- Email: [email protected]
A tool developed in Matlab for multiple correspondence analysis of fuzzy coded data sets: Application to morphometric skull data
c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 9 8 ( 2 0 1 0 ) 66–75
journal homepage: www.intl.elsevierhealth.com/journals/cmpb
A tool developed in Matlab for multiple correspondence analysis of fuzzy coded data sets: Application to morphometric skull data Antonio Pinti ∗ , Fabienne Rambaud, Jean-Louis Griffon, Abdelmalik Taleb Ahmed LAMIH UMR CNRS 8530, Université de Valenciennes, Le Mont Houy, 59313 Valenciennes Cedex 9, France
a r t i c l e
i n f o
a b s t r a c t
Article history:
Multiple Correspondence factorial Analysis is a multivariate method for the exploratory
Received 4 December 2008
study of multidimensional contingency tables. Its use can be extended to the analysis of
Received in revised form
a table of fuzzy coded data resulting from a distribution into fuzzy windows defined by
9 September 2009
linguistic properties. There are few existing software tools that allow performing this type of
Accepted 16 September 2009
analysis on a data table; furthermore these tools are not interactive and do not allow defining and representing fuzzy windowing. This paper presents a software tool, developed with
Keywords:
Matlab, to compute and represent results from multiple correspondence factorial analyses.
Multiple correspondence analysis
Pre-defined membership functions can be selected by the user according to the distribution
Principal component analysis
histograms of the data.
Fuzzy sets
This paper presents an application example of this program onto a data table of mor-
Biplot
phometric parameters of 150 male skulls throughout 5 periods of Egyptian civilization. The
Skulls dataset
results are compared to those of a principal component analysis, which is more often used for the study of experimental data. Our program allows a rapid description of the morphological evolution of skulls over time, notably thanks to a linguistic description of each variable, whereas the results of the latter method are less obvious to observe and require a deeper analysis in order to arrive at the same conclusions. © 2009 Elsevier Ireland Ltd. All rights reserved.
1.
Introduction
Exploratory data analysis is principally concerned with the description of data sets observed mostly during experimentation, measured or recorded for a certain number of subjects – or statistical units – and with the establishment of relations between the different characteristics of these same subjects, without specifying any particular variable [1]. For this, the expert can rely on summaries, graphical representations as well as on multidimensional analysis methods; the choice of method depends on the type of data to study. Among the
∗
multidimensional exploratory methods, one can cite Principal Component Analysis (PCA), used for the study of quantitative data, or factorial Analysis of Multiple Correspondences (MCA), generally used for qualitative data [2,3]. These methods allow a global study to be performed thanks to graphical representations and a data analysis from several viewpoint [4]. They are placed in the framework of descriptive and data exploration methods, for which data are manipulated in order to be transformed into useful information [5]. Although multiple correspondence analysis is used for crisp data, it is possible to apply it on fuzzy coded data where each modality of a variable corresponds to a membership
Corresponding author. Tel.: +33 32 7511429; fax: +33 32 7511316. E-mail address: [email protected] (A. Pinti). 0169-2607/$ – see front matter © 2009 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.cmpb.2009.09.009
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c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 9 8 ( 2 0 1 0 ) 66–75
value to a particular fuzzy window. According to Loslever et al. [6], this method allows to better account for complex relationships existing between the characteristics, to underscore the presence of classes as well as to rapidly show the presence of aberrant data. This analysis function is already present in some free software solutions such as R [7] or commercial ones such as XLStat or S-PLUS [8]. However these solutions do not offer the option to define cutting functions or to directly visualize the projection of results. In order to facilitate the work of data analysis, we have developed under Matlab a complete software including stages of fuzzy cutting of data and graphical representation of results. This software, dedicated to the fuzzy coding CA with crisp data, can be used just as any toolbox available in Matlab. The objective of this paper is to present this new software tool. Section 2 first describes the fuzzy cutting method, and then, the computation principles for the MCA of such fuzzy coded data. The Matlab implementation of the fuzzy cutting of data and the MCA computations as well as the description of the program and of the graphic interface are presented in Sections 3 and 4. Finally, the significance of this new software tool is illustrated in Section 5 by a real example from a scientific study in the context of anthropometric anatomy of the human skull from 4000 years BC to 150 years AD. Discussions and conclusions are in Section 5.
2. Multiple correspondence analysis of fuzzy coded data Multiple correspondence analysis was proposed by Benzécri in 1964 [9] for the study of contingency tables [10,11]. Multiple correspondence analysis (MCA) was subsequently generalized to describe large binary tables. Data studied by MCA are qualitative and describe modalities. However, it is possible to extend the use of this exploratory method to quantitative data previously coded into modalities. To do this, the use of binary coding in the domain of classical ensembles allows conversion of raw data to binary variables (0 or 1); nevertheless, the Boolean variable is ill adapted to the representation of most ordinary phenomena. But in the domain of fuzzy logic, the membership of a variable to a fuzzy ensemble (or set) will have a real value between 0 and 1 [12]. Moreover, the use of fuzzy coding for quantitative data allows minimization of information loss due to passing from the quantitative to the qualitative scale [13,6]. Processing of a table of raw data is performed in two steps:
1. Fuzzy windowing generating a table of membership values; 2. Multiple Correspondence factorial Analysis applied on the resulting table.
2.1.
Fuzzy coding
The fuzzy sub-sets are defined as follows: Let X be a reference ensemble and E a sub-set of X. E is a sub-set of X if its characteristic function associates to each
Fig. 1 – Principle of fuzzy cutting. Three spatial windows are considered with modalities {small; medium; large}.
element x of X a membership value fFE (x) between 0 and 1: fFE : X → [0, 1]
(1)
Fuzzy rules express the fact that the closer one is to a given situation, the more weight is given to the action recommended for this situation. These rules are represented by membership functions that can take varied forms depending on the nature of the object modeled. We will use linguistic variables to characterize fuzzy sub-sets. The membership values for triangular modalities are calculated in the following way: Let Mi be a triangular shaped fuzzy modality, defined by parameters a, b, c. The membership value of point P with coordinates x in modality M is:
Mi (P) = max min
x − a b−a
, 1,
c−x c−b
,0
(2)
Fig. 1 shows a fuzzy cutting in 3 spatial windows. For a given statistic, the output is a vector of 3 membership values corresponding to the 3 modalities of the variable (small, medium, and large). The sum of these membership values must Eq. (1).
