences in January, there are differences in July.
In January, the easterly cluster characteristics remain essentially fixed
in all characteristics while the more variable westerly cluster breaks down
into two clusters and then three. The last breakdown is not significant.
During July, the easterly cluster characteristics remain essentially fixed
throughout subsequent breakdown of the distribution. The westerly group
breaks into three significantly different groups. The difference between
the most westerly group and the first easterly group is 41 m-s"^.
Examination of the standard deviations is revealing during both months. The
ratios of the major axis to minor axis deviation ranges from about five for
the total sample to two for the clusters.
A look at tables 12 and 14 is interesting. These tables present the corre-
lation coefficients among the four variables, height, temperature, zonal wind
component, and meridional wind component. It is difficult to assess the de-
grees of freedom in each case. From the QBO basis, a January or a July is
expected to be consistent within itself. Therefore, a maximum of eleven points
are available for the 1957-1967 period. The correlation coefficients are ex-
pected to be large, Jarge enough to be significant. This appears to be the
case for July but not for January except for the heights and the zonal winds.
The heights and the zonal winds appear to be significantly correlated for
all cluster breakouts except one in each month where the correlation shifts
to the meridional winds. During January there is no definite correlation
between heights and temperatures at the 30-mb level. During July there is a
definite correlation. This does not indicate that there is no correlation
76
during January or July between the heights and the temperature structure
between the surface and 30 mb. This has not been examined here.
The large correlation coefficients and the large ratio of standard devia-
tions for the total samples may be considered to be a necessary indication
but not a sufficient basis for the conclusion of a two-cluster existence.
Conversely, small correlation coefficients and ratio of standard deviations
of one are indications of only one cluster, i.e., the total sample is a
homogeneous cluster.
77
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81
8.3.3 Mid-Latitude Tropospheric Wind Data Set
8.3.3.1 Input Information.
a. Rantoul , Illinois, U.S.A., latitude 40°18' north,' longitude
88°09' west, elevation 227 m.
b. The period of record is the month of October for the years
1950-1955.
c. The data are the zonal and meridional components at the
700-, 500-, and 300-mb levels, m-s"i.
d. The number of variables is six.
e. The number in the sample is 503.
f. The minimum number to be accepted into a cluster is seven,
one more than the number of variates.
g. The null hypotheses are made that (k + 1) clusters are not
significantly different from the k clusters. The decision
probability level selected is 0.01. Rejection of the hypothesis
then permits the assumption of (k + 1) clusters.
h. The first 40 six-dimensional vector entries are set up as
the 40 means of 40 separate and individual clusters. These
are 40 points in six dimensions.
i. The assumption is made that the covariance matrices are not
equal .
8.3.3.2 Tables. Table 15 shows the sequential input data with six variates.
The variates are the zonal and meridional components of the 700-mb wind, the
zonal and meridional components of the 500-mb wind, and the zonal and meridi-
onal components of the 300-mb wind in m-s"^.
Table 16 shows the computer output after the hierarchical grouping and the
subsequent grouping into two clusters after 39 iterations. The two clusters
with proportions of 0.171 and 0.829 comprise the total set (1.000). First,
the characteristics of the total group are shown. This would be the assumption
of a unimodal six-variate model. Then the characteristics of the two clusters
are provided. Then, through the use of the discriminant function, assignments
of each datum to the clusters may be made as shown. The probability of assign-
ment to each cluster is printed. For example, the first datum of table 15 is
assigned to cluster 2 with a probability of 0.935 versus 0.065 for cluster 1.
The previous section illustrated computer plots for discriminant functions.
Table 17 provides, from the same input data, the output for three clusters
rather than two. The number of iterations required in this case is 101. The
probability of the null hypothesis being rejected is 0.00000349. The three-
cluster configuration is judged to be better than a two-cluster configuration.
Therefore, the program continues further to test a four-cluster configuration
versus the three-cluster form.
Table 18 provides, again from the same input data, the output for the four-
cluster configuration. The initial computing output, i.e., the first estimate
of the learning process, is shown as the zero iteration. This set then becomes
the initial estimation in the iteration process leading to 101 iterations be-
fore the results converged. Please note the changes from the initial estimates
82
to the final estimates. For example, for the zeroth iteration, the estimate
for the 700-mb zonal component in the first cluster is 11.2762 and after 101
iterations the estimate is 11.1041. The respective standard deviations are
5.5488 and 7.2047. Though the means do not appear to be yery different, this
is almost a forty percent reduction in variance in the cluster.
The probability of rejection of the null hypothesis in the above case is
0.00980137. This is near the decision level of 0.01. Therefore, no further
calculations are shown here.
83
Table 15 A multivariate (6) set of Rantoul , Illinois, October 1950-55,
upper wind components, zonal and meridional, at the 700-, 500-,
and 300-mb levels. For example, input variables 1 and 2 are
the zonal and meridional components, respectively, at the
700-mb level. The units are m-s"^.
RANTOUL
OCTDnEP
700/500/AND 300MB
X 4N0 V COMPONfNTS
NV8L6S- 6 NSAMPf 503 DIFF (Q\l fATRIx
MiN Cluster size- t hvputhesis test.o.'>oo
INITIAL KMEANS» 40 TIME LIMIT. *n ITER LIMIT. 100
NO. OF TYPES* 1234560100000
STORA&E BEOUIREMENT. 111932 DFCIMJL BYTES
OIMEHSION A( 3951 )
ANTICIPATED EXECUTION TIME • 21.11 MINUTES
PROGRAM IMDRMIX
WQLFJ NORMAL MIXTURE ANALYSIS PROC EDURF < 1 974 REVISION)
INPUT VARIAdLFS
£0
1
2
3
4
5
6
1
7,071
7.071
2.121
-2.121
5.563
2,248
2
4.296
10.126
1.9S4
4.603
10.000
0,000
3
-0.000
14.000
4.2*3
4.243
5,523
-2.344
4
-0.000
10.000
20.398
8.241
8.485
-6.485
5
-0.000
16.000
1.9il4
4.603
11.046
-4.689
6
5.470
12.887
-o.ono
1 1.000
5.657
5.657
7
4,636
1.873
15.000
0.000
24.107
9.740
S
7,000
0.000
19.331
-8.205
20,229
-50.068
9
11,3J4
-11.314
21.172
-8.987
31,297
-13.285
10
12.887
-5.470
14.142
-14.142
16.108
-39,869
11
9,192
9.192
-3.1?6
-7.364
8.285
-3.S17
12
2,828
2.828
4.636
1.873
2.828
2.828
13
3.536
3.536
2.715
6.444
7.778
7,778
14
-1.124
2.782
2.715
6.444
7.778
7.778
IS
1,954
4.603
3. l?6
7.364
34,306
13.860
16
0,000
5.000
12.053
4.870
13.285
31.297
17
4.243
-4.24S
4.243
-4.243
29.000
0,000
18
5.994
-14.835
7,778
-7.778
21.000
0,000
19
-3,126
-7.364
11.3U
-11.314
17.490
-7. 424
20
2,622
-6.490
19.092
-19.092
23.160
9,365
21
0,000
2.000
8.000
0.000
21.000
0.000
22
2.000
0.000
3.000
0.000
18.385
18.385
23
6.490
2.622
11.046
-4.689
30,597
12.362
24
5,523
-2.344
3.371
-8.345
11.046
-4.689
25
7.000
n.ooo
12.000
0.000
21.000
0,000
26
11.314
-11.314
14.7?8
-6.252
-36,000
O.OOU
27
15.762
6.368
20.000
0.000
20,251
-8.596
28
4.636
1.873
12.887
-5.470
28,536
-12.113
29
3,536
-3.536
17.490
•7.424
21,213
-21.213
30
8.285
-3.517
O.UOO
-14.U00
9.740
-24.10"
31
9.272
3.74»
11.046
-4.689
4.870
-12.053
32
17.000
0.000
11.9*7
-5.079
6,364
-6,364
33
11,000
0.000
12.000
0.000
13,908
5.619
34
16,000
0.000
13.808
-5.861
9.000
0.000
35
18,544
7.492
15.649
-6.642
5.523
-2.344
36
9.000
0.000
11.000
0.000
7,364
-3.126
37
4.636
1.873
12.0no
0.000
13.000
0,000
38
5,657
5.6!7
8.000
0.000
13,808
-5,861
39
6,490
2.622
a. 345
3.371
18,000
0,000
40
1,954
4.603
9.272
3.746
19,000
0,000
♦ I
12,728
12.728
10.000
0.000
20,398
8.241
♦ 2
6.364
6.364
8.485
8.485
18,544
7.492
4}
12.981
5.245
17.617
7.118
12.021
12,021
44
11,000
0.000
12.053
4.870
19.471
7,867
45
16.689
6.743
14.835
5.994
22.252
8.991
46
16,000
0.000
le.ono
0.000
32,451
13.111
47
13.908
5.619
15,7*2
6. 368
26.888
10.864
48
0.000
-3.000
7.118
-17,617
18.000
0.000
49
7.071
-7.071
9.900
-9.899
9.192
-9.192
50
6.364
-6.364
10.607
-10.607
12.887
-5.470
51
2,248
-5.563
5,619
-13.908
7. lie
-17,617
52
3,371
-8.345
-6.292
-14.728
0.000
-26.000
53
0.000
-9.000
0.000
-17.000
0,000
-26,000
54
-4.950
-4.950
-20.506
-20.506
-7.815
-18,410
55
0.000
-9.000
0.000
-13.000
5.994
-14,835
56
2.622
-6.490
4. 495
-11.126
6,364
-6.364
57
3.371
-B.345
0.000
•8.000
0,000
-4.000
58
4,243
4.243
0.000
-5.000
0,000
-5.000
59
1.124
-2.782
1.498
-3.709
-11.126
-4,495
60
-1.854
-0.749
0.000
-2.000
-7.417
-2.997
84
Table 15 (continued)
INPUT VARIABLES
SEQ
1
2
3
4
5
6
61
-0.921
0.
391
-0.921
0.
391
-8.3*5
-3,
371
fr2
0.000
5.
000
-2.997
7
417
-0.749
1.
654
63
0.000
5.
000
2.3**
5
523
2.735
6
44*
5<.
0,000
3
000
0.000
2
000
l.*l*
1
*1*
65
12.000
0.
000
10.000
000
15.6*9
-6
6*2
66
7.071
7.
071
11,000
000
11.0*6
-4
689
67
6.36*
6
36*
6.0fl0
000
5.657
-5
657
68
8,3*5
3
371
8,3*5
3
371
11.000
000
69
7.*17
2
997
9. OOO
000
13.000
000
70
8.3*5
3
371
8,3*5
3
371
12.053
4
670
71
3.907
9
205
7.417
2
997
12.981
5
2*5
72
3.907
9
205
7.071
7
071
15.762
6
368
73
3.126
7
36*
12.0<3
4
870
15,000
000
7<,
5.657
5
657
7.071
7
071
13.908
5
619
75
«.3'i5
3
37l
9.899
9
900
16.689
6
7*3
76
8.*85
8
*35
10.607
10
607
13.*35
13
*35
77
7.778
-7
778
13.000
C
000
19.331
-8
205
7a
11.000
000
16.569
-7
033
26,000
000
79
12.981
5
245
18.5*4
7
492
25.000
000
80
11.31*
11
31*
16.6^9
6
7*3
27.816
11
238
61
10.607
10
617
13.908
5
619
1*.1*2
1*
1*2
8a
9.192
9
192
11.314
11
31*
20,398
8
2*1
93
«.*d5
8
*S5
16.263
16
263
30,597
12
362
S'.
8.*S5
-8
*95
22.092
-9
378
31,297
-13
2B5
85
*.603
-1
95*
17,490
-7
*2*
21,172
-8
987
86
11.126
*
*95
12.000
000
22,000
000
37
12.981
5
2*5
13.908
5
619
25,961
10
489
US
e.*85
8
*85
11.314
U
31*
19,*71
7
867
89
11.000
f^Cj
21.000
000
35,000
000
vo
17.000
ooo
25.000
000
49.000
000
91
9.205
-3
907
20.251
-8
596
38,000
000
92
9.000
000
17.000
000
28.743
11
613
93
-2.622
6
*90
13.435
13
*35
9,899
9
900
9<,
6.***
-2
735
4.5*6
1
873
24,000
000
95
*.000
COO
12.887
-5
*70
28.000
000
96
13.000
000
4.2*3
• 4
2*3
18.000
000
97
16.689
6
7*3
13.000
000
10.126
-4
296
98
11.31*
-11
31*
IB, 410
-7
815
24,85*
-10
550
99
9.205
-3
907
14.8*9
-14
8*9
23.933
-10
159
100
13.806
-5
861
15,649
• 6
6*2
17,490
-7
42*
101
1?.806
-5
861
14.728
• 5
252
13,808
-5
651
102
8.*85
-8
4S5
21.000
000
27.816
11
238
103
7.071
-7
07i
15.6*9
-6
6*2
11,987
-29
670
10*
8.*85
-8
*e5
9.192
-9
192
15.649
-6
6*2
105
6.***
-2
735
13.000
000
17.000
000
106
6.36*
-6
36*
7.36*
-3
126
0.000
-12
■ 000
107
2.762
-1
172
0.000
-9
000
-14.8*8
-34
979
103
2.828
-2
828
0.000
-6
000
-7.*2*
-17
*90
109
0.000
-*
000
0,000
-3
000
-*.243
*
2*3
110
l.*98
-3
709
2.762
-1
172
-3.000
000
111
0.000
-*
000
4.9?0
-4
950
10.!99
*
121
112
-3.126
-7
36*
0.000
-6
000
0.000
-5
000
113
-7,773
-7
778
-*.6»9
-11
0*6
-6,36*
-6
36*
lU
-10,607
-10
507
-12.0?1
-12
021
*,*95
-U
126
Us
-3.536
-3
53»
0.000
-6
000
0,000
-2*
000
116
l.*98
-3
709
3,7*6
-9
272
11.31*
-U
31*
117
3.536
-3
53»
11.000
000
13,808
-5
861
iia
6.***
-2
735
2.7*2
-1
172
21.000
000
119
11.0*6
-*
589
10.1?6
-4
298
20.000
000
120
6.***
-2
735
9.000
000
11.0*6
-4
689
121
10.126
-*
298
19.000
000
33,000
.000
122
18.000
000
18.410
-7
815
17.*90
-7
42*
123
18,000
000
22.000
000
28,000
.000
12*
11.000
000
27.000
.000
28,000
,000
125
13.000
.000
30,0(10
.000
30.000
.000
126
16.000
000
22,002
-9
,378
37.000
,000
127
5.657
-5
557
22.0no
.000
21,000
iOoo
128
10.126
-*
298
17.490
-7
.*2*
0,000
-22
.000
129
13.808
-5
861
14.7J8
-6
252
0,000
-28
000
130
O.OOO
-10
000
5.619
-13
.908
15.556
-IS
555
131
*.2*3
-*
.2*3
2.6?2
-6
.*90
0.000
-5
000
132
3,682
-1
563
3.000
000
-10,000
000
133
2.828
-2
828
3,010
.000
-8.*85
8
485
13*
1,873
-*
536
4.603
-1
95*
6,*90
2
622
135
3.000
,000
9.205
-3
907
9.272
3
,746
136
3,000
000
3.5'(5
-3
535
7,*17
2
,997
137
i.285
-3
.517
8.2S5
-3
.517
15.689
6
.7*3
138
8,285
-3
517
11.0*5
• 4
689
*.950
-4
950
139
11,967
-5
.079
11.31*
-11
31*
*.*95
-U
126
1*0
0,000
-9
000
0.000
-13
000
-16,*ll
-3b
661
1*1
0,000
-8
000
0.000
-16
000
-33.9*1
-33
9*1
1*2
9,205
-3
.907
5.2*5
-12
981
4.870
-12
053
1*3
8.*85
-8
.485
6.7*3
-16
.589
14,235
-35
.233
1**
12.887
-5
.*70
7.778
-7
778
11.314
-U
314
1*5
12.887
-5
.470
16,000
.000
15,649
-6
642
1*6
7.778
-7
.778
16,569
• 7
.033
27,615
-11
722
1*7
5,657
-5
.5!7
11.31*
-11
.31*
26.69}
-11
331
1*8
*.2*3
-*
.2*3
14.7S8
•5
.252
23.013
• 9
,766
1*9
8,285
-3
.517
10.1?5
• 4
.298
26.000
.000
150
10,199
*
.121
1*.000
.000
2*. 107
9
,7*0
151
17,000
.000
20.251
• e
.596
22.000
,000
85
Table 15 (continued)
INPUT VARIABLES
SEQ
1
2
3
i
5
6
152
17,000
0.000
1*.7?8
-6.252
30,000
0.000
153
12.72B
-12,728
12.0?1
-12.021
27.615
-11.722
15*
15.6*9
-6.6*2
8.616
-21.325
10.86*
-26.888
155
1*.728
-6.252
13,*55
-13.*35
21,213
-21.213
156
13.808
-5.861
17.*90
.7.*2*
21,172
-8,987
157
13.9C8
5.619
15.7*2
6.368
26.000
0.000
158
1*.000
0.000
l*,000
0.000
19,000
0.000
159
18.*10
-7,815
17,*90
.7.*2*
17,000
0.000
160
2.622
-6.*90
6.***
-2.735
15.6*9
-6.6*2
161
-2.3**
-5.523
*.910
-*.950
13.808
'5.861
162
-6.*90
-2.622
3.536
.3.536
18.*10
-7,815
163
-7.36*
3.12*
3.000
0.000
9.900
-9,899
16*
-1.873
*.63»
6.*90
2.622
11,967
-5,079
165
1,172
2.762
3,536
.3.536
11,0*6
-*,689
166
1.85*
0.7*9
3,7-59
l.*98
12,887
-5,*70
167
*,2*3
*.2*3
*,2*3
*.2*3
17,*90
-7,*2*
168
2.3**
5.523
*,603
-1.95*
16.569
-7.033
169
3.536
3.53*
6.*<»0
2.622
11.967
-5,079
170
5.657
5.657
10.199
*.12l
20.000
0,000
171
7.071
7.071
11.1?6
*.*95
2*. 000
0.000
172
7.071
7.071
5,079
11.967
16.000
0.000
173
7.071
7.071
16,6H9
6.7*3
2*. 107
9,7*0
174
12.000
0.000
15,000
0,000
31,000
0,000
175
3.000
0.000
10.000
0.000
25,961
10,*89
176
6.000
0.000
13,908
5.619
31,52*
12,737
177
8.000
0.000
20.000
0.000
31.52*
12,737
178
8.285
-3.517
9.205
-3.907
18.000
0,000
179
8.*85
-8.*85
0.707
-0.707
7.778
7,778
180
13.*35
-13.*35
26,000
0.000
29.*56
-12,503
IBl
12.021
-12.021
29.*56
-12.503
28.991
-28.991
182
0.000
-15.000
10.11*
-25.03*
20.229
-50,068
183
0.000
-1*.000
11.238
-27.815
16.*8J
-♦0,796
16*
5.2*5
-12.981
10.*89
-25.961
22.627
-22.627
185
*,*95
-11.126
8.*85
•8.*85
19.799
•19.799
186
3.7*6
-9.272
6.7*3
-16.689
10.11*
-25.03*
187
8.285
-3.517
12.8117
.5.*70
20.251
-8,596
188
7.36*
-3.12*
12.000
0.000
21.000
0.000
189
7,000
0.000
13.000
0.000
19.000
0,000
190
6.*90
2.622
9.272
3,7*6
9.192
9.192
191
3.536
-3.53*
11.31*
-11,31*
19.799
-19.799
192
13.808
-5.851
12.728
-12,728
10,11*
-25.03*
193
7.000
0,000
8.000
0.000
7,36*
-3,126
19*
5.563
2.2*8
10.000
0,000
9,205
-3.907
195
7.35*
-3.12*
11.9*7
-5,079
9,205
-3.907
196
*,950
-*.950
0.000
-10,000
-11,331
-26,695
197
5.000
0.000
0.000
-1*,000
0,000
-22,000
198
0.000
-5.000
*.87o
-12,053
0.000
-I*. 000
199
7.36*
-3.12*
10.607
-10,607
9.900
-9,S99
200
*.950
-*.950
11.31*
-11,31*
16.263
•16,263
201
13.808
-5.861
19.331
-8,205
8.991
-22.252
202
7.778
-7.778
23.335
-23,33*
2*. 7*9
-2*, 7*9
203 .
