Data Analysis on Spaces of Positive Definite Matrices with an Application to Dyslexia Detection

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1 Data Analysis on Spaces of Positive Definite Matrices with an Application to Dyslexia Detection David Groisser 1, Daniel Osborne 2, Vic Patrangenaru 2 and Armin Schwartzman 3 1 Department of Mathematics, University of Florida, Gainesville, FL 2 Department of Statistics, Florida State University, Tallahasee, FL 32304, U.S.A. 3 Department of Biostatistics, Harvard University, Boston,MA AMS 1991 subject classifications. Primary 62H11; secondary 62H10. Key words and phrases: Symmetric space, intrinsic mean, fast algorithms, extrinsic mean. Abstract. Short title: Data Analysis on PD Spaces. Research partially supported by National Science Foundation Grant DMS

2 1 Introduction. Statistical inference for distributions on manifolds is a broad discipline with wide ranging applications. Its study has gained momentum, due to wide range of applications in biosciences and medicine, image analysis, electrical engineering, etc. A general framework for nonparametric inference for location was introduced in Bhattacharya and Patrangenaru (2002, 2003, 2005) and furthered in Ellingson et. al. (2011) where, apart from the derivation of some general properties of Fréchet means on general manifolds, including Hilbert manifolds, the problem of consistency of the Freéchet sample indices was explored and a derivation of asymptotic distributions of intrinsic and extrinsic sample means and confidence regions based on them have been provided. It was shown that Fréchet means exist on Riemannian manifolds of nonpositive curvature; nevertheless their computations are quite complicated and in general depending on numerical algorithms. Fast algorithms for intrinsic sample means on manifolds, using the fixed point paradigm, were developed by Groisser (2004). In the simple case of zero curvature, the intrinsic mean can be easily estimated using nonparametric bootstrap (Efron(1982)) that reduces the coverage error presenting regular asymptotics (Babu and Singh (1984), Bhattacharya and Ghosh (1978), Hall (1997)). In particular positive definite matrices with the flat metric are convex, providing an example arising for which nonparametric bootstrap provides a reasonable estimation of the of the Frobenius mean. Schwartman (2007) considered a complete canonical metric yielding a structure of Cartan-Hadamard symmetric space for positive definite matrices. In absence of an equivariant embedding of the canonical metric, in this paper we use iterations to approximate the associated intrinsic sample mean of a distribution on this space. These nonparametric approach is used in a concrete example for dyslexia detection in children from DTI. Following Schwartzman et al. (2008), we consider an experiment in which DTI maps acquired for two groups of subjects, a group of healthy 2

3 children and a group of children suffering from dyslexia, differ on average. 2 SPD matrices In a fixed dimension, the set of SPD matrices is open and convex subset of the space of symmetric matrices. By SVD, a SPD with simple eigenvalues can be expressed in a in terms of its eigenvalues and corresponding eigenvectors, in a unique way, up the the signs of the eigenvectors. Given the eigenvalues and eigenvectors give a better geometric insight on each such SPD matrix, inference problems for eigenvalues and eigenvectors have been formulated and treated for SVD matrices in a Gaussian setting by Schwartzman (2006). This, as has lead Schwartzman (2006), and Schwartzman et. al. (2008) to describe the space of SPD matrices as a manifold, to allow for the modeling for SPD matrices as well as for their marginal distributions of their eigenvalues and eigenvectors. Recall from Chapter 3 that GL(p, R) is the Lie group of p p nonsingular matrices, Sym(p) is the set of p p invertible symmetric matrices wand Sym + (p), the set of p p symmetric positive definite matrices, which can be also identified with the set of scalar products on R p : Sym + (p) = {X Sym(p) : v T Xv > 0, v R p }. (2.1) Recall that the Lie algebra of the orthogonal group O(p) is o(p), set of p p antisymmetric matrices o(p) = A M(p; R) : A T = A. (2.2) The set of p p diagonal matrices with real entries is denoted by Diag(p) and its subset, having only positive entries on the diagonal, by Diag + (p). For distribution theory and asymptotic reasons,it is useful to vectorize matrices, by regarding them as points in a numerical space. The most common vectorization function vec(x), consists in stacking all the columns of X on top of each other. In 3

