Jurnal Ilmu Komputer dan Informasi (Journal of Computer Science and Information). 7/2 (2014), 61-66 
DOI: http://dx.doi.org/10.21609/jiki.v7i2.258 

 

DIVERSITY-BASED ATTRIBUTE WEIGHTING FOR K-MODES CLUSTERING 
 

M Misbachul huda, Dian Rahma Latifa Hayun, and Annisaa Sri I. 
 

Informatics Engineering, information Technology Department 
 Institut Teknologi Sepuluh Nopember Surabaya, East Java, Indonesia 

 
E-mail: annisaaindrawanti@gmail.com, misbachul@mhs.if.its.ac.id 

 
Abstract 

 
Categorical data is a kind of data that is used for computational in computer science. To obtain the 
information from categorical data input, it needs a clustering algorithm. There are so many clustering 
algorithms that are given by the researchers. One of the clustering algorithms for categorical data is k-
modes. K-modes uses a simple matching approach. This simple matching approach uses similarity va-
lues. In K-modes, the two similar objects have similarity value 1, and 0 if it is otherwise. Actually, in 
each attribute, there are some kinds of different attribute value and each kind of attribute value has 
different number. The similarity value 0 and 1 is not enough to represent the real semantic distance 
between a data object and a cluster. Thus in this paper, we generalize a k-modes algorithm for catego-
rical data by adding the weight and diversity value of each attribute value to optimize categorical data 
clustering.  
 
Keywords: categorical data, diversity, K-modes, attribute weighting. 

 
 

Abstrak 
 
Data Kategorial merupakan suatu jenis data perhitungan di ilmu komputer .Untuk mendapatkan infor-
masi dari input data kategorial diperlukan algoritma klastering. Ada berbagai jenis algoritma klas-
tering yang dikembangkan peneliti terdahulu. Salah satunya adalah K-modes. K-modes menggunakan 
pendekatan simple matching. Pendekatan simple matching ini menggunakan nilai similarity. Pada K-
modes, jika dua objek data mirip, maka akan diberi nilai. Jika dua objek data tidak mirip, maka diberi 
nilai 0. Pada kenyataannya, tiap atribut data terdiri dari beberapa jenis nilai atribut dan tiap jenis nilai 
atribut terdiri dari jumlah yang berbeda. Nilai similarity 0 dan 1 kurang merepresentasi jarak antara 
sebuah objek data dan klaster secara nyata. Oleh karena itu, pada paper ini, kami mengembangkan 
algoritma K-modes untuk data kategorial dengan penambahan bobot dan nilai diversity pada setiap 
atribut untuk mengoptimalkan klastering data kategorial. 

 
Kata Kunci: data kategorial, diversity, K-modes, pembobotan atribut. 

 
 
1. Introduction 
 
Computer science is always related to the data. 
All computational processes need data not only 
small data but also big data. There are two data 
types, they are numeric data and categorical data. 
Arithmetic process can be given to numeric data 
but it cannot be given to categorical data. Catego-
rical data is used in some systems or applications. 
For example, categorical data in intrusion detec-
tion systems, population data and customer infor-
mation in online shopping. Example of categorical 
data in intrusion detection system is IP address. IP 
address in each data must be different and it can-
not be arithmetically compared. It will be in popu-
lation data and customer information, too. Popu-
lation data has such categorical data like gender, 
blood type and home address. Costumer informa-
tion, in online shopping, has such categorical data. 

For example: the phone number and the used ba-
nk. The categorical data clustering considers the 
similarity or dissimilarity between data. Similarity 
or dissimilarity can be considered by distance bet-
ween the two data object. The shorter the distance 
between objects, the more similar the objects. Si-
mple matching approaches can be used to calcu-
late the distance. Example of simple matching ap-
proach for categorical data is k-modes [7]. In in-
trusion detection system, there is such new intru-
sion that has not been known earlier. It needs a 
clustering algorithm to cluster and detect the new 
intrusion. Clustering is used to cluster population 
data and customer data to obtain the information 
needed, too.  

