dependent sources

Signal Processing 104 (2014) 319–324

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New blind source separation method of independent/ dependent sources A. Keziou a,n, H. Fenniri b, A. Ghazdali c, E. Moreau d,e a

Laboratoire de Mathématiques de Reims EA 4535, Université de Reims Champagne-Ardenne, France CReSTIC, Université de Reims Champagne-Ardenne, France LAMAI, FST, Université Cadi-Ayyad, Morocco d ENSAM, LSIS, UMR CNRS 7296, Université du Sud-Toulon-Var, France e Aix Marseille Université, France b c

a r t i c l e i n f o

abstract

Article history: Received 24 January 2014 Received in revised form 4 April 2014 Accepted 19 April 2014 Available online 30 April 2014

We introduce a new blind source separation approach, based on modified Kullback– Leibler divergence between copula densities, for both independent and dependent source component signals. In the classical case of independent source components, the proposed method generalizes the mutual information (between probability densities) procedure. Moreover, it has the great advantage to be naturally extensible to separate mixtures of dependent source components. Simulation results are presented showing the convergence and the efficiency of the proposed algorithms. & 2014 Elsevier B.V. All rights reserved.

Keywords: Blind source separation Modified Kullback–Leibler divergence between copulas Mutual information

1. Introduction Blind source separation (BSS) is an instrumental problem in signal processing which has been addressed in the last three decades. We consider an instantaneous linear mixture described by xðtÞ ≔AsðtÞ þnðtÞ A Rp ;

ð1Þ

where A A Rpp is an unknown non-singular mixing matrix, sðtÞ ≔ðs1 ðtÞ; …; sp ðtÞÞ > is the unknown vector of source signals to be estimated from xðtÞ ≔ðx1 ðtÞ; …; xp ðtÞÞ > , the vector of observed signals. The number of sources and the number of observations, for the present work, are assumed to be equal. The presence of additive noise nðtÞ within the mixing model complicates significantly the BSS problem. It is reduced by applying some form of n

Corresponding author. E-mail addresses: [email protected] (A. Keziou), [email protected] (H. Fenniri), [email protected] (A. Ghazdali), [email protected] (E. Moreau). http://dx.doi.org/10.1016/j.sigpro.2014.04.017 0165-1684/& 2014 Elsevier B.V. All rights reserved.

preprocessing such as denoising the observed signals through regularization approach, see e.g. [15]. The goal is to estimate the vector source signals sðtÞ using only the observed signals xðtÞ. The estimate yðtÞ of the source signals sðtÞ can be written as yðtÞ ¼ BxðtÞ;

ð2Þ pp

where B A R is the de-mixing matrix. The question is b which has to be how to obtain the de-mixing matrix B close to the ideal solution A  1 , using only the observed signals xðtÞ? It is well known, by Darmois theorem, that if the source components are mutually independent and at b most one component is Gaussian, a consistent estimate B of A  1 (up to scale and permutation indeterminacies of rows) is the one that makes the components of the vector yðtÞ independent; see e.g. [6]. The corresponding signals b b ðtÞ ≔BxðtÞ y are the estimate of the source signals sðtÞ. Under the above hypotheses, many procedures have been proposed in the literature. Some of these procedures use second or higher order statistics, see e.g. [13,2] and the references therein, others consist of optimizing (on the