4 our case, it will be more convenient to extract diagonal and off-diagonal elements separately. We adopt the following definition. DEFINITION 2.1 Given a p p matrix X, diag (X) is the p 1 column vector formed by the diagonal elements of X, and offdiag(x) is the p(p 1)/2 1 column vector formed by the elements of X above the diagonal taken columnwise. In other words, x 11 x 12 x 1p x X = 21 x 22 x 2p x p1 x p2 x pp x 11 x diag (X) = 22, offdiag(x) =. x pp x 12 x 13 x 23. x (p 1)p DEFINITION 2.2 The p(p+1)/2 1 column vector vecd(x) is the concatenation of the on-diagonal and above-diagonal elements of X, i.e. diag (X) vecd(x) = offdiag(x) To each vector x = (x 1,..., x p ) T, we associate the diagonal p p matrix diag (x) with the diagonal entries given the coordinates of x. Consider a scalar product <, > on a p dimensional real vector space V. To each basis (f 1,..., f p ) of V, we associate the symmetric p p matrix A whose entries a ij are given by a ij =< f i, f j >. Thus we established a one-to-one correspondence between the set of scalar products on R p and the set of SPD matrices. Note that any convex combination <, > a = (1 a) <, > 0 +a <, > 1, a (0, 1) of 4

5 the scalar products <, > 0, <, > 1 is a scalar product as well, thus showing, via the one-to-one correspondence afore mentioned, that in any dimension p, PROPOSITION 2.1 The set Sym + (p) is a convex subset of the linear space Sym(p) of p p symmetric matrices. The immediate consequence of Proposition 2.1 is COROLLARY 2.1 (Estimation of the Euclidean mean SPD) The Euclidean mean of a distributions of SPD matrices is also an SPD matrix, and in particular the sample mean of a sample from such a distribution is an SPD matrix, and the CLT can be applied to distribution of SPD matrices in their vectorized forms. In particular the Euclidean mean of such a probability distribution can be estimated using the studentized version of the CLT, in the case of a large sample, respectively nonparametric bootstrap, if a small random sample from this distribution is available. The Euclidean distance on Sym + (p), when regarded as a trivial Riemannian distance is also called Frobenius metric structure for SPD matrices, resulting in a Riemannian manifold that is flat, but whose geodesics, which are straight lines, are constrained by the boundaries of Sym + (p) as a subset of Sym(p) (see Schwatzman, 2006). 3 A symmetric space structure for SPD matrices Since Sym + (p) with the Frobenius metric structure is not complete, additional metric structure between SPD matrices could been considered. 3.1 A homogeneous space structure Note that any SPD matrix in Sym + (p) is associated with a scalar product on R p, and given that for any given scalar product <, > on R p one may find an or- 5