Table 1 – A table of quantitative data: the skull data set. Epoch
X1 (mm)
X2 (mm)
X3 (mm)
X4 (mm)
−4000 −4000 −4000 ... −3300 −3300 −3300 ... −1850 −1850 −1850 ... −200 −200 −200 ... +150 +150 +150 ...
131 125 131 ... 133 138 148 ... 137 129 132 ... 137 141 141 ... 137 136 128 ...
138 131 132 ... 134 134 129 ... 141 133 138 ... 134 128 130 ... 123 131 126 ...
89 92 99 ... 97 98 104 ... 96 93 87 ... 107 95 87 ... 91 95 91 ...
49 48 50 ... 48 45 51 ... 52 47 48 ... 54 53 49 ... 50 49 57 ...
Min Max
119 148
120 145
81 114
44 60
68
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Fig. 2 – Basic triangular membership functions. Min corresponds to the minimum value of the variable’s data; Max corresponds to the maximum value; the Medium membership function depends on the middle of the segment [Min;Max].
Thus, the fuzzy windowing step consists in classifying the raw data table using membership functions between 0 and 1. Firstly, each system input (or variable) is modeled by curves giving the degrees of membership to the states defined by these inputs. For each variable we define membership functions that will allow the creation of columns corresponding to the membership values for each modality of the variable. The final table will be the concatenation of all modalities for each variable; all values from the original table are described by membership values corresponding to the number of modalities per variable (see Tables 1 and 2). The sum of each line of the table equals the number of variables. Respecting that condition, the table can include fuzzy coded modalities as well as binary coded ones (see Goldfarb’s work [13] for more emphasis on that particular point). We can distinguish two types of windowing: Fig. 4 – Program steps for MCA computation of fuzzy coded data developed under Matlab.
• Using membership functions that do not take into account value distributions (an example with 3 membership values is given in Fig. 2).
• Using membership functions that are adapted to the variables’ distributions (an example with 3 membership values is given in Fig. 3).
To our knowledge, there is no precise criterion allowing deciding if one of the methods is better than the other. Only an expert can efficiently judge which type of function should be used, for example by looking at the data distribution histogram. Depending on the shape of the histogram, the expert decides which cutting procedure is most appropriate.
Fig. 3 – Triangular membership functions. Min corresponds to the minimum value of the variable’s data; Max corresponds to the maximum value; the Medium membership function depends on the average value of the data.
Table 2 – Table 1 after fuzzy coding, the membership functions are presented in Fig. 2. Epoch
−4000 −4000 −4000 ... +150
X1
X2
X3
X4
smal
med
larg
smal
med
larg
smal
med
larg
smal
med
larg
0.17 0.59 0.17 ... 0.00
0.83 0.41 0.83 ... 0.83
0.00 0.00 0.00 ... 0.17
0.00 0.12 0.04 ... 0.00
0.56 0.88 0.96 ... 0.96
0.44 0.00 0.00 ... 0.04
0.51 0.33 0.00 ... 0.03
0.49 0.67 0.91 ... 0.97
0.00 0.00 0.09 ... 0.00
0.37 0.50 0.25 ... 0.13
0.63 0.50 0.75 ... 0.87
0.00 0.00 0.00 ... 0.00
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Fig. 5 – GUI for the selection of the membership functions. For each variable, the interface presents an histogram with an adjustable number of classes. This representation can help the user to choose one of the pre-defined fuzzy cutting functions in order to preserve a large amount of the distributional information. He can also preview, at any time, the repercussions on the MCA projections.
2.2.
Multiple correspondence analysis
Let Z be a fuzzy coded table with n individuals as rows and m modalities as columns. Let v be the variable number. Let m(v) be the number of modalities of variable v. Notice that
m=
v j=1
m(j)
and v =
m(i) v
ij
tions implied by projection. It searches for the most accurate representation of the cloud within the smallest number of dimensions [11]. For that, it is necessary to find a plane in which the cloud’s dispersion, characterized by its inertia, is maximal. In practice, we first construct the relative frequency table F.
(3)
i=1 j=1
with ij the membership value for the jth modality of variable i. From this set a factorial analysis can be performed. The objective of this method applied to fuzzy coded data is to facilitate the analysis by processing data described by linguistic properties. The objective is to represent as accurately as possible the clouds of data points that are formed by the rows and columns in their respective spaces (with p or n dimensions) in order to reveal structure contained in the information from the shape of these clouds. The search for a sub-space with a dimension smaller than that of the initial space while losing the least possible information, and the projection of the data of interest onto this sub-space allow obtaining a representation of the data while simultaneously reducing their dimension. Indeed, it is relatively easy to represent a cloud of data points when it is included in a space with 2 or even 3 dimensions; it is much less so when the cloud extends over spaces with dozens of dimensions. The main idea of factorial analysis is that a simple way to visualize the shape of a cloud of data points is to project it onto lines or even better, planes, while minimizing deforma-
F=
Z k
(4)
with k = sum of elements of Z. We then compute the row-margins Dn and columnmargins Dp , which are the diagonal matrices of the sums of respectively rows and columns of F. The Dp defines the metric of the subject space p , while the diagonal elements of Dn are the masses of the n point-rows. We call row-profile the result of the inner product D−1 n F and column-profile the result of D−1 p F. The distance d(i,i ) considered between two point-lines i and i is that of the 2 : d2 (i, i ) =
p 1 ij j=1
.j
i.
