* Université Montpellier II, I3M - EPS ** Université de Montréal,...

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 2013 1 / 49

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


Introduction

1 Introduction






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.


Introduction

Example of complex random data

Fig.: fMRI data representation


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.


Thus arises the need for a proper test for complex normality.


Introduction

Complex random variables in mathematical statistics

The probability distribution P of X can be characterised by its

characteristic function

ϕX(t) = E(e i(t′X)).

.

The empirical characteristic function

ϕn(t) =1

n

n∑k=1

e i(t′Xk)

of independent copies X1, . . . ,Xn, of X ∼ P is an estimator of the

characteristic function of P and is complex valued.


Complex random vectors

1 Introduction







Complex Random Vectors

If X and Y are random vectors in Rd , Z = X+ iY is a random vector in

Cd .

Ze = (Z′,ZH)

′is the augmented vector associated with Z and

X = MZe , Ze = 2MHX , MMH = 2Id , (1)

with X = (X′,Y

′)′and

M =1

2

(Id Id−i Id i Id

), M−1 =

(Id i IdId −i Id

). (2)

The characteristic function of Z is

ϕZ(ν) = E(e i Re(ν

HZ)). (3)



First and second order parameters for complex random

vectors

Expectation : µ = E(Z) = µX + iµY,Positive semi-de�nite hermitian covariance matrix :

Γ = E[(Z− µ)(Z− µ)H

], (4)

Positive semi-de�nite symmetric relation matix :

P = E[(Z− µ)(Z− µ)

′], (5)

Positive semi-de�nite hermitian covariance-relation matrix

ΓP = E[(Ze − µe )(Ze − µe )H] =

(Γ P

PH Γ∗

). (6)



Complex normal distribution

De�nition (Van den Bos (1995), or Picinbono (1996))

Z = X+ iY ∈ Cd is following the complex normal distribution i�(X

Y

)∼ N

((µX

µY

),

(ΣXX ΣXY

ΣYX ΣYY

)). (7)

We write Z ∼ CNd (µ, Γ,P), or alternatively Ze ∼ CeN2d (µe , ΓP) with

ΣX = MΓPMH , ΓP = M−1ΣX (MH)−1. (8)



Proper normal distribution

Consider Z ∼ CNd (µ, Γ, 0), i.e., Z is a complex normal random vector with

P = 0. Then,

fZ(z) =1

πd |Γ|1/2exp{

(z− µ)HΓ−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.



Proper and improper complex normal data

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−3 −2 −1 0 1 2 3

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23

kn = 0

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−3 −2 −1 0 1 2 3

−3

−2

−1

01

23

kn = 0.3

Partie réelle

Pa

rtie

im

ag

ina

ire

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−3 −2 −1 0 1 2 3

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01

23

kn = 0.6

Partie réelle

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−3 −2 −1 0 1 2 3

−3

−2

−1

01

23

kn = 0.9

Partie réelle

Pa

rtie

im

ag

ina

ire



Characteristic function and density

The characteristic function of Z is

ϕZ(`) = exp

{i Re(`Hµ)− 1

4

(`HΓ`− Re(`HP`∗)

)}. (10)

If ΓP is invertible

fZ(z) =1

πd |ΓP |1/2exp

{−12

((z− µ)′, (z− µ)H)Γ−1P

(z− µ

(z− µ)∗

)}.

(11)



Complex normal distribution � Main results

Conservation by linear transforms

Let Z ∼ CNd (µ, Γ,P) and A ∈ Cm×d , then

AZ ∼ CNm(Aµ,AΓAH,APA′). (12)

Conservation by augmented linear transforms

Let Ze ∼ CeN2d (µe , ΓP), AB ∈ C2m×2d such that

AB =

(A B

B∗ A∗

), (13)

then ABZe ∼ CeN2m(ABµe ,ABΓPAHB ).