The study [7], k-means and k-modes are joi-
ned together to cluster numeric and categorical 
data. K-means is a clustering algorithm for nume-
ric data. K-means clusters the data type that can 

61 
 



62 Jurnal Ilmu Komputer dan Informasi (Journal of Computer Science and Information), Volume 7, Issue 2, 
June 2014 
 

be arithmetically compared. Categorical data is a 
kind of data that cannot be arithmetically compar-
ed each other. K-means cannot be used to cluster 
categorical data. To overcome the categorical da-
ta, it uses k-modes algorithm. K-modes algorithm 
uses a simple matching approach to the process of 
matching dissimilarity clustering of data, replac-
ing the means to modes. In k-means, to update a 
cluster centroid, it is used means formulae. On the 
other hand the k-modes, modes updating on the 
clustering process is determined by the frequency 
of occurrence of data so as to minimize the cost 
function. 

In [3], K-modes algorithm is supposed to ha-
ve some deficiencies. In K-modes, the two similar 
objects have similarity value 1, and 0 if it is oth-
erwise. According to [3], a simple matching app-
roach 0 and 1 are not good enough to represent 
the real semantic distance between a data object 
and a cluster. Thus, in [3] the authors proposed a 
range between 0 and 1 which represents the wei-
ght of similarity. In [8], focused on optimizing the 
k-modes algorithm for clustering categorical data 
with the new dissimilarity approach using the rou-
gh set membership. According to the research, the 
shortcoming of the simple matching approach is 
having a weak intra-similarity. To solve the issue, 
the dissimilarity approach (frequency-based) bet-
ween the two objects in [8] takes into account to 
the distribution of universal attribute values. The 
study [1] explains that sometimes the occurrence 
of low frequency data and high frequency data ha-
ve the same overlap similarity value. With these 
problems, the study [1] approach takes into acco-
unt to the similarity of the frequency distribution 
of the different attributes value to define the simi-
larity between two categorical attribute values. 

In addition, the dissimilarity approach taking 
into account the frequency distribution of differ-
ent attribute values can also be applied to the opti-
mization of categorical data dissimilarity [1]. For 
example, there are two data, AAABB and ABC-
DD. If using the k-modes algorithm [8], both dis-
similarity values are equal to 1. As we can see, the 
diversity level between the first data and the se-
cond data is different because the quantity of each 
letter is different. Thus, in this paper, the research 
focuses on the k-modes clustering with diversity-
based attribute weighting and the research contri-
bution is distance values diversity to optimize the 
categorical data dissimilarity. 

 
2. Methods  
 
Categorical Data 
 
Categorical variables represent types of data whi-
ch may be divided into groups. Analysis of cate-

gorical data generally involves the use of data ta-
bles. A two-way table presents categorical data by 
counting the number of observations that fall into 
each group for two variables, one divided into ro-
ws and the other divided into columns. 

There is no intrinsic ordering to the catego-
ries in categorical data. For example, gender is a 
categorical variable having two categories (male 
and female) and there is no intrinsic ordering to 
the categories. Hair color is also a categorical va-
riable having a number of categories (blonde, bro-
wn, brunette, red, etc.) and again, there is no 
agreed way to order these from highest to lowest.  

So that, computing the similarity and the dis-
similarity between categorical data instances is 
not straightforward owing to the fact that there is 
no explicit notion of ordering between categorical 
values. To tackle this, there are some data-driven 
similarity measures that have been proposed for 
categorical data. In this section, we will describe 
the categorical data characteristics.  

