Research Article | Open Access
Amar Rebbouh, "Clustering Objects Described by Juxtaposition of Binary Data Tables", Advances in Decision Sciences, vol. 2008, Article ID 125797, 13 pages, 2008. https://doi.org/10.1155/2008/125797
Clustering Objects Described by Juxtaposition of Binary Data Tables
This paper seeks to develop an allocation of 0/1 data matrices to physical systems upon a Kullback-Leibler distance between probability distributions. The distributions are estimated from the contents of the data matrices. We discuss an ascending hierarchical classification method, a numerical example and mention an application with survey data concerning the level of development of the departments of a given territory of a country.
The automatic classification of the components of a structure of multiple tables remains a vast field of research and investigation. The components are matrix objects, and the difficulty lies in the definition and the delicate choice of an index of distance between these objects (see Lerman and Tallur ). If the matrix objects are tables of measurements of the same dimension, we introduced in Rebbouh  an index of distance based on the inner product of Hilbert Schmidt and built a classification algorithm of k-means type. In this paper, we are interested by the case when matrix objects are tables gathering the description of the individuals by nominal variables. These objects are transformed into complete disjunctive tables containing 0/1 data (see Agresti ). It is thus a particular structure of multiple data tables frequently encountered in practice each time one is led to carry out several observations on the individuals whom one wishes to classify (see [4, 5]). We quote for example the case when we wish to classify administrative departments according to indices measuring the level of economic and human development which are weighted averages calculated starting from measurements of selected parameters. Each department gathers a number of subregions classified as rural or urban. Each department is thus described by a matrix with columns and lines of positive numbers ranging between and . But the fact that the values of the indices do not have the same sense according to the geographical position of the subregion, and its urban or rural specificity led the specialists to be interested in the membership of the subregion to quintile intervals. Thus, each department will be described by a matrix object with 10 columns and lines. This matrix object is thus a juxtaposition of tables of the same dimension containing only one on each line which corresponds to the class where the subregion is affected and the remainder are . The use of conventional statistical techniques to analyze this kind of data requires a reduction step. Several criteria of good reduction exist. The criterion that gives results easily usable and interpretable is undoubtedly the least squares criterion (see, e.g., ). These summarize each table of data describing each object for each variable in a vector or in subspace. Several mathematical problems arise at this stage:(1) the choice of the value which summarizes the various observations of the individual for each variable, do we take the most frequent value or another value, for example an interval ; why this choice? and which links exist between variables,(2)the second problem concerns the use of homogeneity analysis or multiple correspondence analysis (MCA) to reduce the data tables. We make an MCA for each of the data tables describing, respectively, the individuals. We get factorial axis systems. To compare elements of the structure, we must seek a common system or compromise system of the ones. This issue concerns other mathematical discipline such as differential geometry (see ). The proposed criteria for the choice of the common space are hardly justified (see Bouroche ). This problem is not fully resolved,(3) the problem of the number of observations that may vary from one individual to another. We can use the following procedure to complete the tables. We assume that , for all , is the number of observation of the individual and define as the least common multiple of Hence, there exists such that . Now, we duplicate each table , times, we obtain a new table of dimension , is the number of variables. But if is large, the least common multiple becomes large itself, and the procedure leads to the structure of large tables. Moreover, this completion removes any chronological appearance of data. This cannot be a good process of completion, and it seems more reasonable to carry out the classification without the process of completion.
To overcome all these difficulties with the proposed solutions which are not rigorously justified, we introduce a formalism borrowed from the theory of signal and communication see (Shannon ) and which is used to classify the elements of the data structure . Our approach is based on simple statistical tools and on techniques used in information theory (physical system, entropy, conditional entropy, etc.) and requires the introduction of the concept of discrete physical systems as models for the observations of each individual for the variables which describe them. If we consider that an observation is a realization of a random vector, it appears reasonable to consider that each value of the variable or the random vector represents a state of the system which can be characterized by its frequency or its probability. If the variable is discrete, the number of states is finite, each state will be measured by its frequency or its probability. This approach gives a new version and in the same time an explanation of the distance introduced by Kullback . This index makes it possible to build an indexed hierarchy on the elements of the structure and can be used if the matrix objects do not have the same dimension.