Complex normal distribution � Independence

Theorem : Independence between complex gaussian variables

Let Z = (Z1,Z2) ∼ CN2(µ, Γ,P). Z1 and Z2 are independent if and only if

Γ =

(γ1 0

0 γ2

),

and

P =

(p1 0

0 p2

).



Complex normal distribution � Independence

Corollary : Independence between complex gaussian vectors

Let Z ∼ CNd1+d2(µ, Γ,P), Partition Z as (Z′1,Z

′2)′where Z1 is d1 × 1, Z2

is d2× 2, and likewise µ into (µ1, µ2), Γ in

(Γ1 Γ12

ΓH12 Γ2

)and similarly for P .

Z1 and Z2 are independent if and only if Γ12 = P12 = 0.

Corollary : Independence between components of a complex gaussian vector

Let Z = (Z1, . . . ,Zd )′ ∼ CNd (µ, Γ,P). The components Z1, . . . ,Zd are

independent if and only if Γ and P are diagonal.


Hermitian quadratic forms

1 Introduction








Let Z ∼ CNd (µ, Γ,P). We study positive quadratic forms of the form

ZeHRZe .First, notice that

ZeHRZe = 2X ′SX , (14)

with S = MRM−1. It leads to

R = M−1SM

=

(S11 + S22 + i(S12 − S21) S11 − S22 + i(S12 + S21)S11 − S22 − i(S12 + S21) S11 + S22 − i(S12 − S21)

),

and �nally, because S is a symmetric matrix,

R =

(G K

KH G ∗

). (15)




Theorem

Let

R =

(G K

KH G∗

). (16)

and Z ∼ CNd (µ, Γ,P). Then

ZeHRZe =2d∑k=1

αkχ21(δ2k), (17)

where the χ21(δ2k) are independent, the αk are the eigenvalues of RΓP and

the δk are function of µ, Γ, P and R.




Corollary

Let

R =

(G K

KH G∗

). (18)

and Z ∼ CNd (0, Γ,P). Then

ZeHRZe =2d∑k=1

αkχ21, (19)

where the χ21 are independent and the αk are the eigenvalues of RΓP .



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 6= 0. Then (αA)+ = α−1A+,

2 (A′)+ = (A+)′, (A∗)+ = (A+)∗ and (AH)+ = (A+)H,

3 Let A be m × n complex and B be n × p complex. If AAH = Im or

BHB = Ip, then (AB)+ = B+A+,

4 If A is invertible, then A+ = A−1.



More results on hermitian quadratic forms

Theorem

Let Z ∼ CNd (0, Γ,P). Let Γ+P be the Moore-Penrose inverse of the

covariance-relation matrix ΓP of Z. Then,

ξ = ZeHΓ+PZe ∼ χ2q, (20)

where q ≤ 2d is the rank of ΓP .

Corollary

Let Z ∼ CNd (0, Γ,P), such that ΓP is invertible. Then,

ξ = ZeHΓ−1P Ze ∼ χ22d . (21)




Theorem

Let Z ∼ CNd (µ, Γ,P), Ze =

(Z

Z

), m =

(µµ

), A and B two 2d × 2d

matrices, that share properties with matrix R in (18). Then ZeHAZe and

ZeHBZe are independent if and only if

(i) ΓPAΓPBΓP = 0,

(ii) ΓPAΓPBΓPm = ΓPBΓPAm = 0,

(iii) mHAΓPBΓPm = 0.




Theorem

Let Y ∼ CNd (0, ΓY,PY) and Z ∼ CNd′ (0, ΓZ,PZ), such that Y and Z are

independent. Let ΓP,Y and ΓP,Z be the covariance-relation matrices of Y

and Z respectively. Then,

b

a

YeHΓ+P,YYe

ZeHΓ+P,ZZe

∼ F(a, b),

where a ≤ 2d is the rank of ΓP,Y, b ≤ 2d′is the rank of ΓP,Z and F(a, b)

denotes the Fisher distribution with degrees of freedom a and b.