But first, we will define the notation used in 
later explanation. Let’s consider for a categorical 
data set D containing N objects, defined over a set 
of l attribute where 𝐴𝐴𝑖𝑖 denotes the ith attribute. 
And let the ni be the number of values in each at-
tribute𝐴𝐴𝑖𝑖. The next notation used are following: 
𝑓𝑓𝑖𝑖(𝑥𝑥) = the frequency of value 𝑥𝑥 lies on attribute 
Ai in data set D. 
𝑝𝑝𝑖𝑖(𝑥𝑥) = the probability of attribute 𝐴𝐴𝑖𝑖 to take the 
value 𝑥𝑥 in the data set D, defined as equation(1). 

 
𝑝𝑝𝑖𝑖(𝑥𝑥) =  

𝑓𝑓𝑖𝑖(𝑥𝑥)
𝑁𝑁

   (1) 
 

𝑝𝑝𝑖𝑖
2(𝑥𝑥) = another probability measure of attribute 
𝐴𝐴𝑖𝑖 to take the value 𝑥𝑥 in the data set D, defined as 
equation(2). 

 
𝑝𝑝𝑖𝑖
2(𝑥𝑥) = 𝑓𝑓𝑖𝑖

(𝑥𝑥)(𝑓𝑓𝑖𝑖(𝑥𝑥)−1)
𝑁𝑁(𝑁𝑁−1)

     (2) 
 
Size of the data, N. Most measure are typi-

cally invariant to the size of the data. Number of 
attributes, l. In [1] experiment showed that the nu-
mber of attributes does affect the performance of 
the outlier detection algorithm. 

The frequency of values taken by each attri-
bute, ni. A data set may contain attributes that take 
some values and attributes that only take a few va-
lues. A similarity measure may ignore another at-
tribute while finding the more importance attri-
bute. 

Distribution of 𝑓𝑓𝑖𝑖(𝑥𝑥). It refers to the distri-
bution of frequency of values taken by attribute in 
the given data set. It is possible for a similarity 
measure to give more priority to frequently occu-
rring attribute values or otherwise. 
 



M. Misbachul Huda, et a.l, Diversity-Based Attribute  63 
 

Similarity and Dissimilarity Measure for Cate-
gorical Data in K-Modes 
 
The classical theory of clustering in k-Modes, eit-
her an element belongs to a cluster or it does not. 
The corresponding membership function is the 
characteristic function of the cluster that takes 
values 1 and 0. Suppose that there are 𝑥𝑥,𝑦𝑦 ∈ 𝐷𝐷, 
then the simple matching dissimilarity measure in 
k-Modes is defined following equation(3). 

 

𝐷𝐷𝐷𝐷𝐷𝐷(𝑥𝑥, 𝑦𝑦) =  �
1, 𝐷𝐷𝑓𝑓 𝑥𝑥 ≠ 𝑦𝑦

 0, 𝑜𝑜𝑜𝑜ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝐷𝐷𝐷𝐷𝑒𝑒
 (3) 

 
With this concept, the distance between two 

objects computed often results in cluster with we-
ak intra-similarity and disregards the similarity 
embedded in the categorical values [1]. 

Then, a new dissimilarity measure between 
the mode of a cluster and an object is introduced 
for the k-Modes Algorithm. To obtain such a clus-
ter having the strong intra-similarity, the rough 
membership is used. It takes values between 0 and 
1.  

Taking account of the frequency of mode 
components in the current cluster, Ng et al. [2] in-
troduced a valuable dissimilarity measure into the 
k-Modes clustering algorithm. Let 𝑃𝑃 ⊆ 𝐴𝐴 and 𝑎𝑎 ∈
𝑃𝑃, then the Ng dissimilarity measure 𝐷𝐷𝐷𝐷𝐷𝐷𝑃𝑃(𝑧𝑧𝑙𝑙, 𝑥𝑥𝑖𝑖) 
between a categorical object 𝑥𝑥𝑖𝑖 and the mode of a 
cluster 𝑧𝑧𝑙𝑙 with respect to P is defined as the 
following equation(4). 

