Complex random vectors and Hermitian quadratic forms

Transcription

1 Complex random vectors and Hermitian quadratic forms Gilles Ducharme*, Pierre Lafaye de Micheaux** and Bastien Marchina* * Université Montpellier II, I3M - EPS ** Université de Montréal, DMS 26 march 2013 Bastien Marchina (UM2) Complex random vectors 26 march / 49

2 1 Introduction 2 Complex random vectors 3 Hermitian quadratic forms in complex random vectors 4 Statistics of complex random vectors 5 Applications to goodness-of-t tests Bastien Marchina (UM2) Complex random vectors 26 march / 49

3 Introduction 1 Introduction 2 Complex random vectors 3 Hermitian quadratic forms in complex random vectors 4 Statistics of complex random vectors 5 Applications to goodness-of-t tests Bastien Marchina (UM2) Complex random vectors 26 march / 49

4 Introduction Complex random data Complex data appears in various situations. For instance, periodic signals can be representated by (a collection of) complex numbers. For instance, radar and fmri data are usually aggregated as complex-valued data. Thus arises the need for statistical modeling using complex random elements. Many results have already been established by the signal processing community, but are not well known in the statistical community. Bastien Marchina (UM2) Complex random vectors 26 march / 49

5 Introduction Example of complex random data Fig.: fmri data representation Bastien Marchina (UM2) Complex random vectors 26 march / 49

6 Introduction Motivation : fmri activation with complex data D.B. Rowe and B.R. Logan (2004) une normality assumptions to build a fmri activation model using all the complex signal. Fig.: fmri data representation Thus arises the need for a proper test for complex normality. Bastien Marchina (UM2) Complex random vectors 26 march / 49

7 Introduction Complex random variables in mathematical statistics The probability distribution P of X can be characterised by its characteristic function. ϕ X (t) = E(e i(t X) ). The empirical characteristic function ϕ n (t) = 1 n n k=1 e i(t X k ) of independent copies X 1,..., X n, of X P is an estimator of the characteristic function of P and is complex valued. Bastien Marchina (UM2) Complex random vectors 26 march / 49

8 Complex random vectors 1 Introduction 2 Complex random vectors 3 Hermitian quadratic forms in complex random vectors 4 Statistics of complex random vectors 5 Applications to goodness-of-t tests Bastien Marchina (UM2) Complex random vectors 26 march / 49

9 Complex random vectors Complex Random Vectors If X and Y are random vectors in R d, Z = X + iy is a random vector in C d. Z = (Z, Z H ) is the augmented vector associated with Z and e X = MZ e, Z e = 2M H X, MM H = 2I d, (1) with X = (X, Y ) and M = 1 ( ) ( ) Id I d, M 1 Id ii = d. (2) 2 ii d ii d I d ii d The characteristic function of Z is ϕ Z (ν) = E ( e i Re(νH Z) ). (3) Bastien Marchina (UM2) Complex random vectors 26 march / 49

10 Complex random vectors First and second order parameters for complex random vectors Expectation : µ = E(Z) = µ X + iµ Y, Positive semi-denite hermitian covariance matrix : Γ = E [ (Z µ)(z µ) H], (4) Positive semi-denite symmetric relation matix : [ P = E (Z µ)(z µ) ], (5) Positive semi-denite hermitian covariance-relation matrix ( ) Γ P = E[(Z µ )(Z µ ) H Γ P ] = e e e e P H Γ. (6) Bastien Marchina (UM2) Complex random vectors 26 march / 49

11 Complex random vectors Complex normal distribution Denition (Van den Bos (1995), or Picinbono (1996)) Z = X + iy C d is following the complex normal distribution i ( (( ) ( )) X µx ΣXX Σ N, XY. (7) Y) µ Y Σ YX Σ YY We write Z CN d (µ, Γ, P), or alternatively Z C N 2d (µ, Γ P ) with e e e Σ X = MΓ P M H, Γ P = M 1 Σ X (M H ) 1. (8) Bastien Marchina (UM2) Complex random vectors 26 march / 49