In Section 2, we introduce an adapted formalism and the notion of physical random system as a model of description of the objects. We define in Section 3 a distance between the elements of the structure. The numerical example and an application are presented in Section 4. Concluding remarks are made in Section 5.
2. Adapted Formalism
Let be a finite set of elementary objects, be discrete variables defined over and taking a finite number of values in , respectively, and is the th modality or value taken by We suppose that the observations of the individual for the variable are given in the table and represents the number of the observations of the individual , if the th observation of the individual for the the variable is where . is the vector with components corresponding to the different observations of for
The structure of a juxtaposition of categorical data tables is is a matrix of order . For the sake of simplicity, we transform each vector in a data matrix :
The structure of a juxtaposition of data tables is is a matrix of order where .
is estimated by the relative frequency of the value observed in the th column of the matrix
Let be the random single physical system associated to for :where the symbol means that the system lies in the state , and is the conjunction between events.
In the multidimensional case, the associated multiple random physical system is where The multiple random physical system associated to the marginal distributions is where is the conjunction between single physical systems, and are the single random physical systems given by
3. Distance between Multiple Random Physical Systems
3.1. Entropy as a Measure of Uncertainty of States of a Physical System
For measuring the degree of uncertainty of states of a physical system or a discrete random variable, we use the entropy which is a special characteristic and is widely used in information theory.
3.1.1. Shannon's  Formula for the Entropy
The entropy of the system is the positive quantity:
The function has some elementary properties which justify its use as a characteristic for measuring the uncertainty of a system.
(1) If one of the states is certain such that , then (2) The entropy of a physical system with a finite number of states is maximal if all its states are equiprobable: for all . We have also
The characteristic of the entropy function expresses the fact that probability distribution with the maximum of entropy is the more biased and the more consistent with the information specified by the constraints .
3.2. Entropy of a Multiple Random Physical System
The conditional random physical system is given bywhere is a conditional probability.
The entropy of this system is
The multiple random physical system is written by which implies Hence, The quantity is nonnegative. We have
It is clear that is not a symmetric function, thus it is not a distance in the classical sense but characterizes (from a statistical point of view) the deviation between the distributions and . It should be noted that is symmetrical.
Kullback  explains that the quantity evaluates the average lost information if we use the distribution while the actual distribution is
Let be a set of random physical systems with states
Let dist be the application defined by
and are the multivariate distributions of order governing, respectively, the random physical systems and . is defined as follows: dist verifies (1),(2),(3).
We admit that
dist measures the similarity between physical systems. The smaller the value of dist is, the larger the uncertainty of the systems is. dist represents the lost quantity of average information if we use the distribution () to manage the system while the other distribution is true. dist is nothing else than the Kullback-Leibler distance between the multivariate distributions and Indeed, the Kullback-Leibler distance between and is given by
Developing this expression will give dist.
4. Numerical Application
4.1. Procedure to Estimate the Joint Distribution
In the case where all variables involved in the description of the individuals are discrete, we give a procedure taken from classical techniques of factor analysis to estimate the joint distribution and derive the entropy of the multiple physical system.
Let be a juxtaposition of data tables. For fixed, we have
is the number of simultaneous occurrences of the modalities :
We use an algorithm for ascending hierarchical classification . We call points either the objects to be classified or the clusters of objects generated by the algorithm.
Step 1. There are points to classify (which are the objects).
Step 2. We find the two points and that are closest to one another according to distance dist and clustered in a new artificial point .
Step 4. We again find the two closest points and aggregate them. We calculate the new distances and repeat the process until there is only one point remaining.
In the case of single linkage, the algorithm uses distances in terms of the inequalities between them.
4.3. Numerical Example
Consider 6 individuals described by 2 qualitative variables with, respectively, 2 and 3 modalities. 10 observations for each individual, the observations are grouped in Table 1.