6 thobasis f 1,..., f p with respect to <, >. Assume A Sym + (p) is an arbitrary SPD and <, > is the scalar product associated with A. If f 1,..., f p is an orthobasis for this scalar product and e 1,..., e p is the canonical basis, then e i = p j=1 gj i f j, i = 1,..., p, where G = (g j i ) i,j=1,...,p is the matrix of change of coordinates from the standard basis to the basis f 1,..., f p. This yields the the following GAG T = I p, (3.1) where I p is the identity p p matrix. We are led into considering the group action α : GL(p, R) Sym + (p) Sym + (p) given by α G (X) = GXG T. (3.2) PROPOSITION 3.1 GL(p, R) acts transitively on X, W Sym + (p), and the isotropy group at I p is GL(p, R) Ip = O(p). Moreover the action α is faithful. Indeed given X, W Sym + (p), from equation (3.1) there are two matrices G 1, G 2 GL(p, R) such that G 1 XG T 2 = I p, G 2 W G T 2 = I P. If we set G = G 1 2 G 1 we get α G (X) = GXG T = W. REMARK 3.1 The group action (3.2) can be interpreted as the effect of a change of coordinates at the covariance matrix level as well. Indeed let W be a zeromean random vector in R p with covariance X = E(W W T ) Sym + (p). the marginals of W are interpreted as coordinates with respect to a p-dimensional frame G GL(p, R), where the columns g j, j = 1,..., p of G are the frame vectors with coordinates in an ambient space R p, then the marginals of W in the ambient space are given by the vector W = W 1 g W p g p = GW If The covariance of the transformed vector w is E( W W T ) = E(GW W T G T ) = GXG T 6

7 In other words, the action W = GXG is the expression in ambient coordinates of X, when X is in G-coordinates. Conversely, if W is expressed in ambient coordinates, the expression of W in G-coordinates is given by the action of G 1, X = G 1 W (G T ) 1. By analogy, the same coordinate system interpretation applies to the group action that results on the tangent vectors at X and W respectively. 3.2 The canonical metric. Assume K act transitively on the manifold M with the isotropy group H = K q at the point q M. LEMMA 3.1 A scalar product g q on T q M can be extended to a K-invariant Riemannian metric on M if for any h H, d q α h is an isometry of T q M. To prove lemma 3.2, we consider a point r M, and an element k K such that α k (q) = r, and define g r (u, v) = g q ((dα k ) 1 (u), (dα k ) 1 (v)). Ifk K, is such that k q = k q = r, then h = k 1 k H, and it follows that g r is well defined if and only if d q α h is an isometry of T q M. In our case, Sym + (p) is an open subset of the vector space Sym(p), thus the tangent space T X Sym + (p) is X Sym(p). The group action (3.2) on Sym + (p) yields a canonical isomorphism between the tangent spaces at X and α G (X); given a tangent vector Y T X Sym + (p) at X and fixed G, the differential map of α G is the isomorphism given by dα G (Y ) = GY G T T αg (X)Sym + (p). We now identify T Ip Sym + (p) = I p Sym(p) with Sym(p) and note that the Frobenius ( Euclidean ) scalar product on Sym(p) is g Ip (Y, Z) = tr(y Z T ) = tr(y Z). From Proposition 3.1, the isotropy subgroup of the action (3.2) at I p is the orthogonal group O(p) and for any matrix H O(p) the differential d H α : Sym(p) Sym(p) is given by d H (Y ) = HY H T. (3.3) 7

8 From equation (3.3) we see that g Ip (d H α(y ), d H α(z)) = tr(hy H T HZH T ) = tr(hy ZH T ) = tr(h T HY Z) = tr(y Z) = g Ip (Y, Z), therefore by Lemma, we may extend the scalar product g p to a GL(p, R) invariant Riemannian metric g on Sym + (p) as follows: if Y M, Z M T M Sym + (p) are two tangent vectors at M (symmetric matrices, since T M Sym + (p) was identified above with Sym(p)) then g M (Y M, Z M ) = g Ip ((dα G ) 1 (Y M ), (dα G ) 1 (Z M )). (3.4) Explicitly, in terms of matrix operations, if α G (I p ) = GG T = M, then d G α(y ) = GY G T and (dα G ) 1 (Y M ) = G 1 Y M (G T ) 1 therefore, this GL(p, R)-invariant metric is given by g M (Y M, Z M ) = tr(g 1 (Y M )(G T ) 1 G 1 (Z M )(G T ) 1 ) = tr(m 1 Y M M 1 Z M ), (3.5) and was called canonical inner product, and from Lemma 3.2 we get PROPOSITION 3.2 The canonical inner product is invariant under the group action of GL(p, R). The notation in Schwartzman (2007) for the canonical inner product is g M =, M, and as a result Schwartzman gives an alternate proof of the GL(p, R) invariance, stated as follows: COROLLARY 3.1 Let Y M, Z M T M Sym + (p) be two tangent vectors at M and let G GL(p) be any square root of M (i.e. GG T = M). Let Y Ip = α G 1(Y M ) and Z Ip = α G 1(Z M ) be the translations to the identity of Y M and Z M respectively. Then Y M, Z M M = Y Ip, Z Ip Ip, by noting that conceptually, the point M Sym + (p) is a translation of the identity I p by the group action, M = GI p G T, and this result does not depend on the specific 8