−
i j 2 i .
(5)
We then construct the matrix S to be diagonalized: −1 S = F D−1 n FDp
(6)
where F is the transpose of F. The eigenvalues ˛ of S are called “moments of inertia” of the cloud and satisfy the condition ˛ ∈ [0,1].
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c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 9 8 ( 2 0 1 0 ) 66–75
Fig. 6 – Interface of the Multiple Correspondence Analysis of fuzzy coded data under Matlab.
The corresponding eigenvectors u˛ define the factorial axes in subject space. To obtain the factorial axes v˛ of variable space, one must compute: 1 v˛ = √ FD−1 p u˛ ˛
(7)
u˛ and v˛ must verify the normalization relations u˛ D−1 p u˛ = 1 v = 1. and v˛ D−1 ˛ n Finally, the factorial components ˛ and ϕ˛ , projections of row points and column points on the respective factorial axes are obtained by: −1 D−1 n FDp u˛
(8)
−1 ϕ˛ = D−1 p F Dn v˛
(9)
˛
=
In order to correctly interpret the projections of cloud points on factorial axes, one may define tools that offer supplementary information on the participation of a row point or a column point to the construction of the axes, the contribu-
tion, and on the quality of the representation of the point on a factorial sub-space, the cosine squared.
3. Implementation and description of the program developed under Matlab At the startup of the program developed under Matlab, the data can be loaded from a file with a table format. To help the analysis, each vector can be identified by a class. The program depends on three main functions:
• Fuzzy cutting from histograms (the number of classes can be modified). • Computation of the multiple correspondence analysis on fuzzy coded data. • Numerical representations of contribution tables and graphs of results on axes chosen by the user. Fig. 4 shows the diagram representing the steps of the program.
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Fig. 7 – Four measurements (in mm) performed on skulls from men having lived in the Thebes region of Egypt.
When applying this method, a large part of the time is allocated to the search for membership functions that best correspond to the variable distribution. Here, a graphic interface helps the user in his choice by plotting both the histogram for each variable and the cutting function that will be applied on them. Fig. 5 illustrates this step. A dozen of pre-defined membership functions are currently available. They differ by:
• The number of fuzzy windows (3, 5,. . .); • Their nature (triangular functions, sigmoid, etc.); • The way they adapt themselves to the distribution of the data (mean/median value, histogram’s modes, etc.).
Fig. 9 – MCA of fuzzy coded data, eigenvalues.
At any moment, a pre-visualization of the MCA is possible in order to best adjust the cutting. When the choice of membership functions seems satisfactory, the fuzzy coding can be applied and saved, and a more complete MCA can be started. A new window opens allowing the user to display all desired information (eigenvalues, eigenvectors, contributions, etc.), as well as projections on interesting factorial planes. The graphical representation of results is a figure that allows a visualization of the projections of individuals and variables along two principal axes. Fig. 6 shows that the tool associates a marker (color, pointers of various shapes and sizes) with each vector class to facilitate the data analysis.
Fig. 8 – Histogram and cuttings for variable X1. Up: a basic triangular membership functions, 3 membership values. Down: cutting windows, 5 binary modalities, partition of the [Min;Max] interval into 5 segments of equal size.
4.
Application example
In order to present an example of the use of the program that we created, we analyzed the Skulls database [14,15] that con-
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Table 3 – Table 1 as a contingency table with crisp data (decomposition into 5 modalities: VS: Very Small, S: Small, M: Medium, L: Large, VL: Very Large). Epoch
−4000 −4000 −4000 ... +150
X1
X2
X3
X4
VS
S
M
L
VL
VS
S
M
L
VL
VS
S
M
L
VL
VS
S
M
L
VL
0 0 0
0 1 0
1 0 1
0 0 0
0 0 0
0 0 0
0 0 0
0 1 1
1 0 0
0 0 0
0 0 0
1 1 0
0 0 1
0 0 0
0 0 0
0 0 0
1 1 1
0 0 0
0 0 0
0 0 0
0
0
0
1
0
1
0
0
0
0
0
1
0
0
0
0
1
0
0
0
Fig. 10 – MCA of fuzzy coded data, projections on axes 1 and 2 (resp. abscissa and ordinates).
Fig. 11 – MCA of fuzzy coded data, projections on axes 3 and 4 (resp. abscissa and ordinates).
tains morphometric data (4 variables, Fig. 7) for 150 male skulls from 5 periods of Egyptian civilization:
• 12th and 13th dynasty (1850 BC); • Ptolemaeus period (200 BC); • Roman period (150 AD).
• Ancient pre-dynastic period (4000 BC); • Recent pre-dynastic period (3300 BC);
The data are presented in Table 1.
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Inspection of the histograms for each variable reveals the Gaussian shape of the distributions. A fuzzy cutting into 3 categories of same size (Small S, Medium M, and Large L) is thus chosen (see Fig. 8) [16]. Then, MCA is performed on the fuzzy coded table. The first 4 factorial components represent 68.32% of the total inertia (Fig. 9), therefore the study will take into account the first 4 factorial axes. By studying the variables’ contributions to the axes as well as the cosine squared, one notes that the variable X1 is particularly well represented on axis 1, variables X1 and X4 on axis 2, variable X1 on axis 3 and variable X3 on axis 4. No result is obvious from the study of axes 3 and 4, but the plane formed by axes 1 and 2 is more interesting. Periods are grouped along axis 2, with more ancient skulls having smaller X1 and X4 values, whereas subjects closer to contemporary times have larger values. Axis 1 shows that values of X2, and to a lesser extent of X3, tend to decrease with time. On the whole, groups describe a clockwise movement with the pivot localized at the X4 Large modality. Figs. 10 and 11 illustrate these results. The program allowed us to perform a first quick analysis of a table of 150 × 4 data points using a method that usually requires a non-negligible computer program processing time [6]. A binary coding is also performed in order to compare results with those given by fuzzy coding. 5 binary modalities {VS, S, M, L, VL} are created for each continuous variable according to an equal division of the [Min; Max] interval. Then, a MCA is applied on the table given by the concatenation of all modalities (see Table 3). The goal of the study is to look for one or several evolutionary modes of morphometric data through epochs. Binary coding still facilitates the analysis with the use of linguistic properties, but here, the loss of information in comparison with fuzzy coding is obvious: this surjection associates 4 elev ments with vectors of a finite set of m(j) elements. This j=1 is why 5 modalities per variable were created instead of 3 to avoid too much loss. This also raises an other defect of binary coding: the more modalities we create, the more inertia will be spread along the factorial axes and the less projections on principal axes will be meaningful. In our case, MCA fails to extract remarkable axes of projection. The two principal axes of the MCA represent only 10% of the total inertia each (Fig. 12).