 
𝐷𝐷𝐷𝐷𝐷𝐷𝑃𝑃(𝑧𝑧𝑙𝑙, 𝑥𝑥𝑖𝑖) =  �𝐷𝐷𝐷𝐷𝐷𝐷𝑎𝑎(𝑧𝑧𝑙𝑙, 𝑥𝑥𝑖𝑖),

𝑎𝑎∈𝑃𝑃

 

 
where, 

 

𝐷𝐷𝐷𝐷𝐷𝐷𝑎𝑎(𝑧𝑧𝑙𝑙, 𝑥𝑥𝑖𝑖) =  �
1, 𝐷𝐷𝑓𝑓 𝑓𝑓(𝑧𝑧𝑙𝑙, 𝑎𝑎) ≠  𝑓𝑓(𝑥𝑥𝑖𝑖, 𝑎𝑎),

1 − 𝑚𝑚𝑎𝑎, 𝑜𝑜𝑜𝑜ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝐷𝐷𝐷𝐷𝑒𝑒
 

 
where, 
 

𝑚𝑚𝑎𝑎 =
|{𝑥𝑥𝑖𝑖|𝑓𝑓(𝑥𝑥𝑖𝑖, 𝑎𝑎) = 𝑓𝑓(𝑧𝑧𝑙𝑙, 𝑎𝑎), 𝑥𝑥𝑖𝑖 ∈ 𝑐𝑐𝑙𝑙 }|

|𝑐𝑐𝑙𝑙|
 (4) 

 
And |𝑐𝑐𝑙𝑙| is the number of objects in the lth 

cluster. For the k-Modes algorithm with Ng’s dis-
similarity measure [2], the simple matching dissi-
milarity measure is still used in the first iteration. 
So, Cao et al. in [3] proposed a new dissimilarity 
measure by using 𝑆𝑆𝐷𝐷𝑚𝑚𝑎𝑎(𝑥𝑥, 𝑦𝑦) defined in equation 
(5). 

 

𝑆𝑆𝐷𝐷𝑚𝑚𝑎𝑎(𝑥𝑥, 𝑦𝑦) =  
𝑓𝑓(𝑥𝑥, 𝑎𝑎) ≡ 𝑓𝑓(𝑦𝑦, 𝑎𝑎)

∑ 𝑓𝑓(𝑥𝑥, 𝑎𝑎) ≡ 𝑓𝑓(𝑧𝑧, 𝑎𝑎)𝑧𝑧∈𝐷𝐷
 (5) 

 

 
From the equation(5) formula [3] introduced 

that the new dissimilarity measure is defined as 
the following equation(6). 

 
𝑁𝑁𝐷𝐷𝐷𝐷𝐷𝐷𝑃𝑃(𝑧𝑧𝑙𝑙, 𝑥𝑥𝑖𝑖) =  ∑ 𝑁𝑁𝐷𝐷𝐷𝐷𝐷𝐷𝑎𝑎𝑎𝑎∈𝑃𝑃 (𝑧𝑧𝑙𝑙, 𝑥𝑥𝑖𝑖)       (6) 