5.619
-13.906
9.365
-23,180
8.991
-22,252
20*
0.000
-17.000
17.000
0.000
13.*35
-13, ♦SS
205
.5.080
-11.967
0.000
-12.000
13,808
-5.861
206
0.000
-7.000
7.36*
-3.126
16,000
. 0,000
207
-2.3**
-5.523
2.997
-7,*17
13.000
0.000
208
-3.907
-9.205
2.622
-6.*90
U.OOO
0.000
209
-9.192
-9.192
0.000
-13.000
8.285
-3.J17
210
.6.36*
-6.36*
0.000
-7.000
7.36*
-3,126
211
-5.657
-5.657
-8.*"5
-e.*e5
7.778
-7,778
212
•6,36*
-6.36*
-3.517
-8.285
9.205
-3.907
213
-7.071
-7.071
-3.536
3.536
6.36*
-6.36*
21*
2.622
-6.*90
1.873
.*.636
-8.203
-19,331
215
2.622
■-6.*90
0.000
-9.000
-8.987
-21,172
216
7.778
-7.778
.*.298
-10.126
-16.971
-16.971
217
3.7*6
-9.272
-3.907
-9.205
-13.*35
-13. ♦35
218
*.*95
-11.12*
0.000
-6.000
-12.021
-12,021
219
2.997
-7.*i7
-3.517
-8.285
-*.298
-10,126
220
0,000
-10.000
0.000
-5.000
-3.517
-8,285
221
-1.8*1
0.781
l.*98
-3.709
-1.12*
2.782
222
-l.*l*
-l.*l*
l.*l*
-l.*l*
2.735
6.***
223
2.121
-2.121
3.682
-1.563
3.126
7,»6*
22*
3.536
-3.536
5.523
-2.3**
5.657
5.657
225
*.603
-1.95*
7,000
0.000
*.2*3
*,2*3
3.7*6
226
5.000
0.000
8,000
0,000
9,272
227
6.000
0.000
3.709
1,*98
7.778
7,778
228
2.3**
5.523
8.3*5
3,371
9,000
0,000
229
*.2*3
*.2*3
*.950
*.950
9.000
0.000
230
6.*90
2.622
5.523
-2.3**
20.000
0,000
231
7.*17
2.997
0.921
-0.391
16.000
0.000
232
-0.391
-0.920
3.000
0.000
18.*10
-7,815
233
0.000
3.000
-2.000
0.000
15.6*9
-6,6*2
23*
2.735
6,***
0.000
3.000
12,021
-12,021
235
0.000
2.000
3.000
0,000
10.607
-10,607
236
-2.622
6.*90
l.*l*
-1,*1*
*.870
-12,053
237
0.000
7.000
-1,000
0,000
-*.689
-11,0*6
238
-0.000
9.000
-1,873
*.636
0.000
-12.000
239
-3.7*6
9.272
-1.873
*.636
0.000
-10,000
2*0
-2,997
7.*17
.0,921
0.391
10.607
-10,607
2*1
0,391
0.921
l.*l*
l.*l*
12.728
-12,728
2*2
0,000
5.000
*.*95
-11.126
23.933
-10.J59
86
Table 15 (continued)
INPUT VARIABLES
SEQ 12 3 4 5 6
2«3 «.«S0 4.930 12.000 0.000 19.331 -8.205
244 11.126 4.495 14.000 0.000 26.000 0.000
24J 10.000 0.000 21.000 0.000 24.000 0,000
246 15.000 O.OnO 19.000 0.000 22.092 -9,376
247 10.607 -10.607 23.335 -23.334 53.369 -22,662
248 7.492 -18.544 10.489 -25.961 20.506 -20.506
249 0.000 -14.000 7.492 -18.544 16.971 -16,971
250 0.000 -12.000 7.118 -17.617 15.556 -15,556
251 3.746 -9.272 8.485 -8.485 18.410 -7,815
252 3.000 0.000 8.000 0.000 15.000 0.000
253 4.636 1.873 11.126 4.495 20.000 0.000
254 3.536 3.536 4.950 4.950 11.314 11,314
255 3.517 8.285 14.142 14.142 16.971 16.971
256 1.485 8.485 8.205 19.331 21.920 21.920
257 8,485 8.485 14.142 14,142 23.335 23,335
258 4,243 4.243 11.314 U.314 11.722 27,615
259 0.000 6.000 9,899 9.900 8.987 21,172
260 1.563 3.682 3.90? 9.205 8.967 21.172
261 -3.000 0.000 2,622 -6.490 0,000 -3,000
262 5.657 -5.657 8.000 0.000 12.887 -5,470
263 9,000 0.000 9.272 3.746 10.199 4.121
264 5.657 -5.657 6.490 2.622 4,950 4,950
265 2,997 -7.417 4,603 -1.954 0.000 -7.000
266 -5,080 -11.967 -5.080 -11.967 -10,550 -24.854
267 -6.642 -15.649 -10.550 -24.854 -16.411 -38.661
268 -4.689 -11.04* -10. H9 -23.933 -13.285 -31,297
269 -1.563 -3.682 -8.205 -19.331 -14.848 -34,979
270 0.000 -9.000 0.000 -14.000 0.000 -24,000
271 4,950 -4.950 4.1J1 -10.199 0.000 -18,000
272 2.000 0.000 4.603 -1.954 3.746 -9.272
273 8,000 0.000 8.285 -3.517 6.000 0.000
274 7.000 0.000 4.910 4.950 13,908 5,619
275 9,272 3.746 13.000 0.000 13.908 5,619
276 8,345 3.371 12.053 4. 870 17.678 17.678
277 12,000 0.000 14.655 5.994 16.263 16,263
278 13.000 0.000 17.000 0.000 35.233 14.235
279 9.272 3.746 16.000 0.000 38.000 0.000
280 11.126 4.495 14.835 5.994 34.000 0,000
281 7.778 7.778 15,762 6.368 26.000 0.000
282 7.417 2.997 18.000 0.000 28.000 0.000
283 2.828 2.828 15.000 0.000 33.000 0,000
284 9.000 0.000 16.000 0.000 24.854 -10.550
285 7.000 0,000 15.000 0.000 32.218 -13.676
286 11.000 0.000 15.742 6.368 27.515 -11.722
287 14.835 5.994 12.8§7 -5.470 31.297 -13.285
288 16.000 0.000 15.649 -6.642 28.000 0,000
289 21.325 8.614 21.000 0.000 23.013 -9,768
290 12.887 -5.470 19.000 0.000 26.000 O.OOC
291 22.000 0.000 25.000 0.000 14.835 6.994
292 14.000 0.000 15.649 -6.642 21.000 0.000
293 13.908 5.619 20.251 -8.596 24.107 9.740
294 14,000 0.000 12.053 4.870 29.000 0,000
295 12.981 5.245 15.589 5.743 32.000 0,000
296 18,000 0.000 25.000 0.000 29.670 11.987
297 17.000 0.000 23.013 -9.756 43.000 0,000
298 21.000 0.000 26.000 0.000 27.000 0,000
299 10.607 -10.607 19.000 0.000 34.059 -U.i,57
300 7.071 -7.071 19.331 -8.205 35.820 -15,629
301 4.495 -11.126 19.331 -6.205 38.661 -16.411
302 0,000 -4.000 5.619 -13.908 22.627 -22,627
303 2.000 0.000 11.045 -4.689 18.385 -18,385
304 6.000 0.000 12.000 O.OOO 19.331 =8.205
305 7,417 2.997 21.000 0.000 24.000 0,000
306 18.000 0.000 17.4<S0 -7.424 18.385 -18.385
307 15,649 -5.642 14.7?8 -5.252 15.556 -15,556
308 11.046 -4.689 12.887 -5.470 21.172 -8.987
309 9,272 3.746 13.000 0.000 20.251 -8,596
310 16,000 0.000 16.000 0.000 12.000 0.000
3il 14.728 -6.252 11.967 -5.079 21.000 0,000
312 12.961 5.245 17.000 0.000 17.490 -7,424
313 15.762 6.368 21.000 0.000 18,410 -7,815
314 18.544 7.492 22.000 0.000 16.000 0.000
315 18,544 7.492 24.000 0.000 24.000 0,000
316 24.107 9.740 15.5S6 15.556 32.451 13.111
317 20.398 8.241 28.7*3 11.613 20.506 20.506
318 16.689 5.743 20.308 8.241 25.456 25.456
319 21.213 21.213 21.213 21.213 18.385 18.385
320 15.762 5.368 13.000 0.000 32.451 13,111
321 12.981 5.245 2*.0oo 0.000 26.000 0,000
322 9.192 9.192 34.000 0.000 21.325 8,616
323 5.079 11.957 15.619 6.743 21.325 8.616
324 15.556 15.556 5.470 12.887 21.325 8,616
325 13.435 13.4SS 5.079 11.967 10.159 23.933
326 16.971 16.971 20.506 20.506 12.113 28,536
327 11.385 18.385 29.698 29.699 21.881 51.548
328 10.199 4.121 14.457 34.059 -0.000 78.000
329 9.272 3.746 13.000 0.000 -0.000 72,000
330 7.364 -3.126 5.563 2.248 -23.226 57,485
331 5.657 -5.657 3.371 -8.345 -4.689 -11.046
332 3.371 -B.34S 0.000 -32.000 0.000 -41.000
87
Table 15 (continued)
INPUT VARIABLES
SEQ
1
2
3
4
5
6
333
4,870
-12.053
16.897
-♦1.723
20.229
-90.066
334
12.7J8
-12.728
24.7*9
-2*. 7*9
36.184
-38.184
33S
19-331
-3.205
32,216
-13.676
92.469
-22.272
336
14.728
-6,252
28,536
-12.113
38.184
-38.184
337
12.887
-5,470
23.933
-10.159
49.707
•21.099
33S
7.778
-7,778
19,799
-19.799
26.870
-26,870
339
7.778
-7.778
18.395
-18.385
15.73*
"38.942
3*0
-6.364
-6.364
0.000
-15.000
7.867
-19.471
3*1
0.000
-8.000
0.000
-15.000
!3,*39
-13,435
3*2
.4.2*3
-4.2*3
0.090
-12.000
4.368
-19,762
3«.3
-5-657
-5.697
-3.517
-e.2es
-15.762
-6.366
3*'.
-3.536
-3.53*
-4.990
-4,950
-9,899
-9.899
345
0.000
-6.000
o.oco
.9.000
-8.34J
-3,371
346
0.000
-6,000
•3.126
-7.364
-7.778
-7,776
347
-6.000
0,000
-9.697
-5,657
-5.657
-5.697
348
-1 .«54
-4,603
-1.5f3
-3,682
•7;o7i
-7.071
349
-4.243
-4.243
-6,3»*
-6,364
-17.000
0.000
350
-1.954
-4.603
-7.417
-2.997
•13.906
-5.619
351
-5.000
0.000
.2.826
.2.628
-12.021
-12.021
352
1.172
2,762
-4,603
1.954
-22.252
-8.991
353
0.000
3.000
-0.375
0.927
-12.053
-4,870
354
1.172
2.762
2,3**
5.523
-6.490
•2.622
355
4.2'<3
4.2*3
4.689
11.046
. 3.682
1,563
356
1.563
3.682
5,079
11.967
-3.371
8,3*5
357
2.782
1.12*
J. 907
9.205
-0.000
15.000
339
2.121
2,lSl
4.298
10.126
-0.000
19,000
359
1.563
3.682
6.364
6.364
7.819
18,*10
360
15,000
0,000
9.192
.9,192
30.377
-12,89*
361
10.126
-4.298
13.000
0.000
26.936
-12.113
362
11.000
0.000
22.000
0.000
23.180
9.365
363
22.000
0.000
25.000
0.000
25.961
10. 489
364
18.410
-7. 815
23,013
.9.768
44.000
0.000
365
14.849
-14.649
18,385
-16.385
9.740
-24.107
366
3.746
-9.272
7.071
.7.071
12.887
-5.470
367
10.126
-4.298
11.126
4.495
23.000
0.000
368
7.778
7.778
9.192
9.192
31.52*
12.737
369
12.887
-5.470
14.7J8
•6.252
24.854
-10,550
370
21.000
0,000
30.507
12.362
54.704
22.102
371
19.000
0.000
22.29?
8.991
28.743
11.613
372
12.887
-5.470
8.241
-20.398
12.728
-12.728
373
9.192
-9.192
17.678
-17.678
23,335
-23.334
374
11.046
-4.689
21.172
•8.987
33.138
-14.066
375
6.364
-6.36*
13.808
-5.861
31.297
-13.289
376
6.444
-2,735
10.126
•4.298
15,762
6.368
377
8.285
-3,517
6.489
•8.485
12.887
-5.470
378
1.485
-8.485
7,071
"7.071
11.967
-5.079
379
5.657
-5.657
9.192
.9.192
11.046
-4.689
380
9,192
-9.192
12.021
-12.021
15.556
-15.556
361
14,835
5.99*
26.000
0,000
31.000
0,000
382
17.000
0.000
21,000
0,000
25.000
0,000
383
12,887
-5.470
19,000
0.000
23.000
0.000
384
8,485
-8.485
18,000
0,000
25.000
0.000
385
7.778
-7.778
15,000
0.000
23.000
23.93J
0-000
-10.159
386
7.417
2.997
15,000
0,000
387
9,27?
3.7*4
17.000
0.000
24.654
-10.550
388
5.523
-2.344
18,000
0.000
26.695
-11.331
389
3.682
-1.563
10.126
-4.298
2*. 85*
-10.J50
390
3.709
1.498
14.000
0.000
23.013
-9,769
391
1,485
8.485
14.000
0.000
22.000
0,000
392
7.776
T.778
10.000
0.000
26.000
0,000
393
6,364
6.364
3.536
3.5S6
22.000
0.000
394
9.272
3.746
10.000
0.000
24.000
0,000
395
3.907
9.205
6.000
0.000
28.000
0.000
396
7.071
7.071
7.417
2.997
20.000
0.000
39?
5.657
5.697
13,908
5.61?
21.000
0.000
398
9.192
9.192
16.000
0.000
26.000
0,000
399
12.981
5.2*5
12.981
3.245
16..69f
6.743
400
14.639
5.994
18.410
.7.815
22.000
0,000
401
13,908
5.619
19,000
0,000
16.000
0.000
402
16.000
0.000
12,991
5.245
U'.OOO
0,000
403
17.617
7.118
19.471
7.867
21.325
8,616
404
17.000
O.QOC
16.971
16.971
23.339
23.339
405
22.252
8.991
15.238
35.900
16.41V
38.661
406
14,849
14,849
17.1«2
40.502
35. 35 J
35.355
407
17.617
7.116
21.881
51.548
22,662
93,389
408
21,000
0,000
44.000
0.000
38.18*
36,184
409
U.046
-4,689
56.000
0.000
54.70*
22,102
410
14,1<>2
-14.142
27.619
-U.722
53,000
0.000
'. U
8,485
-8.485
31.297
-13,285
35.900
-15,238
412
7.364
-3.126
21.920
-21.920
22.627
-22,627
413
. 4.243
-4.2*3
7,071
.7.071
22.627
-22,627
414
6.000
0.000
U,967
-5.079
28.536
-12.113
415
8,000
0.000
14.728
.6.292
12.728
-12,T28
416
7.000
0.000
12.887
•5.470
8.48}
-8.499
417
4.000
0.000
11,046
.4.689
6.46!
-8, 485
418
6.490
2.622
4,602
-1.994
5.697
-9.657
419
9.000
0.000
9.000
0,000
4;634
1.873
420
7.364
-3.126
9,000
0.000
10.607
10,607
421
5.563
2.248
7,000
0.000
10.199
4.121
422
9.272
3.7*6
9,000
0.000
10.000
O.QOO
88
Table 15 (continued)
INPUT VARIABLES
SEQ
423
*2«
429
426
♦ 27
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
44S
449
450
451
452
453
454
455
456
457
498
459
460
461
462
463
464
469
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
489
486
487
488
489
490
491
492
493
494
499
496
497
498
499
900
901
902
903
4.835
4.142
5.762
2.292
1.000
2.981
1.0'i6
7.364
7.071
4.000
1.124
-1.95*
-0.927
0.000
0.000
1.854
5.523
•2.828
-5.6'7
0.000
0.000
0.000
0.000
0.000
0.000
3.371
0.000
2.997
3.371
4.870
5.994
9.619
7.071
7.778
11.967
6.444
13.000
12.000
18.000
13.808
6.364
6.444
6.000
7.000
9.000
3.908
2.728
0.398
5.034
32.451
5.000
6.263
1.213
5.556
5.649
2.728
5.649
6.569
9.192
9.192
7.364
9.205
5.657
9.899
14.849
6.364
5.470
4.298
7.071
8.596
9.192
4.000
2.053
1.126
6.689
0.199
2,000
4.728
8.410
4.726
7.490
5.994
14.142
6.368
8.991
0.000
5.245
-4.689
-3.126
-7.071
0.000
-2.782
-4.603
-0.375
5.000
6.000
0.749
-2.344
-2.828
-5.6S7
-9.000
-12.000
-17.000
-13,000
-10.000
-13.000
-8.345
-9.000
-7.417
-8.345
-12.053
-14.835
-13.908
-7.071
-7.778
-5.079
-2.735
0.000
0.000
0.000
-5.861
-6.364
-2.735
0.000
0.000
0.000
5.619
12.728
8.241
10.114
13.111
0.000
•16.263
-21.213
-15.556
-6.642
•12.728
-6.642
-7.033
-9.192
-9.192
-3.126
-3.907
5.657
9.900
14.849
6.364
12.887
10.126
7.071
20.251
9.192
0.000
4.870
4.495
6.743
4.12J
0.000
-6.252
-7.815
-6.252
-7.424
9.899
17.617
10.607
21.920
15.762
20.338
18.544
8.000
8,000
10.000
4.603
.090
.8*1
,3*4
,000
,4*4
,657
,245
,000
0.000
o.o'oo
-5.8*1
•5.3*1
0,000
-7.033
4,495
0.000
0.000
4.870
0.000
0.000
9.3*5
11.314
7.118
10.607
10.607
19.000
19.000
23.000
27,000
6,364
10,000
6,000
10.126
IS.fOB
11.9*7
18.544
29.034
41.723
22.627
49.432
34.648
27.577
22.627
19.799
16.2*3
20.506
16.263
20.291
14.849
10.607
11.9*7
15.7*2
19.956
17". 67 8
17.617
8.205
8.987
12.894
-0.000
5.861
9.899
17.617
8.48!>
9.657
6.3*4
9.900
14.142
20.506
18.385
17.000
9.900
7.118
10.607
21,920
6.36B
8.2*1
7.492
0.000
0,000
0.000
-1.954
0.000
-0,781
•3,126
0,000
-2,735
-5,697
-12.981
-12.000
-13.000
-13.000
-13,808
-13,808
-16.000
-16,569
-11,126
-6.000
-12.000
-12,053
-16,000
-26.000
-23.180
-11.314
-17.617
-10,607
-10,607
0,000
,000
,000
.000
.3H>
,000
.000
,298
,861
.079
.492
.114
•5.
-5.
7.
10,
16.857
22.627
18.356
-34.648
-27.577
-22.627
-19.799
-16,263
-20.906
-16.263
-8.596
-14.849
-10.607
•5.079
6.368
15.596
17.678
7.118
19.331
21.172
30.377
25.000
13.808
9.900
7,118
8.485
5.697
•6.364
•9.899
-14.142
-20.906
-18.385
0.000
16.689
23.000
20.398
18.544
19.799
25.4"56
15.238
12.000
16.000
14.835
12.981
1.873
8.485
10.126
10.607
14.1*2
15.359
31.113
22.627
0.000
-10.199
-37.087
10.489
0.000
5.619
11.0*6
•4.000
0.000
10.489
0.000
11.238
11.238
9.7*0
10.489
16.263
16.569
22.000
28.000
33.000
33.000
26.69S
11.046
13.435
19.649
21.920
27.619
32.000
4 3.000
46.000
48.214
60.104
41.719
28.991
28.284
22.627
24.854
22.627
21.213
19.799
14.142
13.000
17.490
20.000
23.961
21.329
28.74)
26.163
8.996
28.991
-0.000
5.861
5.079
5.470
9.470
1.414
7.778
12.021
18.383
37.477
36.820
39.900
6.743
0.000
8.241
7.492
19.799
25.456
35.900
0.000
0.000
5.994
5.245
-4.636
-8.485
-4,298
-10.607
-14.1*2
-38.015
-31.113
-22.627
-14.000
-4.121
-14.984
-25.961
-22.000
-13.908
-4.689
0.000
-21.000
-25.961
-43.000
-27.815
-27.815
-24.107
-25.961
•16.263
-7.033
0.000
0.000
0.000
0.000
-11.331
-4.689
-13.435
-6.642
-21.920
-U.722
0.000
0.000
0.000
19.480
60.104
•41.719
•28,991
•28,284
-22,627
•10,550
•22.627
-21,213
•19.799
•14.142
0.000
-7.424
0.000
10,489
8.616
11.613
26.163
20.251
2A.991
58.000
13.808
11.967
12.887
12.887
1,414
•7,778
•12,021
•18,185
-37,477
•19,629
-15,238
89
Table 16 Separation of a multivariate (6) set of Rantoul , Illinois,
October 1950-55, upper wind components, zonal and meridional
mixed distribution, at the 700-, 500-, and 300-mb levels into
two separate distributions. The units are m-s"^.
PROGRAM NORMIX
Mifi NQRMiL MIXTURE ANALYSIS PROCgDURE ( 197* REVISION)
SAMPLE • RANTDUL
OCTOBER
70a/500»AND 300MB
X AND Y COMPONENTS
SAMPLE SIZE • S03
NUMBER OF VARIABLES • »
NUMBER HE TYPES • f
ITERATION NUMBER 39
ClKELIHDDO OF 2 TYPES IN THIS SAMPLE • -0.73297066D 0*
CHARACTERISTICS OF THE WHOLE SAMPLE
MEAKS
12 3*56
7.3252 -0.9*86 10.6700 -2.570* 15.*0*9 .3.6999
STANPA»0 DEVIATIONS
1 2 3 * 5 *
6.6262 6. 8921 8.959* 10.31*9 13.70B2 16.0056
CORRELATIONS
1
2
3
*
5
6
I. 0000
0.2215
0.752S
0.2313
0.5589
0.2311
0.2215
1.0000
O.U3«
0.7581
0.1265
0'.5757
0.752!
0.1138
I. 0000
0.1587
0.7513
0.1977
0.2313
0.7581
0.1587
1.0000
0.1200
0.7856
0.5589
C.1265
0.7513
0.1200
I. 0000
0.1266
0.2311
0.5757
0.1977
0.7856
0.1266
I '.0000
CHARACTERISTICS OF TYPE 1
THE PROPORTION OF THE POPULATION FROM THIS TYPE. 0.171
MEANS
12 3*'?
10.*5*5 -2.0123 16.2238 -2.6858 18.763* .3.325*
1 2
STANDARD DEVIATIONS
3*5
5 *
7.*317 9.0069 12.5099 17.*952 20.1227 ?9.783*
CORRELATIONS
lloOOO 0^3578 o!607* 0.**01 0.5639 0.3522
0.3578 1.0000 0.0J97 0.7*69 0.0730 '•""
6076 i.0297 1.0000 0.1637 0.7*20 0.1966
O.**0l 0.7*69' 0.U37 1.0000 0.11*8 °'l*.lt ■
0.5639 0.0730 0.7*20 0,11*8 1.0000 0.0799
o!3522 0.6333 0.1966 0.788* 0.0799 1.0000
EIGENVALUES EIOENVEeTORS 3*56
,164 8027 o!l01* o!l798 0.1382 0.2729 o;.*299 0-"*0
iSV??79 86* .0.0377 0.38*0 -0,17*9 0.7770 .0.*266
!oe *213 1060 0.**67 -0.20*3 0.7783 0.0*92 -O-^Ol
5^7679 **98 -0.0*90 0.7*97 0.199* .0.*398 -0.0070
lo'Jell o'.ull 8592 0.0*71 .o.*e2* -o nss 0.0091
22 0357 8529 -0.161S .0.*76e -0.1293 0.0181 0.0*J7
CHARACTERISTICS OF TYPE 2
THE PROPORTION OF THE POPULATION FROM THIS TYPE. 0.829
MEANS
12 3*56
6. 6783 -0.7287 9.5n8 -2.5*66 1*.7107 -J. 7773
STANDARD DEVIATIONS
12 3*5*
6.25*8 6.3*57 7.5377 8.0693 11.8*22 11.2131
90
Table 16 (continued)
CnnRELATIONS
1.0000
0.2039
0.8038
0.1453
0.5563
0.1878
EICFNVALUES
236.2003
155.9890
31.5369
29.5496
8.3730
6.7008
0.2039
l.oono
0.1937
a.79<.3
0. 1673
0.5'''t9
0.8038
0. 1937
l.UOOO
0.1721
0.7*26
0.2195
EIGENVECTORS
1
0.2*83
0.2529
0.3590
0.3591
0.5713
0.5387
-0.2052
0.2350
-0.3U0
0.4036
-0.5859
0.5457
7943
1673
1721
7625
1
0000
1260
1280
1
0000
7914
1756
}
4
6858
-0
0776
1020
6736
4636
-0
0919
0176
4196
5400
0860
1120
-0
5902
0.5849
0.2195
0.7914
0.1756
1.0000
5
6
0.3252
5605
0.5317
-0
3660
0.4232
-0
6083
0.6195
3846
0.0879
1540
0.2024
-0
1008
SAMPLE • RANTOUL
OCTOBER
700/500*4NB 300MB
X ANO y COMPONENTS
PROBABILITIES
OF TYPE MEMBERSHIP
1 2
0.065 0.935
0.011 0.989
0.015 0.985
0.996 0.004
0.022 0.979
6 0,037
7 0.026
8 1.000
9 0.119
0.851
1 0.981
963
974
000
881
149
019
0.005 0.995
0.007 0.993
0.003 0.997
0.122 0.878
0.976
0.034
8 0.054
9 0.088
1.000
1 0.003
2 0.125
3 0,269
0,011
0.005
0,024
0.966
0.9*6
0.912
0.000
0,997
0,875
0.731
0.989
0.995
1,000 0,000
0.028 0,972
987
706
0.013
0.294
0,218 0.782
0.029 0.971
0.034 0,966
0.011 0,969
0,037 0,963
0,407 0.593
0.011 0.989
37 0.008 0.992
8 O.OOS 0.995
9 0.005 0,995
0,003 0,997
1 0,323 0.677
0.006 0.994
0.0*3 0.957
0.017
0,025
0.071
0,983
0,975
0,929
5
6
7 0,016 0,9S4
8 0,916 0,084
9 0,010 0,990
0.013 0.9J7
1 0,013 0,9S7
2 0,181 0,819
3 0,020 0.960
4 0.996
5 0,005
6 0.009 0,991
7 0.012 0.968
8 0.030 0,970
9 0.028 0,972
0,008 0,992
0.004
0.995
0.008 0.
