Differential geometric properties of textile plot

28/10/2015
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14271

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The textile plot proposed by Kumasaka and Shibata (2008) is a method for data visualization. The method transforms a data matrix in order to draw a parallel coordinate plot. In this paper, we investigate a set of matrices induced by the textile plot, which we call the textile set, from a geometrical viewpoint. It is shown that the textile set is written as the union of two differentiable manifolds if data matrices are restricted to be full-rank.

Differential geometric properties of textile plot

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application/pdf Differential geometric properties of textile plot Tomonari Sei, Ushio Tanaka

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The textile plot proposed by Kumasaka and Shibata (2008) is a method for data visualization. The method transforms a data matrix in order to draw a parallel coordinate plot. In this paper, we investigate a set of matrices induced by the textile plot, which we call the textile set, from a geometrical viewpoint. It is shown that the textile set is written as the union of two differentiable manifolds if data matrices are restricted to be full-rank.

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What is textile plot? Textile set Main result Other results Summary Geometric Properties of textile plot Tomonari SEI and Ushio TANAKA University of Tokyo and Osaka Prefecture University at ´Ecole Polytechnique, Oct 28, 2015 1 / 23 What is textile plot? Textile set Main result Other results Summary Introduction The textile plot proposed by Kumasaka and Shibata (2008) is a method for data visualization. The method transforms a data matrix into another matrix, Rn×p X → Y ∈ Rn×p , in order to draw a parallel coordinate plot. The parallel coordinate plot is a standard 2-dimensional graphical tool for visualizing multivariate data at a glance. In this talk, we investigate a set of matrices induced by the textile plot, which we call the textile set, from a differential geometrical point of view. It is shown that the textile set is written as the union of two differentiable manifolds if data matrices are “generic”. 2 / 23 What is textile plot? Textile set Main result Other results Summary Introduction The textile plot proposed by Kumasaka and Shibata (2008) is a method for data visualization. The method transforms a data matrix into another matrix, Rn×p X → Y ∈ Rn×p , in order to draw a parallel coordinate plot. The parallel coordinate plot is a standard 2-dimensional graphical tool for visualizing multivariate data at a glance. In this talk, we investigate a set of matrices induced by the textile plot, which we call the textile set, from a differential geometrical point of view. It is shown that the textile set is written as the union of two differentiable manifolds if data matrices are “generic”. 