Corollary

Let Z = (Z′1,Z

′2)′ ∼ CNd1+d2(0, Γ,P). Let ΓP,1 and ΓP,2 be the

covariance-relation matrices of the marginal distributions of Z1 and Z2,

with respective ranks a and b and a + b ≤ 2d Then,

b

a

Z1e HΓ+P,1Z1e

Z2e HΓ+P,2Z2e

∼ F(a, b), (22)

if and only if if Z1 and Z2 are independent.


Statistics of complex random vectors

1 Introduction







Central limit theorem for complex random vectors

Theorem

Let W be a random vector in Cd , E(W) = µ, with de�nite covariance and

relation matrices Γ and P . Let W1, . . . ,Wn be independant copies of W.

Then,

√n

1

n

n∑j=1

Wj − µ

CNd (0, Γ,P). (23)



Maximum likelihood and moments method estimators

Z1, . . . ,Zn 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

µ =1

n

n∑k=1

Zk = Z,

Γ =1

n

n∑k=1

(Zk − Z)(Zk − Z)H,

P =1

n

n∑k=1

(Zk − Z)(Zk − Z)′,

ΓP =1

n

n∑k=1

(Zke − Ze )(Zke − Ze )H.



Asymptotics in the case d = 1

In this case,

√n

µ− µγ − γp − p

CN3(0, Γθ,Pθ), (24)

where

Γθ =

γ 0 0

0 γ2 + |p|2 2p∗γ0 2pγ 2γ2

, Pθ =

p 0 0

0 γ2 + |p|2 pγ0 2pγ 2p2

. (25)


Applications to goodness-of-�t tests

1 Introduction







Empirical characteristic function and empirical characteristic

process

Let X1, . . . ,Xn be a random sample with characteristic function ϕX(·). Inorder to test H0 : ϕX(·) = ϕ0(·) it is customary to intoduce the empirical

characteristic process

Un(t) =√n(ϕn(t)− ϕ0(t)). (26)



The empirical characteristic process

Under H0, the empirical characteristic process is such that

E(Un(·)) = 0

C (s, t) = E(Un(s)Un(t)∗) = ϕ0(s− t)− ϕ0(s)ϕ0(t)∗,

P(s, t) = E(Un(s)Un(t)) = ϕ0(s+ t)− ϕ0(s)ϕ0(t) = C (s,−t).

(27)



Koutrouvelis's goodness-of-�t test

Koutrouvelis (1980) gives a test statistic of H0 : P = P0 based on the

evaluation of Un(·) on cleverly chosen points t1, . . . , td .

With Wn = (R1, . . . ,Rd , I1, . . . Id )′, Rk = Re(Un(tk)) and Ik = Im(Un(tk)),

he shows that

W′nΣ−1Wn χ22d . (28)

under H0 with Σ the covariance matrix of Wn under H0.



Use of a complex quadratic form

Using a complex framework, we have the following test statistic

ξn = Une HΓ+PUne χ2q, (29)

with Une = (U′n,U

Hn )

′, Un = (Un(t1), . . . ,Un(td )) and q < 2d is the rank

of

ΓP =

(ΓUn PUnPHUn

Γ∗Un

), (30)

ΓUn =

0B@C(t1, t1) . . . C(t1, tm)...

...C(tm, t1) . . . C(tm, tm)

1CA , PUn =

0B@P(t1, t1) . . . P(t1, tm)...

...P(tm, t1) . . . P(tm, tm)

1CA (31)



Test of a simple hypothesis for complex distributions

A sample of complex random vectors has the following e.c.f.

ϕn(ν) =1

n

n∑k=1

e i Re(νHZk), (32)

and Un(·) has the same properties than in the real case. Once again,

Une HΓ+PUne χ2q, (33)

with Un = (Un(ν1), . . . ,Un(νd )) and q < 2d is the rank of ΓP .



Test for complex normality

Back to our signal processing problematic, we need to test the composite

hypothesis

H0 : P ∈ { CN1(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.




