(1^5'./3:£2)^ n ^V^^^"°"^Q Q ^m^s \ 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 Environmental Data Service Series The Environmental Data Service (EDS) archives and disseminates a broad spectrum of environmental data gathered by the various components of NOAA and by the various coooerating agencies and activities throughout the world. The EDS is a "bank" of worldv/ide environmental data upon which the researcher may draw to study and analyze environmental phenomena and their imoact upon commerce, agriculture, industry, aviation, and other activities of man. The EDS also conducts studies to put environmental phenomena and relations into proper historical and statistical perspective and to provide a basis for assessing changes in the natural environment brought about by man's activities. The EDS series of NOAA Technical Reports is a continuation of the former series, the Environmental Science Services Administration (ESSA) Technical Report, EDS. 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 K/) 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- 1 ' 1 1 1 1 *- ^ 1X3 ° ; 1 i ! i 1 >" 1 , 1 1 1 c .>"■ ' 1 ; ' Jr^ .>1 1 1 J >1 r d' °>1 r ^ BO E2 EH BE EB 70 72 FRTHER ' S HEIGHT < 1 KJCHES ) 7H I BS U1 D I BO U1 X I 7S d X H 7D 3 I BE Figure 2a Regression of sons' statures on the fathers' stature, in. and cm., adapted from Galton (1889) and Pearson and Lee (1903). 7n m UJ I z. BB E7 \rf EE z tn a. in ES EH in E3 ir Ul t- in E2 E 1 in EQ z: It UJ S9 EB 3H HO FDREHRM DF BRDTHEf! C CM 3 M2 HH HE HB SO S2 SH EE EB EZl ;!!!!. ^' [ill ^^-'^ — — — - I \^ h^ ^ 1 1 r ;^ 1^ J, ,rrf ^ V ^ ^^ ^ 1 75: I 7D BE I EO I SE i ED ^ I E IE I *? IB IS SQ 2 I 22 FDREHRM DF HRDTHER C I KJCHES 7 23 2H Figure 2b Regression of sister's span, in. and cm., for given forearm of brother, adapted from Galton (1889) and Pearson and Lee (1903). r= 1 Y a I — h H \ — I 1 r= 1 h — 1- H 1 1 1 r= 1 c i — (- H'^-l i 1 1 ( 1 Y d I — h r=0 m IS t m I — I — I- e I — h 0 Figure 3 General schematic of points and clusters with implied correlations, r. 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 X / / \ \ \ \ o \ / / / (a) . T (b) T f T ' iU \ U M i Mlli ii i 1 X Figure 4 Schematic illustrations of discrimination for two clusters. (a) Two separate and distinct clusters, (b) same two clusters with 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. n o 4-» o • E O O (£) •I- CTv +J I— O O OJ S- ^ Q. U -M -E =3 •I- C_) E o O S- <4- "O OJ 4-J Q. fO ■D I o o •>- S +-) +-> fO c E •!- •r- E C S- o u •1— CO -M -1- ■r- S- E fO ■1- OJ S- E (J •!