 
where, 
 

𝑁𝑁𝐷𝐷𝐷𝐷𝐷𝐷𝑎𝑎(𝑧𝑧𝑙𝑙, 𝑥𝑥𝑖𝑖) = 1 − 𝑆𝑆𝐷𝐷𝑚𝑚𝑎𝑎(𝑧𝑧𝑙𝑙, 𝑥𝑥𝑖𝑖)  × 𝑚𝑚𝑎𝑎 
 

As opposed to Ng’s dissimilarity measure, 
the similarity 𝑆𝑆𝐷𝐷𝑚𝑚𝑎𝑎(𝑧𝑧𝑙𝑙, 𝑥𝑥𝑖𝑖) between object 𝑥𝑥𝑖𝑖and 
cluster 𝑧𝑧𝑙𝑙 is included in the proposed measure 
𝑁𝑁𝐷𝐷𝐷𝐷𝐷𝐷𝑎𝑎(𝑧𝑧𝑙𝑙, 𝑥𝑥𝑖𝑖). In [3] introduced a weighted k-
modes clustering algorithm, by considering that 
the similarity value between two objects is not al-
ways 1, but it can be a value between 0 and 1. A 
pair of object 𝑥𝑥𝑖𝑖, 𝑥𝑥𝑗𝑗 is considered more similar th-
an the second pair(𝑥𝑥𝑠𝑠, 𝑥𝑥𝑡𝑡), if and only if 𝑥𝑥𝑖𝑖and 𝑥𝑥𝑗𝑗 
exhibit a less common attribute value match in the 
population. In other words, similarity among obje-
cts could be decided by the un-commonality of 
their attribute value matches. Based on that, in [3] 
defined the “More Similar Attribute Set” of an at-
tribute value 𝑎𝑎𝑗𝑗

(𝑟𝑟) as the following equation(7). 
 

𝑀𝑀�𝑎𝑎𝑗𝑗
(𝑟𝑟)� = {𝑎𝑎𝑗𝑗

(𝑡𝑡)|𝑓𝑓�𝑎𝑎𝑗𝑗
(𝑡𝑡)|𝐷𝐷� ≤ 𝑓𝑓(𝑎𝑎𝑗𝑗

(𝑟𝑟)|𝐷𝐷)} (7) 
 

where 𝑓𝑓(𝑎𝑎𝑗𝑗
(𝑡𝑡)|𝐷𝐷) is the frequency count of attribute 

𝑎𝑎𝑗𝑗
(𝑡𝑡) in the data set D. This is the set of attribute 

values with lower or equal frequencies of occu-
rrence than that of 𝑎𝑎𝑗𝑗

(𝑟𝑟). Note that a value pair is 
more similar if it has lower frequency of occur-
rence. The weighting function in [3] is defined as 
equation(8). 

 

𝜔𝜔�𝑎𝑎𝑗𝑗
(𝑟𝑟)� = 1 − ∑

𝑓𝑓(𝑎𝑎𝑗𝑗
(𝑡𝑡)|𝐷𝐷)(𝑓𝑓�𝑎𝑎𝑗𝑗

(𝑡𝑡)
�𝐷𝐷�−1)

𝑛𝑛(𝑛𝑛−1)𝑎𝑎𝑗𝑗
(𝑡𝑡)∈𝑀𝑀(𝑎𝑎𝑗𝑗

(𝑟𝑟))
 

(8) 
 
where n is the frequency of objects in the data set 
D. If above function is used, less frequent values 
will make more contributions to the similarity va-
lue. 
 
Diversity Index and Variable Uniqueness 
 
Although the above algorithms can effectively im-
prove the accuracy of the clustering result of the 
k-modes algorithm, it is noted that the k-modes al-
gorithm and its modified version cannot detect the 
diversity of the data in a certain attribute in data 
set. It is easily prove in simple matching similarity 
measure. Because in the value will be 1 if it is ide-
ntical, and 0 if otherwise. In wk-modes, the basic 



64 Jurnal Ilmu Komputer dan Informasi (Journal of Computer Science and Information), Volume 7, Issue 2, 
June 2014 
 

TABLE 1  
DATA SAMPLE 

Dataset/ 
object 𝑥𝑥1 𝑥𝑥2 𝑥𝑥3 𝑥𝑥4 𝑥𝑥5 Div 

𝐷𝐷1 A A A A A 1 

𝐷𝐷2 A A A A B 2 

𝐷𝐷3 A A A B B 2 

𝐷𝐷4 A A B B C 3 

𝐷𝐷5 A A B C D 4 

𝐷𝐷6 A B C C C 3 

𝐷𝐷7 A B C D E 5 

 
concept used is using individual Simpson diver-
sity index (will be explained in the next part). By 
using this information, it is noted that this algo-
rithm cannot detect the diversity of data in data 
set. Let us see Table 1 as the data sample of coun-
ting the dissimilarity between two objects. Let the 
number of attribute in the given data set is only 1, 
and the number of data set is 7. The number of 
object in a dataset is 5. From the table we have the 
diversity value (Div) as variable uniqueness sho-
wn in the table.  