0.008 0,
0.004 0,
0.003 0,
0,017 0,
992
992
996
997
983
0.993
0.995
0,979
,991
.978
,866
,919
.968
,766
PROBABILITIES
OF TYPE MEMBERSHIP
1 2
66 0,010 0,990
67 0,009 0,991
68 0,007 0.993
69 0,005 0.995
70 0.007 0.993
71 0,007 0,993
72 0,004 0.996
73 0.007
74 0.005
75 0.021
76 0.010 0.990
77 0,093 0.907
76 0,024 0.976
79 0,025 0,975
80 0.014 0,986
81 0,036 0.962
8? 0,010 0,990
63 0,050 0.950
84 0,163 0.837
85 0,066 0.934
86 0.008 0,9y2
0,016 0.984
0.009
0.022
0.114
0,081
0,032
0,234
9» 0,048 0,952
95 0,014 0.986
96 0.205 0,795
97 0.031 0.969
98 0.058 0.942
99 D.O55
lOP 0.013
101 0.015
102 0.403
103 0.873
104 0.009
105 0.009
106 0,045
107 0.875
108 0,028 0.972
109 0.014 0,986
110 0,010 0,990
in 0,008 0,992
112 0,004 0,996
113 0.009 0.991
114 0,005 0,995
115 0,026 0,974
116 0,004 0,996
117 0,013 0,987
118 0.037
119 0.013
120 0,009
121 0,034
122 0,023 0.977
123 0.019 0.981
124 0.069 0.931
125 0,685 0,315
126 0.106 0.694
127 0,615 0,385
128 0.811 0,189
129 0,892 0.108
130 0,006 0,994
87
86
89
90
91
92
93
.945
0.987
0.985
0.597
0.122
0.991
0.991
0.955
0.125
0.963
0.987
0.991
0.966
PROBABILITIES
OF TYPE MEMBERSHIP
1 2
131
009
991
132
037
963
133
075
925
134
006
994
135
019
981
136
008
992
137
018
982
136
016
984
139
027
973
140
926
074
141
998
002
142
030
970
143
240
760
144
023
977
145
069
931
146
027
973
147
on
989
146
016
c
984
149
010
990
150
020
980
151
061
939
152
056
944
153
047
953
154
272
728
155
033
967
156
014
986
157
019
981
156
on
989
159
038
962
160
006
994
161
003
997
162
002
996
163
004
996
164
003
997
165
004
996
166
004
996
167
015
985
168
005
995
169
004
996
170
.004
996
171
,005
995
172
092
908
173
on
989
174
0:3
987
175
013
987
176
023
977
177
098
902
178
■ 007
993
179
.311
689
180
998
002
181
997
003
182
977
023
183
.857
1*3
184
.204
.796
185
,043
,957
186
,020
.980
187
.007
.993
138
,008
.992
189
,006
.994
190
,010
.990
191
.020
.980
192
,063
.937
193
.007
.993
194
.006
994
195
.012
988
91
Table 16 (continued)
PROBABILITIES
OF TYPE MEMBERSHIP
1 2
19t, 0,208 0.792
197 0, 121 0.B79
199 0.023 0.977
199 0,015 0.985
200 0.013 0.987
201 0.397 0.603
202 0.983 0.017
203 0.073 0.927
20* 1.000 0.000
205 0.005 0,095
206 0.006 0.99*
207 0.005 0.995
208 0.005 0.995
20' 0.021 0.979
2^0 0,003 0.997
211 O.OOe 0.992
212 0.003 0.997
213 0.022 0.978
214 0.107 0.893
215 0,0*5 0,955
214; 0,371 0.629
217 0.066 0.93*
218 0.131 0.669
219 0.019 0.961
220 0.01* 0.986
221 0.01* 0,986
222 0,007 0,993
223 0,013 0,987
22* 0.011 0.989
225 0.010 0.990
226 0.006 0.99*
227 0.013 0.987
228 0.005 0.995
229 0,005 0,995
230 0.011 0.989
231 0.062 0.938
232 0.00* 0.996
233 0.011 0.989
23* 0.035 0.965
235 0.00* 0.996
236 0.006 0.99*
237 0.015 0.985
238 0.025 0.975
239 0.007 0.993
2*0 0.005 0.995
2*1 0,012 0.988
2*2 0.181 0.819
2*3 0.007 0.993
24* 0.008 0.992
2*5 0.037 0.963
2*6 0.03* 0.966
2*7 0.922 0.078
2*8 0,08* 0,916
2*9 0,020 O,980
250 0,017 0,983
251 0,006 0,99*
252 0,003 0,997
253 0,006 0.99*
25* 0.006 0.99*
255 0,026 0.97*
256 0.213 0.787
257 0,033 0,967
258 0,211 0.769
259 0.063 0.937
260 0.025 0.975
261 0.017 0.983
262 0.016 0.96*
263 0.012 0.988
26* 0.033 0.967
265 C.023 0.977
266 0.028 0,972
267 0,200 0,800
268 0.197 0.803
269 0.6*9 0.351
270 0.013 0.987
271 0.015 0.965
272 0.005 0.995
273 0.011 0.989
27* 0.018 0.982
275 0.012 0.988
276 0.031 0.969
277 0.057 0.9*3
278 0.087 0,913
279 0,017 0.983
280 0,029 0.971
281 0,009 0.991
282 0.013 0.967
283 0.012 0.988
28* 0.023 0,977
283 0.058 0.9*2
PROBABILITIES
OF TYPE MEMBERSHIP
1 2
286 0.566 0.43*
287 0.1*5 0.855
288 0.032 0.968
289 0.056 0,9**
290 0.0*1 0.959
291 0.132 0,868
292 0,020 0,960
293 0.929 0.072
29* 0.1*5 0.855
295 0.028 0.972
296 0.088 0,912
297 0.195 0.805
298 0.039 0.961
299 0.933 0.067
300 0.103 0.997
301 0.*15 0,585
302 0,023 0,977
303 0,02* 0,976
30* 0,008 0,992
305 0,069 0.931
306 0.0*6 0.95*
307 0.051 0.9*9
308 0.010 0.990
309 0,009 0,991
310 0,020 0,980
311 0,030 0,970
312 0,016 0,98*
313 0,038 0.952
31* 0,050 0,950
315 0,035 0,965
316 0,96* 0,036
317 0.*e6 0,51*
318 0,22* 0.776
319 0.171 0.829
320 0.36* 0.636
321 0.166 0.83*
322 1,000 0.000
323 0,023 0,972
32* 0,811 0, 189
325 0,6*3 0,357
326 0,367 0,633
327 0,999 0,001
320 1,000 0,000
329 1.000 0.000
330 1,000 0,000
331 0.018 0.982
332 0.980 0.020
333 1.000 0.000
33* 0,931 0.069
335 0.796 0.20*
336 0.998 0.002
337 0.635 0.*65
338 C.*76 0.52*
339 0.9*0 0.060
3*0 0.009 0.991
3*1 0.011 0.989
3*2 0.005 0.995
3*3 0.037 0.963
3** C.006 0.99*
3*5 0.026 0.972
3*6 0.011 0.989
3*7 0.005 0.995
3*8 0,007 0,993
3*9 0,0*7 0.953
350 0,019 0.981
351 0.013 0.967
352 0.206 0.792
353 0.021 0.979
35* 0.020 0.980
355 0.080 0.920
356 0.053 0,9*7
357 0.036 0.96*
358 0.055 0.9*5
359 0.02* 0.976
360 0.310 0.690
361 0.130 0.870
362 0,111 0.869
363 0,069 0.931
36* 0.139 0.861
365 0,193 0.807
366 0.007 0.993
367 0,101 0,899
368 0,031 0.969
369 0.015 0.965
370 0.95* 0.0*6
371 U.168 0,912
372 0.386 0.612
373 0,090 0.910
37* 0.0*8 0,952
375 0.026 0,97*
PROBABILITIES
OF TYPE MEMBERSHIP
1 2
376 0.016 0-98*
377 0,008 0,992
3/9 0,011 0.989
379 0.010 0.990
380 0.012 0.989
381 0.06* 0.936
382 0,017 0.983
383 0.04* 0,956
38* 0,119 0,881
365 0,0*7 0,953
386 0,01* 0,986
387 0.020 0.980
388 0.107 0.893
389 0.006 0,99*
390 0i0l5 0.985
391 0.012 0.988
392 0,013 0.987
393 0.027 0.973
39* 0,008 0,992
395 0.028 0.972
396 0.007 0.993
397 0.007 0,993
398 0.017 0.983
399 0.011 0.989
*00 0.207 0.793
*01 0.02* 0.976
*02 0.087 0,913
*03 0,021 0,979
*0* 0,9*2 0.056
*05 I. 000 0.000
*06 1,000 0,000
»07 1,000 0.000
*08 1.000 0.000
»09 1.000 0,000
*10 0,816 0,18*
*ll 0,996 0,00*
*12 0,991 0,009
*13 0,0*5 0,955
*1* 0,010 0.990
*15 0.025 0,975
*16 0,020 0,980
*17 0.01* 0.986
*18 0,007 0.993
*19 0,012 0,9S8
*20 0,02* 0,976
*21 0,006 0,99*
*22 0,008 0.992
*23 0,099 0,901
*2* 0,021 0,979
*25 0,157 0.843
*26 0,992 0.008
*27 0,09* 0.906
*2e 0,238 0,762
*29 0,997 0.003
*30 0,008 0.992
*31 0,023 0.977
*32 0.007 0.993
433 0.005 0.995
*3* 0,006 0,99*
*35 0,003 0.997
*36 0,010 0.990
437 0,005 0,9«S
*36 0,006 0,994
4J9 0,980 0.020
4*0 0,292 0,708
441 0,015 0.985
442 0,007 0,993
443 0.062 0.918
444 0,967 0.033
445 0.161 0,839
446 0.011 0,989
447 0.055 0.945
448 0.009 0«'9l
449 0.015 0.985
490 0,015 0,965
451 0,025 0.975
452 0,914 0.086
453 0,183 0,817
4}4 0,061 0,9J9
4J5 0.048 0.9S2
456 0,033 0.967
457 0,015 0,985
453 0.014 0,986
439 0.014 0.986
460 0.012 0,988
461 0,023 0,977
*62 0.37* 0.626
*63 0,028 0.972
464 0,C10 0.990
*65 0.U21 0.979
92
Table 16 (continued)
PROBABILITIES
OF TYPE MEMBERSHIP
1 2
466 0,009
0.995
OftT 0.055
0.945
♦66 0.071
0.929
»69 0.019
0.981
*70 0.265
0.735
'.71 U.999
0.301
*72 1.000
0.000
*73 1.000
0.000
<.7» 1.000
0,000
*73 0.556
0,444
»76 0.241
0,759
477 0.132
0:868
478 0.030
0.970
PROBABILITIES PROBABILITIES
OF TYPE MEMBERSHIP OF TYPE MEMBERSHIP
12 12
47? 0.194 0.806
480 0,037 0.963
481 0.362 0.638
482 0,032 0.968
483 0,051 0,949
46^ 0,007 0.993
485 0.015 0.985
486 0,029 0.971
487 0,067 0,933
48H 0,015 0,985
489 0,054 0.946
490 0.087 0.913 503 0.955 0,045
491 0.998 0.OO2
492
1
000
000
49=<
045
955
494
312
688
495
!71
829
4S6
0.J41
959
497
332
66S
498
035
965
499
023
977
500
.023
977
501
&
,785
215
502
.149
851
LOGARITHM OF LIKELIHOOD RATIO OF J TO I TYPtS • 0.169250330 03
CHI-SOUARE WITH 54 DEGBEES OF FRFETOi'. 372,86
PI<a8'BILITY OF NULL HYPOTHESIS, 0.00000000
9?
Table 17 Separation of a multivariate (6) set of Rantoul , niinois,
October 1950-55, upper wind components, zonal and meridional
mixed distribution, at the 700- , 500- , and 300-mb levels into
three distinct distributions. The units are m-s"^.
PROGRAM NORMIX
WOLFE NORMAL MIXTURE ANALYSIS PROCEDURE (1974 REVISION)
SAMPLE • RANTOUL
OCTOBER
700/500/ANO 300M8
X 4ND Y COMPONENTS
SAMPLE SHE ■ SOS
NUMBER DF VARtiBLFS • 6
NUMBER OF TYPES ■ 3
ITERATION NUMtER 101
LIKELIHOOD OF 3 TYPES IN THIS SAMPLE . -0,727177290 0*
CHARACTERISTICS OF THE WHOLE SAMPLE
MtANS
1
2
3
4
5
6
7.J232
-0.9'i86
10.6700
.2.570*
IJ.4049
.3'. 6999
STANDARD OBVIATIONS
1
2
S
*
5
6
6.62D2
6.6921
8.9S9A
10.31*9
13.70S2
16'.0OS8
CPRRELATIONS
1
2
3
*
5
6
I. 0000
0.2215
0.7525
0.2313
0.5389
0'.231l
0.2215
1.0000
0.113S
0.7581
0.126S
0-.5TJ7
0.7525
0.1138
1.0600
0.1587
0.7513
0'.1977
0.2313
0.7581
0.1587
1.0000
0.1200
0'.7I56
O.SJJ?
0.1265
0.7513
0.1200
1.0000
0.1266
0.2311
U.5757
0.1977
0.7B56
0.1266
r.oooo
CHARACTERISTICS OF TYPE 1
THE PROPORTION OF THE POPULATION FROM THIS TYPE» 0.182
1
2
3
4
J
6
10.5173
-1.7616
16,0**7
-2.8166
19.4572
-3.J912
STAi^DARD DEVIATIONS
1
2
3
4
5
*
7.<.715
9.10*3
12.1651
17.3130
19.5*70
29.2311
correCations
1
2
3
4
5
6
I. 0000
0.2932
0.6046
0.3978
0.5377
0.3250
0.2932
1.0000
-0.0231
0.7480
0.05*0
0.6252
0.60«6
-0.0233
1.0000
0.1336
0.719*
o;iTi9
0.3978
0.7480
0.1S36
1.0000
0.0917
0.790*
0.5377
0.05*0
0.7194
0.0917
1.0000
O'.0S73
0.3250
0.6252
0.1719
0.7906
0,0573
I'.OOOO
EIGENVALUES
EIGENVECTORS
1
2
3
*
3
6
1122.2211
0.0936
0.1893
0.0998
0.3370
o;si3*
0.7539
481. ^OS*
0.1910
-0.0S7J
0.4161
-0.1832
0.72*1
-0.4807
107.0029
0.0873
0.447S
-0.2426
0.733*
0.0*23
-0.4409
59.8619
0.4557
.0.0371
0.7255
0.2552
-0'.*»*4
0.0433
29.3601
0.0895
0.8629
0.0824
.0.*746
.0.1111
0.0303
23.1239
0.8353
.0.128*
-0.4742
-0.1571
0;0274
0.0403
CHARACTERISTIeS OR TYPE 2
THE PRDPCtRTION OF THE POPULATION FROM THI$ TVPB» 0.183
MEANS
^23456
5.9298 1.1741 11.3*42 0.4841 19.9322 .4.8802
4.0432
STANDARD DEVIATIONS
2 3 4 J
6.0597 6.8744 4.7102 8.3419
6
7.2*47
94
Table 17 (continued)
CBRdELATinNS
; ' 1
2
S
4
5
t,
1.0000
•0.3849
7454
-0.0248
0.7052
0'.3055
■ -0.3849
1.0000
-0,
5189
0.6927
-0.5375
0;4247
0.74 J*
-0.91S9
1.
0000
-0.3651
0.8079
0.0263
-0.0248
0.6927
-0.
3451
I. 0000
-0.3992
o;7287
0.70S2
-0.5375
0.
8079
-0.3992
1.0000
-0.1160
0.30S5
0.4247
0.
0263
0.7287
-0.1160
r.oooo
EIGENVALUES
EIOBNVECTOR!
2
3
4
1
6
J37.9714
234S
0.2674
.0
1184
-0.0908
0'
5475
0.7426
71.7321 .0
3902
0.2424
7711
0.3460
-0
0«10
0.2609
16.9294
5133
0.2587
.0
0625
0.7630
1223
-0.2621
9.8691 -0
2481
0.3*73
1214
-0.2410
0'
6562
-0.5478
5.1576
6998
0.2014
4947
-0.4725
-0'
2137
-0.1023
3.2296 -0
1801
0.7921
.0
3577
-0.0925
-0'
4499
0.0351
CHARACTERISTICS Or TYPE 3
THE PROPORTIDN OF THE POPULATION FROM THIS TYPE* 0.635
MEANS
1
2
3
4
5
6
6.8098
-1.3264
8.9194
-3.3796
12.9361
-3.3911
STANDARO DEVIATIONS
1
2
S
4
5
6
6.6829
6.2294
7.6649
8.4692
12.2260
11.9304
CORRELATIONS
1
2
S
4
5
*
I. 0000
0.3441
0.8404
0.1994
0.5799
0.1907
0.3441
I. 0000
0.3814
0.8169
0.2775
0.6420
0.8404
0.3814
1.0000
0.2586
0.7613
0.2987
0.1994
0.81*9
0.258*
1.0000
0.1608
0.8260
0.5799
0.2775
0.7613
0.1608
1.0000
0.2*43
0.1907
0.6420
0.2*87
0.8260
0.2643
1.0000
EIGENVALUES
EIGENVECTORS
1
2
3
4
5
6
278.7235
0.2436
-0.2469
0.6157
-0,2932
0.250*
0.5932
154.5198
0.2676
0.1768
0.2977
0.5868
0.6060
-0.3113
37.4431
0.3437
-0.3103
0'.4015
-0.2494
-0.3559
-0.6599
22.2415
0.3694
0.4038
0.1879
0.4234
-0.6307
0.2970
6.6393
0.5399
-0.6093
-0.4)85
0.2521 .
0'.0212
0.1577
6.1806
0.5683
0.526*
-0.2955
-0.5145
0".2l24
-0.051S
PROCRAH NORMIX
WQLPE NORMAL MIXTURE ANALYSIS PROCEDURE ( 1974 REVISION)
SAMPLE ■ RANTQUL
OCTOBER
700<500/AND 300HB
X AND Y COMPONENTS
PROBABILITIES
OF TYPE MEMBERSHIP
1 2 3
1 0,249 0.001
2 0.004 0.944
3 O.OOl 0.993
4 1.000 0.000
5 0.001 0.998
6 0.015 0.942
7 0.032
8 1.000
9 0.027
lO 0,90*
0.750
0.052
0.005
0.000
0.001
0.043
0.959
0,009
0.000 0.000
0.960 0.013
.. -. __ 0.000 0.094
11 1,000 0,000 0.000
12 0,007 0.062 0.932
0.196
0.006
0.000
0.000
0.000
0.001
0.001
0.000
0,052
0.000
13 0,008
14 0,005
15 0,241
16 0,960
17 0,041
18 0,116
19 0,198
20 1.000
21 0,008
22 0.102
23 0,219
24 0.014
25 0.003
26 1.000
27 0,034
28 0,013
29 0,087
796
990
759
040
959
8(3
.802
000
0.941
0.898
0.000 0.781
0.000 0.986
0.614 0.383
000 0.000
000 0.966
768 0.219
900 0.012
30 0,267 0.000 0.733
PROBABILITIES
OF TYPE MEMBERSHIP
1 2 3
31 0.033 U.OOO 0.967
32 0,028 0.000 0.972
33 0.012 0.011 0.978
34 0.039 0.000 0.9*1
35 0.495 0.000 0.509
36 0,011 0.006 0.983
37 0.006 0,535 0.459
3» 0.005 0.675 0.321
39 0,002 0.772 0.225
40 0.004 0.618 0.377
41 0,740 0.000 0.2*0
42 0.006 0.536 0.458
43 0,037 0,009 0-95*
44 0,014 0.379 0.607
45 0.028 0.005 0.967
46 0,065 0.001 0.934
47 0,016 0.125 0.858
4e 0,907 0.000 0.093
49 0,011 0.000 0.989
50 0.014 0.000 0.986
51 0,014 0.000 0.986
52 0,164 0.000 0.836
53 0.015 0.000 0.985
54 1.000 0.000 0.000
55 0,004 0.000 0.996
56 0.008 0.000 0.992
57 0,011 0.000 0.989
58 0,089 0,000 0.911
59 0,019 0.000 0,981
60 0.008 0.000 0.991
PROBABILITIES
OF TYPE MEMBERSHIP
1 2 3
61 0,009
0.001 0.991
62 0.013
0.379 0.607
63 0,006
0.081 0.913
64 0,006
0,088 0.907
65 0,023
0.014 0.9*3
66 0,017
0.205 0.778
67 0.019
0.038 0.943
68 0,007
0,224 0.7*9
69 0,006
0.183 0.811
70 0,007
0.190 0.803
71 0.007
0.716 0.277
72 0.003
0.722 0.275
73 0.004
0.864 0.133
74 0,004
0,636 0.3*1
75 0,019
0.504 0.477
76 0.011
0.227 0.7*2
77 0,018
0.955 0,047
78 0,024
0.003 0.973
79 0,050
0.206 0.7*3
SO 0,021
0.166 0.813
81 0.053
0.067 0.881
82 0.013
0.»38 0,549
83 0.203
0,000 0.796
84 0,022
0,969 0,009
85 0,031
0,844 0.125
86 0,011
0.106 0.883
87 0,017
0.138 0.849
88 0,013
0.424 0.563
89 0,017
0.754 0.229
90 0,273
0.029 0.»98
95
Table 17 (continued)
PROBABILITIES
OF TYPE MEMBERSHIP
1 2 3
PROBABILITIES
OF TYPE MEMBERSHIP
1 2 3
PROBABILITIES
OF TYPE MEMBERSHIP
1 2 3
91
0,
108
0.
004
0.
386
92
0.
032
0.
008
0.
960
93
0,
663
0.
000
0.
337
9*
0,
0*5
0.
536
0,
419
95
0,
020
0,
on
0.
9»9
96
0.
3*0
0,
000
0,
659
97
0,
030
0.
000
0.
970
98
0,
060
0.
811
0.
129
99
0.
053
0,
000
0.
947
100
0,
020
0.
ooo
0,
980
101
0,
020
0.
000
0.
980
102
0.
773
0.
038
0.
189
103
0,
969
0.
000
0.
on
10*
010
001
989
105
0,
007
629
0,
364
106
0,
0*8
000
952
107
0,
777
000
223
loa
0,
021
000
0.
979
109
0,
012
000
0.
968
110
0.
008
000
991
111
0,
007
0.
000
0.
993
11?
0,
003
0.
000
0.
997
113
0,
006
0.
000
0,
994
u*
0,
003
0.
000
0.
997
115
0,
071
0.
052
0.
877
116
0.
005
0.
006
0.
988
117
0.
009
0.
781
0.
210
ue
0,
052
0.
107
0,
6*1
119
0.
016
0.
009
0.
976
120
0.
009
0.
3*0
0.
652
121
01*
901
OS*
122
019
000
981
123
028
006
9*6
12'.
1*9
018
833
125
284
702
014
126
082
000
918
U7
360
625
015
128
655
000
145
129
868
000
132
130
009
001
991
131
003
000
992
132
025
000
975
133
059
000
9*1
13*
006
.002
.992
135
,021
.005
.974
136
010
001
989
137
018
001
9Pi
138
016
000
984
139
023
000
977
UO
835
000
.165
Ul
985
000
01!
1*2
030
,000
970
1*3
272
,000
.728
14*
02*
,000
976
1*5
166
,050
764
1*6
008
9*3
0*9
1*7
015
.013
.971
1*8
,007
,870
,12?
