Presentation & Motivation Matrix associated with a symmetric graph Case of actual data: number of employees in economic sectors Visual Displays. Some evidence through artificial and real data K. Fern´ andez-Aguirre M.A. Gar´ ın-Mart´ ın J. I. Modro˜ no-Herr´ an karmele.fernandez@ehu.es University of the Basque Country (UPV/EHU), Bilbao, Spain CARME 2011, Rennes, February 8-11 K. Fern´ andez-Aguirre, M.A. Gar´ ın-Mart´ ın, J.I. Modro˜ no Visual Displays
Presentation & Motivation Matrix associated with a symmetric graph Case of actual data: number of employees in economic sectors Contents Presentation & Motivation 1 Matrix associated with a symmetric graph 2 Analytic study Experimental study Case of actual data: number of employees in economic sectors 3 Matrix of quantitative variables: PCA Contingency or frequency table: two-way CA Clustering K. Fern´ andez-Aguirre, M.A. Gar´ ın-Mart´ ın, J.I. Modro˜ no Visual Displays
Presentation & Motivation Matrix associated with a symmetric graph Case of actual data: number of employees in economic sectors Presentation Displaying and exploring data Principal Component Analysis (PCA) (quantitative variables) and simple or multiple Correspondence Analysis (CA) (categorical variables) are useful for the identification of structures in the data through interesting graphical visualizations However, some kinds of data sets could be treated alternatively by PCA or CA. For both methods, the clustering would be complementary in the exploration of the data These methods are applied in almost all areas of knowledge where predilection for each of them is variable K. Fern´ andez-Aguirre, M.A. Gar´ ın-Mart´ ın, J.I. Modro˜ no Visual Displays
Presentation & Motivation Matrix associated with a symmetric graph Case of actual data: number of employees in economic sectors Motivation I Displaying and exploring data In certain areas in particular, it is still frequent the treatment of categorical variables as if they were continuous, due to the great influence of the classic school, see Gifi (1990) A data matrix that contains the number of employees in different economic sectors for the countries of the European Union could be treated alternatively by PCA or two-way CA There are another examples (data from surveys, textual data...) that could be treated alternatively by two or more methods Different results depending on the characteristics and the properties of each method. K. Fern´ andez-Aguirre, M.A. Gar´ ın-Mart´ ın, J.I. Modro˜ no Visual Displays
Presentation & Motivation Matrix associated with a symmetric graph Case of actual data: number of employees in economic sectors Motivation II Displaying and exploring data Our emphasis in the following discussion is on methods, such as PCA and CA, and visual displays.This paper has two parts In the first part, we analitically study the case of a binary matrix M associated to a symmetric graph G (Octagon), also valid for the cases of high dimensionality graphs. Lebart et al. (1998) shows the case of a Chessboard (square lattice grid) In the second part, we present a case of actual data on the distribution of employees in different economic sectors for the countries of the European Union K. Fern´ andez-Aguirre, M.A. Gar´ ın-Mart´ ın, J.I. Modro˜ no Visual Displays
Presentation & Motivation Analytic study Matrix associated with a symmetric graph Experimental study Case of actual data: number of employees in economic sectors Symmetric graphs Symetric undirected graphs with a central vertex We analytically study the Octagonal graph and get a conclusion which is also valuable for the Dodecagonal and the Hexadecagonal graphs And show the superiority of CA for the reconstitution and visualization of a M matrix associated with a G symmetric graph over the visualization obtained with PCA Octagonal shaped graph 1 Dodecagonal shaped graph 2 Hexadecagonal shaped graph 3 K. Fern´ andez-Aguirre, M.A. Gar´ ın-Mart´ ın, J.I. Modro˜ no Visual Displays
Presentation & Motivation Analytic study Matrix associated with a symmetric graph Experimental study Case of actual data: number of employees in economic sectors Octagonal undirected graph ( m ext = 8 + m int = 5) vertices and 23 edges 1 3 4 2 1 8 5 6 7 9 11 12 10 13 K. Fern´ andez-Aguirre, M.A. Gar´ ın-Mart´ ın, J.I. Modro˜ no Visual Displays