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Fig. 12 – MCA of binary coded data, eigenvalues. No axis predominates.
Thus, the first projection plane describes few informations and an investigation through the other axes becomes mandatory. Nevertheless a first tendency can be obtained thanks to projections on principals axes 1 and 2 (Fig. 13). X1 is well represented on axis 2, and variables X2 and X3 on axis 1. Groups can be formed by epochs: as previously, we can observe that X1 value tends to increase with time, but it is barely the only drawable conclusion at first sight. A normalized PCA was applied to the table via the statistical software StatLab [17,1] to compare methods and results. No axis predominates: each of the first 2 axes accounts for 30% of the total inertia, the 2 others for 20%. Inspection of the variables projected on the factorial axes 1 and 2 brings out the strong correlations of variables X2 and X3 with axis 1 and that of X4 with axis 2 (Fig. 14). Projections of individuals on the first two factorial axes reveal results that are less obviously interpretable than those obtained with MCA. Groupings by epochs are present, but are less easily dissociated than with the former method. Furthermore, the time evolution of the morphometric data is less evident. One can however come to the same conclu-
Fig. 13 – MCA of binary coded data, projections on axes 1 (abscissa) and 2 (ordinate).
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Fig. 14 – Normalized PCA. Projections on axes 1 (abscissa) and 2 (ordinate). sions by inspection of the groups: X1 and X4 have a tendency to increase while X2 and X3 decrease. As with the former method, no probing result comes from inspection of axes 3 and 4. The information obtained by the PCA method is the same as that obtained by the developed MCA method, but it is extracted with more difficulty.
5.
Conclusion
The objective of this paper was to present a new complete statistical tool for a method of multidimensional exploration of quantitative data, that is user-friendly and interactive. This analysis method consists in a multiple correspondence factorial analysis, more commonly called MCA, of quantitative data that has been previously fuzzy coded. The method is not well known, probably because it is not directly implemented in current statistical software, and must be performed ‘manually’ with several processing steps and representations. Because of this, it can appear repulsive if the table to be analyzed has a large number of variables. The tool developed for MCA presented in this paper proposes a computerized solution to this analysis and allows a gain in time of data analysis. To show the relevance of this tool, it was used on a morphometric database of human skulls in time. The results of this analysis were compared to those obtained by the principal component analysis (PCA) method performed in StatLab,
which is the method used for multidimensional exploration of numerical data tables. The multiple correspondence analysis on fuzzy coded tables holds interesting properties. By distinguishing the values of a variable by linguistic properties, the analysis is facilitated without loss of the real meaning of the data, meaning that can disappear in normalized principal component analysis because of centering and reduction. The fuzzy coding allows conserving a great amount of information contained in the data distribution [18]. This assertion was verified here by comparing results with those given with a simple binary coding. The graphical results allow, by the inspection of so-called extreme values, to more easily detect groups of individuals and variables by the fuzzy coding MCA method. This new tool is still not complete and it would be of interest to propose a greater choice of cuttings. Also, it should be possible for the users to choose, select and modify the membership functions directly with the mouse (if the histogram has several modes for instance) so that they may interactively visualize the effects of the cuttings on the global analysis.
Software availability The software is available on request free of cost from the corresponding author ([email protected]).
c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 9 8 ( 2 0 1 0 ) 66–75
Conflict of interest None.
Acknowledgments The present research work has been supported by International Campus on Safety and Intermodality in Transportation. The authors gratefully acknowledge the support of these institutions: the Nord-Pas-de-Calais Region, the European Community, the Regional Delegation for Research and Technology, the Ministry of Higher Education and Research, and the National Center for Scientific Research.
references
[1] L. Lebart, A. Morineau, Statistique Exploratoire Multidimensionnelle, Dunod, Paris, 1995. [2] B. Escofier, J. Pagès, Multiple factor analysis (AFMULT package), Computational Statistics & Data Analysis 18 (1994) 121–140. [3] M. Bécue-Bertaut, J. Pagès, Multiple factor analysis and clustering of a mixture of quantitative, categorical and frequency data, Computational Statistics & Data Analysis 52 (2008) 3255–3268. [4] L. Lebart, A. Morineau, K.M. Warwick, Multivariate Descriptive Statistical Analysis, Correspondence Analysis and Related Techniques for Large Matrices, Wiley, New York, 1984. [5] Y. Theodorou, C. Drossos, P. Alevizos, Correspondence analysis with fuzzy data, the fuzzy eigenvalue problem, Fuzzy Sets and Systems 158 (7) (2006) 704–721.