1*9
.012
.030
,958
130
,021
,018
.962
151
,058
.000
.942
152
,05*
.000
.946
153
,097
.001
.902
154
,179
.000
.821
155
,033
.000
.9*7
156
,025
.001
.973
157
,032
.158
.810
158
.013
.010
.977
159
.077
.000
.923
160
.008
.4*8
.54*
161
.005
,007
.988
162
■ Oil
000
.988
163
.039
.123
.833
16*
.006
.723
.271
165
.007
.2*1
.752
166
.002
.77*
.22*
167
,00*
.9*7
.049
168
.00*
.831
.164
169
.002
.837
.1*1
170
.002
.838
.1*0
171
.003
.809
.197
172
.106
.672
.222
173
.015
.292
.692
17*
,013
.399
.588
175
.012
.000
.987
176
,036
.000
.963
177
.115
.003
.881
178
.008
.025
.967
1T9
.435
.000
.564
100
.938
.062
.000
lei l.uuu u.ouu u
182 0.999 0.000
183 0.954 0.000
!64 0.149
185 0.033
186 0.023
167 o.ooa
168 0.005
189 0.004
190 0.010
191 0.042
192 0,049
000
824
000
292
672
635
062
262
000
001
046
S51
.143
,977
700
.323
.3*2
.929
.695
,951
193 0.007 0.042 0.951
194 0,007 0.271
195 0.014
196 0,100
197 0.197
199 0.02*
199 0,016
200 0.021
201 0.556
202 0.996
203 0.070
20*
205
206
207
20B
209
210
000
004
010
005
004
013
003
005
000
000
000
000
025
,000
,000
.000
.000
,000
.n03
722
981
900
603
976
,984
,95*
,444
,00*
.930
.000
,996
,987
,000 0.995
.000 0.996
,000 0.987
.000 0.997
211 0,010 0.000 0,990
212 0,002 0.000 0.998
213 0.073 0.000 0,927
214 0.078 0.000
215 0.026
216 0.255
217 0.039
218 0.084
219 0.017
220 0,012
221 0,019
222 U,007
223 0.012
224 0.010
225 0.010
226 0,006
227 0,015
,000
,000
,000
,000
.000
.000
.000
922
974
745
961
916
983
988
961
,000 0.993
0.988
0.989
0.985
051 0.943
010 0,975
,000
,000
,006
228 0.003 0.708 0,269
229 0,003 0,671 0.326
230 0,019 0.070
231 0,139
232 0,009
233 0.015
234 0.009
235 0,002
236 0.003
237 0,056
238 0,032
239 0,004
240 0.002
,124
,485
,725
,965
9U
737
306
2*0
026
672 0.126
942 0.055
112 0,831
839 0.129
95! 0.0*0
977 0.022
241 0.005 0.927 0,0*9
242 0,563 0.001 0.436
0,002 0.923 0,075
0.206 0.784
0.870
0,003
2*3
2*4 0,010
2*5 0.013
246 0,067
0.957
247
246 0.093
0.000
0.000
0.117
0.930
0.0*3
0.907
2*9
250
0,026
0.019
0.000 0.974
0.000 0.981
251 oiOlO 0,034 0.9J6
252 0.003 0.336 0.6*0
253 0,004 0.7*7 0.2*9
254 0,008 0.027 0,9*4
255 0,050 0.000 0.949
256 0.340 0.000 0.6*0
257 0.035 0.001 0.9*4
258 0,1*9 0.000 0.851
259 0,062 0.000 0.938
260 0,023 0,000 0,977
261 0.025 0,001 0,97*
262 0,012 0.653 0.335
263 0,012 0.108 0.880
26* 0.039 0.015 0.9*6
265 0,026 0,003 0.971
266 0.017 0.000 0.983
267 0.057 0,000 0.9*3
268 0,115 0,000 0,885
269 0,686 0.000 0.31'>
270 0,011 0.000 0.989
271 0.012 0,000 0.988
272 0,007 0,122 0,871
273 O.OU 0.000 0.988
27* 0,015 0,417 0.5*8
275 0.01* 0,036 0.948
276 0,029 0.006 0.9*5
277 0.060 0.032 0.908
276 0,068 0,000 0,932
279 0,036 0,070 0.893
280 0.055 0.464 0.481
281 0.010 0.725 0.2*5
282 0.008 0.784 0,208
263 0,051 0,025 0,925
28* 0,012 0,679 0.108
285
042
907
051
286
865
070
045
287
269
000
731
286
029
000
971
269
041
000
959
290
016
888
096
291
150
000
850
292
020
000
9B0
293
941
000
059
294
105
646
249
295
053
366
582
296
107
008
685
297
132
000
8*6
298
059
000
941
299
310
,690
,000
300
034
945
,021
301 0,292 0.702 0.006
302 0,065 0.026 0.907
303 0.008 0,947 0.0*4
304 0,003 0,902 0.095
305 0.016 0.919 0.0*5
306 0,037 0.000 0.9*3
307 0.076 0.000 0.922
306 0,015 0.058 0.927
309 0,012 0.406 0.582
310 0.019 0.000 0.981
311 0,043 0.000 0.957
312 0,022 0,001 0.977
313 0,043 0,000 0.957
314 0.036 0,000 0.9*4
3l4 0,030 0,000 0.970
316 0.980 0.000 0.020
317 0.39S 0.000 0.604
318 0,175 0.001 0,823
319 0,133 0.000 0.8*7
320 0.432 0,000 0.5*8
321 0,207 0.338 0.4J5
322 1,000 0.000 0.000
323 0.033 0.664 0,303
324 0,977 0,003 0.020
323 0,926 0,002 0.072
326 0.227 0,000 0,773
327 0.997 0,000 0,003
328 1,000 0,000 0.000
929 1,000 0,000 0,000
330 1,000 0,000 0,000
331 0,013 0,000 0.987
332 0.968 0.000 0.032
3J3 1.000 0.000 0.000
334 0.993 0.000 0.007
335 0.989 0.000 0.011
336 1.000 0.000 0.000
337 0.928 0.021 0,050
338 0.773 0,000 0.227
339 0,990 0.000 O.OIO
340 0,015 0.000 0.965
341 0.009 0.000 0.991
342 0.009 0.001 0.991
343 0,024 0,000 0,976
3*4 0.006 0.000 0,994
345 0,022 0.000 0,978
3*6 0,008 0,000 0.992
347 0,010 0,000 0,990
3*8 0,005 0,000 0.995
349 0,041 0,000 0.959
350 0.014 0,000 0.986
351 0,017 0,000 0,982
352 0,142 0,000 0,856
353 0,023 0.000 0.977
354 0,021 0.004 0.975
355 0,079 0.002 0.919
356 0.050 0.001 0.9*9
357 0,029 0,001 0.970
356 0.041 0.000 0.959
359 0,023 0.000 0.977
260 0.380 0.000 0.620
96
Table 17 (continued)
PROBABILITIES
OF TYPE MEMBERSHIP
1 2 3
361 0.032
.937
.031
362 0.117
,295
,588
363 O.oa')
000
916
36* 0.199
000
801
365 O.*30
000
570
366 0.009
oil
980
367 0.020
926
.053
368 0.072
005
922
369 0.027
032
9*0
370 0.991
000
009
371 0.3H
120
565
372 0.»29
000
571
373 0.193
000
802
37* 0.072
611
317
375 O.UU
926
063
376 0.015
001
98^
377 0.009
000
991
378 0.01*
001
986
379 0.010
000
990
3«0 0.018
000
982
381 O.lOO
102
798
382 0.02*
006
970
383 0.031
785
184
38* 0.027
951
022
385 0.019
905
076
386 0.007
888
105
387 0.025
68*
291
388 0.027
961
012
389 0.005
0.
786
209
390 0.005
0.
957
0.
038
391 0.01*
.51*
*72
392 0.025
■ 357
618
393 0.022
.801
177
39* 0.010
.295
695
395 0.1*9
,138
712
396 0.005
.78*
212
397 0.005
813
182
398 0.023
.*81
*95
399 0.012
.o^e
939
*00 0.238
.000
7*2
*01 0.02*
001
975
*02 0.106
,001
89*
*03 0.019
001
980
*0* 0.97*
001
02*
♦05 1.000
000
800
*06 1.000
.000
000
*07 1.000
.000
.000
♦08 I. 000
.000
.000
PROBABILITIES
OF TYPE MEMBERSHIP
1 2 3
♦ 09
1.000 0.000 0.000
*10
0.981 0.000 0.019
*11
0.702 0.298 0.000
*12
0.997 0.000 0.003
*13
0.0*^ 0.772 0.18^
*1*
0.010 0.719 0.271
*15
0.041 0.010 0.9*9
*16
0,026 0.005 0.968
*17
0.021 0.13* 0.e45
*lG
O.Oji 0.008 0.981
O.0I2 0.001 0.988
*19
*20
0.02* 0.002 0.975
*21
0.007 O.OSK 0.935
*22
0,010 0.012 0.978
*23
0.136 0.039 0.825
*2*
0.053 0.004 0.9*3
*25
0.222 0.058 0.720
*26
0.993 0.000 0.007
*27
0.096 0.010 0.89^
♦ 28
0.183 0.002 0.816
*29
0.999 0,000 0.001
*30
0.009 0.158 0.833
*3l
0.022 0.473 0.505
*32
0.008 0.0*7 0.945
*33
0.005 0.001 0.995
*3*
0.008 0.013 0.980
*35
0.003 0.51* 0.483
*36
0.012 0.652 0.336
*37
0.002 0.956 0.043
*3e
0.003 0.878 0.119
*39
0.998 0.000 0.002
**0
O.89I 0.022 0.086
**1
0.072 0.002 0.925
**2
0.005 0.000 0.995
♦ ♦3
0.06* 0.000 0.936
***
0.896 0.000 O.lO*
**5
0.185 0.000 0.81*
**6
0.008 0.000 0.992
*«7
0.038 0.000 0.9*2
♦ *8
0.007 0.000 0.993
*«9
O.Oll 0.000 0.989
*50
0.012 0.000 0.968
*51
0,0*3 0.002 0.955
*52
0.89* 0.000 O.IO6
*51
0.101 0.000 0.899
*5*
0.062 0.000 0.938
♦ 55
0.082 0.000 0.918
♦ 56
0.027 0.000 0.973
PROBABILITIES
OF TYPE MEMBERSHIP
1 2 3
457
.015
■ 000
■ 9(5
45E
.015
.001
.98*
459
.018
■ 197
■ 785
460
,008
■ 656
.335
461
.038
■ 032
■ 9J0
462
.039
■ 957
.00*
*63
.039
■ 30*
■ 658
*6*
,010
■ 353
■ 637
465
.023
.583
■ 394
466
,006
■ 126
868
467
.177
■ 029
.794
468
.135
000
865
469
,075
■ 063
■ 861
470
.577
■ 000
423
471
1
,000
■ 000
000
472
1
,000
000
000
473
1
■ 000
000
000
47*
I
,000
000
000
*75
,967
000
.033
*76
636
000
364
*77
■ 122
000
878
*78
056
000
944
*79
■ 183
000
817
*80
032
000
968
*81
■ 70*
206
090
482
048
000
952
*83
052
000
948
*8*
009
070
921
485
015
766
220
480
091
015
894
487
129
000
871
486
032
024
943
489
123
000
877
490
119
000
881
491
I
000
000
000
492
1
000
oon
000
493
061
089
850
*9*
323
001
476
*95
135
001
864
*96
041
006
952
*97
419
000
581
498
063
000
937
499
023
0.
000
0.
977
50U
020
0.
000
980
501
0,
8*7
0,
000
0,
153
502
0,
113
0.
000
0.
887
503
0,
989
0.
008
0.
003
LQCARITHM OF LIKELIHOQO R4T10 HF 3 TO 2 TYPES ■
CHI-SOOARE WITH 54 0F6HEES OF FRFEeOH» 114.02
PHOBABILITV OF NULL HYPOTHESIS. 0.000003*9
0.579336430 02
97
Table 18 Separation of a multivariate (6) set of Rantoul , Illinois,
October 1950-55, upper wind components, zonal' and meridional
mixed distribution, at the 700-, 500-, and 300-mb levels into
four distributions. The units are m-s~^.
PROCRAM NQRMIX
WDLFf NORMAL MIXTURE ANALYSIS PROCEDURE ( 197<. REVISIONI
SAMPLE • fANTnuL
OCTOBER
700/SOO,AND 300MB
X AND Y COMPONENTS
SAMPLE SIZE • 503
NUMBER OF VARIABLES • 6
NUMBER OF TYPES • ♦
ITERATION NUMBER
LIKELIHOOD OF « TYPES IN THIS SAMPLE • -0.727177290 0*
CHARACTERISTICS Q' THE WHOLE SAMPLE
MEANS
1.0000
0.2215
0.7525
0.2313
0.5589
0.2311
11.2762
5. 5488
1
1.0000
0.1304
0.6960
0.1793
0.*565
0.0233
3.7585
5.5489
-0.9486
6.8921
2
0.221!
1.0000
0.1136
0.7581
0.1265
0.5757
10.6700
-2.570*
STANDARD DEVIATIONS
3 *
8,959* 10.31*9
CBRRELATIONS
3 *
0.7)25 0.2313
0.1139 0.7581
1.0000 0.1567
0.1)87 1.0000
0.7513 0.1200
0.1977 0.7856
19.*0*9
13.7082
5
0.5589
0.1265
0.7513
0.1200
1.0000
0.1266
6
.3.6999
16.0058
6
o;23U
0'.5T57
0'. 1977
0'.78J6
o;i266
1.0000
CHARACTERISTICS OP TfPE 1
THE PROPORTION OF THE POPULATION FROM THIS TYPEp 0.*06
MEANS
2.*46*
0.130*
l.OOOO
0.0569
0.6547
-0.0423
0.4006
1*.1*60
1.2900
STANDARD DEVIATIONS
3 *
7.8*21 8.8618
CORRELATIONS
3 *
0.6960 0.1793
0.0)69 0.65*7
l.OOOO 0.1*98
0.1*98 1,0000
0.6851 0.0*92
0,092* 0.7*51
20.3731
10.7092
5
0.*S65
-0.0*23
0.68S1
0.0*92
l.OOOO
-0,03S*
6
5.8172
6
i3'.*027
0'.0233
0.*006
0.092*
0.7*51
-0.035*
l.OOOO
CHARACTERISTICS Of TYPE 2
THE PROPORTION OF THE POPULATION FROM THIS TYPE* 0.189
MEANS
4.1B69
3
7. 5600
*
2.5*86
1
2
1.0000
0.130*
0.1304
1.0000
0.6960
0.0569
0.1793
0.6547
0.4565
-0.0423
0.0233
0.4006
STANDARD DEVIATIONS
3 *
7,8*21 8.8618
chrreCations
3 *
0,6960 0.1793
0.0)69 0.65*7
l.OOOO 0.1*98
0,1*98 1,0000
0,6851 0.0*92
0.092* 0,7*51
13.1635
10.7092
5
0.*56S
-0.0*23
0.6891
0.0492
l.OOOO
-0.03S*
6
.3.0*06
13.4027
6
0.0233
0.4006
0.092*
0'.7*51
■0.039*
l.OOOO
CHARACTERISTICS OP TYPE 3
THE PROPORTIPN OF THE POPULATION FROM THIS TVPg» 0.310
MEANS
2 3 *
-7.19*0 11.4*81 .9.710*
16.9286
-1*.19*7
98
Table 18 (continued)
5T4NBAK0 OfVIiTIONS
1
2
3
*
5.:<.8e
*.B'?68
7.8*21
8.S6ie
CnRRELATIDNS
1
2
3
*
1.0000
0.130*
0.6'>60
0.1793
0.130*
1.0000
0.0969
0.65*7
Q.bfhO
0.0569
I. 0000
0.1*9B
0.1793
0.65*7
0.1*98
1.0000
0.*565
-0.0*23
0,6851
0.0*92
0.0233
0.*006
0.092*
0.7*51
10.7092
5
0.*565
-0.0*23
0.6831
0.0*92
1.0000
-0.03S*
CHARACTERISTlej Of TYRE *
THE PROPORTION OF THF POPULATION FROM THIS TYPE.
MEANS
5
■10.18*0
5
10.7092
5
0.*565
-0.0*23
0.68S1
0.0*92
1.0000
-0.035*
13.*027
0'.0233
0,*006
0.092*
0.7*51
-0.035*
1.0000
1
2
3
*
0.
7826
-5.2503
-0.*77l
.5.9032
STANOA'D
DEVIATIONS
1
2
3
4
5.
5*88
*.e968
7.6*2
6.8618
CORRELATIONS
1
2
3
*
!■
0000
0.130*
0.6460
0.1T93
0.
130*
1.0000
0.0969
0.65*7
0.
6960
0.0569
I. 0000
0.1*98
0.
1793
0.65*7
0.1*98
i.ooon
0.
*565
-0.0*23
0.6851
0.0*92
0.
0233
0.*006
0.092*
0.7*51
ITERATION
1
LOG
LIKELIHOOD
OF
TYPES
ITERATION
2
LOO
LIKELIHOQB
OF
TYPES
ITERATION
3
LOG
LIKELlHOOB
OF
TYPES
ITERATION
*
LOG
LIKELIHHOD
OF
TYPES
ITERATION
5
LOG
LIKELlHOOB
OF
TYPES
ITERATION
6
LOG
LIKELlHOOB
OF
TYPES
ITERATION
7
LOG
LIKELlHOOB
OF
TYPES
ITERATION
a
LOG
LIKELIHOOD
OF
TYPES
ITERATION
9
LOG
LIKELlHOOB
OF
TYPES
ITERATION
10
LOG
LIKELIHOOD
OF
TYPES
ITERATION
11
LOG
LIKELlHOOB
OF
TYPES
ITERATION
12
LOG
LIKELlHOOB
OF
TYPES
ITERATION
13
LOG
LIKELIHOOD
OF
TYPES
ITERATION
1*
LOG
LIKELIHOOD
OF
TYPES
ITERATION
15
LOG
LIKELIHOOD
OF
TYPES
ITERATION
16
LOG
LIKELIHOOD
OF
TYPES
ITERATION
17
LOG
LIKELIHOOD
OF
TYPES
ITERATION
18
LOG
LIKELlHOOB
OF
TYPES
ITERATION
19
LOG
LIKELIHOOD
OF
TYPES
ITERATION
20
LOG
LIKELIHOOD
OF
TYPES
ITERATION
21
LOG
LIKELIHOOD
OF
TYPES
ITERATION
22
LOG
LIKELIHOOD
OF
TYPES
ITERATION
23
LOG
LIKELIHOOD
OF
TYPES
ITERATION
2*
LOG
LIKELlHOOB
OF
TYPES
ITERATION
25
LOG
LIKELIHOOD
OF
TYPES
ITERATION
26
LOG
LIKELIHOOD
OF
TYPES
ITERATION
27
LOG
LIKELlHOOB
OF
TYPES
ITERATION
28
LOG
LIKELlHOOB
OF
TYPES
ITERATION
29
LOG
LIKELlHOOB
OF
TYPES
ITERATION
30
LOG
LIKELIHOOD
Of
TYPES
ITERATION
31
LOO
LIKELIHOOD
OF
TYPES
ITERATION
32
LOG
LIKELlHOOB
OF
TYPES
ITERATION
33
LOG
LIKELIHOOD
OF
TYPES
ITERATION
3*
LOO
LIKELIHOOD
OF
TYPES
ITERATION
35
LOG
LIKELIHOOD
OF
TYPES
ITERATION
36
LOG
LIKELIHOOD
OF
TYPES
ITERA-ION
37
LOG
LIKELIHOOD
OF
TYPES
ITER flON
38
LOG
LIKELIHOOD
OF
TYPES
ITEf'-TION
39
LOG
LIKELIHOOD
Of
TYPES
ITERATION
*0
LOG
LIKELIHOOD
OF
TYPES
ITERATION
*1
LOO
LIKELIHOOD
OF
TYPES
ITERATION
*2
LOO
LIKELlHOOB
OF
TYPES
ITERATION
*3
LOG
LIKELlHOOB
OF
TYPES
ITERATION
**
LOG
LIKELlHOOB
OF
TYPE?
ITERATION
*5
LOG
LIKELIHOOD
OF
TYPES
ITERATION
*6
LOG
LIKELIHOOD
OF
TY^ES
ITERATION
*7
LOO
LIKELIHOOD
OF
TYPES
ITERATION
*8
LOG
LIKELIHOOD
OF
TYPES
ITERATION
*9
LOG
LIKELIHOOD
OF
TYPES
ITERATION
50
LOG
LIKELIHOOD
OF
TYPES
ITERATION
51
LOO
LIKELlHOOB
OF
TYPES
ITERATION
52
LOG
LIKELIHOOD
OF
TYPES
ITERATION
53
LOG
LIKELIHOOD
OF
TYPES
ITERATION
5*
LOG
LIKELIHOOD
OF
TYPES
ITERATION
55
LOG
LIKELIHOOD
OF
TYPES
ITERATION
56
LOG
LIKELIHOOD
OF
TYPES
ITERATION
57
LOO
LIKELIHOOD
OF
TYPES
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN TnIS
IN THIS
IN This
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
IN THIS
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SaMPlE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
SAMPLE
13.*027
6
0,0233
0,*006
0,092*
0,7*51
.0'.035*
1,0000
-0,752736060 0*
-0,7*2l5g89D 0*
-0,737263560 0*
-0,73*15*930 0*
-0,731713750 0*
-0,729716050 0*
-0,728398620 0*
-0,727597600 0*
-0,727053890 0*
-0.7266*5320 0*
-0.726326910 0*
-0.726071*50 0*
-0.725857860 0*
-0.725673520 0*
-0.72551*010 0*
-0.729378530 0*
-0.72526*750 0*
-0.726*00870 0*
-0.72*927*90 0*
-0,72*790960 0*
-0,725576«60 0*
-0,72*6*5810 0*
-0.72*571510 0*
-0,72*825150 0*
-0.72*469*50 0*
-0.72**23910 0*
-0,72*887810 0*
-0.72*293210 0*
-0.72*216810 0*
-0.72*726580 0*
-0,72*015990 0*
-0,723933290 0*
-0.72****610 0*
-0.723802560 0*
-0.7237*89*0 0*
-0.723803760 0*
-0.7237025*0 0*
-0.723682900 0*
-0.723932*50 0*
-0,7236*1900 0*
-0,72361*200 0*
-0.723769090 0*
-0.7235876*0 0*
-0,723568290 0*
-0,723977760 0*
-0.7235*6010 0*
-0.723520*30 0*
-0.7235*1*60 0*
-0.723*9*060 0*
-0.723*82260 0*
-0.7238991*0 0*
-0.723*28970 0*
-0.7233912*0 0*
-0.723699620 0*
-0.72336567D 0*
-0.7233078*0 0*
-0.7236999*0 0*
99
Table 18 (continued)
iTERiTlQN
53
LOG
LiKELIHCOn
OF
TYPES
IN
THIS
SAMPLE •
ITER4TIDN
59
LOG
LIKELIHCOn
OF
TYPES
IN
THIS
SAMPLE •
ITERATION
60
LOG
LIKELIHDOC
OF
TYPES
IN
THIS
SAMPLE •
ITERATION
61
LOG
LIKELIHOOD
OF
TYPES
IN
THIS
SAMPLE •
ITERATION
62
LOG
LIKELIHCOe
BF
TYPES
IN
THIS
SAMPLE •
ITERATION
63
LOG
LIKELIHPOB
OF
TYPES
IN
THIS
SAMPLE ■
ITERATION
64
LOG
LIHELIHOQO
OF
TYPES
IN
THIS
SAMPLE •
ITERATION
65
LOG
LIKELIHCOB
OF
TYPES
IN
THIS
SAMPLE •
ITERATION
66
LOG
LIKELIHHOD
OF
TYPES
IN
THIS
SAMPLE •
ITERATION
67
LOG
LIKELIHOOD
OF
TYPES
IN
THIS
SAMPLE .
ITERATION
68
LOG
LIKELIHOOD
OF
TYPES
IN
THIS
SAMPLE •
ITERATION
69
LOG
LIKELIHDOn
OF
TYPES
IN
THIS
SAMPLE .
ITERATION
70
LOG
LIKELIHOOD
OF
TYPES
IN
THIS
SAMPLE •
ITERATION
71
LOG
LIKELIHOOD
OF
TYPES
IN
THIS
SAHPLt •
ITERATION
72
LOG
LIKELIHOOD
OF
TYPES
IN
THIS
SAMPLE •
ITERATION
73
LOG
LIKELIHOOD
OF
TYPES
IN
THIS
SAMPLE •
ITERATION
T,
LOG
LIKELIHOOD
OF
TYPES
IN
THIS
SAMPLE •
ITERATION
75
LOG
LIKELIHOOD
OF
TYPES
IN
THIS
SAMPLE •
ITERATION
76
LOG
LIKELIHOOD
OF
TYPES
IN
THIS
SAMPLE •
ITERATION
77
LOG
LIKELIHOOD
OF
TYPES
IN
THIS
SAMPLE •
ITERATION
78
LOG
LIKELIHOOD
OF
TYPES
IN
THIS
SAMPLE •
ITERATION
79
LOG
LIKELIHOOD
OF
TYPES
IN
THIS
SAMPLE .
ITERATION
80
LOG
LIKELIHOOD
OF
TYPES
IN
THIS
SAMPLE .
ITERATION
81
LOG
LIKELIHOOD
OF
TYPES
IN
THIS
SAMPLE •
ITERATION
62
LOG
LIKELIHOOD
OF
TYPES
IN
THIS
SAMPLE ■
ITERATION
83
LOG
LIKELIHOOD
OF
TYPES
IN
THIS
SAMPLE ■
ITERATION
8<i
LOG
LIKELIHOOD
OF
TYPES
IN
THIS
SAMPLE •
ITERATION
B5
LOG
LIKELIHOOD
OF
TYPES
IN
THIS
SAMPLE •
ITERATION
86
LOG
LIKELIHOOD
OF
Types
IN
THIS
SAMPLE •
ITERATION
87
LOG
LIKELIHOOD
OF
TYPES
IN
THIS
SAMPLE .