- 00 I — TD S- O f- <4- o CO E T- O X ■I- ra +-> ra fO fO E OJ JO OJ ■a fo • -C c +-> E r-H -M «3 fO S Q_ • T3 O * — ^ c I/) -E o o cu OJ CD t/1 +-> UD -C (_> -E cr> 5 E +-> 00 -O r— fO •r- o--- OJ •I- +-> E E S- fO rt c:7i s- •1 — fO -E o •r- O) 1 — > +-> 00 -E r> 1/1 o y — » r— O E 03 ■!-> -o ra i/i O =J O) =3 OJ s- +J cr-o s- S- C_) -M E -o +-> •>- • 1— C_) o — ^ :3 s- s- s- JD (T3 u M- M- c: cu O o 00 E +J XJ E -1- E +J O) +-> ro 1— 03 <4- 4-> u cu E QJ a. c E _oo •r— r— 03 13 E X) 4- 1 — OJ C CT"'- 00 • fO O) ^-) • 1— O +j O a n3 r— •r- XJ 00 -E na " -IJ CU •1 — E *-— ^ O E -o •> O X3 CU CD CN-i— QJ •i-O-r- O) Q. O 00 .c E ■(-> S- 00 +-> XJ CU O Q. ra c E t3 4- fO T- >».-^ r- O) O ■o 1 — >- C/1 <: (c c qT cb '( — o s X c_> • r- "+J ro • •4-> CO S 4- n3 E QJ Q O O 4-> O -E s- E •I- -M QJ S- -M O) -M x: n -E 00 03 14- +-) C7> OJ 1 — 1 — 1 QJ -1- S- 23 o x: s_ Q. CL +-> +J OJ O • E Q) s- Q. 00 O 4- -E E O) -a o +-> n3 (T3 +-> ftJ 4-> na •1 — 'r- o 0) E S- S- > o Ol o o -r- O s- SZ. .d oo -Q U cn.-)-> 1— UD O) s- 3 CD 13 Figure 7 Schematic "D" illustration of constellations of annual temperature climates for North American stations. The constellations and clusters are based on the ao, ai, and a2 harmonic coefficients- Dispersion matrices within constellations are not significantly different but are different between constellations. Crutcher (1960) 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' = (' ' 1/5 , r> O) I— f-^ CO o) r— fO 3 • 1 .c JD E E +J 1— 1— 03 O to CU 03 s- s- -a JZ s- S- O 3 O) M- O -M o CU E O u s- JE -E o cnl— 0) E 4J jr O S- ■o i- +-> •i- O E 3 • • 4-> E ITS O E^^ w U -C S CM •1— CU CU •» O ' to •r^^ LO ^ ■o 0) CU 4-> tn o ■^ -o ^ s- cy> ro rO E 4-> E 1— O) O a. to o CU s- " E j3 " >, -P M- 1— o •4-> j= OJ CO i. >, I/) C, CU 1 4- E CU ^— •■- s_ 1 — fO S I — -D 03 o 3 E s- o E O) E * •1- to 0) I— •I- x: o r^ 4-> ja ■*-> M- O I— -ii E o •- CD o O CM to CU -M o E E O CvJ •I- E u o 03 O >, O t~^ S- O O OJ E " to O) 1 — C CM to O o •<- -E CU •1 — • • 03 O o 4J > JD o o •1- O O Q. 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O) •r- o (U s: i/> o LU \ 4J O yD CM in o CO lO «o CTi CO O O CM vo s- 00 in cy> cyi o *^ en fO 00 >5r CM sj- o "^l- 00 -C CO • • o • CM I— cr> 1— o o o 1 1 c O o ■»-> •r- o +J 0) S- CO CO o in > «£> 00 o r- U3 O O vo CM Q. en vn o cTi '!t O r^ 00 o •* 00 ^ ^ r-^ r-^ Q CD CD CM 0) CI >l 1— XI M- Q) O 3 C «/) •1— O ■M •1— C -M o to o •r— s_ • to r^ Ln CM (U > cu cy> 1^ +-> cu 3 vo 00 «:d- ■— (t3 (O fO -o &- CD 1— -C O) -»-> o •r- — ' o s: oo o LU 39 OCT SrC ' TYPES SfiW JUflN SAME CQVRR. lU 1 y OCT SFC 3 TYPES r — r — SflN JUfiN SfiHE COVflR. r -10 ^ OCT SFC 2 TYPES SRN JUHN DCFF COVHR. r -20 — OCT SFC 3 TYPES 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 o +j CL E >1 «4_ O) a; -E O N OJ E •I— s- o I— T3 — 0) O O) Q. r^ OJ a. 