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Asymptotic Performance of Complex M-estimators for
Multivariate Location and Scatter Estimation
Bruno Meriaux, Chengfang Ren, Mohammed Nabil El Korso, Arnaud Breloy,
Philippe Forster
To cite this version:
Bruno Meriaux, Chengfang Ren, Mohammed Nabil El Korso, Arnaud Breloy, Philippe Forster.
Asymptotic Performance of Complex M-estimators for Multivariate Location and Scatter
Estima-tion. IEEE Signal Processing Letters, Institute of Electrical and Electronics Engineers, 2019, 26 (2),
pp.367-371. �10.1109/lsp.2019.2891201�. �hal-01968961�
Asymptotic Performance of Complex M -estimators
for Multivariate Location and Scatter Estimation
Bruno M´
eriaux, Student Member, IEEE, Chengfang Ren, Member, IEEE,
Mohammed Nabil El Korso, Arnaud Breloy, Member, IEEE, and Philippe Forster
Abstract—The joint estimation of means and scat-ter matrices is often a core problem in multivariate analysis. In order to overcome robustness issues, such as outliers from Gaussian assumption, M -estimators are now preferred to the traditional sample mean and sample covariance matrix. These estimators are well established and studied in the real case since the sev-enties. Their extension to the complex case has drawn recent interest. In this letter, we derive the asymptotic performance of complex M -estimators for multivariate location and scatter matrix estimation.
Keywords—Complex observations, Robust estimation of multivariate location and scatter, Complex El liptical ly Symmetric distributions.
I. Introduction
S
EVERAL classical methods in multivariate analysis require the estimation of means and scatter matrices from collected observations [1]–[5]. In pratice, the Sample Mean and the Sample Covariance Matrix are classically used in such procedures. Indeed, they coincide with the Maximum Likelihood Estimators (MLE) for multivariate Gaussian data. However, they are neither robust to devi-ation from Gaussianity assumption nor to the presence of outliers, which could lead to a dramatic performance loss. To overcome these problems, several approaches have been proposed in the literature [6], [7] improving the estimator’s behavior under contamination through diverse criteria such as the breakdown point, the contamination bias, the finite-sample efficiency while preserving the computa-tionally feasibility for high dimension. Among the most frequently used robust affine equivariant estimators [8]– [17], we focus on the M -estimators, which have been first introduced within the framework of elliptical distributions [18]. The latters encompass a large number of classical distributions as for instance the Gaussian one but also non-Gaussian heavy-tailed distributions such as the t-, K-and W -distributions [19]. One of the main interests of the real M -estimators is to possess, under mild conditions, good asymptotic properties, namely weak consistency and asymptotic normality over the whole class of elliptical distributions [8], [20]–[22]. Their extension to the complex B. M´eriaux and C. Ren are with SONDRA, CentraleSup´elec, France. M.N. El Korso and A. Breloy are with LEME, Paris-Nanterre University, France. P. Forster is with SATIE, Paris-Nanterre Univer-sity, France.This work is financed by the Direction G´en´erale de l’Armement as well as the ANR ASTRID referenced ANR-17-ASTR-0015.
case has drawn recent interest [23], notably with the class of Complex Elliptically Symmetric (CES) distributions [24]. In the context of known mean, their asymptotic prop-erties are established in [25], [26]. In this letter, we extend this result for multivariate location and scatter matrix
M -estimates under CES distributed data. The achieved
outcome is analogous to its real-counterpart [21] and may be used for deriving the performance of adaptive processes in non-zero mean non Gaussian distributed observations [1].
In the following, the notation =,d → andd → indicate re-P spectively equality in distribution, convergence in law and in probability. The symbol ⊥ refers to statistical indepen-dence. The operator vec(A) stacks all columns of A, des-ignated by (a1, . . . , ai, . . .) into a vector. The operator ⊗
refers to Kronecker product. The notation GCN (0, Σ, Ω) refers to the zero mean non-circular complex Gaussian dis-tribution, where Σ (respectively Ω) denotes the covariance matrix (respectively pseudo-covariance matrix) [27], [28].