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de-mixing matrix space) an estimate of some measure of dependency structure of the components of the vector yðtÞ. As measures of dependence used in BSS, we find in the literature the criterion of mutual information (MI) [14,7], the criteria of α, β and Renyi's-divergences [5,19], and the criteria of ϕ-divergences [15]. The procedures based on minimizing estimates of MI are considered as the most efficient, since this criterion can be estimated efficiently, other procedures using divergences lead to robust method for appropriate choice of divergence criterion [15]. In this paper, we will focus on the criterion of MI (called also modified Kullback–Leibler divergence), viewed as measure of difference between copula densities, and we will use it to propose a new BSS approach that applies in both cases of independent or dependent source components. In the following, we will show that the mutual information of a random vector Y ≔ðY 1 ; …; Y p Þ > can be written as the modified Kullback–Leibler divergence (KLm-divergence) between the copula of independence and the copula of the vector. Then, we propose a separation procedure based on minimizing an appropriate estimate of KLm-divergence between the copula density of independence and the copula density of the vector. This approach applies in the standard case, and we will show that the proposed criterion can be naturally extended to separate mixture of dependent source components. The proposed approach can be adapted also to separate complex-valued signals. In all the sequel, we assume that at most one source is Gaussian, and we will treat separately the case of independent source components, and then the case of dependent source components. Chen et al. [3] proposed a BSS algorithm (for independent source components) based on minimizing a distance between the parameter of the copula of the estimated source and the value of the parameter corresponding to copula of independence. Ma and Sun [9] proposed a different criterion combining the MI between probability densities and Shannon entropy of semiparametric models of copulas. 2. Brief recalls on copulas Consider a random vector Y ≔ðY 1 ; …; Y p Þ > A Rp , p Z1, with joint distribution function (d.f.) FY ðÞ: y A Rp ↦FY ðyÞ ≔ FY ðy1 ; …; yp Þ ≔PðY 1 ry1 ; …; Y p ryp Þ, and continuous marginal d.f.'s F Y j ðÞ: yj A R↦F Y j ðyj Þ ≔PðY j r yj Þ, 8 j ¼ 1; …; p. The characterization theorem of Sklar [17] shows that there exists a unique p-variate function CY ðÞ: ½0; 1p ↦½0; 1, such that, FY ðyÞ ¼ CY ðF Y 1 ðy1 Þ; …; F Y p ðyp ÞÞ, 8 y ≔ðy1 ; …; yp Þ > A Rp . The function CY ðÞ is called a copula and it is in itself a joint d.f. on ½0; 1p with uniform marginals. We have for all u ≔ðu1 ; …; up Þ > A ½0; 1p , CY ðuÞ ¼ PðF Y 1 ðY 1 Þ r u1 ; …; F Y p ðY p Þ r up Þ. Conversely, for any marginal d.f.'s F 1 ðÞ; …; F p ðÞ, and any copula function CðÞ, the function CðF 1 ðy1 Þ; …; F p ðyp ÞÞ is a multivariate d.f. on Rp . On the other hand, since the marginal d.f.'s F Y j ðÞ, j ¼ 1; …; p, are assumed to be continuous, then the random variables F Y 1 ðY 1 Þ; …; F Y p ðY p Þ are uniformly distributed on the interval [0, 1]. So, if the components Y 1 ; …; Y p are statistically independent, then the corresponding copula writes C0 ðuÞ ≔∏pj¼ 1 uj ; 8 u A ½0; 1p . It is called the copula of independence. Define, when it exists, the copula density (of the random vector Y)

cY ðuÞ ≔ð∂p =∂u1 ⋯∂up ÞCY ðuÞ; 8 u A ½0; 1p . Hence, the copula density of independence c0 ðÞ is the function taking the value 1 on ½0; 1p and zero otherwise, namely, c0 ðuÞ ≔1½0;1p ðuÞ;

8 u A ½0; 1p :

ð3Þ

Let f Y ðÞ, if it exists, be the probability density on Rp of the random vector Y ≔ðY 1 ; …; Y p Þ > , and, respectively, f Y 1 ðÞ; …; f Y p ðÞ, the marginal probability densities of the random variables Y 1 ; …; Y p . Then, a straightforward computation shows that, for all y A Rp , we have p

f Y ðyÞ ¼ ∏ f Y j ðyj ÞcY ðuÞ;