From the circularized empirical characteristic process

Un,Y (ν) =√n(ϕn,Y (ν)− ϕ0(ν)), (35)

where

ϕ0(ν) is the c.f. of a CN1(0, 1, 0),

Ye = Γ−1/2P (Ze − µe), and therefore Y ∼ CN1(0, 1, 0).

we build

Un,Y (ν) =

√n(ϕ

n,Y (ν)− ϕ0(ν)), (36)

where Y is obtained through the m.l.e. estimates of the parameters.




The modi�ed ξn test statistic follows

ξn = UeHn,Y ΓP(ν)−1Ue n,Y . (37)

Here

UHn,Y

= (Un,Y (ν1), . . . ,U

n,Y (νm))′

ΓP(ν), albeit complicated, does not depend on the true value of the

parameters, but only on the choice of points.



Simulation : case m = 1

Tab.: Quantiles of the distribution of ξn based on 30 000 repetitions with m = 1

n E0 V0 Q90 Q95 Q99

50 2.02 3.27 3.82 5.82 14.55

250 1.98 2.38 4.33 5.80 11.42

500 1.98 2.19 4.44 5.93 10.41

1000 1.99 2.12 4.48 5.87 9.97

χ22 2 2 4.60 5.99 9.21



P-P plot for the case m = 1

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

p−p plot for xi(1) vs. a chi2(2) distribition

theoretical cumulative distribution

empi

rical

cum

ulat

ive d

istrib

utio

n



Simulation : case m = 3

Tab.: Quantiles of the distribution of ξn based on 30 000 repetitions with m = 3

n E0 V0 Q90 Q95 Q99

50 6.07 5.45 10.87 14.64 27.10

250 6.00 4.07 10.74 13.37 20.85

500 6.00 3.85 10.68 13.04 19.16

1000 6.00 3.66 10.68 12.83 17.93

χ26 6 3.46 10.64 12.59 16.81



P-P plot for the case m = 3

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

p−p plot for xi(3) vs. a chi2(6) distribition

theoretical cumulative distribution

empi

rical

cum

ulat

ive d

istrib

utio

n



Back to the real data

Fig.: ξn applied to the 16384 datasets

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0 5000 10000 15000

010

0020

0030

0040

0050

0060

00Obersved value for xi(3) using the FMRI dataset

Sample number

Obs

erve

d va

lue




Examining closely sets associated with high ξn value, they show unusually

high values in the �rst and last observations. Here is the 5050-th dataset,

ξn = 4676.


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0 50 100 150 200 250

0.0

0.1

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0.4

0.5

5050−th dataset : Modulus

Index

Mod

ulus




We can build an image of the brain slice with colors depending on ξn.

30 40 50 60 70 80 90 100

3040

5060

7080

9010

0

Brain representation using xi(3) on the FMRI dataset

x

y




Alternatively, and using

√ξn,.

30 40 50 60 70 80 90 100

3040

5060

7080

9010

0Brain representation using sqrt(xi(3)) on the FMRI dataset

x

y



1 T. Adali, P. J. Schreier, L. L. Scharf �Complex-Valued Signal Processing : TheProper Way to Deal With Impropriety�, IEEE Transactions on Signal Processing,vol. 59, n.11, pp.5101�5124, November 2011.

2 Rowe, D. B. and Logan, B. R. �A complex way to compute fMRI activation�,NeuroImage, vol. 23, 1078�1092, 2004.

3 I.A. Koutrouvelis, �A Goodness-of-�t Test of Simple Hypotheses Based on theEmpirical Characteristic Function�, Biometrika, vol. 67, n. 1, pp. 238�240.

4 A. van den Bos, �The Multivariate Complex Normal Distribution - aGeneralization�, IEEE Transactions on Information Theory, vol. 31, pp.537�539,1995.

5 R.A. Wooding, �The Multivariate Complex Distribution of Complex Normal

Variables�, Biometrika, vol. 43, pp.212�215, 1956.



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