12 Complex random vectors Proper normal distribution Consider Z CN d (µ, Γ, 0), i.e., Z is a complex normal random vector with P = 0. Then, f Z (z) = 1 π d Γ 1/2 exp { (z µ) H Γ 1 (z µ) }. (9) This is the density of the complex normal distribution with two parameter introduced by Wooding (1956). It is usually refered as the proper or circular complex normal distribution. Bastien Marchina (UM2) Complex random vectors 26 march / 49

13 Complex random vectors Proper and improper complex normal data kn = 0 Partie réelle Partie imaginaire kn = 0.3 Partie réelle Partie imaginaire kn = 0.6 Partie réelle Partie imaginaire kn = 0.9 Partie réelle Partie imaginaire Bastien Marchina (UM2) Complex random vectors 26 march / 49

14 Complex random vectors Characteristic function and density The characteristic function of Z is { ϕ Z (l) = exp i Re(l H µ) 1 ( l H Γl Re(l H Pl ) )}. (10) 4 If Γ P is invertible 1 f Z (z) = π d Γ P 1/2 exp { 1 ( 2 ((z µ), (z µ) H z µ )Γ 1 P (z µ) )}. (11) Bastien Marchina (UM2) Complex random vectors 26 march / 49

15 Complex random vectors Complex normal distribution Main results Conservation by linear transforms Let Z CN d (µ, Γ, P) and A C m d, then AZ CN m (Aµ, AΓA H, APA ). (12) Conservation by augmented linear transforms Let Z C N 2d (µ, Γ P ), A B C 2m 2d such that e e e ( ) A B A B =, (13) then A B Z C N 2m (A B µ, A B Γ P A H). B e e e B A Bastien Marchina (UM2) Complex random vectors 26 march / 49

16 Complex random vectors Complex normal distribution Independence Theorem : Independence between complex gaussian variables Let Z = (Z 1, Z 2 ) CN 2 (µ, Γ, P). Z 1 and Z 2 are independent if and only if ( ) γ1 0 Γ =, 0 γ 2 and ( ) p1 0 P =. 0 p 2 Bastien Marchina (UM2) Complex random vectors 26 march / 49

17 Complex random vectors Complex normal distribution Independence Corollary : Independence between complex gaussian vectors Let Z CN d1 +d 2 (µ, Γ, P), Partition Z as (Z ( 1, Z 2 ) where ) Z 1 is d 1 1, Z 2 Γ1 Γ is d 2 2, and likewise µ into (µ 1, µ 2 ), Γ 12 in Γ H and similarly for P. 12 Γ 2 Z 1 and Z 2 are independent if and only if Γ 12 = P 12 = 0. Corollary : Independence between components of a complex gaussian vector Let Z = (Z 1,..., Z d ) CN d (µ, Γ, P). The components Z 1,..., Z d are independent if and only if Γ and P are diagonal. Bastien Marchina (UM2) Complex random vectors 26 march / 49

18 Hermitian quadratic forms 1 Introduction 2 Complex random vectors 3 Hermitian quadratic forms in complex random vectors 4 Statistics of complex random vectors 5 Applications to goodness-of-t tests Bastien Marchina (UM2) Complex random vectors 26 march / 49

19 Hermitian quadratic forms Hermitian quadratic forms Let Z CN d (µ, Γ, P). We study positive quadratic forms of the form Z e H RZ e. First, notice that with S = MRM 1. It leads to Z e H RZ e = 2X SX, (14) R = M 1 SM ( ) S11 + S = 22 + i(s 12 S 21 ) S 11 S 22 + i(s 12 + S 21 ), S 11 S 22 i(s 12 + S 21 ) S 11 + S 22 i(s 12 S 21 ) and nally, because S is a symmetric matrix, ( ) G K R =. (15) K H G Bastien Marchina (UM2) Complex random vectors 26 march / 49

20 Hermitian quadratic forms Hermitian quadratic forms Theorem Let and Z CN d (µ, Γ, P). Then ( G K R = Z e H RZ e = K H 2d k=1 G ). (16) α k χ 2 1(δ 2 k ), (17) where the χ 2 1 (δ2 k ) are independent, the α k are the eigenvalues of RΓ P and the δ k are function of µ, Γ, P and R. Bastien Marchina (UM2) Complex random vectors 26 march / 49