4.3.1. Procedure to Build a Hierarchy on these Objects
The empirical distributions which represent the individuals are given by Table 2.
The program is carried out on this numerical example. We obtain the following results (Table 3).
Step 1. From the similarity matrix, using the single linkage of Sneath (4.3), we obtainThen, the objects and are aggregated into the artificial object which is placed at the last line, and the rows and columns corresponding to the objects and are removed in the similarity matrix.
Step 2. From the new similarity matrix, we obtainThe objects and are aggregated into the artificial object
Step 3. The objects and are aggregated into the artificial object .
Step 4. The objects and are clustered in the new object The object is aggregated with the object and (see Figure 1)
In Figure 1 it can be seen that two separated classes appear in the graph by simply cutting the hierarchy on the landing above the individual . In this algorithm, we started by incorporating the two closest objects using the index of distance between corresponding physical systems. The higher the construction of the hierarchy is, the more dubious the obtained states of the mixed system are. The example shows that the index of Kullback-Leibler and the index of aggregation of the minimum bound (single linkage) lead to the construction of a system with a maximum of entropy, and thus lead to a system for which all the states are equiprobable.
If the total number of modalities of the various criteria is large compared with the number of observations, the frequency of choosing a set of modalities becomes small, and a lot of frequencies are zero. The set of modalities whose frequency is zero will be disregarded and does not intervene in the calculation of the distances. This can lead to the impossibility of comparing the systems.
4.3.2. Classification of the Six Objects after Reduction
If each object is described by the highest frequencies “mode,” we obtain the following table:
This table contradicts the fact that in our procedure, the objects and are very close while and are not the same. This shows that the classification after reduction, for this type of data, can lead to contradictory results.
The data come from a survey concerning the level of development of departments of a country. The aim is to search the less developed subregions in order to establish programs of adequate development. Every department is constituted of subregions . For every and , we measured the composite economic development index and the composite human development index These two composite indices are weighted means of variables measuring the degree of economic and human development, developed by experts of the program of development of the United Nations for ranking countries. These indices depend on the geographical situation and on the specificity of the subregions.
For every , and The closer to the value of the index is, the more the economic or human development is judged to be satisfactory. However, these indices are not calculated in the same manner. They depend on whether the subregions are classified as farming or urban zone. The ordering of the subregions according to each of the indices do not have sense anymore. The structure of data in entry is for every :
The structure is not exploitable in this form. It is therefore necessary to transform the tables in a more tractable form. The specialists of the programs of development cut the observations of each index in quintile intervals and affect each of the subregions to the corresponding quintile. We thus determine for the series of observations of the two indices the various corresponding quintiles:The quintile intervals areFor every , the table is transformed into a table of data.
The problem is to build a hierarchy on all departments of the territory in order to observe the level of development of each of the subregions according to the two indices and thus to make comparisons. The observations are summarized in tables which constitute a structure of juxtaposition of data matrices. These data presented are from a study of municipalities involved in Algeria. The municipalities are gathered in departments. The departments do not have the same number of municipalities which have not the same specificities: size, rural, urban, and the municipalities do not have the same locations: mountain, plain, costal, and so forth. We have to build typologies of departments according to their economic level and human development according to the United Nations Organization standards.
The result of the study made it possible to gather the great departments (cities) which have large and old universities and the municipalities which have a long existence. Another group emerged which includes enough new departments of the last administrative cutting and develops activities and services of small and middle companies. The other groups are distinguished by great disparities between municipalities in their economic level and human development and according to surface and importance.
In this paper, the definition of the entropy is that stated by Shannon . This definition is still used in the theory of signal and information. The suggested formalism gives an explanation and a practical use of the distance of Kullback-Leibler as an index of distance between representative elements of a structure of tables of categorical data. It is possible to extend these results to the case of a structure of data tables of measurements and to adapt an algorithm of classification to the case of functional data.
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Copyright © 2008 Amar Rebbouh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.