The value of Div show us the number of dis-
tinct value in a dataset. We assume that by using 
this Div value it will increase the dissimilarity 
between clusters based on the diversity of attribu-
te value. 

From the Table 1, we will get 1 as the value 
of dissimilarity because each data is different. 
But, as we can see in the real in each data, there 
are some similar data in it. But, they have differ-
rent diversity. For AAAAA, the diversity value is 
1 because there’s just a single object data. But for 
ABCDE, the diversity value is 5 because actually 
there are 5 different data. Based on this diversity, 
we propose a new method to count the dissimi-
larity measure to count the distance between two 
distinct objects. We use this diversity value to de-
crease the inter-similar cluster value.  

Diversity index is formerly a concept in bio-
logical area. It is a mathematical measure of spe-
cies diversity in a community. Diversity indices 
provide more information about community com-
position than simply species richness (i.e., the 
number of species present); they also take the re-
lative abundances of different species into account 
[4]. Diversity indices provide important informati-
on about rarity and commonness of species in a 
community. The ability to quantify diversity in th-
is way is an important tool for biologists trying to 
understand community structure.  

One of the commonly used diversity index 
measures is Simpson Diversity Index (SDI). The 
basic concept of SDI represents the measurement 
of dissimilarity by frequency based. Simpson's Di-

versity Index is a measure of diversity. In ecolo-
gy, it is often used to quantify the biodiversity of a 
habitat. It takes into account the number of speci-
es present, as well as the abundance of each spe-
cies. Simpson's Index (D) measures the probabi-
lity that two individuals randomly selected from a 
sample will belong to the same species (or some 
category other than species). The formula is defi-
ned as below. 
 

𝐷𝐷 =  
∑𝑛𝑛(𝑛𝑛 − 1)
𝑁𝑁(𝑁𝑁 − 1)

 (9) 

 
where n is the total number of organisms of a par-
ticular species, and N is the total number of orga-
nisms of all species. Simpson's Index gives more 
weight to the more abundant species in a sample. 
The addition of rare species to a sample causes 
only small changes in the value of D [5]. In clus-
tering theory, the property n can be defined as the 
number of object 𝑥𝑥𝑖𝑖 in a dataset D, and the pro-
perty N can be defined as the number of all ob-
jects in data set D. 
 
Diversity-based Attribute Weighting for K-
Modes Clustering 
 
Taking into account of the intra and inter cluster 
information, we introduce the new dissimilarity 
measure in the equation(10). 
 

𝐷𝐷𝐷𝐷𝐷𝐷𝑃𝑃�𝑥𝑥𝑖𝑖,𝑦𝑦𝑗𝑗� =
𝑆𝑆𝐷𝐷𝑆𝑆𝑖𝑖 ×  𝐷𝐷𝐷𝐷𝐷𝐷 ×  𝑛𝑛𝑖𝑖

𝑁𝑁
×
𝑆𝑆𝐷𝐷𝑆𝑆𝑗𝑗 ×  𝐷𝐷𝐷𝐷𝐷𝐷 ×  𝑛𝑛𝑗𝑗

𝑁𝑁
 

(10) 

 
where, 𝑆𝑆𝐷𝐷𝑆𝑆𝑖𝑖 is simpson diversity index in data set 
D for object 𝑥𝑥𝑖𝑖, 𝑆𝑆𝐷𝐷𝑆𝑆𝑗𝑗 is simpson diversity index in 
data set D for object 𝑦𝑦𝑗𝑗, 𝐷𝐷𝐷𝐷𝐷𝐷 is the number of 
category in data set D, 𝑛𝑛𝑖𝑖 is the frequency of 𝑥𝑥𝑖𝑖 in 
a cluster, 𝑛𝑛𝑗𝑗 the frequency of 𝑦𝑦𝑗𝑗 in a cluster, 𝑁𝑁 is 
the number of data in data set D. 