2 / 23 What is textile plot? Textile set Main result Other results Summary 1 What is textile plot? 2 Textile set 3 Main result 4 Other results 5 Summary 3 / 23 What is textile plot? Textile set Main result Other results Summary Textile plot Example (Kumasaka and Shibata, 2008) Textile plot for the iris data. (150 cases, 5 attributes) Each variate is transformed by a location-scale transformation. Categorical data is quantified. Missing data is admitted. Order of axes can be maintained. Specie s Sepal.Length Sepal.W id th Petal.Length Petal.W id th setosa versicolor virginica 4.3 7.9 2 4.4 1 6.9 0.1 2.5 4 / 23 What is textile plot? Textile set Main result Other results Summary Textile plot Example (Kumasaka and Shibata, 2008) Textile plot for the iris data. (150 cases, 5 attributes) Each variate is transformed by a location-scale transformation. Categorical data is quantified. Missing data is admitted. Order of axes can be maintained. Specie s Sepal.Length Sepal.W id th Petal.Length Petal.W id th setosa versicolor virginica 4.3 7.9 2 4.4 1 6.9 0.1 2.5 4 / 23 What is textile plot? Textile set Main result Other results Summary Textile plot Let us recall the method of the textile plot. For simplicity, we assume no categorical variate and no missing value. Let X = (x1, . . . , xp) ∈ Rn×p be the data matrix. Without loss of generality, assume the sample mean and sample variance of each xj are 0 and 1, respectively. The data is transformed into Y = (y1, . . . , yp), where yj = aj + bj xj , aj , bj ∈ R, j = 1, . . . , p. The coefficients aj and bj are determined by the following procedure. 5 / 23 What is textile plot? Textile set Main result Other results Summary Textile plot Let us recall the method of the textile plot. For simplicity, we assume no categorical variate and no missing value. Let X = (x1, . . . , xp) ∈ Rn×p be the data matrix. Without loss of generality, assume the sample mean and sample variance of each xj are 0 and 1, respectively. The data is transformed into Y = (y1, . . . , yp), where yj = aj + bj xj , aj , bj ∈ R, j = 1, . . . , p. The coefficients aj and bj are determined by the following procedure. 5 / 23 What is textile plot? Textile set Main result Other results Summary Textile plot Let us recall the method of the textile plot. For simplicity, we assume no categorical variate and no missing value. Let X = (x1, . . . , xp) ∈ Rn×p be the data matrix. Without loss of generality, assume the sample mean and sample variance of each xj are 0 and 1, respectively. The data is transformed into Y = (y1, . . . , yp), where yj = aj + bj xj , aj , bj ∈ R, j = 1, . . . , p. The coefficients aj and bj are determined by the following procedure. 5 / 23 What is textile plot? Textile set Main result Other results Summary Textile plot Coefficients a = (aj ) and b = (bj ) are the solution of the following minimization problem: Minimize a,b n∑ t=1 p∑ j=1 (ytj − ¯yt·)2 subject to yj = aj + bj xj , p∑ j=1 yj 2 = 1. Intuition: as horizontal as possible. Solution: a = 0 and b is the eigenvector corresponding to the maximum eigenvalue of the covariance matrix of X. yt1 yt2 yt3 yt4 yt5 yt. 6 / 23 What is textile plot? Textile set Main result Other results Summary Example (n = 100, p = 4) X ∈ R100×4. Each row ∼ N(0, Σ), Σ =   1 −0.6 0.5 0.1 −0.6 1 −0.6 −0.2 0.5 −0.6 1 0.0 0.1 −0.2 0.0 1  . −2.71 2.98 −3.93 3.27 −2.72 2.43 −2.58 2.23 −2.71 2.98 −3.93 3.27 −2.72 2.43 −2.58 2.23 (a) raw data X (b) textile plot Y 7 / 23 What is textile plot? Textile set Main result Other results Summary Our motivation The textile plot transforms the data matrix X into Y. Denote the map by Y = τ(X). What is the image τ(Rn×p)? We can show that Y ∈ τ(Rn×p) satisfies two conditions: ∃λ ≥ 0, ∀i = 1, . . . , p, p∑ j=1 yi yj = λ yi 2 and p∑ j=1 yj 2 = 1. This motivates the following definition of the textile set. 8 / 23 What is textile plot? Textile set Main result Other results Summary Our motivation The