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Q. > x: E 1 — +-> O) (0 CU ra -E OO -E c H- 1- H- O O . -a E • JD • "1 — o - E 1 1 cu s- E O) , •r— CO O 03 E to (J • LO U) •1 — CU E cu to 'r-> 00 cu x: cu CU 0) -r- ^ -M x: S- S- -M 4-) fO 1— * — ^ o to S- — ' >j -»-> +J cu fO -E to •r— 1 — -o u c OO c 1 — •I— r3 cu cu fO 1 — -a s- o 3 E 3 •r— «< E •r— • CO S- +J * :^ 00 -i-> 00 4- 00 .^ • cu «o cu O CTi 00 to rD s- E S CO ^ • I— Q- >> 00 C s- "O cu cu +y CM 1^ cu c cu u ^ •I— o cr> •4-> »o ^ C ^-J 1 — O CM to 1— ro • r- o o ro • •r- E X2 • • ct s- o ro o o E ro S- JD cC to <^ > »+- O • VO O S- .'' — ^ :d cr. o cu Q. ro JD • > ^ — ■ CU ^ 1 cu 'r- a. -O Kt SZ +^ . r. >> C LD 4-) •!- • 4-> ro CTi to If- r^ n— ■♦-> o • x: (/) ro Q. • T3 u >-H >>jc: jE ro s- +-> «> +J «=a- CU C CO +-> 3 O 3 00 C O x: O 4-> e •1- CU to +J +J c: fo E "r— (0 "-D E O cu s ■o tj O CL^ cu M- •■- E 4J cu '^ "^ E s- o +-> o s- CM r^ CD o a. o E ro • • 4- 00 E O 3 1 — o 00 -C 3 -o s- cr CM 1 — to • to -i-J CO E '^ to ro CU o c to •■- 1 • 1 — •r- O ro 2 cu • r— ^ — * ^ — * 4-> r— a. -i-> E > je ro JD E CU E to CU I— • r- o s._^ N ^ 3 > ro •r- CU SZ ro 4-> O CU CO -M ^ 1— C •1— E r- <0 ■!-> O to • • rO CU -!-> N O -o OO CO 1 — 00 to • £2. o CO CTi .— O o •r— cx) cu o CM tn ro • x: T3 CO x: ** -E LD 00 E O 2 c -o I— +-> ■4-> •1 — "^^ CTi O •I- S- E 1 — 00 ■sd- •p- CU cu s o to '— ~ >^ >^ •1— CJ 4-> +-> CU to •r— «+- -E x: 4-> 00 E o o to o S- JZZ +J +-> •I- Q. cu O^ 4-> >i +-> •r" CM CO ro -E o 0) S- ■o ro t— ro JD 1— i. ^ CD - — » ' — . E r- ro .•o O ro -Q rO 3 JE H- _J > — ' ■ _J E O CM 1— r^ o o CO Ln 00 00 Ln CO o CM 00 00 CM r^ .— CO O CO CM kO O CO 1^ o o in cT> o O CO CM CO O I— CTi CO O O CM 00 O 00 Ln I— O CM <;;J- CM O O CM CM I— CD 1^ Ln o "^J- I— o 1^ 00 o CM O O CO o I— "^ o Ln Ln o I— O O 00 CT> O O to I— o > cu o to E ro CU C/1 s- o o o. 3 o S- • cs > cu E Q •f— • x: • s- +-> -o s- •1— +-> o 3 tn o ro -o XI E ro «vf CO CO CM cr> 00 Kt CO I — ';:f -^ I— O O O I— C\J CO Ln CO cr> Ln f^ CO CO Ln o o CTi «vi- r- •* I— CO Ln r^ CO CO r— I 00 CM CO to on 1 I— TM CO Ln tn cr> to <:d- CM E I— O O . . t ^ o o o c: 3 1— CM +-> E cu cu ro f— r^ E JQ XJ •1— ro ro (- •r- 'r— S- s- S- ro ro o > > to 54 ■o o to (U (U ro C S_ I— -I- .