ð4Þ

j¼1

where u ≔ðu1 ; …; up Þ > ≔ðF Y 1 ðy1 Þ; …; F Y p ðyp ÞÞ > . In the monographs by [11,8], the reader may find detailed ingredients of the modeling theory as well as surveys of the commonly used semiparametric copulas. 3. Mutual information and copulas The MI of a random vector Y ≔ðY 1 ; …; Y p Þ > A Rp is defined by Z ∏pj¼ 1 f Y j ðyj Þ f Y ðyÞ dy1 ⋯dyp : MI ðY Þ ≔  log ð5Þ f Y ðyÞ Rp It is called also the modified Kullback–Leibler divergence (KLm-divergence) between the product of the marginal densities and the joint density of the vector. Note also that MIðYÞ≕KLm ð∏pj¼ 1 f Y j ; f Y Þ is nonnegative and achieves its minimum value zero if and only if (iff) f Y ðÞ ¼ ∏pj¼ 1 f Y j ðÞ, i.e., iff the components of the random vector Y are statistically independent. An equivalent formula of (5) is ! ∏pj¼ 1 f Y j ðY j Þ MI ðY Þ ≔E log ; ð6Þ f Y ðYÞ where EðÞ is the mathematical expectation. Using the relation (4), and applying the change variable formula for multiple integrals, we can show that MIðYÞ can be written as   Z 1 cY ðuÞ du≕KLm ðc0 ; cY Þ  log MI ðY Þ ¼ cY ðuÞ ½0;1p ¼ Eðlog cY ðF Y 1 ðY 1 Þ; …; F Y p ðY p ÞÞÞ≕ HðcY Þ; R where HðcY Þ ≔ ½0;1p logðcY ðuÞÞcY ðuÞ du is the Shannon entropy of the copula density cY ðÞ. The relation above means that the MI of the random vector Y can be seen as the KLm-divergence between the copula density of independent c0 ðÞ, see (3), and the copula density cY ðÞ of the random vector Y. We summarize the above results in the following proposition. Proposition 1. Let Y A Rp be any random vector with continuous marginal distribution functions. Then, the MI of Y can be written as the KLm-divergence between the copula density c0 of independence and the copula density of the vector Y:   Z 1 MI ðY Þ ¼ cY ðuÞ du≕KLm ðc0 ; cY Þ  log cY ðuÞ ½0;1p ¼ Eðlog cY ðF Y 1 ðY 1 Þ; …; F Y p ðY p ÞÞÞ:

ð7Þ

A. Keziou et al. / Signal Processing 104 (2014) 319–324

Moreover, KLm ðc0 ; cY Þ is nonnegative and takes the value zero iff cY ðuÞ ¼ c0 ðuÞ≕1½0;1p ðuÞ, 8 u A ½0; 1p , i.e., iff the components of the vector Y are statistically independent.

4. A separation procedure for mixtures of independent sources through copulas In this section, we describe our approach based on minimizing a nonparametric estimate of the KLm-divergence KLm ðc0 ; cY Þ, assuming that the source components are independent. Denote by S ≔ðS1 ; …; Sp Þ > the random source vector, X ≔AS the observed random vector and Y ≔BX the estimated random source vector. The discrete (noise free) version of the mixture model (1) writes xðnÞ ≔ AsðnÞ; n ¼ 1; …; N. The source signals sðnÞ; n ¼ 1; …; N, will be considered as N i.i.d. copies of the random source vector S, and then xðnÞ; yðnÞ ≔BxðnÞ; n ¼ 1; …; N, are, respectively, N i.i.d. copies of the random vectors X and Y ≔BX. When the source signal data sðnÞ; n ¼ 1; …; N, are not i.i.d., such as cyclo-stationary sources, see e.g. [10], we can return to the above i.i.d. case, by developing the observed signals xðnÞ; n ¼ 1; …; N, following some harmonic or trigonometric basis, and then separate the error vector components ϵ ≔ðϵ1 ; …; ϵp Þ > from the data ϵðnÞ; n ¼ 1; …; N, which can be assumed i.i.d. copies of ϵ. Note that, by the linearity of the problem, the separating matrix will be the same for the original observed signals xðÞ or the errors signals ϵðÞ. The same methodology can be used for Section 5 below. In view of Proposition 1, the function B↦KLm ðc0 ; cY Þ is non-