21 Hermitian quadratic forms Hermitian quadratic forms Corollary Let and Z CN d (0, Γ, P). Then ( G K R = Z e H RZ e = K H G ). (18) 2d α k χ 2 1, (19) where the χ 2 1 are independent and the α k are the eigenvalues of RΓ P. k=1 Bastien Marchina (UM2) Complex random vectors 26 march / 49

22 Hermitian quadratic forms Moore-Penrose inverse The Moore-Penrose inverse of A is the only matrix such that AA + A = A A + AA + = A + (AA + ) H = AA + (A + A) H = A + A, A few of the properties of the Moore-Penrose are 1 Let α 0. Then (αa) + = α 1 A +, 2 (A ) + = (A + ), (A ) + = (A + ) and (A H ) + = (A + ) H, 3 Let A be m n complex and B be n p complex. If AA H = I m or B H B = I p, then (AB) + = B + A +, 4 If A is invertible, then A + = A 1. Bastien Marchina (UM2) Complex random vectors 26 march / 49

23 Hermitian quadratic forms More results on hermitian quadratic forms Theorem Let Z CN d (0, Γ, P). Let Γ + be the Moore-Penrose inverse of the P covariance-relation matrix Γ P of Z. Then, ξ = Z e H Γ + P Z e χ2 q, (20) where q 2d is the rank of Γ P. Corollary Let Z CN d (0, Γ, P), such that Γ P is invertible. Then, ξ = Z e H Γ 1 P Z e χ2 2d. (21) Bastien Marchina (UM2) Complex random vectors 26 march / 49

24 Hermitian quadratic forms More results on hermitian quadratic forms Theorem Let Z CN d (µ, Γ, P), Z e = ( ) Z, m = Z ( ) µ, A and B two 2d 2d µ matrices, that share properties with matrix R in (18). Then Z e H AZ e and Z e H BZ e are independent if and only if (i) Γ P AΓ P BΓ P = 0, (ii) Γ P AΓ P BΓ P m = Γ P BΓ P Am = 0, (iii) m H AΓ P BΓ P m = 0. Bastien Marchina (UM2) Complex random vectors 26 march / 49

25 Hermitian quadratic forms More results on hermitian quadratic forms Theorem Let Y CN d (0, Γ Y, P Y ) and Z CN d (0, Γ Z, P Z ), such that Y and Z are independent. Let Γ P,Y and Γ P,Z be the covariance-relation matrices of Y and Z respectively. Then, Y b H Γ + Y P,Y e e a Z H Γ + Z P,Z e e F(a, b), where a 2d is the rank of Γ P,Y, b 2d is the rank of Γ P,Z and F(a, b) denotes the Fisher distribution with degrees of freedom a and b. Bastien Marchina (UM2) Complex random vectors 26 march / 49

26 Hermitian quadratic forms More results on hermitian quadratic forms Corollary Let Z = (Z 1, Z 2 ) CN d1 +d 2 (0, Γ, P). Let Γ P,1 and Γ P,2 be the covariance-relation matrices of the marginal distributions of Z 1 and Z 2, with respective ranks a and b and a + b 2d Then, if and only if if Z 1 and Z 2 are independent. b a Z H 1 Γ + Z P,1 1 e e Z H 2 Γ + Z F(a, b), (22) P,2 2 e e Bastien Marchina (UM2) Complex random vectors 26 march / 49

27 Statistics of complex random vectors 1 Introduction 2 Complex random vectors 3 Hermitian quadratic forms in complex random vectors 4 Statistics of complex random vectors 5 Applications to goodness-of-t tests Bastien Marchina (UM2) Complex random vectors 26 march / 49

28 Statistics of complex random vectors Central limit theorem for complex random vectors Theorem Let W be a random vector in C d, E(W) = µ, with denite covariance and relation matrices Γ and P. Let W 1,..., W n be independant copies of W. Then, n 1 n W j µ CN d (0, Γ, P). (23) n j=1 Bastien Marchina (UM2) Complex random vectors 26 march / 49