ITERATION
88
LOG
LIKELIHOOD
OF
TYPES
IN
THIS
SAMPLE •
ITERATION
89
LOG
LIKELIHOOD
OF
TYPES
IN
THIS
SAMPLE •
ITERATION
90
LOG
LIKELIHOOD
OF
TYPES
IN
THIS
SAMPLE .
ITERATION
91
LOG
LIKELIHOOD
OF
TYPES
IN
THIS
SAMPLE •
ITERATION
92
LOG
LIKELIHOOD
OF
TYPES
IN
THIS
SAMPLE •
ITERATION
93
LOG
LIKELIHOOD
OF
TYPES
IN
THIS
Sample •
ITERATION
94
LOG
LIKELIHOUD
OF
TYPES
IN
THIS
SAMPLE •
Iteration
93
LOO
LIKELIHOOD
Of
TYPES
IN
THIS
SAMPLE .
ITERATION
96
LOG
LIKELIHOOD
OF
TYPES
IN
THIS
SAMPLE •
iteration
97
LOG
LIKELIHOOD
OF
TYPES
IN
THIS
SAMPLE .
ITERATION
98
LOG
LIKELIHOOD
OF
TYPES
IN
THIS
SAMPLE •
ITERATION
99
LOG
LIKELIHOOD
OF
TYPES
IN
This
SAMPLE .
ITERATION
100
LOG
LIKELIHOOD
OF
TYPES
IN
THIS
SAMPLE •
-0,72327*170 0*
-0.723179610 0*
-0.723911560 04
-0.723298950 04
-0.72317J96D 04
-0.72312275D 04
-0.7237J290D 04
-0.723256210 04
-0.723155650 04
-0.723106720 04
-0.72308061D 04
-0.723416780 04
-0.7231463O0 0^>
-0.723087620 04
-0.72306692D 04
-0.723058890 0»
-0,723055380 04
-0.723052980 04
-0.723059780 04
-0.7230SJ590 04
-0.7230J249D 04
-0.723052110 04
-0.723054140 04
-0.72J052290 04
-0.723051950 04
•'•0.723051830 04
-0.72S05197D 04
-0.723051730 04
-0.723051690 04
-0.723051720 04
-0.723051640 04
-0.723051600 04
-0.723051610 04
-0.723051520 04
-0,723051470 04
-0.723051800 04
-0.723051420 04
-0.723051280 04
-0.723051560 04
-0.723051120 04
-0.723050940 04
-0.723050950 04
-0.723030570 04
ITERATION TIME •
TOTAL TIME USED •
3)43.00 SECONDS.
11S62.01 SECONDS.
ITERATION NUMBER 101
LIKELIHOOD OF 4 TYPES IN THIS SAMPLE • -0.72305057D 04
CHARACTERISTIes OF THE WHOLE SAMPLE
7.3232
11.1041
1
1.0000
0.2647
0.5390
0.3324
0.5169
0.2525
MEANS
STANDARD DfVIATIONS
3 4
8.9594 10.3149
CORPELATIONS
3 4
0.7525 0.2313
0.1138 0.7581
1.000b 0.1587
0.1587 1.0000
0.7515 0.1200
0.1977 0.7856
CHAPACTERISTIes OP TYPE I
THE PROPORTION OF TH( POPULATION FROM
MIANS
1
2
1.0000
0.2215
0.2215
1.0000
0.7525
0.1138
0.2313
0.7581
0.5589
0.1265
0.2311
0.5757
15.4049
13.7082
3
0.5589
0.1265
0.7513
0.1200
1.0000
0.1266
-3;6999
6
16'. 003 8
6
0.2311
0'.J757
O; 1977
0'.7e36
0,1266
1,0000
THIS TYPE. 0.193
2
-0.3525
2
9.2886
0.2647
1.0000
-0.1035
0.7443
-0.1470
0.6119
16.6560
4
■1.3465
STANDARD DfVlATIONS
3 4
11.3528 17.2976
CORRELATIONS
3 4
0.5390 0.3324
-0.1035 0.7443
l.OOOP 0.0730
0.0730 1.0000
0.7250 -0.0241
0.0848 0.7911
22.8836
14.8718
5
0.5169
-0.1470
0.72S0
-0.0241
1.0000
-0.1123
6
.0'.6482
27.5962
6
0.2525
0'.6119
0;0948
o'.79ai
■ 0'.lt23
I'. 0000
100
Table 18 (continued)
EIGENVALUES
EIGENVECTORS
3
O.ll"
0.5*60
1021.7*63
0.0*66
0.251''
0.1879
326.0660
0.2057
-O.OSl'!
0.4355
-0.0397
0.6673
104,5090
0.0270
0.5!2«
-0.1959
0.7'i83
-0.0513
'•0.0002
0.<.61T
0.0*27
0.7053
0.1472
-0.<.735
33.*5*5
-0.0'i90
0.7903
0.0619
-0.5999
-0.0621
23.2217
O.B'.7»
-0.0U8
-0.5063
-0,147<,
0'.0608
CH4R4CTFRISTleS QP TvPE 2
THE PROPORTION OF THE POPULATION fPQM T-HIS TYPE. 0.293
CHARACTERISTIf S OP TYPE 3
THE PROPORTION DP THE POPULATION FROM THIS TYPE» 0,<.21
MEANS
12 3 4 5*
6.3830 -2.6212 8.3100 -5.2*61 10.5901 -4.5*39
STANBARD DEVIATIONS
12 3 4 5 6
6.6569 5.5893 7.5718 7.9899 n.9976 12.0984
C8RPELATI0N5
CHARACTERISTICS OF TYPE 4
THE PROPORTION OF THI POPULATION FROM THIS TYPE" 0.094
MEANS
1
2
i
4
5
6
5.6571
-3.2181
6.8328
.3.4952
7.2228
^6.5235
STANDARD DEVIATIONS
1
2
3
4
5
6
6.3812
6.6990
8.7872
8.7555
IB. 2393
16.5218
CORRELATIONS
, 1
2
3
4
5
*
X.OOOO
0.3470
0.9558
0.3597
0.4762
0'.1332
0.3470
1.0000
0.4558
0.9402
0.6151
0.6943
0.9558
0.4550
1.0000
0.4338
0.4895
0'.2214
0.3597
0.9402
0,413»
1.0000
0,5464
0.8140
0.4762
0.6151
0.4895
0.5464
1,0000
0'.3738
0.1332
0.6943
0.2J14
0.8140
0,3738
l.OftOO
6
0.7497
-0.5643
.0.3054
0.1439
-0.0739
0.0017
MEANS
1
2
3
4
5
6
6.7197
1.7879
11.3394
0.7926
20.0058
-3.5488
STANDARD DEVIATIONS
1
2
3
4
5
*
5.2062
5.7219
6.7337
5.1816
8.3725
8.0912
CHRRELATIONS
1
?
3
4
5
*
I. 0000
-0.0490
0.7322
0.2324
0.5236
0;4*35
-0.0490
1.0000
-0.2795
0.6794
-0.3196
0;4835
0.7322
-0.2795
1.0000
-0.1278
0.7437
0'il412
0.2324
0.6794
-0.1278
1.0000
-0.2774
0.7870
0.523*
-0.319*
0.7437
-0.2774
I. 0000
-0.0414
0.4435
0.4635
0.1412
0.7870
-0.0414
1.0000
EIGENVALUES
EIGENVECTORS
1
2
3
4
5
6
120.3379
0.3038
0.2911
-0.2839
0.3728
0.6914
.0.3541
102.2809
-0.2543
0.3305
0.7201
0,462*
-0.1243
•0.2796
20.2525
0.5440
0.1*97
.0.2136
0.5162
.0'.5472
0.2525
13.1203
-0.1758
0.4J19
0.0912
0.0344
0'.3317
0.8145
6.870*
0.7153
0.0421
0.5396
-0.4197
0'. 1338
0.0349
4.7389
-0.0663
0.7*77
.0.2360
-0.4509
-0'.2815
-0.2609
1.0000
0.3045
0.8419
0.1154
0.606S
0'. 1489
0.3045
1.0000
0.3*15
0,7889
0.1293
0.6263
0.8419
0.3415
1.0800
0.1965
0.8310
0.3252
0.1154
0.7889
0.1965
1.0000
0.0264
0.8370
0.6065
0.1293
0,8318
0.0264
1.0000
0'.3023
0.1489
0.6263
0.3252
0.8370
0.3023
1,0000
EIGENVALUES
EIGENVECTORS
1
2
3
4
5
*
266.8513
0.2422
0.2551
0.6124
-0.4722
-0.1*61
0.5007
159.8561
0.2087
-0.1989
0.4059
0.6391
-0.5800
.0.0858
37.9134
0.3663
0.2998
0,3293
-0.1248
0.2361
-0.7721
12.6521
0.3106
-0.4448
0.2560
0.2736
0.7070
0.2558
5.5168
0.5554
0.5782
-0.4125
0.3347
0',0417
0.2707
4.2504
0.5989
-0.5228
-0.3407
-0.4076
-o'.2eo5
-0.0841
101
Table 18 (continued)
EIGENVALUES
EIGENVECTORS
529.7700
20*. 9783
80.2752
2J.726'.
3.3614
1.0011
1370
2'.52
2120
3267
6787
5552
2
-0.1791
0.0753
-0.201S
0.1956
-0.6n8*
0.716*
3
0.5272
0.0656
0.7391
0.1260
-0.3929
-0.0352
-0.1521
0.6255
-0, >3<i7
0.6207
-0.1170
-0.4105
5
0',5265
.0.5210
-0'.<il52
0,5243
0.0120
-0'.063«
PROBABILITIES
PROBABILITIES
OF TYPE MEMBERSHIP
OF TYPE MEMBERSHIP
2 3 4
1
2 3
4
,6091
,5164
.4215
,4232
,0310
06d!
1 0.622 0.051 0.327 n.OOO
2 0.007 0.992 0.001 0.000
3 O.Ooe 0.991 0.002 0.000
4 1.000 0.000 0.000 0.000
5 0,004 0.996 0.000 0,000
6 0,022 0.978 0.000 0.000
7 0.036 0.031 0.933 0.001
8 1.000 0.000 0.000 0.000
9 0.060 0.935 0.005 0.000
10 0,699 0.000 0.008 0.293
U 1.000 0.000 0.000 0.000
12 0.006 0.072 0.837 0.085
13 0.018 0.469 0.321 O-l'l
14 0.015 0.127 0.839 0.019
15 0.002 0.000 0.000 0.998
16 0.991 0,000 0.008 0.000
17 0.017 0.014 0.004 0.96?
18 0.223 0,013 0.7»4 0.000
19 0.366 0.046 0.969 0.000
20 1.000 0.000 0.000 0.000
21 0,011 0.841 0.147 0.001
22 0.059 O.OOO 0.158 0.783
23 0.307 O.OOO 0.692 0.001
24 0,030 0.002 0.9*7 0.000
25 0.006 0.758 0.195 0.041
26 0.000 0.000 O.OOn 1.000
27 0.132 0.124 0.743 0,001
28 0.011 0.984 0.005 0.001
29 0.155 0.844 0.001 0.000
30 0.820 0.000 0.180 0.000
31 0.025 0.00? 0.972 0.000
32 0,015 0.000 0.985 0.000
33 0.C13 0.061 0.69* 0.032
34 0.052 O.OOO 0.948 0.000
35 0.729 0.000 0.271 0.000
36 0.008 0.06"; 0.724 0.164
37 0.011 0,497 0.478 O.OU
38 0,007 0,937 0.054 0,001
39 0.005 0.861 0.024 0.091
40 0.005 0,968 0.024 0.003
41 0.944 0,044 0.012 0,000
42 0.012 0.817 0.025 0.146
43 0.038 0.153 0.580 0.229
44 0.021 0.638 0.236 0.106
45 0,056 0,720 0.224 0.000
46 0.096 0,022 0.869 0,OlS
47 0,032 0.757 0.166 0.025
48 0.962 0.000 0.038 0.000
49 0.009 O.OOI 0.876 0.113
50 0,016 0.001 0,956 0.027
51 0,012 O.OOO O.oeo 0.007
52 0.424 0.000 0.J76 0.000
53 0.007 0,000 0,943 0,050
54 1,000 0.000 0.000 0,000
55 0.004 0.000 0,8*3 0,132
56 0,008 0.000 0.946 0.046
57 0,008 0.000 0.780 0.212
58 0.147 0.007 0.846 0.000
59 0,004 0.000 0,451 0,545
60 0,004 0,001 0.842 0.153
61 0.003 0.003 0.859 0.135
62 0.025 0.826 0.103 0.046
63 0.009 0.151 0.721 0.H9
64 0.009 0,176 0,704 0,111
65 0,045 0.660 0,271 0.024
66 0.020 0.651 0,329 0,000
67 0.027 0,399 0,574 0,000
68 0,009 0.694 0.270 0.027
69 0,009 0,424 0.555 0.012
70 0,011 0.488 0.480 0.020
71 0.015 0.866 0.118 0.000
72 0.007 0.960 0.021 0,013
73 0.010 0.950 0,038 0,002
74 0,008 0,855 0.059 0.078
75 0.038 0.867 0,045 0.050
76 0,025 0.546 0.370 0.059
77 0,035 0.961 0.004 0.000
78 0,028 0.017 0.955 0.0 ''
79 0.051 0.696 0,019 0.
80 0,041 0.834 0,124 0- il
81 0,100 0,255 0,645 0.000
82 0,027 0,872 0.012 0.088
63 0.974 0.020 0.004 0,003
84 0,064 0,933 0,003 0,000
85 0.077 0,838 0,084 0.000
86 0,020 0,786 0.193 0.000
87 0,036 0,786 0.163 0,015
88 0.028 0,803 0,019 0,150
89 0.024 0.932 0.043 0.000
90 0,375 0,406 0.016 0.203
91 0,079 0,029 0,893 0.000
92 0,029 0,019 0.942 0.010
93 0.973 0,001 0.;27 0.000
94 0.044 0,671 0.001 0,264
95 0,032 0.169 0.797 -^.OOI
96 0,945 0,012 0,042 0,000
97 0,040 0.003 0.957 O.OOO
98 0,064 0,893 0.043 0,000
99 0.056 O.OOO 0.943 0,000
100 0.018 0.026 0,845 0,111
101 0,016 0,008 0.881 0.094
102 0,847 0.047 0,106 0.000
103 0,996 0.003 O.OOI 0.000
104 O.OU 0.005 0.864 0.100
105 0.012 0.459 0.328 0.001
106 0.016 0,009 0.689 0,286
107 0,097 0,000 0.882 0,021
108 0,004 0,000 0.957 0.039
109 0,007 0,000 0.529 0.464
HO 0,004 0.002 0.833 0.161
111 0.008 0.000 0.978 0.013
112 0.003 0,001 0,995 0,002
113 0.005 0,000 0.978 0,015
114 0,023 0,000 0.403 0,574
115 0,128 0,414 0,400 0.059
116 0,009 0,035 0,899 0,05''
117 0.014 0,901 0.085 0,000
118 0.083 0.447 0.011 0.458
119 0.031 0.078 0.815 0,076
120 0,010 0,568 0.352 0,070
121 0,028 0,950 0,022 0,000
122 0,019 0,000 0.981 0.000
123 0.032 0.463 0.4iJ 0,071
124 0,093 0,661 0.U2 0.084
125 0,416 0.563 0.001 0.000
125 0.073 0.001 0,926 0.000
127 0.549 0.450 0.001 0.000
128 0.031 0.000 0,002 0,967
129 0,005 0,000 0,002 0,993
130 0,014 0,012 0,973 0.000
131 0.005 0.000 0,949 0.046
132 0.005 0.000 0.544 0.450
133 0.012 0.000 0.109 0.i79
134 0.006 0,007 0,958 0.030
135 0,030 0,007 0.9;)8 0,02*
136 0,014 0,005 0.956 0,024
137 0,026 0,004 0.875 0,095
138 0.009 0,005 0,653 0,333
139 0,019 0,000 0.981 0.000
140 0,003 0,000 0,015 0,963
141 0,000 0,000 0.000 1.000
142 0,024 0.000 0,975 0,000
143 0.540 0.000 0.237 0.203
144 0.030 0,001 0.9*9 0.000
1*5 0.071 0.758 0.143 0.028
1*6 0.017 0.964 0.018 0.000
147 0.037 O.I43 0.808 0.012
148 0.013 0.931 0.055 0.000
149 0.029 0.202 0.458 0.312
150 0.037 0.074 0.887 0.003
151 0,096 0.000 0.904 O.OOO
152 0.236 0.008 0.755 0.000
153 0.258 0.035 0,9*4 u,143
154 0.103 0.000 0,897 0,000
155 0.064 0.000 0,936 0,000
156 0.028 0.073 0.777 0.122
157 0.037 0.871 O.Oll 0.0B2
158 0.016 0.353 0.615 0.014
159 0.U4 0.001 0.880 0.005
160 0,012 0.837 0.151 0.000
PROBABILITIES
OF TYPE MEMBERSHIP
12 3 4
161 0.011 0,228 0.761 0,000
162 0,021 0,891 0.088 0.000
163 0.009 0.984 0,007 0.000
164 0.006 0,978 0.015 0,000
165 0.015 0.593 0.363 0,010
165 0,004 0,919 0.015 0,062
167 0,011 0.879 0,000 0.109
168 0,005 0,989 0,005 0.000
169 0.004 0.919 0.027 0.050
170 0.004 0.952 0.012 0.03?