density. A more appropriate choice of the kernel kðÞ, for estimating the copula density, can be done according to [12], which copes with the boundary effect. The bandwidth parameters H 1 ; …; H p in (9) and h1 ; …; hp in (10) will be chosen according to Silverman's rule of thumb, see e.g. [16], i.e., for all j ¼ 1; …; p, we take  1=ðp þ 4Þ  1=5 4 b j N  1=ðp þ 4Þ and hj ¼ 4 b j N  1=5 ; Hj ¼ s Σ p þ2 3 b j and s b j are, respectively, the empirical standard where Σ deviation of the data Fb Y ðy ð1ÞÞ; …; Fb Y ðy ðNÞÞ and j

     1 N d KL ∑ log b c Y Fb Y 1 y1 ðnÞ ; …; Fb Y p yp ðnÞ ; m ðc0 ; cY Þ ≔ Nn¼1

ð8Þ

j

j

from the proper definitions of the estimates as follows:      d b b b d dKL 1 N dB c Y F Y 1 y1 ðnÞ ; …; F Y p yp ðnÞ m ðc0 ; cY Þ ¼ ∑ ; dB Nn¼1 b c YðFb Y 1 ðy1 ðnÞÞ; …; Fb Y p ðyp ðnÞÞÞ ð11Þ where d=dB ≔ð∂=∂Bij Þij , i; j ¼ 1; …; p, and ∂b c Y ðFb Y 1 ðy1 ðnÞÞ; …; Fb Y p ðyp ðnÞÞÞ ∂Bij ¼

N 1 ∑ NH 1 ⋯H p m ¼ 1

0

negative and achieves its minimum value zero iff B ¼ A (up to scale and permutation indeterminacies). In other

b b ðnÞ ¼ BxðnÞ; mated source signals y n ¼ 1; …; N. Based on the equality (7), we propose to estimate the KLm-divergence KLm ðc0 ; cY Þ by a “plug-in” type procedure. We obtain then the estimate

j

b the estimate of the yj ð1Þ; …; yj ðNÞ. In order to compute B, de-mixing matrix, we can use a gradient descent algorithm taking as initial matrix B0 ¼ I p , the p  p identity d matrix. The gradient in B of KL m ðc0 ; cY Þ can be explicated

1

words, we have A  1 ¼ arg inf B KLm ðc0 ; cY Þ. Hence, to achieve separation, the idea is to minimize on B some d statistical estimate KL m ðc0 ; cY Þ, of KLm ðc0 ; cY Þ, constructed from the data yð1Þ; …; yðnÞ. The separation matrix is then d b ≔arg inf B KL estimated by B m ðc0 ; cY Þ, leading to the esti-

321

k

p

∏

j ¼ 1;j a i

k

! Fb Y j ðyj ðmÞÞ  Fb Y j ðyj ðnÞÞ Hj

 1 ! b b Fb Y i ðyi ðmÞÞ Fb Y i ðyi ðnÞÞ 1 ∂ F Y i ðyi ðmÞÞ  F Y i ðyi ðnÞÞ A ; ∂Bij Hi Hi

with    ∂Fb Y i ðyi ðmÞÞ 1 N y ðℓÞ yi ðmÞ  ¼ ∑ k i xj ðℓÞ  xj ðmÞ ∂Bij Nhi ℓ ¼ 1 hi and    ∂Fb Y i ðyi ðnÞÞ 1 N y ðℓÞ  yi ðnÞ  ¼ ∑ k i xj ðℓÞ  xj ðnÞ : ∂Bij Nhi ℓ ¼ 1 hi