29 Statistics of complex random vectors Maximum likelihood and moments method estimators Z 1,..., Z n is a random sample of CN(µ, Γ, P). Method of moments and maximum likelihood estimators for µ, Γ and P are identical. Parameter estimates ˆµ = 1 n Z k = Z, n ˆΓ = 1 n ˆP = 1 n ˆΓ P = 1 n k=1 n (Z k Z)(Z k Z) H, k=1 n (Z k Z)(Z k Z), k=1 n (Z k k=1 e Z e )(Z k e Z e ) H. Bastien Marchina (UM2) Complex random vectors 26 march / 49

30 Statistics of complex random vectors Asymptotics in the case d = 1 In this case, ˆµ µ n ˆγ γ CN 3 (0, Γ θ, P θ ), (24) ˆp p where γ 0 0 p 0 0 Γ θ = 0 γ 2 + p 2 2p γ, P θ = 0 γ 2 + p 2 pγ. (25) 0 2pγ 2γ 2 0 2pγ 2p 2 Bastien Marchina (UM2) Complex random vectors 26 march / 49

31 Applications to goodness-of-t tests 1 Introduction 2 Complex random vectors 3 Hermitian quadratic forms in complex random vectors 4 Statistics of complex random vectors 5 Applications to goodness-of-t tests Bastien Marchina (UM2) Complex random vectors 26 march / 49

32 Applications to goodness-of-t tests Empirical characteristic function and empirical characteristic process Let X 1,..., X n be a random sample with characteristic function ϕ X ( ). In order to test H 0 : ϕ X ( ) = ϕ 0 ( ) it is customary to intoduce the empirical characteristic process U n (t) = n(ϕ n (t) ϕ 0 (t)). (26) Bastien Marchina (UM2) Complex random vectors 26 march / 49

33 Applications to goodness-of-t tests The empirical characteristic process Under H 0, the empirical characteristic process is such that E(U n ( )) = 0 C(s, t) = E(U n (s)u n (t) ) = ϕ 0 (s t) ϕ 0 (s)ϕ 0 (t), (27) P(s, t) = E(U n (s)u n (t)) = ϕ 0 (s + t) ϕ 0 (s)ϕ 0 (t) = C(s, t). Bastien Marchina (UM2) Complex random vectors 26 march / 49

34 Applications to goodness-of-t tests Koutrouvelis's goodness-of-t test Koutrouvelis (1980) gives a test statistic of H 0 : P = P 0 based on the evaluation of U n ( ) on cleverly chosen points t 1,..., t d. With W n = (R 1,..., R d, I 1,... I d ), R k = Re(U n (t k )) and I k = Im(U n (t k )), he shows that W nσ 1 W n χ 2 2d. (28) under H 0 with Σ the covariance matrix of W n under H 0. Bastien Marchina (UM2) Complex random vectors 26 march / 49

35 Applications to goodness-of-t tests Use of a complex quadratic form Using a complex framework, we have the following test statistic ξ n = U n e H Γ + P U n e χ 2 q, (29) with U n = (U n, U H n ), U n = (U n (t 1 ),..., U n (t d )) and q < 2d is the rank of e Γ Un = Γ P = ( ΓUn P Un P H U n 0 1 C(t1, t1)... C(t1, tm) C(tm, t1)... C(tm, tm) C A, P Un = ), (30) Γ U n 0 1 P(t1, t1)... P(t1, tm) P(tm, t1)... P(tm, tm) C A (31) Bastien Marchina (UM2) Complex random vectors 26 march / 49

36 Applications to goodness-of-t tests Test of a simple hypothesis for complex distributions A sample of complex random vectors has the following e.c.f. ϕ n (ν) = 1 n n e i Re(νH Z ) k, (32) k=1 and U n ( ) has the same properties than in the real case. Once again, U n e H Γ + P U n e χ 2 q, (33) with U n = (U n (ν 1 ),..., U n (ν d )) and q < 2d is the rank of Γ P. Bastien Marchina (UM2) Complex random vectors 26 march / 49

37 Applications to goodness-of-t tests Test for complex normality Back to our signal processing problematic, we need to test the composite hypothesis H 0 : P { CN 1 (µ, γ, p), µ C, γ R, p C such that p < γ }. (34) The parameters are unknown and must be estimated in order to build a test statistic. Bastien Marchina (UM2) Complex random vectors 26 march / 49