textile plot transforms the data matrix X into Y. Denote the map by Y = τ(X). What is the image τ(Rn×p)? We can show that Y ∈ τ(Rn×p) satisfies two conditions: ∃λ ≥ 0, ∀i = 1, . . . , p, p∑ j=1 yi yj = λ yi 2 and p∑ j=1 yj 2 = 1. This motivates the following definition of the textile set. 8 / 23 What is textile plot? Textile set Main result Other results Summary Our motivation The textile plot transforms the data matrix X into Y. Denote the map by Y = τ(X). What is the image τ(Rn×p)? We can show that Y ∈ τ(Rn×p) satisfies two conditions: ∃λ ≥ 0, ∀i = 1, . . . , p, p∑ j=1 yi yj = λ yi 2 and p∑ j=1 yj 2 = 1. This motivates the following definition of the textile set. 8 / 23 What is textile plot? Textile set Main result Other results Summary Textile set Definition The textile set is defined by Tn,p = { Y ∈ Rn×p | ∃λ ≥ 0, ∀i, ∑ j yi yj = λ yi 2 , ∑ j yj 2 = 1 }, The unnormalized textile set is defined by Un,p = { Y ∈ Rn×p | ∃λ ≥ 0, ∀i, ∑ j yi yj = λ yi 2 }. We are interested in mathematical properties of Tn,p and Un,p. Bad news: statistical implication such is a future work. Let us begin with small p case. 9 / 23 What is textile plot? Textile set Main result Other results Summary Textile set Definition The textile set is defined by Tn,p = { Y ∈ Rn×p | ∃λ ≥ 0, ∀i, ∑ j yi yj = λ yi 2 , ∑ j yj 2 = 1 }, The unnormalized textile set is defined by Un,p = { Y ∈ Rn×p | ∃λ ≥ 0, ∀i, ∑ j yi yj = λ yi 2 }. We are interested in mathematical properties of Tn,p and Un,p. Bad news: statistical implication such is a future work. Let us begin with small p case. 9 / 23 What is textile plot? Textile set Main result Other results Summary Textile set Definition The textile set is defined by Tn,p = { Y ∈ Rn×p | ∃λ ≥ 0, ∀i, ∑ j yi yj = λ yi 2 , ∑ j yj 2 = 1 }, The unnormalized textile set is defined by Un,p = { Y ∈ Rn×p | ∃λ ≥ 0, ∀i, ∑ j yi yj = λ yi 2 }. We are interested in mathematical properties of Tn,p and Un,p. Bad news: statistical implication such is a future work. Let us begin with small p case. 9 / 23 What is textile plot? Textile set Main result Other results Summary Textile set Definition The textile set is defined by Tn,p = { Y ∈ Rn×p | ∃λ ≥ 0, ∀i, ∑ j yi yj = λ yi 2 , ∑ j yj 2 = 1 }, The unnormalized textile set is defined by Un,p = { Y ∈ Rn×p | ∃λ ≥ 0, ∀i, ∑ j yi yj = λ yi 2 }. We are interested in mathematical properties of Tn,p and Un,p. Bad news: statistical implication such is a future work. Let us begin with small p case. 9 / 23 What is textile plot? Textile set Main result Other results Summary Textile set Definition The textile set is defined by Tn,p = { Y ∈ Rn×p | ∃λ ≥ 0, ∀i, ∑ j yi yj = λ yi 2 , ∑ j yj 2 = 1 }, The unnormalized textile set is defined by Un,p = { Y ∈ Rn×p | ∃λ ≥ 0, ∀i, ∑ j yi yj = λ yi 2 }. We are interested in mathematical properties of Tn,p and Un,p. Bad news: statistical implication such is a future work. Let us begin with small p case. 9 / 23 What is textile plot? Textile set Main result Other results Summary Tn,p with small p Lemma (p = 1) Tn,1 = Sn−1, the unit sphere. Lemma (p = 2) Tn,2 = A ∪ B, where A = {(y1, y2) | y1 = y2 = 1/ √ 2}, B = {(y1, y2) | y1 − y2 = y1 + y2 = 1}, each of which is diffeomorphic to Sn−1 × Sn−1. Their intersection A ∩ B is diffeomorphic to the Stiefel manifold Vn,2. → See next slide for n = p = 2 case. 10 / 23 What is textile plot? Textile set Main result Other results Summary Tn,p with small p Lemma (p = 1) Tn,1 = Sn−1, the unit sphere. Lemma (p = 2) Tn,2 = A ∪ B, where A = {(y1, y2) | y1 = y2 = 1/ √ 2}, B = {(y1, y2) | y1 − y2 = y1 + y2 = 1}, each of which is diffeomorphic to Sn−1 × Sn−1. Their intersection A ∩ B is diffeomorphic to the Stiefel manifold Vn,2. → See next slide for n = p = 2 case. 10 / 23 What is textile plot? Textile set Main result Other results Summary Example (n = p = 2) T2,2 ⊂ R4 is the union of two tori, glued along O(2). θ φ ξ η T2,2 = { 1 √ 2 ( cos θ cos φ sin θ sin φ )} ∪ { 1 2 ( cos ξ + cos η cos ξ − cos η sin ξ + sin η sin ξ − sin η )} 11 / 23 What is textile plot? Textile set Main result Other results Summary For general dimension p To state our main result, we define two concepts: noncompact Stiefel manifold and canonical form. Definition (e.g. Absil et al. (2008)) Let n ≥ p. Denote by V ∗ the set of all column full-rank matrices: V ∗ := { Y ∈ Rn×p | rank(Y) = p }. V ∗ is called the noncompact Stiefel manifold. Note that dim(V ∗) = np and V ∗ = Rn×p. The orthogonal group O(n) acts on V ∗. By the Gram-Schmidt orthonormalization, the quotient space V ∗/O(n) is identified with upper-triangular matrices with positive diagonals. → see next slide. 12 / 23 What is textile plot? Textile set Main result Other results Summary For general dimension p To state our main result, we define two concepts: noncompact Stiefel manifold and canonical form. Definition (e.g. Absil et al. (2008)) Let n ≥ p. Denote by V ∗ the set of all column full-rank matrices: V ∗ := { Y ∈ Rn×p | rank(Y) = p }. V ∗ is called the noncompact Stiefel manifold. Note that dim(V ∗) = np and V ∗ = Rn×p. The orthogonal group O(n) acts on V ∗. By the Gram-Schmidt orthonormalization, the quotient space V ∗/O(n) is identified with upper-triangular matrices with positive diagonals. → see next slide. 12 / 23 What is textile plot? Textile set Main result Other results Summary For general dimension p To state our main result, we define two concepts: noncompact Stiefel manifold and canonical form. Definition (e.g. Absil et al. (2008)) Let n ≥ p. Denote by V ∗ the set of all column full-rank matrices: V ∗ := { Y ∈ Rn×p | rank(Y) = p }. V ∗ is called the noncompact Stiefel manifold. Note that dim(V ∗) = np and V ∗ = Rn×p. The orthogonal group O(n) acts on V ∗. By the Gram-Schmidt orthonormalization, the quotient space V ∗/O(n) is identified with upper-triangular matrices with positive diagonals. → see next slide. 12 / 23 What is textile plot? Textile set Main result Other results Summary Noncompact Stiefel manifold and canonical form Definition (Canonical form) Let us denote by V ∗∗ the set of all matrices written as            y11 · · · y1p 0 ... ... ... ... ypp 0 · · · 0 ... ... 0 · · · 0            , yii > 0, 1 ≤ i ≤ p. We call it a canonical form. Note that V ∗∗ ⊂ V ∗ and V ∗/O(n) V ∗∗. 13 / 23 What is textile plot? Textile set Main result Other results Summary Noncompact Stiefel manifold and canonical form Definition (Canonical form) Let us denote by V ∗∗ the set of all matrices written as            y11 · · · y1p 0 ... ... ... ... ypp 0 · · · 0 ... ... 0 · · · 0            , yii > 0, 1 ≤ i ≤ p. We call it a canonical form. Note that V ∗∗ ⊂ V ∗ and V ∗/O(n) V ∗∗. 