|- r- O) CO ITJ I— -CO -Hi • S- I O I— O) to LT) ro E to O" Ol r- a; -c C 4-> 13 $- I— 03 to • to S- +-> i»s: O) 4-> (o • S- fO O) rD Q. E S "o o) oj a» E x: o -c us I— c +-> ro • «* oo to • C C (O to r3 S- O o <+- to to -tJ u c •I- o +-> E to •I- JC <0 4-> +-> to to •f— T3 C T3 •1- S- 2 O o S_ O) OJ i- Ol ClM- ID O u- o > M- o > (U -r- -E +-> -M •■- to +-> o 03 Q. • JZ -E +-> " +-> +J 3 to E O •r- O) to E E O O) O Q.-E •.- E •4-' +J O Q. O E E O 3 -O S- 00 E M- tO •!- fO S O) > O) I— ^ eo - I— E -r- O to N O Q. 00 I— +-> 4-> E •I- 1— E O O) Q. N OJ E •I- i. O to n3 O to to O) sz 4-> O Q. <4- O >^ +-> XI O s- «4- x: +-> s- 3 cr to I 1^ I— I— o 1^ o r>. o o ro ^ I- o -D O O cu +-> 4- O CO CVJ o "* r— KD m JD r^ CM *^ r»» I— LO CT> to ^ 00 cr> CM o CO CM to to (U O) C2. Q. ==? >? 4-> +-> I— CM O O -f-> +-> to (O E Q. U O x: o CL+-> O 03 s- Q. to OJ 00 JE OJ +-> .E +-> to O •I- Q. 03 -E XJ JD I— E •— 03 3 _J E CM ^ r-^ , — o CM CT> r^ o r^ o cr> 1 — "^ 1 — o cr> 1— CM to CM o CO CTi • • • • • • . — N O CO o 1 — o o oo 1 1 r>. CO (/) CD CM o ■M O 00 CO o 1 — > Cn OJ O) r-> a^ o r-^ to CTi CL CM "* o r— r— O^ >^ •=d- ■^ O CM cr> CO cr> "^ I— o o o to cy> LO O) r— CO 3 CO CM I I — oo 1^ E 03 • • Ol >• I— I— CD CM I — CU r— CL CM cn "vf o o O LO 00 o o r-~ CM o to CM O O O CO O I— 03 to OJ o sz cu sz +J o to u •r- 4-> to il en CO o o CM CO O 00 LO I — O CM ^ CM O O CM OJ 1— O > O) (O Q • E i_ 03 -a 1- OJ -p o s: oo o o O s_ O- lO (U ex .>1 I— CM CTi O VO OJ I— CO O t^ Q. CO O O r— >> LO to O CO ^^ LO LO "d- o cr> 1 — cr> • • o 1 o to s- o ■(-> o O) > LO r— ^ LO 1 — Cn CT> I— O o o • to I— to > cu cr> CM <. CO t^ 03 "O s- cn CTi -. "^ cxj o CO CM ^- o <* UD r-~ o 1^ CO o o • • • • OJ I— o ■— I— cr> CO CTi o cr> o o to s- o ■»-> o 0) (£> Cvj O C\J 00 O O r— — o I ID r^ crs 1— CTi O-j cr. o o o I in CO 00 rs CO st I I— cy^ «^ E ro • • cu > CO n CD C\J CM CO -1-^ S- O) o Ci. CL >. ^-^ o h- •u s- cu D. i:: .C •1— ■4-> -(-> •1— c 3 o o CO Cl o >v ■~ f— O) a» ^— O) -Q i- fO .c 1— ■^ uo CO o <^ r- CO O I — OO CO o en C3^ CM o O CM O r- CO CO O CO I — i — O 0"> I— r-^ o <.o r^ CO o CM O CO r— I Ln en UD ur> (Ti en O CT> o o 1 to s- o +J o O) > CO ID ID vo cr, (Ti CTt o o o . to CM -^ > oj 00 cr> cu 3 r^ CM Q 1 1— 00 CT> to • C fC • • c • S- O) > cr> r->. (O -a s- CD r^ O) ^-J •1— 2: oo (_> LjJ r-- Ln CO CM p— r~^ ro "^ ro CvJ -— CM CO , 00 1 CM CM CO Q. 1^ r- O CO r^ o o r^ o r^ o =^ o «=a- o CM CO 1^ Ol CO •^ CO en U) %. ■!