This formula is derived from the basic 
concept of probability count of object 𝑥𝑥𝑖𝑖 as 
defined below. 
 

𝑝𝑝(𝑥𝑥𝑖𝑖) =  
𝑁𝑁𝑥𝑥𝑖𝑖
𝑁𝑁

   (11) 
 
where, 𝑝𝑝�𝑥𝑥𝑖𝑖,� is the probability of 𝑥𝑥𝑖𝑖, 𝑁𝑁𝑥𝑥𝐷𝐷 is the 
frequency of 𝑥𝑥𝑖𝑖, and 𝑁𝑁 is the number of data in 
data set D. 

In order to count the dissimilarity between 
two objects 𝑥𝑥𝑖𝑖,𝑦𝑦𝑗𝑗 , we modify the above formula 
as defined in equation(12). 

 

𝐷𝐷𝐷𝐷𝐷𝐷𝑃𝑃�𝑥𝑥𝑖𝑖, 𝑦𝑦𝑗𝑗� =  
𝑁𝑁𝑥𝑥𝑖𝑖
𝑁𝑁

×  
𝑁𝑁𝑦𝑦𝑗𝑗
𝑁𝑁

              (12) 



M. Misbachul Huda, et a.l, Diversity-Based Attribute  65 
 

TABLE 2  
SCALABILITY EXPERIMENTAL RESULT 

Pengujian Scalability 

Number of 
objects 

Diversity 
based 

Time(ms) 

Original 
Time 
(ms) 

10 4 3 
50 10 10 
100 18 18 
500 135 124 

1000 737 585 
5000 38010 18984 

 
TABLE 3.  

DATA SET CHARACTERISTIC 

Dataset 
Class 

number 
Number 
of data 

Number 
of 

attributes 

Voting 2 435 16 

Mushroom 2 8124 23 

Soybean 4 47 36 
 

TABLE 4  
CLUSTERING EFFICIENCY RESULT 

 Voting Mushroom Soybean Average 
Original 
K-Modes 0.8592 0.7381 0.8177 0.805 

wk-
Modes 0.8651 0.7905 0.897 0.850867 

Zengyou 
k-modes 0.8734 0.7644 0.86 0.8326 

Diversity 
k-modes 0.8861 0.7849 0.887 0.852667 

 

 
 

Figure 1. Scalability Graphic between original k-modes 
and diversity-based k-modes 

 
 
where, 𝐷𝐷𝐷𝐷𝐷𝐷𝑃𝑃�𝑥𝑥𝑖𝑖, 𝑦𝑦𝑗𝑗�is the dissimilarity of 𝑥𝑥𝑖𝑖, 𝑦𝑦𝑗𝑗in 
attribute P, 𝑁𝑁𝑥𝑥𝑖𝑖is the number of object 𝑥𝑥𝑖𝑖in data 
set D, 𝑁𝑁𝑦𝑦𝑗𝑗is the number of object 𝑦𝑦𝑗𝑗in data set D 
and 𝑁𝑁is the number of data in data set D. 

To increase the accuracy of clustering pro-
cess, we add the Simpson diversity index that has 
defined in the previous to the formula above. Our 
new proposing dissimilarity measure is defined as 
the following equation(13). 
 