171 0.006 0.965 0.003 0.027
172 0.232 0,707 0,000 0,062
17i 0,037 0,519 0,332 D,0l2
174 0.017 0,778 0,041 0,164
175 0,022 0.005 0.9*0 0.013
176 0,171 0,060 0,749 0,000
177 0,087 0.010 0,902 0,000
178 0,012 0,078 0,774 0^135
179 0,796 0,005 0.188 0,OlO
160 0,949 0,051 0,000 0,000
181 0,997 0,003 0.000 0,000
182 1.000 0.000 0.000 0.000
183 0,988 0,000 0,012 0,000
184 0,157 0,000 0,843 0,000
185 0.054 0,934 0.012 0.000
186 0.021 0,000 0,935 0,044
167 0,012 0.933 0.390 0.065
188 0.010 0.803 0.160 0.007
189 0.007 0.588 0.284 0,022
190 0,010 0,065 0.778 0,146
191 0,072 0,668 0.258 0.002
19? 0,024 0,000 0.975 0.001
193 0,006 0,158 0,697 0,139
194 0,008 0,411 0,502 0.079
195 0.013 0.024 0.815 0,148
196 0,005 0,000 0,992 0,003
197 0,153 0,000 0,847 0.000
198 0.023 0.000 0.951 0.026
199 0.016 0.001 0.981 0,002
200 0,032 0,095 0.8*1 0,011
201 0,025 0,000 0.011 0.963
202 1.000 0.000 0.000 0.000
203 0.139 O.OOO 0,855 0,006
204 1.000 0.000 0.000 O.OOO
205 0.006 0,000 0,994 0.000
205 0.017 0.056 0.917 0.000
207 0.006 0.000 0,992 0,002
208 0,007 0,000 0,993 0,000
209 0,018 0,000 0.982 0,000
210 0.004 0.001 0,995 0,000
211 0,006 0,001 0,017 0,976
212 0,007 0,001 0.977 0,0l6
213 0,510 0,420 0.070 0.000
214 0,008 0,000 0,943 0,448
215 0,002 0.000 0.734 0.263
215 0.018 0.000 0.982 0.000
217 0,006 0,000 0,982 0,012
218 0,009 0,000 0,592 0,298
219 0.011 0.000 0,973 0,01*
220 0.007 0,001 0,968 0,004
221 0,023 0,001 0.92* 0.052
222 0,007 0,000 0,875 0.118
223 0,010 0,000 0.699 0,291
224 0,009 0,001 0.770 0.220
225 0,007 0.010 0.786 0.1«6
226 0.006 0.054 0.814 0,12*
227 0.028 0.0*2 0.8*3 0.067
228 0.008 0.758 0.217 0.018
229 0.006 0.851 0.077 0.067
230 0.068 0,765 0.1*5 0.002
231 0,240 0,754 0.005 0,000
232 0,005 0,987 0,002 0,006
233 0,008 0,917 0,000 0,07*
234 0,029 0,969 0,000 0,002
235 0,004 0,959 0,008 0,030
235 0,007 0,963 0,010 0,001
237 0,063 0,397 0.540 0,000
236 0.053 0.944 0.002 0.000
239 0.011 0.985 0.002 0.002
240 0.002 0.997 0.000 0.001
102
PROBABILITIES
OF TYPE MEMBERSHIP
1 2
2<,1 0.010 0.923
2*2 Oj79'3 0.177
2*3 0.005 0.982
2** 0,016 0.B63
2*5 0.028 0.901
2*6 0.0$5 0.316
2*7 0.972 0,000 0.02S 0.000
0.000 0.8**
0.000 0.9**
0.000 0.975
3 4
0.000 0.067
0.02* 0,000
0.009 0.00*
0,119 0,003
0.071 0.000
111 0.518
2*8 0.155
2*9 0,036
250 0.025
251 0.017 0. 12*
232 0.006 0.552
253 0.009 0.962
858
0,*15
0,023
001
000
000
001
026
001
25* 0,016 0,085 0,60* 0.295
255 0,236 0,017 0.7*7 0.000
256 0.933 0.012
257 0.10* 0.012
25fl 0.183 0.000
259 0.096 0.000
260 0,03* 0.000 0.711
0.002 0.958
0.863
0.31*
0,063
261 0,032
262 0,017
263 0.011
26* 0,035
,119
265 0,017 0,033 0.9*5
i66 0.00* 0.000 0.979
0.00* 0.000 0,67*
7*7 0,000
003 0,052
5*8 0.337
803 O.OU
90* 0,000
0.25*
0.008
0.001
551 0.123
898 0.005
267
268 0.032 0.000
269 0.1*3 0.000
270 0.005 0.000
271 0,00*
272 0.007
273 0,010
27* 0,029
275 0,018
276 0.033
277 0,060
278 0.063
279 0,0*3
280 0.039
281 0.019
,968
,857
,766
,966
.005
,016
.323
.000
.000
.229
.029
622 0.15*
982 0.008
0.121
0.005
7*7 0.1*1
699 0.180
786 0.129
0.1*6
088
883
0.013
0.000
0.217
0.005
0.762
0.095
0.009
0.061
0.006
0.798
0.»20 0.000 0.5*0
_. . - 0.950 0.006 0.026
282 0.013 0.916 0.070 0.000
283 0.0*3 0.91* 0.0*3 0.000
28* 0.022 0.965 0.005 O.OOB
285 0.091 0.909 0.000 0.001
286 0.9*7 0.053 0.000 0.001
287 0.961 0.035 0.00* 0.000
288 0.067 0.OO5 0.929 0.000
0.001
289 0.272
290 0.022 0,936
291 0,*12 0,015
292 0,025
293 0.9B9
29* 0.109
295 0.0*9 _ . _
296 0.122 0,081
297 0.12* 0,001
298 0,078
0,003
0,000
0,832
0,523
727
0*2
525
972
Oil
000
001
751
875
561
0,000
0,000
0.0*9
0.000
0.000
0.059
0.*27
0.0*6
0.000
0.123
_ . 0.238 . _
299 0.806 0.19* 0.000 0.000
300 0,0*2 0.958 0.001 0.000
301 0.125 0.875 0.000 0.000
302 0.162 0.737 0.093 0.008
303 0.018 0.980 0.003 0.000
30* 0.006 0.972 0.016 0.006
305 0,052 0.907 0.0*0 0.000
306 0.059 0.000 0.9*1 0.000
307 0.07* 0.009 0.65* 0.263
0.357 0.*20 0.203
,032
,3**
308 0.020
309 0.01* 0.908
310 0.016 0.026
311 0,083
312 0.051
313 0.126 0.080
31* 0.070
315 0.075
316 0,995 0.005
317 0.299 0.037
318 0.2*5 0.010
319 0.887
320 0.9*3
321 c.ieo
1.000
322
323
32*
325
0.073 0.8*1
0.806 0.19*
_ 0.93*
326 0.5*5
327 0.98*
328 1.000
036 0.0*2
953 0.005
878 0.007
599 0.006
792 0.002
0.008 0.922 0.000
0.066 0.859 0.000
0.000 0.000
0,070 0,593
0,735 0,010
0.096 0.000
0.0*9 0.000
0.057 0.001
0,000 0.000
0.085 0.000
0.000 0.000
0.001 0.000
0,**1 0.000
0.000 0.0l6
0.000 O.OQO
0.017
0.009
0.761
0.000
0.065
0.01*
0.000
0.000
Table 18 (continued)
PROBABILITIES
OF TYPE MEMBERSHIP
12 3 4
329 1.000 0.000 0.000 0.000
330 1.000 O.OOn 0.000 0.000
331 0.00* O.OOO 0.950 0.0*6
33? 0.919 0.000 n.OBl 0.000
333 I. 000 0.000 0.000 0.000
J3* 0.999 0.000 0.001 0.000
335 0.99* 0.006 0.000 0.000
336 1.000 0.000 0.000 0.000
337 0.890 0.110 0.000 0.000
338 0.925 0.00* 0.071 0.000
339 0.999 0.000 0.001 0.000
3*0 0.027 0.009 0.96* 0.000
3*1 0.02* 0.000 0.911 0.065
3*2 0.01* O.Ol* 0.9*6 0.007
3*3 O.OU 0.000 0.*33 0.556
3** 0.002 0.000 0.820 0.178
3*5 0.01* 0.000 0,*02 0.585
3*6 0.002 0.000 0.67* 0.32*
3*7 0.010 0.002 0,9*9 0.0*0
3*8 0.002 0.001 0.883 O.ll*
3*9 0.01* 0.000 0,258 0.729
350 0.00* 0.000 0.882 0.11*
351 0.005 0.002 0.921 0.072
352 0.005 0.000
353 0.005 0.001
35* 0.009 0.027
355 0.07* 0.060
350 0.0*6 0.012
357 0.026
,995
,957
,85*
,82*
,9*1
.000
.036
.110
.0*1
.001
,027
0.003 0.9**
358 0.039 0.001 0.924 0.036
359 0.026 0.001 0.836 0.137
360 0.99* 0.002 0.00*
361 0,099 0,89* 0,000
0,253 0.500
0.0*0
0.392
0.000
0.772
0.000
0.006
829
805
.0*3
,951
.003
,000
,000
,006
,00*
,002
,005
836
003
,000
,895
,328 0.238
,001 0.000
058 o.on
,339 0.000
,655 0.001
,05* 0.000
.002 0.000
913 0.06B
362 0.2*3
36:t 0.129
36* 0.186 0.00*
365 0.120 0.000
36o O.OU 0.03*
367 0.0*9 0.9*8
368 0,037 0.068
369 0,0*2
370 0,999
371 0.158
372 0.661
373 0.338
37* 0.076 0.870
375 0.016 0,982
376 0.017 0.002
377 0.009 0.002 0.987 0.002
378 0.01* 0.007 0.829 0.150
379 0,010 0.001 0.9*2 0.0*6
380 0.018 0.005 0.870 0.107
381 0.101 0.736 0.161 0.002
382 0,025 0.*2* 0.*»9 0.082
383 0.026 0.910 0.0»* 0.000
38* 0.067 0.921
385 0.038 0.920
386 0.013 0.971
387 0.028 0.933
388 0,066 0.93*
389 0.005
390 0.010
391 0.018
392 0.022
393 0.021
39* 0,017
395 0,0*1 0.959
396 0.006 0.990
397 0,011 0,967 0,018 0.00*
398 0,02* 0,922 0,05* 0,000
399 0,016 0,616 0.3*6 0.001
*00 0.*03 0.001 0.596 0.000
*01 0.039 0.139 0.822 O.OOl
0.583
0.501
0.012 0.000
0.0*2 0.000
0.005 O.Oll
0.009 0.029
0.000 0.000
0.98* 0.008 0.003
0.988 0.002 0.000
019 0.000
009 0.000
000 0.000
0*7 0.00*
000 0.000
003 0.000
0.893
0.970
0.979
0.932
*02 0.116
*03 0.033
*0* 0.986 0.011
*05 1.000
*06 1.000
*07 1,000 0.000
*08 1.000
*09 1.000
*10 0,9**
*ll 0,9*1
0,29* 0,007
0.*** 0,002
0.003 0.000
0.000 0.000 0.000
0.000 0.000 0.000
000 0.000
000
000
052
000
0.000
0.000
0.00*
.. -. - 0.059 _ _ ,
*12 1,000 0.000 0.000 ,
*13 0.065 0.686 0.001 0.2*8
*l* 0.011 0.967 0.005 0.017
*15 0,056 0,162 0.735 0.0*6
*16 0.031 0.05* 0.8*3 0.051
0.000
0.000
0.000
0.000
0.000
PROBABILITIES
OF TYPE MEMBERSHIP
*17 0.026
*ie 0.017
*19 0.D08
*20 0.022
*21 0.010
*22 0.013
*23 0.098
*2* 0.136
*25 0.151
( 4
713 0.026
8*6 0.000
93* 0.0*1
71* 0.261
8** 0,028
159 0,827 0,000
891 0,011 0,000
0.03* 0.000
0.003 0.000
0.000 0.000
2
0.23*
0.U7
0.017
Ot003
0.097
0,
830
8*6
*2» 1.000 0.000
*27 0.100 0.013 0.797 0.089
*28 0.162 0.002 0.633 0.203
0.000
0.288
*29 1.000
*30 O.OIO
*31 0.037
*32 0.010
*33 0,007
*3* 0,010
*35 0,005
0,000 0,000
0.550 0.152
0.730 0.230 0.002
0.061 0.906 0.02*
0.007 0.93* 0.051
0.097 0.893 0.000
0.886 0.079 0.0J5
*36 0.025 0.69* 0.281 O.OOO
*37 0.00* 0.990 0.005 0.002
*38 0.006 0.963 O.Oll 0.020
*39 0,*32 0.000 0.000 0.568
**0 0.319 0.680 0.000 0.000
**1 0.0*1 0.958 O.OOl 0.001
**2 0.003 0.000 0.8*7 0.130
**3 0,009 0.000 0.016 0.979
*** 0.000 0.000 0.000 l.OOO
**5 0.0*9 0.001 0.001 0.9*9
**6 0.00* 0.000 0.896 O.IOO
**7 0,158 0.000 0.515 0.327
**8 0.008 0,000 0.915 0.07t
**9 0.009 0.000 0.7*7 0,22*
*50 0,006 0.000
*51 0.0*8 0.021
*52 0.058 0.000 0.005
*53 0.180 0,000 0.820
*5* 0,065 0.000 0.929
*55 0.056 0.005 0.*91
*56 0.019 0.000 0.981 0,000
*57 0,022 0,001 0.975 0.001
*58 0.017 0.003 0.979 0.001
*59 0.015
*60 0.012
*61 0.038
*62 O.IU
*63 0.03*
.92* 0.071
0.306 0.62*
0.9i7
0.000
0.006
0.**9
0.999 0,J08 0,079
0,8*6 0.122 0.020
0.657 0.215 0.090
0.887 0.001 0.000
0.**8 0.00* 0.51*
*6* 0.011 0.583 0.3*7 0.039
*65 0.02* 0.650 0.008 0.319
0.330 0.62i 0.0)6
0.288 0,022 <).*61
. . _ 0.133 O.03* 0.000
*69 0.151 0.8*2 0.002 0.005
*70 0.315 0.002 0.000 0.682
0.000 0.000 0.000
0.000 0.000 0.000
0.000 0.000 0.000
0.000 0.000 0.000
0.000 0.002 0.000
*76 0.822 0.000 0.177 0.001
*77 0.13* 0.000 0.8*6 0.000
*78 0.066 O.OOO 0.927 0.007
*79 0.231 0.000 0.7*9 0.000
*80 0.028 0.000 0.9T2 0.000
*81 0.*13 0.582 0.005 0.000
*82 0.080 0.000 0.8(7 0.03)
0.000 0.906 0.001
0.238 0.607 0.1**
0.9*2 0.025 0.000
. . 0.188 0.011 0.32*
*87 0.859 0.076 0.01* 0.052
*88 0.105 0,335 0,959 0,001
*89 0.119 0.000 0,001 0,8S*
*90 0,637 0.002 0.3*1 0.000
*91 1.000 OtOOO 0.000 0<000
*92 1.000 0.000 0.000 0.000
*93 0.097 0.723 O.IIO O.OOO
*9* 0.2*5 0.222 0.5)1 0.002
*95 0.063 0.017 O.lZt 0.791
*96 0.0*6 0.156 0.796 0.002
*97 0.601 0.012 0.3(7 0.000
*98 0,106 0.002 0.893 0.000
*99 0.02* 0.000 0.976 0.000
500 0.018 0.000 0.912 0.000
501 0.995 C.OOO 0.009 O.COO
502 0.1*5 0.000 0.855 0.000
503 0.98* 0.019 0.000 0.001
*66 0,010
*67 0.228
*68 0.833
*71 1,000
*72 l.OOO
*73 1.000
*7* 1.000
*79 0.998
*83 0.093
*8* O.OU
*85 0.032
*e6 0.*78
LOeARITHM OF LIKELIHnOB RATIO OF * TO 3 TYPES •
CHI>50UARE WITH 9* DEGREES OF FREEOONa 81.1*
PHOBABILITY OF NULL HYPOTHESIS- 0.00986137
0.*12672(*D 02
103
8.3.3.3 Figures and Discussion. Figures 20(a) through (f) illustrate the
tabular output of table 16. In (a), under the assumption of only one cluster
at each of three levels, i.e., a unimodal bivariate distribution, the ellipse
shows the relative size of the distribution at the 700- , 500-, and 300-mb
levels. The 0.50 probability ellipses all have more or less the same orienta-
tions. In (b), the 700-mb unimodal assumption is illustrated along with the
two clusters in the assumed bimodal bivariate distribution. The same procedure
is followed in (c) and (d) for the 500- and 300-mb levels. For further com-
parison, (e) and (f) show the level-to-level comparison for the cluster type 1
and the cluster type 2.
Figures 21(a) through (g) illustrate the tabular output of table 17. Figure
21(a) is the same as figure 20(a). Figure 21(b) shows the 700-mb unimodal
distribution of figure 21(a) with the added three-cluster breakout at the 700-
mb level. The same procedure follows through figures 21(c) and 21(d). Figures
21(e), (f), and (g) show the comparison of type distributions through the three
levels. For example, figure 21(e) shows roughly the same orientation but
greatly increased variance with altitudes. The same is true for figures 21(f)
and (g).
Figures 22(a) through (h) follow the same procedural pattern as the previous
figures. Its comparable table is table 18. Here there are four clusters.
Again the initial unimodal distribution is shown in figure 22(a) while (b),
(c), and (d) show the four clusters at each level. The following four show
the cluster types at the three levels. Though the small numbers may be diffi-
cult to read, they appear also in the tables. Of major importance here is the
pictorial display of the orientations and sizes of the ellipses or clusters.
The change in orientation with altitude is noted. Some strong change in
orientation between types is also noted.
Further work will be done with these distributions to determine the relation-
ship among these clusters, their orientation and dispersion and concurrent
weather.
104
I "/rh-^. 1
(a)
■M
tf^l
■ ■ta
(c)
Figure 20 Bivariate distributions of winds in m-s"^ at Rantoul, niinois,
October 1950-1955 at the 700- , 500- , and 300-mb levels. Two
cluster types (1 and 2) are assumed in the total mixed observed
distribution (0.171 + 0.829 = 1.000). (a) Total distribution,
(b) 700-mb mixed, (c) 500-mb mixed, (d) 300-mb mixed, (e) Type 1,
and (f) Type 2.
105
[M
-"
■
(e)
::
;'
Figure 20 (continued)
106
■m
\
1 . ^
(b)
[
/ / "">W"'"/
)
.
(d)
1
Figure 21 Bivariate distributions of winds in m-s"^ at Rantoul , Illinois,
October 1950-1955 at the 700- , 500- , and 300-mb levels. Three
cluster types (1, 2, and 3) are assumed in the total mixed ob-
served distribution (0.182 + 0.183 + 0.635 = 1.000). (a) Total
distribution, (b) 700-mb mixed, (c) 500-mb mixed, (d) 300-mb
mixed, (e) Type 1, (f) Type 2, and (g) Type 3.
107
<'.
■
(0
■ m
Figure 21 (continued)
108
m
-.fa
r
(b)
.„
,
^
)
■ T ^F
/ . - -
■ ■ \
(c)
'K^-7;,iti;. '•%.,
Figure 22
Bivariate distributions of winds in m-s"^ at Rantoul , Illinois,
October 1950-1955 at the 700- , 500- , and 300-mb levels. Four
cluster types (1, 2, 3, and 4) are assumed in the total mixed
observed distribution (0.094 + 0.193 + 0.293 + 0.421 - 1.000).
(a) Total distribution, (b) 700-mb mixed, (c) 500-mb mixed,
(d) 300-mb mixed, (e) Type 1, (f) Type 2, (g) Type 3, and
(h) Type 4.
109
■
(f)
..
f .(.fm
(g)
/• ' ' '
Figure 22 (continued)
110
8.3.4 Mountain Pass Wind Data Set
8.3.4.1 Input Information.
a. Stampede Pass, Easton, WA, U.S.A. - latitude 47°17' north,
longitude 121°20' west, elevation 1206 m.
b. The period of record is the month of December for the years
1966-1970. The local standard time hours are 0700 and 1300.
c. The data are surface winds (m-s"M and temperatures °C.
d. The number of variables is two then three. The first two
are the zonal and meridional components of the wind, positive
from the west and south, then third is the temperature in °C.
e. The number in the sample is 310, 155 from 0700 and 155 from
1300 l.s.t.
f. The minimum number to be accepted into any cluster is one
more than the number of variates.
g. The null hypotheses are made that (k + 1) clusters are not
significantly different from the k clusters. The decision
probability level selected is 0.01. Rejection of the hypothesis
then permits the assumption of (k + 1) clusters.
h. The first 40 two-dimensional vector entries are set up as the
40 means of 40 separate and individual clusters. These are
40 points in two or three dimensions.
i. The assumption is made that the covariance matrices are not
equal .
8.3.4.2 Tables. Table 19 provides the output data for the two-variable wind
component distributions taken in tabular form from the Wolfe (1971b) NORMIX-
NORMAP computer routine. Table 20 provides the output data for the three-
variable temperature and wind components.
In both tables the mixture proportions, by cluster type, the means, standard
deviations, correlation matrices, and the eigenvalue-eigenvector matrices are
given. An asterisk indicates the rejection of the null hypothesis that the
(k + 1) type is not significantly different from the (k) types.
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115
8.3.4.3 Figures and Discussion. Figure 23a shows the single cluster distri-
bution versus the two-cluster breakout, each cluster assumed to be bivariate
normal. In essence, these assumptions are the unimodal versus the bimodal
bivariate distributions. Table 19 provides the statistics for this figure.
Clearly, it is seen that the total distribution is not well represented by the
single unimodal elliptical bivariate normal distribution with the mean at
(-0.4915, 1.1879) m-s"^ with east-west and north-south component standard de-
viations of 5.7123 and 1.6058 m-s"^, respectively. The ratio is almost four
to one.
The total distribution breaks out into two separate distributions, each
assumed to be unimodal. The attempt to determine whether the distribution
might actually be trimodal rather than bimodal met with no success. There-
fore, it is assumed that the bimodal bivariate distribution is a better
representation than a unimodal or trimodal bivariate representation. The two
modes are east-southeast and west-southwest.
Let us now discuss the location and terrain features of Stampede Pass.
Stampede Pass is located in mountainous terrain on the Main Cascade Divide at
latitude 47°17' north and longitude 121 °20' west. The elevation of the ground
at the station is about 1206 m. The wind instruments are approximately 11 m
higher.
East of the station the ground drops abruptly into the Yakima River Valley,
about 600 m down and a little over three km distant. This land and valley
fall towards the southeast. The lowest part of Stampede Pass is 5/8 km north
and 30 m lower. There is a ridge 1-1/4 km south of the station and about 200 m
higher. West of the station the land drops rapidly about 600 m over a distance
of 6.4 km to the Green River Valley. This land and valley fall toward the
southwest. To the north, from west northwest through east, there are ridges
and peaks which rise to 1.2- and 2.7-km.
General winds from the west will be channeled from the west southwest up and
around the ridge nose then turning to the east southeast and thence southeast.
General winds from the east will traverse this same channel but in the opposite
sense. It would seem then that a wind distribution through the pass would have
to be elliptical. In addition, it would seem that traveling weather systems
would create one distribution from the west southwest and one from the east
southeast. This agrees with the tabular values (table 19) and the illustration
(figure 23a).
Though not presented in either the table or the figure, it should be mentioned
that the computer routine did attempt to converge to a solution for the trimodal
assumption. The last estimates did indicate a tendency for the east southeast
mode to break down into two east southeast modes located on the major axis of
the mode shown but centered, one further to the east southeast and one to the
west northwest.
Table 20 and figures 23b, c, and d present the situation when the temperature
arguments are added to those of the wind. The singularity problem noted above
is resolved and the computations indicate definitely that four cluster types
are present. The null hypothesis would have been rejected at the 0.00000264
116
level. The selected decision level was 0.01. Therefore, there is a good
probability that five or even six clusters would have been isolated if the
computer had been allowed to continue. The option selected was to examine
the structure through only four groups.
Figure 23b provides the two-cluster breakout with an illustration of the
0.25 error ellipses of the wind. The mixture proportion as well as the mean
temperature (°C) associated with each group is shown. As progress is made
through the next two subfigures, note that the western group changes only
slightly moving farther away as the breakdown continues. Please note the
breakout of the eastern group into two groups with the easternmost group being
extremely cold (=-25°C). Note that it then remains relatively fixed while the
central group of (b) breaks down into two clusters.
Note again that the easternmost group which comprises only two percent of
the total is extremely cold. This leads to two conjectures:
(1) This group is an outlier and the data are bad.
(2) This group is an outlier but is a valid cluster. The output of the
data permits identification of the individual datum. The two-percent cluster
occurred in the same period of time ciiid actually composed a string of data.
The records were checked. The data are correct. Figure 24 is a copy of a
National Meteorological Center surface analysis chart for 1200 Greenwich time
on December 30. A star marks the approximate location of Stampede Pass. Note
the incursion of the cold air mass from Canada. With its increasing cold and
speed as seen on prior maps, it is worthwhile to read the comments by Phillips
(1969) published in the Climatological Data publication of the National Oceanic
and Atmospheric Administration. The temperatures mentioned are in degrees
Fahrenheit.
"Washington - December 1968
"Special Weather Summary
"Until near the end of the month, weather systems from over the
Pacific moved across the State at frequent intervals. In western
Washington, this resulted in measurable precipitation on 23 to 28
days and on 12 to 18 days in eastern Washington.
"West of the Cascades, rather heavy precipitation was recorded on
several days, falling as rain in lowlands during the first half
of the month, and as snow and rain the latter half. In the moun-
tains, most precipitation fell as snow, with near record depths
for December on the ground at the end of the month.
"East of the Cascades, snow began accumulating on the ground the
first of the month. After the middle of the month, most agricul-
tural areas in southern counties were covered with 1 to 3 inches
of snow. Temperatures were near or above normal for the first
half of the month and slightly below normal from the 15th to the
25th.
117
"An outbreak of yery cold arctic air accompanied by strong
northerly and northeasterly winds began moving into northern
valleys of eastern Washington, and other localities near the
Canadian State on the 26th, spreading over most of the State on
the 27th. Temperatures continued to fall for 3 days, with the
lowest occurring on the 30th. In many respects, this was the
most severe outbreak of cold air since the winter of 1949-50.
Minimum temperatures dropped below previous records at several
stations in eastern Washington and in the Cascades. The -48°
recorded at Mazama and Winthrop 1 SW is a new record for the
State and -43° at Chesaw 4 NNW is also below previous record of
-42° at Deer Park on January 20, 1937. Other stations where
minimums dropped below previously recorded low temperatures were:
Anatone -32°, Chelan -18°, Colfax 1 NW - 33°, Colville Airport
-33°, Dayton 1 SW -25°, Holden Village -32°, Lacrosse 3 ESE -34°,
Leavenworth 3 S -36°, Methow -37°, Pomeroy -27°, Pullman 2 NW -32°,
Republic -38°, Rosalia -29°, Snoqualmie Pass -19°, Stampede Pass
-21°, Stevens Pass -25°, Stehekin 3 NW -21°, and Waterville -33°.
"On the 30th, a warmer moist airmass from over the Pacific began
moving inland over the colder air near the surface. Snow began
falling during the day, becoming heavy at night and continuing
through the 31st. West of the Cascades, snow depths in the lowlands
ranged from 8 to 1 5 inches and 24 to 36 inches or more in foothills.
In numerous localities, highway traffic was at a near standstill on
the 31st and many offices and businesses remained closed.
"Preliminary reports from fruit producing areas indicate the low
temperatures caused extensive damage to stone fruits and perhaps
some damage to other fruit trees. Most of the winter wheat section
was covered with 1 to 3 inches of snow, thus very little freeze
damage is expected."
Note the fourth paragraph. Ludlum (1969) disucsses this particular feature
in Weatherwise on pages 36-37 of the February issue. Dickey and Wing (1963)
also discuss the problem of arctic air flowing into the Pacific Northwest.
The above example and discussion point out the usefulness of a program such
as this for editing and quality control of multivariate data; i.e., data groups
other than one element at a time. Here, an outlier group was isolated but it
was a valid group. The authors did not realize that this particular outlier
group was embedded in the data set used. Simply, a location and a period were
selected where it was thought that the program would successfully and pointedly
demonstrate its capability.
118
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(d)
Figure 23 Stampede Pass, Easton, Washington, U.S.A.; winds (m-s"M and
temperatures (°C), December 1966-1970, showing breakdown of
winds only into groups 2 and 3 from group 1 (a) and breakdown
of wind-temperature combination into 2, 3, and 4 groups (b, c
and d).
119
1200Z DEC. 30 1968
69. NMC SFC ANALYSIS
ASUS 69.
Figure 24 Selected area of North American chart, 1200Z, Monday, December 3,
1958, NMC analysis. The star represents the approximate position
of Stampede Pass, Easton, Washington.
120
8.3.5 Marine Surface Data Set
8.3.5.1 Input Information.
a. OSV "C," 52°45' north latitude, 35°30' north longitude
b. The period of record is the month of February for the years
1964 through 1972.
c. The data are the 1200 G.C.T., pressures, temperatures, dew
points, and surface winds.
d. The number of variables is five. The wind is broken into the
zonal and meridional components. The units are mb, °C, and m-s"^.
e. The number in the sample is 251.
f. The minimum number to be accepted into a sample is six.
g. The null hypotheses are made that (k + 1) clusters are not
significantly different from the k clusters. The decision
probability level selected is 0.01. Rejection of the hypothesis
then permits the assumption of (k + 1) clusters.
h. The first 40 five-dimensional vector entries are set up as the
40 means of 40 separate and individual clusters. These are 40
points in five dimensions.
i. Non-equality of covariances is assumed.
8.3.5.2 Tables - Output Information. The program computes the necessary
statistics for two clusters versus one cluster, three clusters versus two
clusters, and four clusters versus three clusters. These are taken from the
tabular outform of the Wolfe (1971b) NORMIX computer routine.
The output statistics are now shown for the above in tables 21, 22, and 23
even though the null hypothesis is rejected for the last case.
The first variable is the surface atmospheric pressure in mb, the second
variable is the dry-bulb air temperature in °C, the third variable is the
dew point temperature in °C, while the fourth and fifth variables are the
zonal and meridional components of the wind in m-s"^.
121
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8.3.5.3 Figures and Discussion. Figure 25 is an attempt to illustrate a part
of the output of table 21. The basic two-dimensional representation shews the
decomposition of the assumed unimodal distribution into two modes or groups.
The probability ellipses represent the area of the central 25 percent of the
wind vector origins if the assumption which they represent is valid. As shown
in table 21, the probability of the null hypothesis for two groups versus one
group not being rejected is wery low, i.e., 0.00000002. Therefore, the more
valid assumption is that two groups better represent the data set than does
one group. The single group centered at (2.0414, 1.0315) is shown for reference.
At each point, the following concurrent values are printed, the mixture
proportion, the mean pressure, the mean temperature, and the mean dew point.