Presentation & Motivation Analytic study Matrix associated with a symmetric graph Experimental study Case of actual data: number of employees in economic sectors Octagonal undirected graph Numerical coding of the graph Two vertices are adjacent if there is an edge joining them We consider each vertex as adjacent to itself The associated M binary matrix contains the value 1 in position ( i , j ) if vertices i and j are adjacent and 0 otherwise Since the graph is undirected each pair of adjacent vertices appears twice K. Fern´ andez-Aguirre, M.A. Gar´ ın-Mart´ ın, J.I. Modro˜ no Visual Displays
Presentation & Motivation Analytic study Matrix associated with a symmetric graph Experimental study Case of actual data: number of employees in economic sectors Octagonal graph Associated M binary ( m ext + m int , m ext + m int )=(13, 13) matrix 1 3 4 2 1 8 5 6 7 9 11 12 10 13 K. Fern´ andez-Aguirre, M.A. Gar´ ın-Mart´ ın, J.I. Modro˜ no Visual Displays
Presentation & Motivation Analytic study Matrix associated with a symmetric graph Experimental study Case of actual data: number of employees in economic sectors CA of ( m ext + m int , m ext + m int )=(13, 13) M binary matrix Eigenvalues and eigenvectors: ( MN − 1 ) 2 u s = λ s u s where N is a same order diagonal matrix as M with n ii (adjacency degree) as diagonal elements and N − 1 M is the row or column profile matrix tr( N − 1 M ) 2 = m ext + m int + 1 � = λ s 5 s ≥ 1 In the analysis relative to the center of gravity: λ s = m ext + m int − 4 � 5 s > 1 K. Fern´ andez-Aguirre, M.A. Gar´ ın-Mart´ ın, J.I. Modro˜ no Visual Displays
Presentation & Motivation Analytic study Matrix associated with a symmetric graph Experimental study Case of actual data: number of employees in economic sectors Octagonal graph: CA of ( m ext + m int , m ext + m int ) binary matrix The inertia rate explained by the s -th axis: 5 λ s τ s = = m ext + m int − 4 λ s , 0 < λ s < 1 � λ s s > 1 Conclusion: The inertia explained by the subspace that approximates the initial structure can be made as small as needed, simply by increasing the number of vertices of the graph K. Fern´ andez-Aguirre, M.A. Gar´ ın-Mart´ ın, J.I. Modro˜ no Visual Displays
Presentation & Motivation Analytic study Matrix associated with a symmetric graph Experimental study Case of actual data: number of employees in economic sectors Octagonal undirected graph ( m ext = 8 + m int = 5) vertices and 23 edges 1 3 4 2 1 8 5 6 7 9 11 12 10 13 K. Fern´ andez-Aguirre, M.A. Gar´ ın-Mart´ ın, J.I. Modro˜ no Visual Displays
Presentation & Motivation Analytic study Matrix associated with a symmetric graph Experimental study Case of actual data: number of employees in economic sectors Octagonal graph Reconstitution and visualization: PCA versus CA Figure: PCA Figure: CA K. Fern´ andez-Aguirre, M.A. Gar´ ın-Mart´ ın, J.I. Modro˜ no Visual Displays
Presentation & Motivation Analytic study Matrix associated with a symmetric graph Experimental study Case of actual data: number of employees in economic sectors Dodecagonal undirected graph edges and vertices 1 2 3 4 5 6 7 8 9 12 13 14 15 16 10 11 17 18 19 20 21 22 23 24 25 K. Fern´ andez-Aguirre, M.A. Gar´ ın-Mart´ ın, J.I. Modro˜ no Visual Displays
Presentation & Motivation Analytic study Matrix associated with a symmetric graph Experimental study Case of actual data: number of employees in economic sectors Dodecagonal graph Reconstitution and visualization: PCA versus CA Figure: PCA Figure: CA K. Fern´ andez-Aguirre, M.A. Gar´ ın-Mart´ ın, J.I. Modro˜ no Visual Displays
Presentation & Motivation Analytic study Matrix associated with a symmetric graph Experimental study Case of actual data: number of employees in economic sectors Hexadecagonal undirected graph edges and vertices 1 3 4 2 8 9 5 6 7 12 10 11 13 14 15 16 17 19 25 18 20 22 21 23 24 27 28 29 30 31 32 26 33 34 36 37 35 38 39 40 41 K. Fern´ andez-Aguirre, M.A. Gar´ ın-Mart´ ın, J.I. Modro˜ no Visual Displays
Presentation & Motivation Analytic study Matrix associated with a symmetric graph Experimental study Case of actual data: number of employees in economic sectors Hexadecagonal graph Reconstitution and visualization: PCA versus CA Figure: PCA Figure: CA K. Fern´ andez-Aguirre, M.A. Gar´ ın-Mart´ ın, J.I. Modro˜ no Visual Displays
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