II. Problem setup
Let ZN , (z1, . . . , zN) be N i.i.d samples of
m-dimensional vectors, following a complex elliptical distri-bution, which is denoted by zn ∼ CESm(te, Λ, gz) and
whose p.d.f. is proportional to
pz(zn; te, Λ, gz) ∝ det(Λ)−1gz (zn− te)HΛ−1(zn− te). (1)
The vector te is the location parameter, Λ denotes the
scatter matrix and the function gzis the density generator
[29]. We aim to estimate jointly the location parameter and the scatter matrix from the observations. The complex joint M -estimators ˆtN, cMN are solutions of the system
SysN(ZN, u1, u2) [23]: 1 N N X n=1 u1(d (zn, t; M)) (zn− t) = 0 1 N N X n=1 u2 d2(zn, t; M)(zn− t) (zn− t)H | {z } H(ZN,t,M) = M (2) (3)
with d2(z, t; M) = (z − t)HM−1(z − t). Let us consider te, Me a solution related to the system Sys∞(z1, u1, u2):
E [u1(d (z1, t; M)) (z1− t)] = 0
Eu2 d2(z1, t; M)(z1− t) (z1− t)H, H∞(t, M) = M
(4) (5)
The functions u1(·) and u2(·) verify the conditions given
in [8] for the real case (for the special case, where u1(s) =
u2(s2), [30] provides more general conditions). The proofs
of Lemmas 1 and 2 and Theorems 1-3 of [8], addressing the existence and uniqueness of M -estimates, are transposable to the complex field, by following the same methodology as in [8]. The derivations of these proofs require the same conditions on u1(·) and u2(·) as the ones needed in the
real case. Thus, this ensures the existence of ˆtN,cMN and te, Me as well as the uniqueness of te, Me. In
this letter, we derive the statistical performance of the complex joint M -estimators ˆtN,cMN, namely consistency
and asymptotic distribution.
III. Consistency of the joint M -estimator Let ˆtN,McN
be a solution of SysN(ZN, u1, u2) and
te, Me be the solution of the system Sys∞(z1, u1, u2).
Theorem III.1. The complex joint M -estimators are
consistent, i.e.
ˆ tN,MbN
P
→ te, Me. (6)
with Me = σ−1Λ in which σ is the solution of
Eψ2 σ|ζ|2
= m, for ζ ∼ CESm(0m, Im, gz) and ψ2(s) = su2(s).
Proof: First, let us define θT=htT, vec (M)Ti and the functionΨN(θ) = " Ψ1,N(θ) = 1 N N P n=1 u1(d (zn, t; M)) (zn− t) Ψ2,N(θ) = vec (H (ZN, t, M) − M) # . The Strong Law of Large Numbers (SLLN) gives
∀ θ ∈ Θ ΨN(θ) P → Ψ (θ) (7) with ∀ θ ∈ Θ, Ψ (θ) =h Ψ1(θ) = E [u1(d (z1, t; M)) (z1− t)] Ψ2(θ) = vec (H∞(t, M) − M) i . According to the Theorem 5.9 [31, Chap. 5] and uniqueness of solution, we can show that any bθ
T N = h ˆtT N, vec McN Ti solution of ΨN b θN = 0 converges in probability to θTe = tTe, vec (Me) T solution of Ψ (θe) = 0, yielding the intended outcome. Furthermore, the matrix Me is
proportional to Λ through a scale factor σ−1 [32, Chap. 6]. Multiplying (5) by M−1 and taking the trace yields Eψ2 σ|ζ|2 = m of which σ is the solution.
IV. Asymptotic distribution of the joint
M -estimators A. Main theorem
We consider ˆtN,cMN a solution of SysN(ZN, u1, u2) as
well as te, Me the solution of Sys∞(z1, u1, u2).
Theorem IV.1. Assuming that sψi0(s) (i = 1, 2) are bounded and Ehψ10 (√σ|ζ|)i > 0, the asymptotic
distri-bution of ˆtN, cMN is given by
√
N
ˆtN− te, veccMN− Me d
→ (a, b) with a ⊥ b and
a ∼ CN 0, Σt=
α β2Me
, b ∼ GCN (0, ΣM, ΩM = ΣMKm)
where Km is the commutation matrix satisfying
Kmvec (A) = vec AT [33] and ΣM is obtained by
ΣM = σ1MTe ⊗ Me+ σ2vec (Me) vec (Me)H, (8)
in which α = m−1 Eψ12( √ σ|ζ|) β = E 1 − (2m)−1 u1( √ σ|ζ|) + (2m)−1ψ0 1 √ σ|ζ| σ1= a1(m + 1)2 (a2+ m)2 , σ2= a−22 h (a1− 1) − a1(a2− 1) m + (m + 2)a2 (a2+ m)2 i a1= Eψ2 2 σ|ζ|2 m(m + 1) , a2= Eσ|ζ|2ψ0 2 σ|ζ|2 m with σ solution of Eψ2 σ|ζ|2 = m in which ζ ∼ CE Sm(0m, Im, gz).