-> O) >■ r^ CO ^ CO cr> CO en "^ o o o . to ^ r*>. > CU OJ 3 1— 1"-- Q 1 I— CO r^ (/) C (O • . C • S- O) > 1— en 03 T3 J- C7) CM cu +-> •1— s: 00 LlI 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 X X x X i X X X X X ^Xx X X X >< X "* '« X ( X X X X X X X X ' ''xX X X i 1 -4 X c ;«< >^x X - J* X X xX K xxfd'^ \'k xX i X ,< « -,^x ' ^ X > X ^ ?! u x-^l X c ^ X ^ X [ 11 jj „x X X X **\^> x"x x' X X X X Xx^x X X X j X X X X ' X X X X X r >i< X (af ' 1 1 ?Q e0703 40 -12 -16 -20 -24 -2S -32 -36 -4« ' j i 1 1 I t 1 : 1 ^ j 1 r i 1 X V X X i X 1 " x!x 1 X X X X X i« j ^1 ' 1 Ix , 1 > I " M X X ' ^ V . i » X [X 1 I 1 1 X * X x X X 1 " 1 V ' ' 1 K -- XXXx^J xlx X x' ■ X ■ j( X X ^ j 1 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 ! i j i 1 1 1 — -^ f ~t -- 1 1 i 1 X X X < X X X X 1 >* x~^ 1 ^ ^ 1 ix > L. 1 X h "" ^ X 1 > X X X , i X ^ X 1 xi . X 1 1 .1 L X 1 <^ h I X X X , X ixjxl J^ 1 X Ij^^l J xi 1 X X 1 1 i 1 : ^ X 1 1 1 ; X i 1 X X *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 X X x^ X X X X < *l X - X X p< * V X V X X X X X P""^ >^ X ,, x"" ^ X >: >: X ''x X X < X K X V fxM '• X ^ * X X X ** ^ X it < X Xx X X X X X / >< 1^ ^$x X X X X X Xv ^x X X ^ X X X V X X ! X X < < X X xX X X X X X (a) 60703 1 30 24 20 16 12 -16 20 24 -28 -36 -40 1 j 1 ! ■ 1 1 ! 1 1 i ! 1 i ' 1 x! ' 1 i ' X ! V ^ -x < X < X X X i 1 i i ! j ( < X ! >; x .. ., X 1 ^ , ^ X >, X ^, X > X 1 1 ' x>^x >^x ix i ^ ^ X - 1 X X T X " f i M^ f xx i ! M^ X : XX K " X < y : X X X ^^' "x 1 X) X X X : X i \ \ X ^n X A X X X 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. 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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 17.42 -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 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 -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, >350UARE 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. Ill +-> , — 13 • OJ (0 fO E 4-> -C E -C E CO t— cu 4-> O cu (0 .i«C "r- -E cu 03 W T3 +-> • +-> "r- •I— <4- O 1 o E cu o (/) 1 — O E x: • CO •1 — E -M E +J "O o t. U) CL E •>- o d) -r- E fO • -M E s- 13 CO o fc cu I/) 1 — • •I— cu cu N CO ra >> CO •>-oxi CO •!- fO E cu O) -M +-> 1/1 o "oJ -E s- •1— cu M > +-> E C CU JZ 'r- o CO O ZD I— ,■ — cu 4-> CL cu i- Q. XI u >> -»-> M- E +-> 0) x: 03 • n3 . 