9 choice of G. At this point, we should recall from Chapter?? that GL(p, R) is a p 2 dimensional Lie group, whose Lie algebra is gl(p, R) = M(p; R). The isotropy group H of the action (3.2) of GL(p, R) on Sym + (p) is the orthogonal group O(p), having the Lie algebra h = so(p). On the other hand, the orbit of I p for this group action is Sym + (p) and the tangent space of Sym + (p) at I p is m = Sym(p). The linear idempotent isomorphism s : gl gl given by x x h s(x) = (3.6) x x m is the differential at I p of an automorphism σ of GL(p, R), such that GL(p, R), O(p), s) is a symmetric triple, and since GL(p, R) is is a transitive group of isometries of M = Sym + (p) with the canonical metric, by Theorem it follows that any geodesic of this homogeneous space starting at I p of tangent vector d Ip π(w ), W gl(p, R), is of the form α exp(tw ) (I p ) = exp(tw )exp(tw ) T = exp(tw )exp(tw T ) = exp(t(w + W T ), (3.7) where t exp(ta) is a one parameter subgroup of GL(p, R), with A gl(p, R). Note that d Ip π(w ) is the tangent vector at t = 0 of π(e tw ) = e tw e tw T at t = 0, that is d Ip π(w ) = W + W T, and from (3.7), we obtain the following PROPOSITION 3.3 The geodesic γ on Sym + (p) with the canonical metric, with γ(0) = I p, dγ (0) = Y m is given by dt γ(t) = exp(ty ). (3.8) From Proposition 3.3 we obtain the following 9

10 COROLLARY 3.2 Given a tangent vector Y T Ip Sym + (p) = Sym(p) = m, the Riemannian exponential map Exp : T Ip Sym + (p) Sym + (p) with respect to the canonical metric is given by Exp Ip (Y ) = exp(y ). (3.9) Given that in general the exponential as a function from the Lie algebra of a Lie group to the Lie group is a local diffeomorphism, it has a local inverse, defined in a neighborhood of the identity of the Lie group, called logarithm. Given that by the singular value decomposition (SVD), an SPD matrix X Sym + (p) can be expressed as a product X = V ΛV T, V O(p), Λ Diag + (p), it follows that X = V exp(log Λ)V T = exp(v log ΛV T ), where log Λ Diag(p) has as diagonal entries, the logarithms of the diagonal entries of Λ. From this observation and from Corollary 3.1 we conclude that PROPOSITION 3.4 If X Sym + (p), X = V ΛV T with V O(p) and ΛinDiag + (p), then log(x) = V log(λ)v T. The Riemannian exponential map Exp : T Ip Sym + (p) Sym + (p) is a diffeomorphism. In particular, the canonical metric on Sym + (p) is complete. In addition from Theorem 4.2. in Helgasson (2001, p. 180), it follows that Sym + (p) with the canonical inner product has non-positive sectional curvature. A simply-connected complete Riemannian manifold with non-positive sectional curvature is called a Cartan- Hadamard manifold. The space Sym + (p) with the canonical metric is therefore a Cartan-Hadamard manifold. THEOREM 3.1 The Riemannian homogeneous space Sym + (p) with the canonical metric is a symmetric space. PROOF 3.1 Let d : Syp + (p) Sym + (p) R be the canonical geodesic distance. From Remark , it suffices to show that the geodesic symmetry at I p is 10