38 Applications to goodness-of-t tests Test for complex normality From the circularized empirical characteristic process U n,y (ν) = n(ϕ n,y (ν) ϕ 0 (ν)), (35) where ϕ 0 (ν) is the c.f. of a CN 1 (0, 1, 0), Y = Γ 1/2 (Z µ ), and therefore Y CN P 1 (0, 1, 0). e e e we build U (ν) = n,ŷ n(ϕ (ν) ϕ n,ŷ 0(ν)), (36) where Ŷ is obtained through the m.l.e. estimates of the parameters. Bastien Marchina (UM2) Complex random vectors 26 march / 49

39 Applications to goodness-of-t tests Test for complex normality The modied ξ n test statistic follows Here U H n,ŷ = (U n,ŷ (ν 1),..., U n,ŷ (ν m)) ˆξ n = U e H n,ŷ Γ P(ν) 1 U e n,ŷ. (37) Γ P (ν), albeit complicated, does not depend on the true value of the parameters, but only on the choice of points. Bastien Marchina (UM2) Complex random vectors 26 march / 49

40 Applications to goodness-of-t tests Simulation : case m = 1 Tab.: Quantiles of the distribution of ˆξ n based on repetitions with m = 1 n E 0 V 0 Q 90 Q 95 Q χ Bastien Marchina (UM2) Complex random vectors 26 march / 49

41 Applications to goodness-of-t tests P-P plot for the case m = 1 p p plot for xi(1) vs. a chi2(2) distribition empirical cumulative distribution theoretical cumulative distribution Bastien Marchina (UM2) Complex random vectors 26 march / 49

42 Applications to goodness-of-t tests Simulation : case m = 3 Tab.: Quantiles of the distribution of ˆξ n based on repetitions with m = 3 n E 0 V 0 Q 90 Q 95 Q χ Bastien Marchina (UM2) Complex random vectors 26 march / 49

43 Applications to goodness-of-t tests P-P plot for the case m = 3 p p plot for xi(3) vs. a chi2(6) distribition empirical cumulative distribution theoretical cumulative distribution Bastien Marchina (UM2) Complex random vectors 26 march / 49

44 Applications to goodness-of-t tests Back to the real data Fig.: ˆξn applied to the datasets Obersved value for xi(3) using the FMRI dataset Sample number Observed value Bastien Marchina (UM2) Complex random vectors 26 march / 49

45 Applications to goodness-of-t tests Back to the real data Examining closely sets associated with high ˆξ n value, they show unusually high values in the rst and last observations. Here is the 5050-th dataset, ˆξ n = Fig.: fmri data representation 5050 th dataset : Modulus Modulus Bastien Marchina (UM2) Complex random vectors 26 march / 49

46 Applications to goodness-of-t tests Back to the real data We can build an image of the brain slice with colors depending on ˆξ n. Brain representation using xi(3) on the FMRI dataset y x Bastien Marchina (UM2) Complex random vectors 26 march / 49

47 Applications to goodness-of-t tests Back to the real data Alternatively, and using ˆξ n,. Brain representation using sqrt(xi(3)) on the FMRI dataset y x Bastien Marchina (UM2) Complex random vectors 26 march / 49

48 Applications to goodness-of-t tests 1 T. Adali, P. J. Schreier, L. L. Scharf Complex-Valued Signal Processing : The Proper Way to Deal With Impropriety, IEEE Transactions on Signal Processing, vol. 59, n.11, pp , November Rowe, D. B. and Logan, B. R. A complex way to compute fmri activation, NeuroImage, vol. 23, , I.A. Koutrouvelis, A Goodness-of-t Test of Simple Hypotheses Based on the Empirical Characteristic Function, Biometrika, vol. 67, n. 1, pp A. van den Bos, The Multivariate Complex Normal Distribution - a Generalization, IEEE Transactions on Information Theory, vol. 31, pp , R.A. Wooding, The Multivariate Complex Distribution of Complex Normal Variables, Biometrika, vol. 43, pp , Bastien Marchina (UM2) Complex random vectors 26 march / 49

49 Applications to goodness-of-t tests Thank you for your attention Bastien Marchina (UM2) Complex random vectors 26 march / 49