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[6] P. Loslever, P. Cloup, J. Delacroix, Mariage d’un système vidéo-informatique -SAGA3- et d’une méthode statistique exploratoire multidimensionnelle -ACM- pour l’étude de données gestuelles, I. Methodologie, ITBM 25 (2) (2004) 75–106. ´ M.J. Greenacre, Correspondence analysis in R, [7] O. Nenadic, with two- and three-dimensional graphics: the ca package, Journal of Statistical Software 20 (2007) 1–13. [8] F. Ambrogi, E. Biganzoli, P. Boracchi, Multiple correspondence analysis in S-PLUS, Computer Methods and Programs in Biomedicine 79 (2005) 161–167. [9] J.P. Benzécri, Cours de linguistique mathématique, Publication Mimeo, Faculté des sciences, Rennes, France, 1964. [10] J.P. Benzécri, L’Analyse Des Données: Analyse des correspondances & classification, Dunod, Paris, France, 1984. [11] J.P. Benzécri, et al., L’Analyse des Données Tome2: L’Analyse des Correspondances, Dunod, Paris, France, 1973. [12] L. Gacogne, Eléments de logique floue, Hermès, Paris, France, 1997. [13] B. Goldfarb, C. Pardoux, Etude de données multidimensionnelles évolutives et comparaison de codages par l’Analyse Factorielle Multiple, Revue de Statistique Appliquée 49 (1) (2001) 97–117. [14] B. Manly, Multivariate Statistical Methods: A Primer, vol. 6, second edition, Chapman & Hall, London, 1994. [15] A. Thompson, R. Randall-Maciver, The Ancient Races of the Thebaid, Oxford University Press, London, 1905. [16] P. Loslever, S. Bouilland, Multiple correspondence analysis of biomechanical signals characterized through fuzzy histograms, Journal of Biomechanics 7 (1998) 663–666. [17] J.L. Hodges, R.S. Crutchfield, D. Krech, STATlab: introduction empirique à la statistique, Economica, Paris, France, 1979. [18] B. Goldfarb, C.M. Pardoux, Exploring series of multivariate censored temporal data through fuzzy coding and correspondence analysis, Statistics in Medicine 25 (2004) 1741–1750.
journal homepage: www.intl.elsevierhealth.com/journals/cmpb
A tool developed in Matlab for multiple correspondence analysis of fuzzy coded data sets: Application to morphometric skull data Antonio Pinti ∗ , Fabienne Rambaud, Jean-Louis Griffon, Abdelmalik Taleb Ahmed LAMIH UMR CNRS 8530, Université de Valenciennes, Le Mont Houy, 59313 Valenciennes Cedex 9, France
a r t i c l e
i n f o
a b s t r a c t
Article history:
Multiple Correspondence factorial Analysis is a multivariate method for the exploratory
Received 4 December 2008
study of multidimensional contingency tables. Its use can be extended to the analysis of
Received in revised form
a table of fuzzy coded data resulting from a distribution into fuzzy windows defined by
9 September 2009
linguistic properties. There are few existing software tools that allow performing this type of
Accepted 16 September 2009
analysis on a data table; furthermore these tools are not interactive and do not allow defining and representing fuzzy windowing. This paper presents a software tool, developed with
Keywords:
Matlab, to compute and represent results from multiple correspondence factorial analyses.
Multiple correspondence analysis
Pre-defined membership functions can be selected by the user according to the distribution
Principal component analysis
histograms of the data.
Fuzzy sets
This paper presents an application example of this program onto a data table of mor-
Biplot
phometric parameters of 150 male skulls throughout 5 periods of Egyptian civilization. The
Skulls dataset
results are compared to those of a principal component analysis, which is more often used for the study of experimental data. Our program allows a rapid description of the morphological evolution of skulls over time, notably thanks to a linguistic description of each variable, whereas the results of the latter method are less obvious to observe and require a deeper analysis in order to arrive at the same conclusions. © 2009 Elsevier Ireland Ltd. All rights reserved.
1.
Introduction
Exploratory data analysis is principally concerned with the description of data sets observed mostly during experimentation, measured or recorded for a certain number of subjects – or statistical units – and with the establishment of relations between the different characteristics of these same subjects, without specifying any particular variable [1]. For this, the expert can rely on summaries, graphical representations as well as on multidimensional analysis methods; the choice of method depends on the type of data to study. Among the
∗
multidimensional exploratory methods, one can cite Principal Component Analysis (PCA), used for the study of quantitative data, or factorial Analysis of Multiple Correspondences (MCA), generally used for qualitative data [2,3]. These methods allow a global study to be performed thanks to graphical representations and a data analysis from several viewpoint [4]. They are placed in the framework of descriptive and data exploration methods, for which data are manipulated in order to be transformed into useful information [5]. Although multiple correspondence analysis is used for crisp data, it is possible to apply it on fuzzy coded data where each modality of a variable corresponds to a membership
Corresponding author. Tel.: +33 32 7511429; fax: +33 32 7511316. E-mail address: [email protected] (A. Pinti). 0169-2607/$ – see front matter © 2009 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.cmpb.2009.09.009
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value to a particular fuzzy window. According to Loslever et al. [6], this method allows to better account for complex relationships existing between the characteristics, to underscore the presence of classes as well as to rapidly show the presence of aberrant data. This analysis function is already present in some free software solutions such as R [7] or commercial ones such as XLStat or S-PLUS [8]. However these solutions do not offer the option to define cutting functions or to directly visualize the projection of results. In order to facilitate the work of data analysis, we have developed under Matlab a complete software including stages of fuzzy cutting of data and graphical representation of results. This software, dedicated to the fuzzy coding CA with crisp data, can be used just as any toolbox available in Matlab. The objective of this paper is to present this new software tool. Section 2 first describes the fuzzy cutting method, and then, the computation principles for the MCA of such fuzzy coded data. The Matlab implementation of the fuzzy cutting of data and the MCA computations as well as the description of the program and of the graphic interface are presented in Sections 3 and 4. Finally, the significance of this new software tool is illustrated in Section 5 by a real example from a scientific study in the context of anthropometric anatomy of the human skull from 4000 years BC to 150 years AD. Discussions and conclusions are in Section 5.