11 an isometry, that is for any pair (A, B) of SPD s, d(a, B) = d(a 1, B 1 ). (3.10) Note that the distance d is invariant under the action (3.2)of GL(p, R), therefore d(a, B) = d(i p, A 1 2 BA 1 2 and d(a 1, B 1 ) = d(i p, A 1 2 B 1 A 1 2. If we set C = A 1 2 BA 1 2, then equation (3.10) is equivalent to showing that d(i p, C) = d(i p, C 1. By Proposition 3.4, there is a symmetric matrix Y such that C = Exp Ip (Y ) = exp(y ), and since the Riemannian exponential is a radial isometry, d(i p, C) = Y. Note that C 1 = exp(y ) 1 = exp( Y ) = Exp Ip ( Y ), therefore d(i p, C 1 = Y = Y = d(i p, C), done. 3.3 The convex cone of positive definite matrices As an illustration we consider here the case p = 2, that is the case of the sample space for the CBR. Positive definite matrices are symmetric matrices with the restriction that their eigenvalues have to be positive. This restriction can be translated into constraints on the values that the entries of the matrix can take. For illustration, let X = a c c b be a 2 2 symmetric matrix. X is positive definite if and only if the diagonal elements are positive (a > 0, b > 0) and the determinant is positive (ab c 2 > 0). The set of triplets (a, b, c) that result in positive definite matrices is an open subset of R 3 and has the shape of a cone. This is illustrated in Figure 1. Valid triplets (a, b, c) lay inside the cone. For example, the matrices X 1 = diag (1, 0.1) and X 2 = diag (0.1, 1) are represented by the triplets (0.9, 0.1, 0) and (0.1, 0.9, 0) respectively. Notice that the cone is convex, so interpolation between any two points of the cone is permitted. This implies averages of positive definite matrices are 11

12 positive definite, as illustrated by the midpoint point X = diag (0.5, 0.5). Extrapolation, however, might result in matrices that are not positive definite: in Figure 1, the straight line connecting X 1 and X 2 extends beyond the boundaries of the cone. The positive definite constraints are eliminated by the log transformation. Applying this transformation to X 1, X 2 Sym + (2) results in symmetric matrices Y 1, Y 2 Sym(p). Tracing a straight line through Y 1 and Y 2 in the log-space and then taking the matrix exponential results in the hyperbola connecting X 1 and X 2 in Figure 1. This hyperbola is entirely inside the cone by definition. The midpoint of this line is X = diag (0.3, 0.3). For positive numbers, the exponential of the average of the logs of two numbers is the same as the geometric mean. Analogously, X can be thought of as a geometric mean of X 1 and X 2. Both the pure straight line and the hyperbola are straight lines, depending on whether it is traced directly or on the log-space. In fact, both lines are geodesics corresponding to two different geometries defined on the cone. Describing these two different geometries was the goal of this section. 4 Nonparametric methods for estimation of Frobenius means and canonical means. For testing H 0 : µ 1 µ 2 = δ 0, in a vector space, assume the two samples are i.i.d.r.vec. s X a,ja µ a, Σ a, j a = 1, n a, a = 1, 2 from two independent multivariate populations, and the total sample size n = n 1 +n 2, s.t. n 1 n q (0, 1), as n, and 1 q Σ q Σ 2 > 0, Then from the C.L.T and Slutsky s theorem, under H 0, it follows that T 2 = (X 1 X 2 δ 0 ) T ( 1 n 1 S n 2 S 2 ) 1 (X 1 X 2 δ 0 ) d C χ p, (4.1) 12