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CM ►^ CU t-^ S- J2 3 ro O r— CO "vT o r>>. o cr> CO o LO 00 CM VO O CO O r— O O CO LO CM O I — O I I I CM LO O "vf O «* CM O CO 00 1 — LO O LO CM CM 00 O O LO LO CO I o o to CM l£> r^ S- LO I— «n1- O CM CO I— 4-> CM CO ^ u • • • QJ O O O > I o t^ en CTl CM CO CO 00 CO CTl O CO o o o • to 1^ oo o > OJ r^ CO «;*■ LO LO CM to -o s_ O) 1— o "r— s: oo o LU O O CO O) Q. CO I— CTl O >— CO "^^^ I— o r^ r^ r^ 00 o CO CO CM O O r— O CO o I— o 00 LO O en 00 CO LD O I— I— o r^ o 00 I CM 1— O LO CO CM ^ O^ «vl- O CTl O d (~> C2 . lO CTl r— O > O) CM «;a- 1— • O CM O fO "a s- CT> r— O) +J o • r— s: oo o LU 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 - \ -IB IB -IB c^^- i^^io 5TIWEDE Pfi55/ MR SmWX UINDS yil«5= K-S ' [ " SaSiP » S DBS. UITHIN 1 IBB 2 51.6 3 HB.M PRDB. LEva= .S , . ...... (a) V -IB • IB -IB ' ' <^r^ 1 '^ "\ ' U 5Tffl1PEI>E Pfl55/ HH swrnce mim>5 m vcm. ■ HIND5= US"' TW5= m { WIW* X DBS. HEflN TW. 1 3B.I -SB 2 E3.9 -2.B PRIH. LEYEL= .2S IB (b) V ■ -IB IB -IB .^3^ ^^^C^ u STfflKK Pta m aSFKL HIWSflND TOR. HIND5= H 5"' TUP5= DEE C EMUP* 8 DBS. fCBN TW. 1 HB.H -H.H 2 1.3 -2B.2 3 H9.7 -2.2 PRDB. LE¥B.= .25 IB 1 (c) V -28 ■ ■ -IB ■ 2B IB -IB -a ■ u ETfWEBE PflSS/ Id SURFBa MINDS AND TEBP5. IB ■ HIND5= B 5'' Ttt1P5=0EEC WiFt % DBS. 1 2H.3 2 32.E 3 l.H H 41.2 PROB. L£Va» .S low TOP. -5.2 -3.M -2B.2 -2.1 2B (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. 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I — s- CO cTi r^ CO o o 1 — -^ CM c\i cr^ +-> O O I— O CTi I o • • • • • C 0)00000 QJ > LU 00 CO LO r- O r-^ O CM <^ LO , — CO 00 r-^ r- O O O CM CO o CO ■vf 1 — VD CM CM r--. csj r-^ o r^ r— S- o Q. o s_ Q_ O r- o •— UD 00 CO "=:^ 1 — "^ «^ vo LO CO o o <^ LO O t^ ';:t- cn to 'd- O ) - — ^ I— -a O) H- rs O c ■r— (/5 +J a C •i — - - o 4-> u to S- • - LO <:3- O r-. CO r- CO 0) > 0) LO CO kO KO CM cvj +-> ■ o •1 — co 1— o s: t/1 o LlJ 128 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 "-a (O c O fO J3 1 — •'00 (73 ^--f— CQ CJ> OJ O > S-— ' C JZl 03 -r- E • +■> O 1 CM m Q.O ■o o -o S LO c OJ OJ fO ^ -O "O -^ c 1 — - (O • • H-'-- C/1 CO > O <-> " O) •1— <=!- OJ CO CM CO O I— CTv CD IT) 1— (T> ID CM >5f 1— CO CM O cn O LO O 1 Q. 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S/1 .>> {A >> to •r- f- (/) •t— 1— •p— s- • • •p- s- • • X cu CU > cu > X CU cu > cu > cC 4-> u cu o cu ■=c 4-> u CU cj cu o E Q E Q o E Q E O s- 03 03 03 s- 03 03 03 o S- E •r- • E •r— • o s- E -t- • E -r- . •>-> 03 03 s- -o 03 s- -a E 03 03 S- -O (O s- -o 03 JZ CU 03 -M cu 03 +-> •1— -E CU 03 +J CU 03 +J s: o s: > OO s > oo s: CJ s: > OO s: > OO 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. 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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. PENN STATE UNIVERSITY LIBRARIES ADQDD72D E3 QSl NOAA— S/T 76-1912