2. Multiple correspondence analysis of fuzzy coded data Multiple correspondence analysis was proposed by Benzécri in 1964 [9] for the study of contingency tables [10,11]. Multiple correspondence analysis (MCA) was subsequently generalized to describe large binary tables. Data studied by MCA are qualitative and describe modalities. However, it is possible to extend the use of this exploratory method to quantitative data previously coded into modalities. To do this, the use of binary coding in the domain of classical ensembles allows conversion of raw data to binary variables (0 or 1); nevertheless, the Boolean variable is ill adapted to the representation of most ordinary phenomena. But in the domain of fuzzy logic, the membership of a variable to a fuzzy ensemble (or set) will have a real value between 0 and 1 [12]. Moreover, the use of fuzzy coding for quantitative data allows minimization of information loss due to passing from the quantitative to the qualitative scale [13,6]. Processing of a table of raw data is performed in two steps:
1. Fuzzy windowing generating a table of membership values; 2. Multiple Correspondence factorial Analysis applied on the resulting table.
2.1.
Fuzzy coding
The fuzzy sub-sets are defined as follows: Let X be a reference ensemble and E a sub-set of X. E is a sub-set of X if its characteristic function associates to each
Fig. 1 – Principle of fuzzy cutting. Three spatial windows are considered with modalities {small; medium; large}.
element x of X a membership value fFE (x) between 0 and 1: fFE : X → [0, 1]
(1)
Fuzzy rules express the fact that the closer one is to a given situation, the more weight is given to the action recommended for this situation. These rules are represented by membership functions that can take varied forms depending on the nature of the object modeled. We will use linguistic variables to characterize fuzzy sub-sets. The membership values for triangular modalities are calculated in the following way: Let Mi be a triangular shaped fuzzy modality, defined by parameters a, b, c. The membership value of point P with coordinates x in modality M is:
Mi (P) = max min
x − a b−a
, 1,
c−x c−b
,0
(2)
Fig. 1 shows a fuzzy cutting in 3 spatial windows. For a given statistic, the output is a vector of 3 membership values corresponding to the 3 modalities of the variable (small, medium, and large). The sum of these membership values must Eq. (1).
Table 1 – A table of quantitative data: the skull data set. Epoch
X1 (mm)
X2 (mm)
X3 (mm)
X4 (mm)
−4000 −4000 −4000 ... −3300 −3300 −3300 ... −1850 −1850 −1850 ... −200 −200 −200 ... +150 +150 +150 ...
131 125 131 ... 133 138 148 ... 137 129 132 ... 137 141 141 ... 137 136 128 ...
138 131 132 ... 134 134 129 ... 141 133 138 ... 134 128 130 ... 123 131 126 ...
89 92 99 ... 97 98 104 ... 96 93 87 ... 107 95 87 ... 91 95 91 ...
49 48 50 ... 48 45 51 ... 52 47 48 ... 54 53 49 ... 50 49 57 ...
Min Max
119 148
120 145
81 114
44 60
68
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Fig. 2 – Basic triangular membership functions. Min corresponds to the minimum value of the variable’s data; Max corresponds to the maximum value; the Medium membership function depends on the middle of the segment [Min;Max].
Thus, the fuzzy windowing step consists in classifying the raw data table using membership functions between 0 and 1. Firstly, each system input (or variable) is modeled by curves giving the degrees of membership to the states defined by these inputs. For each variable we define membership functions that will allow the creation of columns corresponding to the membership values for each modality of the variable. The final table will be the concatenation of all modalities for each variable; all values from the original table are described by membership values corresponding to the number of modalities per variable (see Tables 1 and 2). The sum of each line of the table equals the number of variables. Respecting that condition, the table can include fuzzy coded modalities as well as binary coded ones (see Goldfarb’s work [13] for more emphasis on that particular point). We can distinguish two types of windowing: Fig. 4 – Program steps for MCA computation of fuzzy coded data developed under Matlab.
• Using membership functions that do not take into account value distributions (an example with 3 membership values is given in Fig. 2).
• Using membership functions that are adapted to the variables’ distributions (an example with 3 membership values is given in Fig. 3).
To our knowledge, there is no precise criterion allowing deciding if one of the methods is better than the other. Only an expert can efficiently judge which type of function should be used, for example by looking at the data distribution histogram. Depending on the shape of the histogram, the expert decides which cutting procedure is most appropriate.
Fig. 3 – Triangular membership functions. Min corresponds to the minimum value of the variable’s data; Max corresponds to the maximum value; the Medium membership function depends on the average value of the data.
Table 2 – Table 1 after fuzzy coding, the membership functions are presented in Fig. 2. Epoch
−4000 −4000 −4000 ... +150
X1
X2
X3
X4
smal
med
larg
smal
med
larg
smal
med
larg
smal
med
larg
0.17 0.59 0.17 ... 0.00
0.83 0.41 0.83 ... 0.83
0.00 0.00 0.00 ... 0.17
0.00 0.12 0.04 ... 0.00
0.56 0.88 0.96 ... 0.96
0.44 0.00 0.00 ... 0.04
0.51 0.33 0.00 ... 0.03
0.49 0.67 0.91 ... 0.97
0.00 0.00 0.09 ... 0.00
0.37 0.50 0.25 ... 0.13
0.63 0.50 0.75 ... 0.87
0.00 0.00 0.00 ... 0.00
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Fig. 5 – GUI for the selection of the membership functions. For each variable, the interface presents an histogram with an adjustable number of classes. This representation can help the user to choose one of the pre-defined fuzzy cutting functions in order to preserve a large amount of the distributional information. He can also preview, at any time, the repercussions on the MCA projections.
2.2.
Multiple correspondence analysis
Let Z be a fuzzy coded table with n individuals as rows and m modalities as columns. Let v be the variable number. Let m(v) be the number of modalities of variable v. Notice that
m=
v j=1
m(j)
and v =
m(i) v
ij
tions implied by projection. It searches for the most accurate representation of the cloud within the smallest number of dimensions [11]. For that, it is necessary to find a plane in which the cloud’s dispersion, characterized by its inertia, is maximal. In practice, we first construct the relative frequency table F.