13 / 23 What is textile plot? Textile set Main result Other results Summary Restriction of unnormalized textile set V ∗: non-compact Stiefel manifold, V ∗∗: set of canonical forms. Definition Denote the restriction of Un,p to V ∗ and V ∗∗ by U∗ n,p = Un,p ∩ V ∗ , U∗∗ n,p = Un,p ∩ V ∗∗ , respectively. The group O(n) acts on U∗ n,p. The quotient space U∗ n,p/O(n) is identified with U∗∗ n,p. So it is essential to study U∗∗ n,p. 14 / 23 What is textile plot? Textile set Main result Other results Summary Restriction of unnormalized textile set V ∗: non-compact Stiefel manifold, V ∗∗: set of canonical forms. Definition Denote the restriction of Un,p to V ∗ and V ∗∗ by U∗ n,p = Un,p ∩ V ∗ , U∗∗ n,p = Un,p ∩ V ∗∗ , respectively. The group O(n) acts on U∗ n,p. The quotient space U∗ n,p/O(n) is identified with U∗∗ n,p. So it is essential to study U∗∗ n,p. 14 / 23 What is textile plot? Textile set Main result Other results Summary Restriction of unnormalized textile set V ∗: non-compact Stiefel manifold, V ∗∗: set of canonical forms. Definition Denote the restriction of Un,p to V ∗ and V ∗∗ by U∗ n,p = Un,p ∩ V ∗ , U∗∗ n,p = Un,p ∩ V ∗∗ , respectively. The group O(n) acts on U∗ n,p. The quotient space U∗ n,p/O(n) is identified with U∗∗ n,p. So it is essential to study U∗∗ n,p. 14 / 23 What is textile plot? Textile set Main result Other results Summary U∗∗ n,p for small p Let us check examples. Example (n = p = 1) U∗∗ 1,1 = {(1)}. Example (n = p = 2) Let Y = ( y11 y12 0 y22 ) with y11, y22 > 0. Then U∗∗ 2,2 = {y12 = 0} ∪ {y2 11 = y2 12 + y2 22}, union of a plane and a cone. 15 / 23 What is textile plot? Textile set Main result Other results Summary U∗∗ n,p for small p Let us check examples. Example (n = p = 1) U∗∗ 1,1 = {(1)}. Example (n = p = 2) Let Y = ( y11 y12 0 y22 ) with y11, y22 > 0. Then U∗∗ 2,2 = {y12 = 0} ∪ {y2 11 = y2 12 + y2 22}, union of a plane and a cone. 15 / 23 What is textile plot? Textile set Main result Other results Summary Main theorem The differential geometrical property of U∗∗ n,p is given as follows: Theorem Let n ≥ p ≥ 3. Then we have the following decomposition U∗∗ n,p = M1 ∪ M2, where each Mi is a differentiable manifold, the dimensions of which are given by dim M1 = p(p + 1) 2 − (p − 1), dim M2 = p(p + 1) 2 − p, respectively. M2 is connected while M1 may not. 16 / 23 What is textile plot? Textile set Main result Other results Summary Example U∗∗ 3,3 is the union of 4-dim and 3-dim manifolds. We look at a cross section with y11 = y22 = 1: y12 y13 y33 Union of a surface and a vertical line. 17 / 23 What is textile plot? Textile set Main result Other results Summary Corollary Let n ≥ p ≥ 3. Then we have U∗ n,p = π−1 (M1) ∪ π−1 (M2), where π denotes the map of Gram-Schmidt orthonormalization. The dimensions are dim π−1 (M1) = np − (p − 1), dim π−1 (M2) = np − p. 18 / 23 What is textile plot? Textile set Main result Other results Summary Other results We state other results. First we have n = 1 case. Lemma If n = 1, then the textile set T1,p is the union of a (p − 2)-dimensional manifold and 2(2p − 1) isolated points. Example U∗∗ 1,3 consists of a circle and 14 points: U∗∗ 1,3 = (S2 ∩ {y1 + y2 + y3 = 1}) ∪ {±( 1√ 3 , 1√ 3 , 1√ 3 ), ±( 1√ 2 , 1√ 2 , 0), ±( 1√ 2 , 0, 1√ 2 ), ±(0, 1√ 2 , 1√ 2 ), ± (1, 0, 0), ±(0, 1, 0), ±(0, 0, 1)} . 19 / 23 What is textile plot? Textile set Main result Other results Summary Other results We state other results. First we have n = 1 case. Lemma If n = 1, then the textile set T1,p is the union of a (p − 2)-dimensional manifold and 2(2p − 1) isolated points. Example U∗∗ 1,3 consists of a circle and 14 points: U∗∗ 1,3 = (S2 ∩ {y1 + y2 + y3 = 1}) ∪ {±( 1√ 3 , 1√ 3 , 1√ 3 ), ±( 1√ 2 , 1√ 2 , 0), ±( 1√ 2 , 0, 1√ 2 ), ±(0, 1√ 2 , 1√ 2 ), ± (1, 0, 0), ±(0, 1, 0), ±(0, 0, 1)} . 