B. Proof of Theorem IV.1
The starting point of the proof is to map the complex joint M -estimators ˆtN, cMN into real-ones, then to study
the asymptotic behavior of the latter, and finally to relate the latter to the asymptotic distribution of the complex joint M -estimators ˆtN, cMN.
1) Complex vector space isomorphism: Let us first
intro-duce functions h : Cm→ Rp
and f : Cm×m→ Rp×p with
p = 2m defined byh (a) = < (a)T, = (a)TT and
f (A) =1
2
< (A) −= (A) = (A) < (A)
In addition, let P ∈ Rp×p be the matrix P = 0
m×m −Im
Im 0m×m
. Some useful properties of the previous functions are given in [25]. Furthermore, we set ˆtR
N = h ˆtN , cMRN = f c MN
, MR = f (Me) and tR = h (te). In
addition, let us define un= h (zn) ∼ E Sp(tue, ΛR, gz) and
vn = Pun ∼ ESp(tve, ΛR, gz) for zn ∼ CESm(te, Λ, gz),
where tue = h (te), tve = Ptue and ΛR = f (Λ). The
notation E S refers to real elliptical distributions [18]. Moreover, there exist another relation between the vectors
zn, un and vn, for any Hermitian matrix, A ∈ Cm×m[25]:
2zHnA −1 zn= uTnf (A) −1 un= vTnf (A) −1 vn (9)
Let us apply the function f (·) to the equation (3) (respec-tively h(·) to (2)), we obtain b MR N= 1 2N N X n=1 u2,R d2 un, ˆtRN;MbRN un− ˆtRN un− ˆtRN T (10) + 1 2N N X n=1 u2,R d2 vn, PˆtRN;MbRN vn− PˆtRN vn− PˆtRN T and 1 N N X n=1 u1,R d un, ˆtRN;MbRN un− ˆtRN = 0 (11)
where u2,R(s) = u2 2−1s
and u1,R(s) = u1 2−1/2s
according to (9). Let ψi,R(·) be the functions related to
ui,R(·) by ψi,R(s) = sui,R(s), i = 1, 2. Finally, we introduce the two following real joint M -estimators ˆtu
N, cMuN and
ˆtv N, cMvN
respectively solution of SysN(UN, u1,R, u2,R)
and SysN(VN, u1,R, u2,R). From the results in the real
case on the consistency [8], [21], we obtain ( ˆtu N,Mb u N P → (tu, Mu) = tue, σ −1 R ΛR ˆtv N,MbvN P → (tv, Mv) = tve, σ −1 R ΛR (12)
in which (tu, Mu) and (tv, Mv) are solutions of
Sys∞(u1, u1,R, u2,R) and Sys∞(v1, u1,R, u2,R) and
σR is the solution of Eψ2,R σR|u|2
= p = 2m,
u ∼ E Sp(0, Ip, gz). Thus, we have Mu= σ−1R ΛR= Mv.
Finally, by applying P to the system
SysN(UN, u1,R, u2,R), we obtain cMvN = P cMuNP T
and ˆtv N = P ˆt
u
N. Moreover, since vn = Pun, ∀ n, we have
Mu= Mv= PMuPT and tv = Ptu.
2) Link between asymptotic behaviors of ˆtu N,cMuN, ˆtv N,cM v N and ˆtRN,cMRN: Lemma IV.1. ˆtR
N andˆtuN (respectively cMRN and
1 2 Mb u N+ b Mv
N) share the same asymptotic Gaussian law.
Proof: See Appendix.