(3)
i=1 j=1
with ij the membership value for the jth modality of variable i. From this set a factorial analysis can be performed. The objective of this method applied to fuzzy coded data is to facilitate the analysis by processing data described by linguistic properties. The objective is to represent as accurately as possible the clouds of data points that are formed by the rows and columns in their respective spaces (with p or n dimensions) in order to reveal structure contained in the information from the shape of these clouds. The search for a sub-space with a dimension smaller than that of the initial space while losing the least possible information, and the projection of the data of interest onto this sub-space allow obtaining a representation of the data while simultaneously reducing their dimension. Indeed, it is relatively easy to represent a cloud of data points when it is included in a space with 2 or even 3 dimensions; it is much less so when the cloud extends over spaces with dozens of dimensions. The main idea of factorial analysis is that a simple way to visualize the shape of a cloud of data points is to project it onto lines or even better, planes, while minimizing deforma-
F=
Z k
(4)
with k = sum of elements of Z. We then compute the row-margins Dn and columnmargins Dp , which are the diagonal matrices of the sums of respectively rows and columns of F. The Dp defines the metric of the subject space p , while the diagonal elements of Dn are the masses of the n point-rows. We call row-profile the result of the inner product D−1 n F and column-profile the result of D−1 p F. The distance d(i,i ) considered between two point-lines i and i is that of the 2 : d2 (i, i ) =
p 1 ij j=1
.j
i.
−
i j 2 i .
(5)
We then construct the matrix S to be diagonalized: −1 S = F D−1 n FDp
(6)
where F is the transpose of F. The eigenvalues ˛ of S are called “moments of inertia” of the cloud and satisfy the condition ˛ ∈ [0,1].
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Fig. 6 – Interface of the Multiple Correspondence Analysis of fuzzy coded data under Matlab.
The corresponding eigenvectors u˛ define the factorial axes in subject space. To obtain the factorial axes v˛ of variable space, one must compute: 1 v˛ = √ FD−1 p u˛ ˛
(7)
u˛ and v˛ must verify the normalization relations u˛ D−1 p u˛ = 1 v = 1. and v˛ D−1 ˛ n Finally, the factorial components ˛ and ϕ˛ , projections of row points and column points on the respective factorial axes are obtained by: −1 D−1 n FDp u˛
(8)
−1 ϕ˛ = D−1 p F Dn v˛
(9)
˛
=
In order to correctly interpret the projections of cloud points on factorial axes, one may define tools that offer supplementary information on the participation of a row point or a column point to the construction of the axes, the contribu-
tion, and on the quality of the representation of the point on a factorial sub-space, the cosine squared.
3. Implementation and description of the program developed under Matlab At the startup of the program developed under Matlab, the data can be loaded from a file with a table format. To help the analysis, each vector can be identified by a class. The program depends on three main functions:
• Fuzzy cutting from histograms (the number of classes can be modified). • Computation of the multiple correspondence analysis on fuzzy coded data. • Numerical representations of contribution tables and graphs of results on axes chosen by the user. Fig. 4 shows the diagram representing the steps of the program.
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Fig. 7 – Four measurements (in mm) performed on skulls from men having lived in the Thebes region of Egypt.
When applying this method, a large part of the time is allocated to the search for membership functions that best correspond to the variable distribution. Here, a graphic interface helps the user in his choice by plotting both the histogram for each variable and the cutting function that will be applied on them. Fig. 5 illustrates this step. A dozen of pre-defined membership functions are currently available. They differ by:
• The number of fuzzy windows (3, 5,. . .); • Their nature (triangular functions, sigmoid, etc.); • The way they adapt themselves to the distribution of the data (mean/median value, histogram’s modes, etc.).
Fig. 9 – MCA of fuzzy coded data, eigenvalues.
At any moment, a pre-visualization of the MCA is possible in order to best adjust the cutting. When the choice of membership functions seems satisfactory, the fuzzy coding can be applied and saved, and a more complete MCA can be started. A new window opens allowing the user to display all desired information (eigenvalues, eigenvectors, contributions, etc.), as well as projections on interesting factorial planes. The graphical representation of results is a figure that allows a visualization of the projections of individuals and variables along two principal axes. Fig. 6 shows that the tool associates a marker (color, pointers of various shapes and sizes) with each vector class to facilitate the data analysis.
Fig. 8 – Histogram and cuttings for variable X1. Up: a basic triangular membership functions, 3 membership values. Down: cutting windows, 5 binary modalities, partition of the [Min;Max] interval into 5 segments of equal size.
4.
Application example
In order to present an example of the use of the program that we created, we analyzed the Skulls database [14,15] that con-
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Table 3 – Table 1 as a contingency table with crisp data (decomposition into 5 modalities: VS: Very Small, S: Small, M: Medium, L: Large, VL: Very Large). Epoch
−4000 −4000 −4000 ... +150
X1
X2
X3
X4
VS
S
M
L
VL
VS
S
M
L
VL
VS
S
M
L
VL
VS
S
M
L
VL
0 0 0
0 1 0
1 0 1
0 0 0
0 0 0
0 0 0
0 0 0
0 1 1
1 0 0
0 0 0
0 0 0
1 1 0
0 0 1
0 0 0
0 0 0
0 0 0
1 1 1
0 0 0
0 0 0
0 0 0
0
0
0
1
0
1
0
0
0
0
0
1
0
0
0
0
1
0
0
0
Fig. 10 – MCA of fuzzy coded data, projections on axes 1 and 2 (resp. abscissa and ordinates).
Fig. 11 – MCA of fuzzy coded data, projections on axes 3 and 4 (resp. abscissa and ordinates).
tains morphometric data (4 variables, Fig. 7) for 150 male skulls from 5 periods of Egyptian civilization:
• 12th and 13th dynasty (1850 BC); • Ptolemaeus period (200 BC); • Roman period (150 AD).