The two groups appear to have not too different mean pressures but considerably
different temperature and dew points.
Figure 26 prepared from table 22 similarly portrays the decomposition of the
total group into three groups. It is quite apparent that group 1 is not much
different from group 1 of figure 25, It is just as apparent that group 2 of
figure 25 really breaks down into groups 2 and 3 as shown in figure 26. The
pressures in groups 2 and 3 are quite different though the temperatures and dew
points are not too different. The appearance here implies, as does the proba-
bility of non-rejection of the null hypothesis being small, that the trimodal
representation is better than the bimodal which is in turn better than the
unimodal representation.
Figure 27 prepared from table 23 depicts the further decomposition of the
data set into four groups. Again, group 1 retains essentially the same mixture
proportions and characteristics. The northerly group (group 4) remains almost
the same as in the last decomposition. It is group 2 of figure 26 that now
breaks down into groups 2 and 3 of figure 27. The pressure differences are
remarkable, providing the maximum and minimum pressures for the entire four-
cluster configuration. Group 3 has the highest temperatures and dew points of
the entire ensemble. The 10-percent mixture proportion of group 2 leaves the
impression that this is a real group though the null hypothesis is not rejected.
This non-rejection probability of 0.12 implies that at our decision level of
0.01 probability, the trimodal representation is a better representation than
the four-cluster configuration. However, it is interesting to conjecture that
the differences shown are the difference between storm and non-storm situations
with winds from the southeast quadrant.
No other representations are made though any pairing or triplets could be
graphed in two and three space, respectively, and could be labeled with the
fourth and fifth variable mean values.
The authors consider this to be a good representation of the clustering
techniques of the Wolfe NORMIX (1971b) computer routine. It was hoped that
this routine would isolate a cluster of six or more data which might be con-
sidered to be an outlier cluster which could be examined. There were 251 data.
Ten percent is essentially 25 or 26 data in the clusters. The instructions
provided to the computer were to collect no less than six data in a cluster.
This is a default option which sets the minimum number in any cluster to be
one more than the number of variables. In this case, five plus one is six.
129
Therefore, (a) single, doublet or triplet outlier(s) would not be isolated.
In this case, if the mixture proportion ran as low as 2.5 percent, the group
would be considered as an outlier group and would be examined to see whether
the data were bad because of bad sensors, bad recording, bad entry of data
into the archives, or simply a group of valid observations out of a yery rare
weather situation.
130
-2D
LEGEND
D5Y G FEB. 19EH-72/ Ml DBS.
BlYRRIflTE mmi DI5TR1BUT!DN5 DF U mh Y
PRESSURE IN MB
TEMPERBTURE IN 5EBREE5 (
m POINT IM DEGREES (
MIKTURE RflllD
la
-10
Figure 25 OSV "C" surface distribution of pressure (mb), temperature (°C),
dew point (°C), and wind components (m-s~M» February, 1200 G.C.T.,
1964 through 1972; n = 251. Covariances are assumed to be unequal.
The 0.25 probability ellipses are shown for the wind distribution.
The total distribution and the breakout into two clusters are shown
The mixture proportion and the averages, the pressure, the tempera-
ture, and the dew point data within each cluster are shown.
131
1-20
Figure 26 OSV "C" surface distribution of pressure (mb), temperature (°C),
dew point (°C), and wind components (m-s"M> February, 1200 G.C.T.:
1964 through 1972; n = 251 . Covariances are assumed to be unequal.
The 0.25 probability ellipses are shown for the wind distribution.
The total distribution and the breakout into three clusters are
shown. The mixture proportion and the averages, the pressure, the
temperature, and the dew point data within each cluster are shown.
132
Figure 27 OSV "C" surface distribution of pressure (mb), temperature (°C),
dew point (°C), and wind components (m-s"^), February, 1200 G.C.T.,
1964 through 1972; n = 251 . Covariances are assumed to be unequal,
The 0.25 probability ellipses are shown for the wind distribution.
The total distribution and the breakout into four clusters are
shown. The mixture proportion and the averages, the pressure, the
temperature, and the dew point data within each cluster are shown.
133
8.3.6 Radiosonde and Rawinsonde Data Set
8.3.6.1 Input Information.
a. Balboa (Albrook Field), Canal Zone
b. The period of record is the month of July 1961-1970.
c. The data are pressures or height (mb or m) temperatures (°C),
dew points (''C), east-west (u) wind components, and north-south
(v) wind components, m-s"^ positive from the west and south.
d. The number of variables are 20; the above elements for the
surface and the 850- , 700- , and 500-mb levels.
e. The number in the sample is 259.
f. The minimum number to be accepted into a sample is 21.
g. The null hypotheses are made that (k + 1) clusters are not
significantly different from the k clusters. The decision
probability level selected is 0.01. Rejection of the hypothesis
then permits the assumption of (k + 1) clusters.
h. The first 40 twenty-dimensional vector entries are set up as the
40 vector means of 40 separate and individual clusters. These
are 40 points in twenty dimensions,
i. Two assumptions are made. The first assumption is the equality
of covariance matrices. The second assumption is the non-equality
of the covariance matrices.
8.3.6.2 Tables and Discussion. Table 24 provides selected output data of the
Wolfe (NORMIX) computer routine. The logarithm of the likelihood ratio of 2
to 1 types is 676.30485, the chi-square with 40 degrees of freedom is 1240.33,
and the probability of the null hypothesis not being rejected is to seven
decimals, 0.0000000. Therefore, the two-cluster configuration is not rejected.
The computation failed to converge for the three-cluster versus the two-cluster
configuration. Convergence under the assumption of unequal covariance matrices
also failed. No figures are provided here as the number of variables is too
high. A number of two-dimensional figures, however, could be made from the
statistics provided.
In order to determine the capability of the NCC computer (a Univac Series 70)
and in view of the previous work, a multivariate problem with 40 element vectors
was chosen for Balboa, C.Z. There were eight levels of each radiosonde with
five elements, pressure (or height), temperature, dew points, and the east-west
and north-south components of the winds. The levels were the surface and the
950- , 900- , 850- , 800- , 700- , 600- , and 500-mb levels. The period chosen was
the months of July during 1951-1970. The assumption was made that the covari-
ance matrices were different. The computation failed to converge for a two-
group separation. The number of vector elements was reduced to 20. Again the
computation failed to converge.
The assumption of different covariance matrices was then replaced by the
assumption of equal covariance matrices. The computation converged for a
breakout of two groups but failed on three groups when the vectors were com-
posed of 20 elemeits, five elements from each of four of the levels above,
namely the surf a _., and the 850- , 700- , and 500-mb levels as these were the
134
only elements believed to have moisture measurements in sufficient quantities
to assure enough input data.
Examination of table 24 shows that though there does not appear to be too
much difference, there is some. About 95 percent of the data comprise one
(Type 1) cluster while 5 percent of the data comprise the other (Type 2)
cluster. Cluster 1 is cooler than cluster 2 at all levels through 500 mb, an
altitude of roughly 5854 m. Cluster 1 is more moist than cluster 2 at alti-
tudes above about 3,100 m and probably above 2000 m. Cluster 1 exhibits lower
wind speeds than does cluster 2 from the surface upwards. The difference
ranges from 1.2 m-s"^ at 1500 m to 1.6 m-s~^ at 5854 m. Cluster 1 winds shift
from east northeast to east by southeast at the highest level while the winds
in cluster 2 remain east by northeast throughout the layer. The surface pres-
sure of cluster 1 is about 0.6 mb higher than that of cluster 2.
The types of weather that accompany these groups have not been investigated
here. It would be interesting to do so. Balboa, C.Z., data were selected
(1) to provide an insight into the lower level atmospheric characteristics,
(2) to further understanding of the capability of this program, adopted for
use on the Univac 70 at the NCC, to handle multidimensional problems, and
(3) to demonstrate the utilization of such a program to consider each radio-
sonde observation as a point in multidimensional space.
From the above experience it appears that the program, restricted by the
present Univac 70 configuration, can handle 300 input 20-dimensional data
problems. The first assumption made should be the assumption of equal co-
variance matrices.
Use of the program for data at other stations where greater differences may
be expected may permit the use of more input data, greater dimensions, and the
assumption of unequal covariance matrices.
The minimum number to be accepted into a cluster is always one more than the
number of dimensions. In this case the minimum number is 21. This is about
8.1 percent. Cluster Type 2 has only about 5.4 percent. Therefore, it would
be advisable to check those observations assigned to Type 2 by the discrimina-
tion function for the possibility of all or some of these being outliers.
This is not done here as this is the basis for work beyond the scope of this
paper.
135
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9. MULTIVARIATE QUALITY ASSURANCE AND CONTROL
The capability of the NORMIX program permits a quick elementary view of
principal component analysis and of multivariate quality assurance and control.
The techniques of assurance and control are the same. Assurance is for in-
coming data or material, while control is for data or material processing.
Essentially, these are filtering techniques.
Here, we look at the separation of homogeneous subsets out of mixed sets
which permits the isolation of "outliers" for examination.
Figure 28 (Crutcher, 1966) is an example of a two-tailed Gaussian filter
operating on a set of univariate heterogeneous data to isolate outlying data.
This essentially sets up two or more groups or subsets. The progress of the
sub-figures (a) through (e) schematically shows how the procedure works on a
univariate distribution. The alpha level of rejection is 0.05. In this
illustration the main group is isolated and the statistics estimated. The
other groups have been isolated for further study as to whether they are valid
data.
The NORMIX program clustering techniques essentially go through the same
technique. However, the decision (alpha) level chosen for the previous
examples was 0.01 probability rather than 0.05.
Use can be made of the univariate Gaussian filter or the NORMIX filter can
operate on any univariate set of mixed normal distributions. Refer to the
Canton Island, July, 30-mb data set. Figure 15 illustrates by means of 0.50
probability ellipses the breakdown of the total set of data into two subsets
with quite different characteristics. (See section 8.3.2.3.) A univariate
Gaussian filter could be used on the zonal or meridional components. To
demonstrate the application of the NORMIX clustering technique as well as
principal component techniques, zonal and meridional components are transformed
to uncorrelated pairwise data along the major and minor axes, the two principal
axes of the distribution. Prior to and after the rotation, the components are
standardized to a zero mean and a variance of 1. See table 25.
Figures 29a and 29b show the frequency distribution computer printer plot of
the components along the two principal axes. Figure 29a clearly delineates
the two clusters. In the latter case, separation would be most difficult on
a one by one basis, but at least the mean and variance can be estimated. The
existence of groups with equal mean and variance would not be indicated.
Clearly indicated in Figures 29a and 29b are the potential invalid outliers at
+1.95 on the major axis and beyond 4.16 on the minor axis. These are simply
pointed out to the reader but are not examined for validity.
Figures 30a, 30b, and 30c (Crutcher, 1966) illustrate a Gaussian filter
technique in two dimensions. Figure 30a illustrates a sample drawn from a
homogeneous bivariate distribution contaminated by three subsets of data. One
of the subsets consisting of only a datum is first isolated, eliminated from
further immediate consideration, but set aside for investigation as to its
validity.
137
The clear separation along the major axis exhibited by the frequency diagram
of figure 30a is shown by the above statistics. The separation along the minor
axis is not so clearly demonstrated.
Refer to figure 29a and tables 25 and 26. Table 25 provides the statistics
for the components along the major axis of the distribution of winds at Canton
Island, U.S.A. and U.K., for July 1954-1964. These are standardized trans-
formed variables and as such are dimensionless in the terms of units as in all
figures in this section. The mean of the total distribution is zero and the
variance is about 1.0006 to four decimals. As this is a standardized set of
data, the mean should be zero and the variance one. The error in the fourth
place is a numerical rounding error. The set breaks down into two clusters.
The mean of the first group is -1.0420 with a standard deviation of 0.2442.
The respective values for the second group are +0.7923 and 0.5127.
The output from the Cohen-Falls (1967) program provides the data in table
26. Note the output from ungrouped data as used in the NORMIX program, table
25. There are slight differences which are attributed to the use of ungrouped
data and then grouped data.
The one outlier near 1.95 in figure 30a would not be set aside as a cluster
or outlier by the NORMIX program. As indicated elsewhere, the minimum cluster
size is always one more than the number of variates.
Table 26 also shows the output of the Cohen-Falls program for the dimension-
less wind components along the minor axis as illustrated in figure 29b. It is
recognized that, with the exception of the outliers, the mixed distribution has
degenerated into a single standardized univariate unimodal normal distribution
with a zero mean and variance of one. Note the proportion of two groups 0.998
and 0.002 with only 244 observations. Note the mean of -0.009617 in the first
group with the extraordinarily large mean of 7.410603 for the second group.
Remember the standard deviation of the entire set is only one. Then look at
the untenable negative variance obtained which is printed as x.xxxxx. However,
this negative variance indicates the difficulties in arriving at a solution
and clearly implies that it is the simple case of a symmetric mixed or compound
distribution with equal means and equal variances with proportions equal.
Therefore, the decision will be to use the estimates for the total sample as
estimates of the two groups {a mean of zero and a variance of one in each case).
This is the trivial degenerate case which does provide some computing difficulty
but no interpretative difficulty.
The elimination of the outliers along the minor axis and the re-standardiza-
tion of the data will provide a mean of zero and a variance of one rather than
the values given in table 26.
Figures 31a through 31c show the frequency distributions of the components
along three principal axes of the Stampede Pass, Easton, Washington, wind and
temperature standardized data. These refer to the figure 23. The existence
of outliers is clearly demonstrated. Outliers lie beyond -2.56 on the first
principal axis, beyond -1.68 on the second principal axis, and in both tails
of the third principal axis. Undoubtedly, some of these belong to the extreme
low temperature cluster isolated and discussed in section 8.3.4. The flatness
138
of the distribution on all three axes and the more extreme bimodality exhibited
on the third axis show the existence of the groups or clusters already isolated.
Extension of the Gaussian filter to more than one dimension at a time essen-
tially is the standardization of all components along the major axes so that
the distribution may be assumed to be spherical (Crutcher, 1966). The dis-
tribution of the standardized vectors in n-dimensions then is chi -square with
n degrees of freedom. The appropriate decision probability levels then are
obtained from any standard text book containing chi-square tables. Alterna-
tively, there are computer routines to compute the required values.
For example, the magnitude of the standardized vector in the one-dimensional
case for the rejection of the top and bottom 0.5 percent (0.005 probability)
implying non-rejection of the central 99 percent (0.99 probability) will be
the square root of the 0.99 chi-square value divided by 1 (the degrees of
freedom). This is 2.576 and is, of course, the t-distribution value for a
large sample. In the case of a two-dimensional distribution, the magnitude
of the standardized bivariate vector is the square root of 9.21/2 or 2.146.
For the three-dimensional case, the vector radius would be 1.944.
The point made here is that a datum lying on the main principal axis, the
major axis, has a better chance of being valid than an outlier on only one
of the original coordinate axes. The chance for an observation to be in-
correct in two variables is much less than the chance for an error in the
observation of one variable. An outlying datum on one axis is, of course,
projected as zero on the other transformed axes so that an outlier on one
axis will not be associated with an outlier on another axis. In fact, its
effect is to increase the frequency count at the mean or zero of the other
axes. Another way of viewing this is that the rejection ellipse has a better
chance of enclosing a possible outlier than the rejection bounds on either of
the two original correlated axes.
Figure 30b shows the effect of computing a new filter ellipse once the
outlier shown in figure 30a has been eliminated. Now the two small contami-
nating subsets undetected in the first operation are isolated. Figure 30c
then illustrates the last step where all data points remaining are inside the
filter ellipse. All rejected data are set aside for investigation as to their
validity. Before this check is made, however, these data may be processed to
see whether they constitute one or more groups or clusters.
The procedure, of course, brings up the possible censoring or truncation
problem. Adequate adjustments and better estimates of the parameters can be
made.
There is no discussion here of the applications of higher moments to the
problem of outlier detection, isolation, and removal.
Figures 32a through 32t illustrate the frequency distributions of the radio-
sonde data discussed in section 8.3.6. This discussion indicated only a slight
separation of clusters as contrasted with the stratospheric data of section
8.3.2. The frequency diagrams of figures 32a through 32t also show only a
slight separation potential in the flat of platykurtic curves.
139
Figures 32a through 32t are shown more to demonstrate the outlier problem
than the clustering problem. The mathematical requirements of the clustering
techniques always specify that the minimum cluster size is one more than the
number of variates. Thus, with twenty variables the smallest cluster size is
twenty-one. A cluster of this size would really indicate a true valid cluster
created by weather conditions or a continuing bias in the procedures or operat-
ing conditions. The lone outlier(s) in multi-space might go undetected. How-
ever, the frequency distribution along a principal axis or a clustering tech-
nique along each of the axes would isolate, identify, and permit removal of
the questionable data. Therefore, the frequency distributions of figures 29a
and b and 32a through t can be used to look at the few individual outliers.
Clustering techniques would have a better chance to show this outlier as there
is only one variable or a cluster size of two.
Alt et al . (1973) discuss quality assurance and control in their paper on
the use of control charts for multivariable data. Crutcher and Falls (1976)
discuss the testing of data sets for multivariate normality. The above two
reports and this present report add to the rather meager literature on the
subject of multivariate quality assurance and control.
140
20r
H. 10
(a)
Main body of sample
Outlying subsample
dllh
Outlier
10
20
30
40
50
60
70
80
30-
20-
10
(b)
Mean
=
29.
37
Variance
=
177.
74
Standard deviation
=
13.
33
n
=
81
d. f.
=
80
.050
t
.020
=
1.
99
=
2.
37
.025
30
70
80
Figure 28 Example of a two-tailed Gaussian filter operating on a set of
heterogeneous data to isolate, set aside, and eliminate outlying
data. The dark areas show the rejection; level, 0.05 (0.025 in
each tail). Crutcher (1966).
(a)
(b)
(c)
(d)
(e)
Here are the data in histogram form.
Here are the data with the theoretical fitting curve under
the assumption of normality and independence of data. The
rejection areas under each tail are shown beyond the 0.025
and 0.975 points. The observation at 75 is rejected.
Here is the theoretical fitting curve of the data which
passed the filter in (b). Again the rejection areas under
each tail beyond the 0.025 and 0.975 points are shown.
Data beyond 52 are rejected.
Here is the theoretical fitting curve of the data which
passed the filter in (c). Again the rejection areas under
each tail beyond the 0.025 and 0.975 points are shown.
Data beyond 47 are rejected.
Here is the theoretical fitting curve for the data which
passed the filter in (d). Although the rejection areas
are not shown, the singleton counts below 21 and above 25
would be rejected. This rejection is an unwanted rejection
but is a penalty which must be accepted at this point. The
next filtering step would result in no rejection.
l-ai
Mean =
28.
80
Variance =
139.
62
Standard deviation =
11.
81
n =
80
d. f. =
79
'.050 =
1.
99
70
80
30
20
10
(d)
Mean
=
27.
17
Variance
=
104.
06
Standard deviation
=
10.
20
n
=
75
d. f.
.050
74
1.99
60
— I —
70
80
30
20
10-
(e)
20
Mean
=
23.
00
Variance
=
1.
52
Standard deviation
=
1.
23
n
=
64
d. f.
=
63
k.
— 1 —
30
.050
— I —
60
2.000
10
40
50
Figure 28 (continued)
142
SAMPLE = CANTON ISLAND
JULY 30 MB
X AND Y WIND COMPONENTS
PkFQI^FNC Y DTSTRlRn Tlnil AinNf, &»!■;
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Figure 29 Distribution of wind standardized components along the two
principal axes of the Canton Island, U.S.A. and U.K., July,
30 mb.
143
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M6
STAMPEDE PASS, WASHINGTON
SURFACE TEMPERATURE (DEGREE C), U AND V WIND COMPONENTS (MPS)
DECEMBER 1966, 1967, 1968, 1969, 1970
HOURS 07 AND 13 LOCAL TIME (1ST 155 INPUTS ARE HR 07)
^REQUENI-Y UlblRlPUrUM ALONU AXIS 1
SACH X REPRESENTS — 1 SAMPuefSJ
CLASS Interval- o.32oS6
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61 (b)
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Figure 31 Distribution of wind and temperature standardized components
along the three principal axes of the Stampede Pass, Easton,
Washington, December 1968-1970 data.
147
FreOUeNCY DISTRIBUTIflN AlOnG AxH i
EACH X Represents — i samPle(S>
CLASS INTERVAL* 0.32000
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Figure 31 (continued)
148
ALBROOK AFG CANAL ZONE RADIO!
(PRESSURE AT SUHFACei, TEMPERATURE, OEW POIN
AT SURFACE. 850Me. 70CMB. WOMB
JULY 12Z 1961 THRU 1970
I AND V WIND COMP
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154
10. PREDICTION
In the multimodal multivariate case, prediction first should be to the modes
or clusters. Once a mixed multivariate data set has been separated into its
various homogeneous parts, prediction within the cluster can be made. The
characteristics of the group are known and the appropriate regression equation
for each homogeneous group can be developed.
In a time series which may be composed of two or more periodic or aperiodic
parts, it will be necessary to determine which components are active at the
moment. Prediction is then again straightforward.
The problem still remains as to the deterministic regime operating at the
moment. This may require prediction into the appropriate cluster and then
prediction within the cluster.
The problem of prediction is an important one considered to be beyond the
scope of the present paper. It is one to which the authors intend to return
and to provide the necessary procedures to forecast to the cluster then within
the cluster. Briefly, this has been touched on in section 3 on Discriminant
Techniques.
155
SUMMARY
Clustering techniques to separate mixed distributions of meteorological data
are presented. The techniques and data sets are restricted, to multimodal
multivariate distributions. Dichotomous or polychotomous and non-normal dis-
tributions have not been discussed.
The major element considered is wind. Three specific examples include the
situations of the land and sea breeze, the quasi-biennial oscillation, and
winds in a pass. Three other examples are continental tropospheric winds, a
location on the ocean, and a tropical troposphere. One of the examples does
include temperature; another includes temperatures, dew points, and heights
of pressure surfaces; another includes pressure, temperature, and dew point;
while another includes heights of the pressure surfaces and temperatures.
The electronic computer program available to make the separation of mixtures
into their homogeneous parts for weather data effectively does the job.
The technique will be useful in establishing weather or climate groups which
can be set aside for study and for use in guidance and forecasting.
156
REFERENCES
Alt, F. B. , 1973: Variable control charts for multivariable data. School of
Industrial and Systems Engineering, Georgia Institute of Technology,
Atlanta, 42 pp.
Anderson, Edgar, 1935: The irises of the Gaspe Pennisula. Bulletin of the
American Iris Society , 59, 2-5.
Anderson, T. W. , 1958: An Introduction to Multivariate Statistical Analyses .
John Wiley and Sons, New York, 272-279.
Anderson, T. W. , and Rubin, H., 1956: Statistical inference in factor analysis.
Proceedings of the Third Berkeley Symposium on Mathematical Statistics
and Probability , University of California Press, Berkeley and Los Angeles,
5, 111-150.
Andrews, D. F., Gnanadesikan, R., and Warner, J. L., 1971: Transformations
of multivariate data. Biometrics , 27, 825-840.
Angell, J. K. , and Korshover, J., 1962: The biennial wind and temperature
oscillations of the equatorial stratosphere and their possible extension
to higher latitudes. Monthly Weather Review , 90, 127-132.
Aversen, J. N., and McCabe, G. P., Jr., 1975: Subset selection problems for
variances with applications to regression analysis. Journal of the
American Statistical Association , 70 (349), 166-170.
Baker, Frank B., and Hubert, Lawrence J., 1975: Measuring the power of hier-
archical cluster analysis. Journal of the American Statistical Associa-
tion, 70 (349), 31-38.
Bannerjee, K. S. , and Carr, R. N., 1971: A comment on ridge regression, biased
estimates for non-orthogonal problems. Technometrics , 15 (4), 895-898.
Barnard, M. M. , 1935: The secular variations of skull characters in four
series of Egyptian skulls. Annals of Eugenics , London, 6, 352-371.
Bartlett, M. S., 1950: Tests of significance in factor analysis. British
Journal of Psychology , 3, 77-85.
Bartlett, M. S., 1951: A further test note on tests of significance in factor
analysis. British Journal of Psychology , Statistical Section, 4, 1-2.