19 / 23 What is textile plot? Textile set Main result Other results Summary Differential geometrical characterization of fλ −1 (O) Fix λ ≥ 0 arbitrarily. We define the map fλ : Rn×p → Rp+1 by fλ(y1, . . . , yp) :=       ∑ j y1 yj − λ y1 2 ... ∑ j yp yj − λ yp 2 ∑ j yj 2 − 1       . Lemma We have a classification of Tn,p, namely Tn,p = λ≥0 fλ −1 (O) = 0≤λ≤n fλ −1 (O). 20 / 23 What is textile plot? Textile set Main result Other results Summary Differential geometrical characterization of fλ −1 (O) Fix λ ≥ 0 arbitrarily. We define the map fλ : Rn×p → Rp+1 by fλ(y1, . . . , yp) :=       ∑ j y1 yj − λ y1 2 ... ∑ j yp yj − λ yp 2 ∑ j yj 2 − 1       . Lemma We have a classification of Tn,p, namely Tn,p = λ≥0 fλ −1 (O) = 0≤λ≤n fλ −1 (O). 20 / 23 What is textile plot? Textile set Main result Other results Summary Differential geometrical characterization of fλ −1 (O) Lastly, we state a characterization of fλ −1 (O) from the viewpoint of differential geometry. Theorem Let λ ≥ 0. fλ −1 (O) is a regular sub-manifold of Rn×p with codimension p + 1 whenever λ > 0, y11yjj − y1j yj1 = 0, j = 2, . . . , p, ∃ ∈ { 2, . . . , p }; p∑ j=2 yij + yi (1 − 2λ) = 0, i = 1, . . . , n. 21 / 23 What is textile plot? Textile set Main result Other results Summary Present and future study Summary: We defined the textile set Tn,p and find its geometric properties. Present and future study: . 1 Characterize the classification fλ −1 (O) with induced Riemannian metric from Rnp by (global) Riemannian geometry: geodesic, curvature etc. . 2 Investigate differential geometrical and topological properties of Tn,p and fλ −1 (O), including its group action. 3 Can one find statistical implication such as sample distribution theory? Merci beaucoup! 22 / 23 What is textile plot? Textile set Main result Other results Summary Present and future study Summary: We defined the textile set Tn,p and find its geometric properties. Present and future study: . 1 Characterize the classification fλ −1 (O) with induced Riemannian metric from Rnp by (global) Riemannian geometry: geodesic, curvature etc. . 2 Investigate differential geometrical and topological properties of Tn,p and fλ −1 (O), including its group action. 3 Can one find statistical implication such as sample distribution theory? Merci beaucoup! 22 / 23 What is textile plot? Textile set Main result Other results Summary Present and future study Summary: We defined the textile set Tn,p and find its geometric properties. Present and future study: . 1 Characterize the classification fλ −1 (O) with induced Riemannian metric from Rnp by (global) Riemannian geometry: geodesic, curvature etc. . 2 Investigate differential geometrical and topological properties of Tn,p and fλ −1 (O), including its group action. 3 Can one find statistical implication such as sample distribution theory? Merci beaucoup! 22 / 23 What is textile plot? Textile set Main result Other results Summary References . 1 Absil, P.-A., Mahony, R., and Sepulchre, R. (2008), Optimization Algorithms on Matrix Manifolds, Princeton University Press. . 2 Honda, K. and Nakano, J. (2007), 3 dimensional parallel coordinate plot, Proceedings of the Institute of Statistical Mathematics, 55, 69–83. . 3 Inselberg, A. (2009), Parallel Coordinates: VISUAL Multidimensional Geometry and its Applications, Springer. 4 Kumasaka, N. and Shibata, R. (2008), High-dimensional data visualisation: The textile plot, Computational Statistics and Data Analysis, 52, 3616–3644. 23 / 23