𝐷𝐷𝐷𝐷𝐷𝐷𝑃𝑃�𝑥𝑥𝑖𝑖,𝑦𝑦𝑗𝑗� =
𝑆𝑆𝐷𝐷𝑆𝑆𝑖𝑖 ×  𝐷𝐷𝐷𝐷𝐷𝐷 ×  𝑛𝑛𝑖𝑖

𝑁𝑁
×
𝑆𝑆𝐷𝐷𝑆𝑆𝑗𝑗 ×  𝐷𝐷𝐷𝐷𝐷𝐷 ×  𝑛𝑛𝑗𝑗

𝑁𝑁
 

(13) 

 
where, 𝑆𝑆𝐷𝐷𝑆𝑆𝑖𝑖is simpson diversity index in data set 
D for object 𝑥𝑥𝑖𝑖, 𝑆𝑆𝐷𝐷𝑆𝑆𝑗𝑗is simpson diversity index in 
data set D for object 𝑦𝑦𝑗𝑗, 𝐷𝐷𝐷𝐷𝐷𝐷is the number of 
category in data set D, 𝑁𝑁is the number of data in 
data set D. 

 
3. Results and Analysis 

 
In this part, we will discuss about the experiment-
tal result of scalability and efficiency of clustering 
algorithms. It employs three different algorithms. 
In first part of this section, we will explain the ex-
periment environment and evaluation index. In se-
cond part, we will explain the experimental result 
of scalability of original k-modes and our propos-
ed method. And in the last part, we will show the 
experimental result of clustering efficiency of four 
different modified k-modes algorithm include our 
proposed method.  

 
Experiment Environment and Evaluation In-
dex 
 
The experiment was done in core i3 (1.4 GHz), 
4GB RAM, and windows 8.1x64 based computer. 
The implementation of the algorithm uses Java as 
the programming language. To evaluate the accu-
racy of the algorithm, we use the following equa-
tion(14). 
 

𝑎𝑎𝑐𝑐𝑐𝑐 =
∑ 𝑎𝑎𝑖𝑖𝑘𝑘𝑖𝑖=1
𝑛𝑛

 (14) 

 
where k is the number of known categories, 𝑎𝑎𝑖𝑖 is 
the number of object that lies in the right cluster 
in 𝐶𝐶𝑖𝑖(1 ≤ 𝐷𝐷 ≤ 𝑘𝑘). 
 
Scalability Evaluation 
 
To compare the scalability of original k-modes 
and our proposed method, we use synthetic data 
with the variant number of objects between 10 
and 5000. This synthetic data have 23 different at-
tributes. To decrease the random effect of k-mo-
des algorithm, we did 100 times experiment for 
every scenario.  

For each experimental result shown in Table 
2, is the average value of the 100 times experi-
mental result. From these experiments, we get a 
scalability graphic shown in Figure 1. 

The experimental result of runtime from di-
versity-based k-modes show that it needs longer 
time than the original k-modes to evaluate the gi-



66 Jurnal Ilmu Komputer dan Informasi (Journal of Computer Science and Information), Volume 7, Issue 2, 
June 2014 
 

ven data set. It is caused by added step to compute 
the diversity of the data set. 
 
Clustering Efficiency Evaluation  
 
In this section, to compare the efficiency cluste-
ring algorithms, we use four different modified k-
modes algorithm include our proposed method. 
They are Simple-Matching k-modes, Zengyou’s 
k-modes, wk-Modes and our new proposed me-
thod diversity-based k-modes. We use tree differ-
rent data set from UCI Machine Learning [6]. 
Table 3 shows the characteristic of the data set. 

In experiments, the missing value is ignored. 
To decrease the random effect in k-modes, every 
experiment was conducted 100 times. Table 4 sh-
ows the result of clustering efficiency evaluation 
between the four algorithms. Every value is the 
average of 100 times experiments result. 

 
4. Conclusion  

 
The k-modes algorithm is widely used for clus-
tering categorical data. Dissimilarity and simila-
rity measure play crucial rules in this area. In this 
paper the limitation of the former algorithm on the 
usage of between cluster information is solved by 
using simpson diversity index to extend the value 
of the intra similarity index. The experimental re-
sult shows that our proposed algorithm give better 
result. 
 

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