Bartlett, M. S. , 1953: Factor analysis in psychology as a statistician sees
it. Uppsala Symposium on Psychological Factor Analysis , Uppsala ,
Almqvist Wiksell , 23-24.
Berlage, H. P., 1956: The Southern Oscillation, a 2-3 year fundamental
oscillation of worldwide significance. lUGG 10th General Assembly , Rome,
1954 , Scientific Proceedings International Association of Meteorology ,
London, 336-345.
157
Boehm, Albert, 1974: Modeled conditional climatology using multiple pre-
dictors. Paper presented at the American Meteorological Society ,
Climatology Conference and Workshop at Asheville , North Carolina , on
October 8, 1974 , 36 pp.
Bose, R. C, and Roy, S. N., 1938: The distribution of the Studentized D^
Statistic. Sankhya , 4 (Part 1), 19-38.
Box, G. E. P., and Ti dwell, P. W., 1962: Transformation of the independent
variables. Technometrics , 4, 531-550.
Brooks, C. E. P., and Carruthers, N., 1950: Upper winds over the world.
Great Britain Meteorological Office, Air Ministry, Geophysical Memoirs
No^ 85 , Her Majesty's Stationery Office, London, 150 pp.
Brown, George W. , 1947: Discriminant functions. Annals of Mathematical
Statistics , 18, 514-528.
Burrau, C, 1934: The half- invariants of two typical laws of errors with an
application to the problem of dissecting a frequency curve into compo-
nents. Skandinavisk Aktuarietidskrift , 17, 1-6.
Cattell, Raymond B., 1952: Factor Analysis . Harper and Brothers, New York,
462 pp.
Charlier, C. V. L., 1906: Researches into the theory of probability.
Meddelanden fran Lunds Astronomiska Observatorium, Lunds Universitets
Arsskrift, N.S., Asd2, Band 1, nr. 5, Series 2, Band 1, No. 4.
Charlier, C. V. L., and Wicksell, S. D., 1924: On the dissection of frequency
functions. Arkiv for Matematik, Astronomi och Fysik, Bd. 18, No. 6.
Clayton, H. H., 1885: A lately discovered meteorological cycle. American
Meteorological Journal , 1 , pp. 130 and 528.
Cohen, A. C, 1965: Estimation in mixtures of two normal distributions.
University of Georgia Technical Report , No . 13, 38 pp .
Cohen, A. C, and Falls, L. W., 1967: Estimation in mixed frequency dis-
tributions. NASA TN D-4033 , National Aeronautics and Space Administra-
tion, George C. Marshall Space Flight Center, Huntsville, Alabama, 90 pp.
Cox, Gertrude M., and Martin, William P., 1937: Use of a discriminant
function for differentiating soils with different azotobacter populations.
Iowa State College, Journal of Science , 11 (3), 323-331.
Crutcher, Harold L., 1957: On the standard vector deviation wind rose.
Journal of Meteorology , 14 (1), 28-33.
Crutcher, Harold L., 1958: Upper wind distribution statistical parameter
estimates. Technical Paper No. 34, U.S. Department of Commerce, Weather
Bureau, Asheville, North Carolina, 59 pp.
158
Crutcher, Harold L., 1959: Upper wind statistics of the Northern Hemisphere.
Issued by the Office of the Chief of Naval Operations, NAVAER 50-1C-535 ,
Vol. I, 35 pp. with 168 charts and 54 overlays, Vol. II, 35 pp. with 168
charts and 54 overlays, and Vol. Ill, 23 pp. with 56 charts.
Crutcher, Harold L., 1960: Statistical grouping of climates and the statisti-
cal discrimination among climate groups. A dissertation in the Department
of Meteorology and Oceanography submitted to the faculty of the Graduate
. School of Arts and Sciences in partial fulfillment of the requirements
for the degree of Doctor of Philosophy at New York University, University
Microfilms, Ann Arbor, Michigan, 462 pp.
Crutcher, Harold L., 1966: Gaussian filters for quality control of vector
data. (Unpublished), 42 pp.
Crutcher, Harold L., and Clutter, Jerome L., 1962: Heterogeneous wind distri-
butions. North Carolina Operations Analysis Standby Unit, Chapel Hill,
North Carolina, (unpublished manuscript), 56 pp. and appendices.
Crutcher, H. L., Wagner, A. C. , and Arnett, J. S., 1966: Components of the
lOOO-mb winds of the Northern Hemisphere. Issued by the Office of the
Chief of Naval Operations, NAVAIR 50-1 C-51 , 4 pp. with 74 charts.
Crutcher, Harold L., and Falls, Lee W., 1976: Multivariate normality. NASA
TN D-8226 , National Aeronautics and Space Administration, George C.
Marshall Space Flight Center, Huntsville, Alabama, 273 pp.
Davies, Owen L. , 1956: The Design and Analysis of Industrial Experiments .
Hafner Publishing Company, New York, 637 pp.
Dempster, A. P., 1973: Aspects of the Multinomial Logit Model . Edited by
P. R. Krishnaiah, Academic Press, Inc., New York, 129-142.
Dickey, Woodrow W. , and Wing, Robert, N. , 1963: The unusual: Arctic air
into the Pacific Northwest. Weatherwise , 16 (6), 259-263.
Draper, N. R. , 1963: Ridge analysis of response surfaces. Technometrics ,
5 (4), 469-479.
Duda, Richard 0., and Hart, Peter E., 1973: Pattern Classification and Scene
Analysis . John Wiley and Sons, Inc., New York, 481 pp.
Ebdon, R. A., 1960:^ Notes on the wind flow of 50 mb in tropical and sub-
tropical regions in January 1957 and January 1958. Quarterly Journal of
the Royal Meteorological Society , 86, 540-542.
Essenwanger, Oskar, 1954: Neue Methode der Zerlegung von Haufigkeitsverteil-
ungen in Gausische Normal kurven und ihre Anwendung in der Meteorologie.
Berichte des Deutschen Wetterdienstes Nr. 10, DK 551.501.45, Bad
Kissengen, 1-11.
159
Finney, D. J., 1952: Probit Analysis , A Statistical Treatment of the Sigmoid
Response Curve , Second Edition, Cambridge University Press, New York,
318 pp.
Fisher, R. A., 1928: The general sampling distribution of the multiple corre-
lation coefficient. Proceedings of the Royal Society , Vol. A121, 654 pp.
Fisher, R. A., 1936: The use of multiple measurements in taxonomic problems.
Annals of Eugenics , 7 (2), 179-188.
Fix, E. , and Hodges, J. L., 1952: Discriminatory analysis - non-parametric
discrimination: consistency properties. Report No . 4, USAF School of
Aviation Medicine, Randolph Field, Texas, 21 pp.
Friedman, H. P., and Rubin J., 1967: On some invariant criteria for grouping
data. Journal of the American Statistical Association , 62 (320), 1159-
1178.
Fruchter, B. , 1954: Introduction to Factor Analysis . D. Van Nostrand Co.,
New York, 280 pp.
Galton, Francis, 1889: Natural Inheritance . MacMillan and Company, London,
259 pp.
Girschik, M. A., 1936: Principal components. Journal of the American
Statistical Association , 31, 519-528.
Good, I. J., 1965: Categorization of classification. Mathematics and
Computer Science in Biology and Medicine , Her Majesty's Stationery Office,
London, 115-125.
Graybill, Franklin A., 1961: An Introduction to Linear Statistical Models ,
Vol . 1. McGraw-Hill, New York, 463 pp.
Graystone, P., 1959: Tropical meteorology. Meteorological Office Discussion,
Meteorology Magazine , 88, 113-119.
Gupta, S. S., and Sobel , M., 1962: On selecting a subset containing the
population with the smallest variance. Biometrika , 49 (3, 6), 495-507.
Hald, A., 1952: Statistical Theory With Engineering Applications , John Wiley
and Sons, Inc., New York, 602-606.
Harman, Harry H. , 1967: Modern Factor Analysis , The University of Chicago
Press, Chicago, Illinois, Library of Congress Card 67-20572, 1967, 474 pp.
Hartigan, J. A., 1975: Clustering Algorithms , John Wiley and Sons, Inc., New
York, 351 pp.
Hartley, Herman 0., 1959: Two-component mixtures of circularly symmetric wind
roses. Preliminary Technical Report No . 1_, Statistical Laboratory, Iowa
State College, Ames, Iowa, 25 pp.
160
Heastie, H. , and Stephenson, P. M., 1960: Upper winds over the world, Parts
1 and 2. Great Britain Meteorological Office, Air Ministry, Geophysical
Memoirs No. 103 , Third Number, Vol. XIII, M.O. 631c, Her Majesty's Sta-
tionery Office, London, 217 pp.
Hoerl , A. E. , 1962: Applications of ridge analysis to regression problems.
Chemical Engineering Progress , 58, 54-59.
Hoerl, A. E. , and Kennard, R. W., 1970(a): Ridge regression: biased estima-
tion for non-orthogonal problems. Technometrics , 12 (1), 55-67.
Hoerl, A. E. , and Kennard, R. W., 1970(b): Ridge regression: application to
non-orthogonal problems. Technometrics , 12 (1), 68-82.
Hotelling, Harold, 1931: The generalization of Student's ratio. Annals of
Mathematical Statistics , 2, 360-378.
Hotelling, Harold, 1933: Analysis of a complex of statistical variables into
principal components. Journal of Educational Psychology , 24, pp. 417-441
and 498-520.
Hotelling, Harold, 1936(a): Relations between two sets of variates.
Biometrika , 28, 321-377.
Hotelling, Harold, 1936(b): Simplified calculation of principal components.
Psychometrika , 1, 27-35.
Hotelling, Harold, 1954: Multivariate analysis. In Statistics and Mathematics
in Biology , (edited by 0. Kempthorne et al.), The Iowa State College Press,
Ames, Iowa, 67-80.
Hotelling, Harold, 1957: The relations of the newer multivariate statistical
methods to factor analysis. British Journal of Statistical Psychology ,
10, 69-79.
Hsu, P. L. , 1938: Notes on Hotelling's generalized T. Annals of Mathematical
Statistics , 9, 231-243.
Kendall, M. G. , and Smith, B. Babington, 1950: Factor analysis. Journal of
the Royal Statistical Society , B12, 60-94.
Kendall, Maurice G., and Stuart, Alan, 1968: The Advanced Theory of Statistics ,
Volume 3, Design and Analysis, And Time-Series, Second Edition, Hafner
Publishing Company, New York, 557 pp.
Kullback, Solomon, 1968: Information Theory and Statistics . Dover Publica-
tions, Inc., New York, 199 pp.
Landsberg, H. E. , Mitchell, J. M., Crutcher, Harold L., and Quinlan, F. T.,
1963: Surface signs of the biennial atmospheric pulse. Monthly Weather
Review , 91, Contributions in memory of Harry Wexler, U '^. Department of
Commerce, Weather Bureau, 549-556.
161
Lawley, D. N. , and Maxwell , A. E. , 1963: Factor Analysis as a Statistical
Method . Butterworths, London, 117 pp.
Ludlum, David, 1969: The snowfall season of 1967-68. Weatherwise , 22 (1),
26-31 .
MacQueen, J., 1967: Some methods for classification and analysis of multi-
variate observations. Proceedings of the Fifth Berkeley Symposium of
Mathematical Statistics and Probability, University of California Press,
Vol. 1, ^81-297:
McCreary, F. E., Jr., 1959: Stratospheric winds over the tropical Pacific
Ocean. Presented at the Conference on Stratospheric Meteorology , American
Meteorological Society , Minneapolis , Minnesota , September , 10 pp.
Mahalanobis, P. C, 1936: On the generalized distance in statistics.
Proceedings of the National Institute of Science , India , 12, 45-55.
Miller, Robert G., 1962: Statistical pred^'ction by discriminant analysis.
Meteorological Monographs , 4 (25), American Meteorological Society,
Boston, 54 pp.
Moore, P. G., 1957: Transformations to normality using fractional powers of
the variable. Journal of the American Statistical Association , 52 (278),
237-246.
Mulaik, Stanley A., 1972: The Foundation of Fa ctor Analysis , McGraw-Hill
Book Company, New York, 453 pp.
Myers, Raymond H., 1971: Response Surface Methodology . Allyn and Bacon, Inc.,
246 pp.
Newell, Reginald E., Kidson, J. W. , Vincent, Dayton G., and Boer, George J.,
1974: The General Circulation of the Tropical Atmosphere and Interactions
with Extra tropical Latitudes , MIT Press, Cambridge, Massachusetts, 2,
371 pp.
Pearson, Karl, 1894: Contributions to the mathematical theory of evolution.
Philosophical Transactions of the^ Royal Society of London , Series A,
185, 71-110 + 5 plates.
Pearson, Karl, 1901: On lines and planes of closest fit to systems of points
in space. Philosophical Magazine , 6, 559-572.
Pearson, Karl, and Lee, Alice, 1903: On the laws of inheritance of man.
Biometrika , 2, 357-462. '■
Phillips, Earl L., 1969: Washington-December 1968, special weather summary,
climatological data. Environmental Data Service, Environmental Science
Services Administration, U.S. Department of Commerce, 72 (12), 198 pp.
162
Rao, C. R., 1950: A note on the distribution of D^-D^ and some computational
aspects of p + q the D statistic and discriminant function. Sankhya ,
10, 257-268.
Rao, C. R. , 1952: Advanced Statistical Methods in Biometric Re search . John
Wiley and Sons, Inc., New York, 390 pp.
Reed, R. J., 1960: The structure and dynamics of the 26-month oscillation.
Presented at the 40th Anniversary Meeting of the American Meteorological
Society , Boston , 393-402.
Reed, R. J., and Rogers, D. G., 1962: The circulation of the tropical strato-
sphere in the years 1954-1960. Journal of the Atmospheric Sciences , 19
(2), 127-135.
Reed, R. J., Campbell, W. J., Rasmussen, L. A., and Rogers, D. J., 1961:
Evidence of a downward-propagating, annual wind reversal in the equatorial
stratosphere. Journal of Geophysical Research , 66, 813-818.
Roy, S. N. , 1939: p-statistics of some generalizations in analysis of variance
appropriate to multivariate problems. Sankhya , 4, 381-396.
Roy, S. N. , 1942(a): The sampling distribution of p-statistics and certain
allied statistics on the non-null hypothesis. Sankhya , 6, 15-34.
Roy, S. N. , 1942(b): Analysis of variance for multivariate normal populations,
the sampling distributions of the requisite p-statistics on the null and
non-null hypothesis. Sankhya , 6, 35-40.
Schneider-Carius, K. , and Essenwanger, 0., 1955: Eigentiimlichkeiten der
Niederschlagsverhaltnisse im Norden und Su'den der Schweizer Alpen,
dargestellt durch die Niederschlagswahrscheinlichkeit von Basel, St.
Gotthard und Lugano. Archiv fiir Meteorologie , Geophysik und Bioklimatol-
ogie , Serie B, Allegemeine und Biologische Klimatologie, Band 7, 1. Heft,
32-48.
Smith, C. A. B. , 1947: Some examples of discrimination. Annals of Eugenics ,
14, 272-282.
Snedecor, George W. , and Cochran, William G., 1967: Statistical Methods -
Applied to Experiments in Agriculture and Biology . Iowa State Col 1 ege
Collegiate Press, Ames, Iowa, 534 pp.
Sokal , Robert R., and Sneath, P. H. A., 1963: Principles of Numerical
Taxomony . W. H. Freeman and Company, San Francisco, Library of Congress
Card Number 63-13914, 359 pp.
Spearman, C. , 1904: General intelligence, objectively determined and measured.
American Journal of Psychology , 15, 201-293.
Spearman, C. , 1926: The Abilities of Man . MacMillan Company, New York,
416 pp.
163
Stromgren, B., 1934: Tables and diagrams for dissecting a frequency curve
into components by the half-invariant method. Skandinavisk Aktuarietid-
skrift, 17, 7-54.
Student (W. S. Gosset), 1925: New tables for testing the significance of
observations, Metrometrics , 5 (3), 105-120.
Tatsuoka, M. M. , 1971: Multivariate Analysis . John Wiley and Sons, Inc., New
York, 300 pp.
Thurstone, L. L., 1947: Multiple Factor Analysis . The University of Chicago
Press, Chicago, Illinois, 535 pp.
Tryon, Robert C. , 1959: Domain sampling formulation of cluster and factor
analysis. Psychometrika , 24, 113-135.
Tryon, Robert C, and Bailey, Daniel E. , 1970: Cluster Analysis . McGraw-Hill
Book Company, New York, 347 pp.
Tucker, G. B. , 1960: Upper winds over the world. Part 3. Great Britain
Meteorological Office, Air Ministry, Geophysical Memoirs No . 105 , Fifth
Number, Volume XIII, M.O. 631e, Her Majesty's Stationery Office, London,
101 pp.
Tukey, John W. , 1967: On the comparative anatomy of transformations. Annals
of Mathematical Statistics , 28, 602-632.
Veryard, R. G., 1960: Contribution to "Discussion of synoptic and dynamic
aspects of climatic changes," by E. B. Kraus. Quarterly Journal of the
Royal Meteorological Society , 86, 569-570.
Veryard, R. G. , and Ebdon, R. A., 1961(a): Fluctuations in equatorial strato-
spheric winds. Nature , 189, 791-793.
Veryard, R. G. , and Ebdon, R. A., 1961(b): Fluctuations in tropical strato-
spheric winds. Meteorological Magazine , 90 (1066), 125-143.
Veryard, R. G. , and Ebdon, R. A., 1963: The 26-month tropical stratospheric
wind oscillation and possible causes. Meteoroloqische Abhandlungen ,
Freien Universitat Berlin, 36, 225-244.
Ward, Joe H., Jr., Hall, K. , and Buchhorn, J., 1967: PERSUB Reference Manual
PRL-TR-67-3 (II) . Personnel Research Laboratory, Lackland Air Force
Base, Texas, 60 pp.
Wilks, S. S., 1932: Certain generalizations in the analysis of variance.
Biometrika , 24, 471-494.
Wolfe, John H. , 1965: A computer program for the maximum likelihood analysis
of types. Technical Bulletin 65-15 , U.S. Naval Personnel Research
Activity, San Diego, (Defense Documentation Center, AD 620 020), 52 pp.
164
Wolfe, John H., 1967(a): NORMIX cluster and pattern analysis of normal
mixtures. F0CU$-G7-U$NP-N0RMIX , FORTRAN COOP-COOP Monitor , U.S. Naval
Personnel Research Activity, San Diego, 82 pp.
Wolfe, John H. , 1967(b): NORMIX: computational methods for estimating the
parameters of multivariate normal mixtures of distributions. Research
Memorandum SRM-2 , U.S. Naval Personnel Research Activity, San Diego,
(Defense Documentation Center, AD 656 588), 30 pp.
Wolfe, John H., 1968: NORMAP: cluster and pattern analysis of normal mixtures
with common covariances. FORTRAN 63 C OOP-COOP MONITOR-1604 , U.S. Naval
Personnel Research Activity, San Diego, 88 pp.
Wolfe, John H., 1970: Pattern clustering by multivariate analysis.
Multivariate Behavioral Research , 5, 329-350.
Wolfe, John H. , 1971(a): A Monte Carlo study of the sampling distribution of
the likelihood ratio for mixtures of multinormal distributions. Technical
Bulletin STB-72-2 , Naval Personnel and Training Research Laboratory, San
Diego, (Defense Documentation Center, AD 731 039), 7 pp.
Wolfe, John H. , 1971(b): NORMIX 360 computer program. Research Memorandum
SRM 72-4 , Naval Personnel and Training Research Laboratory, San Diego,
125 pp.
165
AUTHOR INDEX
Alt, F. B., 140
Anderson, Edgar, 29
Anderson, T. W., 15, 16, 18, 19
Andrews, D. F. , 22
Angel 1, J. K. , 33
Arnett, J. S., 41
Aversen, J. N. , 9
Bailey, Daniel E., 15, 19
Baker, Frank B., 20
Banarjee, K. S. , 18
Barnard, M. M. , 10
Bartlett, M. S., 15, 16, 17
Berlage, H. P., 33
Boehm, Albert, 21
Bose, R. C, 2, 17
Box, G. E. P., 22
Brooks, C. E. P., 32
Brown, George W. , 10
Buchhorn, J. , 27
Burrau, C. , 23
Carr, R. N., 18
Carruthers, N. , 32
Cattell, Raymond B., 15
Charlier, C. V. L., 23
Clayton, H. H. , 33
Clutter, Jerome, L., 23, 33
Cochran, William G., 6, 81
Cohen, A. C. , 23, 138, 154
Cox, G. M., 29
Crutcher, Harold L., 10, 12, 13=
14, 23, 32, 33, 41, 57, 137,
139, 140, 141, 144, 145, 146
Davies, Owen L. , 18
Dempster, A. P. ,18
Dickey, Woodrow W. , 118
Draper, N. R., 18
Duda, Richard 0., 19
Ebdon, R. A., 32, 33, 76
Essenwanger, Oskar, 23
Falls, L. W., 23, 138, 140, 154
Finney, D. J., 18
Fisher, R. A., 10, 15, 17, 19, 26, 29
Fix, E., 10
Friedman, H. P., 2
Fruchter, B. , 15
Gal ton, Francis, 6, 7
Girschik, M. A. , 16
Good, I . J. , 3
Graybill, Franklin A., 21, 22
Graystone, P., 32
Gupta, S. S., 9
Hald, A., 23
Hall, K., 27
Harman, Harry H. , 15
Hart, Peter E., 19
Hartigan, J. A., 19
Hartley, H. 0., 23, 33
Heastie, H., 32
Hodges, J. I., 10
Hoerl, A. E. , 18
Hotelling, Harold, 15, 16, 17
Hsu, P. L., 17
Hubert, Lawrence J., 20
Kendall, M. G., 15, 18
Kennard, R. W. , 18
Korshover, J. , 33
166
Kullback, Solomon, 16
Landsberg, H. E. , 33
Lawley, D. N., 15
Lee, Alice, 5, 7
Ludlum, David, 118
MacQueen, J., 28, 30
McCabe, G. P., Jr. , 9
McCreary, F. E. , Jr. , 32
Mahalanobis, P. C, 15, 17
Martin, W. P., 29
Maxwell, A. E., 15
Miller, Robert G., 10
Moore, P. G. , 22
Mulaik, Stanley A,, 15, 16
Myers, Raymond H. , 18
Newell, Reginald E., 32, 33
Pearson, Karl, 6, 7, 10, 16, 23
Phillips, Earl L., 117
Rao, C. R., 10
Reed, R. J., 32, 33
Rogers, D. G., 32, 33
Roy, S. N., 2, 17
Rubin, H., 15
Rubin, J., 2
Schneider-Carius, K. , 23
Smith, B. Babington, 15
Smith, C. A. B., 10
Sneath, P. H. A., 15, 19
Snedecor, George W., 6, 81
Sobel, M., 9
Sokal, Robert R., 15, 19
Spearman, C. , 15
Stephenson, P. M. , 32
Stromgren, B, , 23
Stuart, A., 18
Student (W. S. Gosset), 17
Tatsuoka, M. M., 10, 17
Thurstone, L. L., 15
Tidwell, P. W., 22
Tryon, Robert C, 15, 19
Tucker, G. B., 32
Tukey, John W., 22
Veryard, R. G. , 32, 33, 76
Wagner, A. C, 41
Ward, Joe H., Jr., 27
Wicksell, S. D.. 23
Wilks, S. S., 17
Wing, Robert N., 118
Wolfe, John H., 20, 26, 28, 29,
30, 31, 35, 36, 41, 42, 111,
121, 129, 134, 153
'J'U.S. GOVERNMENT PRINTING OFFICEi 1977-240-848/117
167
(Continued from inside front cover)
EDS 16 NGSDC 1 - Data Description and Quality Assessment of Ionospheric Electron Density Profiles for
ARPA Modeling Project. Raymond 0. Conkright, in press, 1976.
EDS 17 GATE Convection Subprogram Data Center: Analysis of Ship Surface Meteorological Data Obtained
During GATE Intercomparison Periods. Fredric A. Godshall, Ward R. Seauin, and Paul Sabol,
October 1976.
EDS 18 GATE Convection Subprogram Data Center: Shipboard Precipitation Data. Ward R. Seguin and Paul
Sabol , November 1976.
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