3) Asymptotic behavior of ˆtN,McN: Sincevec McR
N and
ˆtR
N have an asymptotic Gaussian distribution according
to Lemma IV.1, and vec cMN
= gTm⊗ gH m vec cMRN and ˆtN = gH
mˆtRN withgm= (Im, −jIm)
T. Consequently, ˆt
N and
vec McN have a non-circular complex Gaussian
distribu-tion [27]. Addidistribu-tionally, using the same approach as in [25], we obtain
√
N vec MbN− Me d
→ b ∼ GCN (0, ΣM, ΩM) (13)
Regarding the location estimate, we have
√ N ˆtN− te d → GCN (0, Σt, Ωt) , where (14) Σt= gHmN E h ˆtu N− tu ˆtuN− tu Ti gm= α β2g H mMRgm= α β2Me in which α = (2m)−1 Eψ21,R( √ σ|x|)= m−1 Eψ12( √ σ|ζ|) β = E 1 − (2m)−1u1( √ σ|ζ|) + (2m)−1ψ01 √σ|ζ|
with x ∼ E Sp(02m, I2m, gz) and ζ ∼ CESm(0m, Im, gz)
hence |ζ| = 2d −1/2|x|. Furthermore, we have the relations
ψ1,R(s) = √2ψ1 2−1/2s and ψ1,R0 (s) = ψ01 2−1/2s. Moreover, we have Ωt= gHmN E h ˆtR N− tR ˆ tR N− tR Ti g∗m= α β2g H mMRg ∗ m= 0.
Thus, we prove that √N ˆtN− te d
→ a ∼ CN (0, Σt). Applying Lemma IV.1, we have ω ⊥ χ and consequently, ( a= gd HmM 1/2 R D −1 χ b=d 1 2 g T m⊗ g H m Ip2+ (P ⊗ P) M1/2 R ⊗ M 1/2 R A−1ω ,
thus we prove that a ⊥ b.
V. Simulations
In order to illustrate our theoretical results, some sim-ulations results are presented. Two scenarios have been considered for the simulations. For m = 3, the true location parameter is te= (1 + 0.5i, 2 + i, 3 + 1.5i)T and the true
scatter matrix is Λ = Im, due to the affine equivariance
propertie of the M -estimators, there is no loss of generality. • Case 1 : the data are generated under a t-distribution
with d = 4 degrees of freedom [34].
• Case 2 : the data are generated under a K-distribution with shape parameter ν = 4 and scale parameter θ = 1/ν [26].
The complex joint M -estimators is obtained with u(s) ,
u2(s) =
d + m d + s = u1(
√
s) and the reweighting algorithm
of [30], whose convergence is established.
The first case coincides with the MLE unlike the case 2, which is a general complex joint M -estimator.
101 102 10−1 100 Number of samples N T r { MSE } (dB) Tr (ΣM) case 1 TrMSE MbN case 1 Tr (ΣM) case 2 TrMSE MbN case 2 Tr (Σt) case 1 TrMSE btN case 1 Tr (Σt) case 2 TrMSE btN
case 2
Fig. 1: Second order moment simulations
In Fig. 1, we plot the trace of the Mean Squared Error of the estimates of the location and the scatter matrix as well as the trace of the theoretical asymptotic covariance matrices Σt and ΣM. The results are validated since the
drawn quantities are identical when N → ∞. Moreover these quantities tend asymptotically to zero, which illus-trate the consistency.
VI. Conclusion
In this letter, we established the asymptotic performance of the joint M -estimators for the complex multivariate location and scatter matrix. This statistical study high-lights a better understanding on the performance of the
M -estimators of the complex multivariate location and
scatter matrix. Again, the obtained results may be used for conducting a performance analysis of adaptive processes involving non-zero mean observations.