• Ancient pre-dynastic period (4000 BC); • Recent pre-dynastic period (3300 BC);
The data are presented in Table 1.
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Inspection of the histograms for each variable reveals the Gaussian shape of the distributions. A fuzzy cutting into 3 categories of same size (Small S, Medium M, and Large L) is thus chosen (see Fig. 8) [16]. Then, MCA is performed on the fuzzy coded table. The first 4 factorial components represent 68.32% of the total inertia (Fig. 9), therefore the study will take into account the first 4 factorial axes. By studying the variables’ contributions to the axes as well as the cosine squared, one notes that the variable X1 is particularly well represented on axis 1, variables X1 and X4 on axis 2, variable X1 on axis 3 and variable X3 on axis 4. No result is obvious from the study of axes 3 and 4, but the plane formed by axes 1 and 2 is more interesting. Periods are grouped along axis 2, with more ancient skulls having smaller X1 and X4 values, whereas subjects closer to contemporary times have larger values. Axis 1 shows that values of X2, and to a lesser extent of X3, tend to decrease with time. On the whole, groups describe a clockwise movement with the pivot localized at the X4 Large modality. Figs. 10 and 11 illustrate these results. The program allowed us to perform a first quick analysis of a table of 150 × 4 data points using a method that usually requires a non-negligible computer program processing time [6]. A binary coding is also performed in order to compare results with those given by fuzzy coding. 5 binary modalities {VS, S, M, L, VL} are created for each continuous variable according to an equal division of the [Min; Max] interval. Then, a MCA is applied on the table given by the concatenation of all modalities (see Table 3). The goal of the study is to look for one or several evolutionary modes of morphometric data through epochs. Binary coding still facilitates the analysis with the use of linguistic properties, but here, the loss of information in comparison with fuzzy coding is obvious: this surjection associates 4 elev ments with vectors of a finite set of m(j) elements. This j=1 is why 5 modalities per variable were created instead of 3 to avoid too much loss. This also raises an other defect of binary coding: the more modalities we create, the more inertia will be spread along the factorial axes and the less projections on principal axes will be meaningful. In our case, MCA fails to extract remarkable axes of projection. The two principal axes of the MCA represent only 10% of the total inertia each (Fig. 12).
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Fig. 12 – MCA of binary coded data, eigenvalues. No axis predominates.
Thus, the first projection plane describes few informations and an investigation through the other axes becomes mandatory. Nevertheless a first tendency can be obtained thanks to projections on principals axes 1 and 2 (Fig. 13). X1 is well represented on axis 2, and variables X2 and X3 on axis 1. Groups can be formed by epochs: as previously, we can observe that X1 value tends to increase with time, but it is barely the only drawable conclusion at first sight. A normalized PCA was applied to the table via the statistical software StatLab [17,1] to compare methods and results. No axis predominates: each of the first 2 axes accounts for 30% of the total inertia, the 2 others for 20%. Inspection of the variables projected on the factorial axes 1 and 2 brings out the strong correlations of variables X2 and X3 with axis 1 and that of X4 with axis 2 (Fig. 14). Projections of individuals on the first two factorial axes reveal results that are less obviously interpretable than those obtained with MCA. Groupings by epochs are present, but are less easily dissociated than with the former method. Furthermore, the time evolution of the morphometric data is less evident. One can however come to the same conclu-
Fig. 13 – MCA of binary coded data, projections on axes 1 (abscissa) and 2 (ordinate).
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Fig. 14 – Normalized PCA. Projections on axes 1 (abscissa) and 2 (ordinate). sions by inspection of the groups: X1 and X4 have a tendency to increase while X2 and X3 decrease. As with the former method, no probing result comes from inspection of axes 3 and 4. The information obtained by the PCA method is the same as that obtained by the developed MCA method, but it is extracted with more difficulty.
5.
Conclusion
The objective of this paper was to present a new complete statistical tool for a method of multidimensional exploration of quantitative data, that is user-friendly and interactive. This analysis method consists in a multiple correspondence factorial analysis, more commonly called MCA, of quantitative data that has been previously fuzzy coded. The method is not well known, probably because it is not directly implemented in current statistical software, and must be performed ‘manually’ with several processing steps and representations. Because of this, it can appear repulsive if the table to be analyzed has a large number of variables. The tool developed for MCA presented in this paper proposes a computerized solution to this analysis and allows a gain in time of data analysis. To show the relevance of this tool, it was used on a morphometric database of human skulls in time. The results of this analysis were compared to those obtained by the principal component analysis (PCA) method performed in StatLab,
which is the method used for multidimensional exploration of numerical data tables. The multiple correspondence analysis on fuzzy coded tables holds interesting properties. By distinguishing the values of a variable by linguistic properties, the analysis is facilitated without loss of the real meaning of the data, meaning that can disappear in normalized principal component analysis because of centering and reduction. The fuzzy coding allows conserving a great amount of information contained in the data distribution [18]. This assertion was verified here by comparing results with those given with a simple binary coding. The graphical results allow, by the inspection of so-called extreme values, to more easily detect groups of individuals and variables by the fuzzy coding MCA method. This new tool is still not complete and it would be of interest to propose a greater choice of cuttings. Also, it should be possible for the users to choose, select and modify the membership functions directly with the mouse (if the histogram has several modes for instance) so that they may interactively visualize the effects of the cuttings on the global analysis.
Software availability The software is available on request free of cost from the corresponding author ([email protected]).
c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 9 8 ( 2 0 1 0 ) 66–75
Conflict of interest None.
Acknowledgments The present research work has been supported by International Campus on Safety and Intermodality in Transportation. The authors gratefully acknowledge the support of these institutions: the Nord-Pas-de-Calais Region, the European Community, the Regional Delegation for Research and Technology, the Ministry of Higher Education and Research, and the National Center for Scientific Research.
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