13 Figure 1: Geodesics joining 2 2 positive definite matrices relative to the Frobenius metric (green) and to the canonical metric (yellow). as n, therefore we reject H 0 at level α if (X 1 X 2 δ 0 ) T ( 1 S S 2 ) 1 (X 1 X 2 δ 0 ) d C > χ p (α). (4.2) n 1 n 2 If the distributions are unknown and the samples are small, for testing, we use nonparametric bootstrap. 5 DTI data Diffusion Tensor Imaging (DTI) is a more sophisticated MR imaging that enables to visualize white matter fibers in the brain and can trace subtle changes in the white matter associated with unusual brain wiring like in dyslexia or in schizophrenia, or brain diseases including multiple sclerosis or epilepsy. In DTI one stores informa- 13

14 Figure 2: DTI slice images of a control subject (left) and of a dyslexia subject (right). tion about the diffusion of the water molecules in the brain. The displacement of a water molecule at a certain location has probability distribution in space, and its diffusion D is half the covariance matrix of that distribution, which is a symmetric d 11 d 12 d 13 positive semidefinite matrix. The diffusion matrix D = d 21 d 22 d 23 at any d 31 d 32 d 33 given voxel in stored in the form of a column (d 11 d 22 d 33 d 12 d 13 d 22 d 23 ) T, and as a result in the imaging of white matter, the location, orientation, and anisotropy of the fiber tracts can be measured. The anisotropy is the departure from a spherical diffusion, and the degree of anisotropy in children brains might offer an explanation for a specific reading disability referred to as dyslexia, which is characterized by a delay in the age at which a child begins to read. Approximately 10 to 15 percent of school-age children have this disorder ( Larson et. al. (1996)). In Figure 2 we display DTI slices including a given voxel recorded in a control subject and a dyslexia subject ( first subject and seventh in the table 1). 14

15 Table 1: DTI data in a group of control (columns 1-6) and dyslexia (columns 7-12) Comparison of Frobenius means Bootstrap Distribution of the Sample Means in the Control Group Quantiles Min Q 1 Median Q 3 Max x 1c x 2c x 3c x 4c x 5c x 6c

16 Figure 3: Marginals of the bootstrap distribution : clinically normal (red) vs dyslexia (blue) Bootstrap Distribution of the Sample Means in the Dyslexia Group Quantiles Min Q1 Median Q3 Max x 1d x 2d x 3d x 4d x 5d x 6d Differences between the Groups Bootstrap Distribution of the Sample Means Quantiles Min Q1 Median Q3 Max x 1c x 1d x 2c x 2d x 3c x 3d x 4c x 4d x 5c x 5d x 6c x 6d

17 x1c x1d x4c x4d x2c x2d x5c x5d x3c x3d x6c x6d Figure 4: Bootstrap distribution for the difference of means of marginal SPD matrices of clinically normal children vs dyslexia children. References [1] Vincent Arsigny, Pierre Fillard, Xavier Pennec, and Nicholas Ayache. (2006). Geometric means in a novel vector space structure on symmetric positivedefinite matrices. SIAM Matrix Anal Appl, 29(1): [2] Babu, G.J. and Singh, K. (1984) On one term Edgeworth correction by Efron s bootstrap, Sankhya Ser. A. 46, [3] Bhattacharya, R. N.; Ghosh, J. K. (1978) On the validity of the formal Edgeworth expansion, Ann. Statist. 6, no. 2, [4] Bhattacharya, R.N. and Patrangenaru, V. (2003), Large sample theory of intrinsic and extrinsic sample means on manifolds, I, Ann. Statist [5] Bhattacharya, R.N and Patrangenaru, V. (2002) Non parametric estimation of location and dispersion on Riemannian manifolds,jspi, volume in honor of the 80th birthday of C.R.Rao, (in press). [6] Efron, Bradley (1982) The Jackknife, the Bootstrap and Other Resampling Plans. CBMS-NSF Regional Conference Series in Applied Mathematics, 38. SIAM, Philadelphia, Pa. 17

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