TABLE I: Equations for the proof of Lemma IV.1 Ip+ ∆Mu= 1 N N P n=1 an knkTn− kn∆t T uM −1/2 u − M −1/2 u ∆tuk T n + bnkTn∆Mukn+ 2kTnM −1/2 u ∆tu knkTn (15) 0 = 1 N N P n=1 cn kn− M−1/2u ∆tu + 1 N N P n=1 dn kTn∆Mukn 2|kn| + kT nM −1/2 u ∆tu |kn| kn (16) AN= Ip2− 1 N N P n=1 bn(kn⊗ kn) (kn⊗ kn)T→ A = IP p2+ 1 p(p + 2)E |k1|4u0 2,R |k1| 2
Ip2+ Kp+ vec (Ip) vec (Ip)T (17)
BN= 1 N N P n=1 (kn⊗ Ip) anIp− bnknkTn P → E|k1|u2,R |k1|2 E [(κ ⊗ Ip)] + E|k1|3u 0 2,R |k1|2 E(κ ⊗ Ip) κκT = 0 (18) CN= − 1 N N P n=1 dn knkTn 2|kn| (Ip⊗ kn)T P→ E u01,R(|k1|) k1kT1 2|k1| (Ip⊗ k1)T =1 2E |k1|2u 0 1,R(|k1|) EκκT(I p⊗ κ) = 0 (19) DN= 1 N N P n=1 cnIp− dn knkTn |kn| P → D = E [u1,R(|k1|)] + 1 pE |k1|u01,R(|k1|) Ip (20) Ip2+ (P ⊗ P)ωN/ √ N = AN+ (P ⊗ P) AN(P ⊗ P)T vec (∆MR) + Ip2+ Kp Ip2+ (P ⊗ P)BNM −1/2 R ∆tR (21) EωχT= E|k1|3u2,R |k1|2 u1,R(|k1|)E(κ ⊗ κ) κT− E|k1|u2,R |k1|2 u1,R(|k1|)Evec (Ip) κT = 0 (22) Appendix Proof of Lemma IV.1
A. Asymptotic behavior of ˆtu
N,cMuN and ˆtvN,cMvN
First of all, let us define Wcu = M
−1/2 u cMuNM −1/2 u and kn= M −1/2
u (un− tu). Since ˆtuN,cM
u N
P
→ (tu, Mu), we can write for N → ∞,cWu= Ip+ ∆MuandˆtuN = tu+ ∆tu. Let
us notean= u2,R |kn|2 , bn= −u02,R |kn|2 , cn= u1,R(|kn|) and dn = −u 0
1,R(|kn|) and use first order expansions for
N sufficiently large, then we obtain (15) and (16) from
SysN(UN, u1,R, u2,R). By vectorizing (15) and after some
calculus, we obtain ωN= AN √ N vec (∆Mu) + Ip2+ KpBN √ N M−1/2u ∆tu χN= CN √ N vec (∆Mu) + DN √ N M−1/2u ∆tu (23) where √N ωN= N P n=1
an(kn⊗ kn)−vec (Ip)and √ N χN= N P n=1 cnkn.
Remark. Note that kn ∼ ESp(0, σRIp, gz). Let be κ = k1
|k1|
, thus we have κ ⊥ |k1| and E [κ] = 0, E
κκT
= Ip
p, all 3rd-order moments vanish and the only non-vanishing 4th-order moments are Eκ4i
= 3 p(p + 2) and E κ2iκ2j −1 = p(p + 2) for i , j where κ = [κ1, . . . , κp] T .
The SLLN yields to (17)–(20). Furthermore, since
N−1
N
P
n=1
an(kn⊗ kn) → vec (Ip)P and N−1 N
P
n=1
cnkn → 0pP , it
yields from the central limit theorem that ωN d
→ ω and
χN → χ with ω and χ zero-mean Gaussian distributed.d Applying Slutsky’s lemma [32], it comes
√ N ˆtu N− tu d = √ N ∆tu d → M1/2u D−1χ √ N vec Mb u N− Mu d → M1/2u ⊗ M1/2u A−1ω
In the same way, we obtain √ N vec Mb v N− Mv d → (P ⊗ P) M1/2u ⊗ M1/2u A−1ω √ N ˆtv N− tv d → PM1/2u D−1χ B. Asymptotic behavior of ˆtR N,cMRN
With the results of Theorem III.1, the continuous map-ping theorem implies
ˆ tR N,Mb R N = h ˆtN , f MbN P
→ (h (te) , f (Me)) = (tR, MR) and (h (te) , f (Me)) = h (te) , σ−1f (Λ)= h (te) , σ−1ΛR= tu, σR−1ΛR Let us define cWR= M −1/2 R Mb R NM −1/2 R . SinceMR= Mu and tR = tu, kn becomes kn = M −1/2
R (un− tR) and satisfies Pkn= M
−1/2
R (vn− PtR). For N → ∞, we can writeWcR= Ip+ ∆MR andˆtRN= tR+ ∆tR. As previously, from (10) we
obtain (21). Since AN+ (P ⊗ P) AN(P ⊗ P)T
→ 2A, theP
Slutsky’s lemma leads to
√ N vec MbR N− MR d →1 2 Ip2+ P ⊗ P M1/2u ⊗ M1/2u A−1ω
Similarly, from (11) we obtain χN= CN √ N vec (∆MR) + DN √ N M−1/2 R ∆tR and thus √N ˆtR N− tR d
→ M1/2R D−1χ, which means that
ˆ tR
N and ˆt u
N have the same asymptotic distribution.
Lastly, we also introduce ξn =
ha
n(kn⊗ kn) − vec (Ip)
cnkn
i , which are zero mean and i.i.d., then
ω N χN = √1 N N X n=1 ξn d → ξ =ω χ ∼ N (0, Var (ξ1))
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