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
Bastien Marchina (UM2) Complex random vectors 26 march 2013 2 / 49
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 2013 3 / 49
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 2013 4 / 49
Introduction
Example of complex random data
Fig.: fMRI data representation
Bastien Marchina (UM2) Complex random vectors 26 march 2013 5 / 49
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 2013 6 / 49
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.
Bastien Marchina (UM2) Complex random vectors 26 march 2013 7 / 49
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 2013 8 / 49
Complex random vectors
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)
Bastien Marchina (UM2) Complex random vectors 26 march 2013 9 / 49
Complex random vectors
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)
Bastien Marchina (UM2) Complex random vectors 26 march 2013 10 / 49
Complex random vectors
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)
Bastien Marchina (UM2) Complex random vectors 26 march 2013 11 / 49
Complex random vectors
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.
Bastien Marchina (UM2) Complex random vectors 26 march 2013 12 / 49
Complex random vectors
Proper and improper complex normal data
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−3 −2 −1 0 1 2 3
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01
23
kn = 0
Partie réelle
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−3 −2 −1 0 1 2 3
−3
−2
−1
01
23
kn = 0.3
Partie réelle
Pa
rtie
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ag
ina
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−3 −2 −1 0 1 2 3
−3
−2
−1
01
23
kn = 0.6
Partie réelle
Pa
rtie
im
ag
ina
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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
Bastien Marchina (UM2) Complex random vectors 26 march 2013 13 / 49
Complex random vectors
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)
Bastien Marchina (UM2) Complex random vectors 26 march 2013 14 / 49
Complex random vectors
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 ).
Bastien Marchina (UM2) Complex random vectors 26 march 2013 15 / 49
Complex random vectors
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
).
Bastien Marchina (UM2) Complex random vectors 26 march 2013 16 / 49
Complex random vectors
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.
Bastien Marchina (UM2) Complex random vectors 26 march 2013 17 / 49
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 2013 18 / 49
Hermitian quadratic forms
Hermitian quadratic forms
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)
Bastien Marchina (UM2) Complex random vectors 26 march 2013 19 / 49
Hermitian quadratic forms
Hermitian quadratic forms
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.
Bastien Marchina (UM2) Complex random vectors 26 march 2013 20 / 49
Hermitian quadratic forms
Hermitian quadratic forms
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 .
Bastien Marchina (UM2) Complex random vectors 26 march 2013 21 / 49
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 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.
Bastien Marchina (UM2) Complex random vectors 26 march 2013 22 / 49
Hermitian quadratic forms
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)
Bastien Marchina (UM2) Complex random vectors 26 march 2013 23 / 49
Hermitian quadratic forms
More results on hermitian quadratic forms
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.
Bastien Marchina (UM2) Complex random vectors 26 march 2013 24 / 49
Hermitian quadratic forms
More results on hermitian quadratic forms
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.
Bastien Marchina (UM2) Complex random vectors 26 march 2013 25 / 49
Hermitian quadratic forms
More results on hermitian quadratic forms
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.
Bastien Marchina (UM2) Complex random vectors 26 march 2013 26 / 49
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 2013 27 / 49
Statistics of complex random vectors
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)
Bastien Marchina (UM2) Complex random vectors 26 march 2013 28 / 49
Statistics of complex random vectors
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.
Bastien Marchina (UM2) Complex random vectors 26 march 2013 29 / 49
Statistics of complex random vectors
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)
Bastien Marchina (UM2) Complex random vectors 26 march 2013 30 / 49
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 2013 31 / 49
Applications to goodness-of-�t tests
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)
Bastien Marchina (UM2) Complex random vectors 26 march 2013 32 / 49
Applications to goodness-of-�t tests
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)
Bastien Marchina (UM2) Complex random vectors 26 march 2013 33 / 49
Applications to goodness-of-�t tests
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.
Bastien Marchina (UM2) Complex random vectors 26 march 2013 34 / 49
Applications to goodness-of-�t tests
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)
Bastien Marchina (UM2) Complex random vectors 26 march 2013 35 / 49
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.
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 .
Bastien Marchina (UM2) Complex random vectors 26 march 2013 36 / 49
Applications to goodness-of-�t tests
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.
Bastien Marchina (UM2) Complex random vectors 26 march 2013 37 / 49
Applications to goodness-of-�t tests
Test for complex normality
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.
Bastien Marchina (UM2) Complex random vectors 26 march 2013 38 / 49
Applications to goodness-of-�t tests
Test for complex normality
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.
Bastien Marchina (UM2) Complex random vectors 26 march 2013 39 / 49
Applications to goodness-of-�t tests
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
Bastien Marchina (UM2) Complex random vectors 26 march 2013 40 / 49
Applications to goodness-of-�t tests
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
Bastien Marchina (UM2) Complex random vectors 26 march 2013 41 / 49
Applications to goodness-of-�t tests
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
Bastien Marchina (UM2) Complex random vectors 26 march 2013 42 / 49
Applications to goodness-of-�t tests
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
Bastien Marchina (UM2) Complex random vectors 26 march 2013 43 / 49
Applications to goodness-of-�t tests
Back to the real data
Fig.: ξn applied to the 16384 datasets
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00Obersved value for xi(3) using the FMRI dataset
Sample number
Obs
erve
d va
lue
Bastien Marchina (UM2) Complex random vectors 26 march 2013 44 / 49
Applications to goodness-of-�t tests
Back to the real data
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.
Fig.: fMRI data representation
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5050−th dataset : Modulus
Index
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ulus
Bastien Marchina (UM2) Complex random vectors 26 march 2013 45 / 49
Applications to goodness-of-�t tests
Back to the real data
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
Bastien Marchina (UM2) Complex random vectors 26 march 2013 46 / 49
Applications to goodness-of-�t tests
Back to the real data
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
Bastien Marchina (UM2) Complex random vectors 26 march 2013 47 / 49
Applications to goodness-of-�t tests
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.
Bastien Marchina (UM2) Complex random vectors 26 march 2013 48 / 49
Applications to goodness-of-�t tests
Thank you for your attention
Bastien Marchina (UM2) Complex random vectors 26 march 2013 49 / 49