5. A solution to the BSS for dependent source components

where ! p N Fb Y j ðyj ðmÞÞ uj 1 b c Y ð uÞ ≔ ∑ ∏ k ; NH 1 ⋯H p m ¼ 1 j ¼ 1 Hj

ð9Þ

for all u A ½0; 1p , is the kernel estimate of the copula density cY ðÞ, and Fb Y j ðxÞ, 8 j ¼ 1; …; p, is the smoothed estimate of the marginal distribution function F Y j ðxÞ of the random variable Yj, at any real value x A R, defined by   yj ðℓÞ  x 1 N Fb Y j ðxÞ ≔ ∑ K ; ð10Þ Nℓ¼1 hj where KðÞ is the primitive of a kernel kðÞ, a symmetric centered probability density. In our forthcoming simulation study, we will take as kernel kðÞ a standard Gaussian

We give in the following subsections a solution to the BSS problem for mixtures of dependent component sources. The method depends on the knowledge about the dependency structure of the source components. 5.1. When both the model and the parameter are known Assume that we dispose of some prior information about the density copula of the random source vector S. Note that this is possible for many practical problems, it can be done, from realizations of S, by a model selection procedure in semiparametric copula density models fcθ ðÞ; θ A Θ  Rd g, typically indexed by a multivariate parameter θ, see e.g. [4]. The parameter θ can be estimated

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using maximum semiparametric likelihood, see [18,1]. Denote by θ0 the obtained value of θ and cθ0 ðÞ the copula density modeling the dependency structure of the source components. Obviously, since the source components are assumed to be dependent, cθ0 ðÞ is different from the density copula of independence c0 ðÞ. Hence, we naturally replace, in the above criterion function B↦KLm ðc0 ; cY Þ, c0 by cθ0 . Moreover, we can show that the criterion function B↦KLm ðcθ0 ; cY Þ is nonnegative and achieves its minimum value zero iff B ¼ A  1 (up to scale and permutation indeterminacies), i.e., A  1 ¼ arg inf B KLm ðcθ0 ; cY Þ, provided that the copula density cθ0 ðÞ of S satisfies the following assumption (C.A): for any regular matrix M, if the copula density of MS is equal to cθ0 ðÞ, then M ¼ DP, where D is diagonal and P is a permutation. Note also that the criterion KLm ðcθ0 ; cY Þ can be written as !!   cY ðF Y 1 ðY 1 Þ; …; F Y p ðY p ÞÞ KLm cθ0 ; cY ¼ E log : cθ0 ðF Y 1 ðY 1 Þ; …; F Y p ðY p ÞÞ So as before, we propose to estimate the de-mixing matrix d b ≔arg inf B KL by B m ðcθ0 ; cY Þ, where bY ðFb Y 1 ðy1 ðnÞÞ; …; Fb Y p ðyp ðnÞÞÞ   1 N c d KL ∑ log : m cθ0 ; cY ≔ Nn ¼ 1 cθ0 ðFb Y 1 ðy1 ðnÞÞ; …; Fb Y p ðyp ðnÞÞÞ The estimates of copula density and the marginal distribub can be tion functions are defined as before. The solution B computed by a descent gradient algorithm. The gradient in B can be explicitly computed in a similar way as in (11) b b ðnÞ ¼ BxðnÞ; Section 4. The estimated source signals are y

indeterminacies), provided that the copula density model fcθ ðÞ; θA Θ  Rd g of S satisfies the following identifiability assumption (C.B): for any regular matrix M, if the copula density of MS A fcθ ðÞ; θ A Θ  Rd g, then M ¼ DP, where D is diagonal and P is a permutation. 5.3. When both the model and the parameter are unknown In this case, we can use the following methodology. We consider a class of L models of copula densities of the source components, denoted M1 ≔fc1θ1 ðÞ; θ1 A Θ1 g; …; M L ≔ fcLθL ðÞ; θL A ΘL g. Here, the goal is to separate the source signal using the true unknown model of dependency of the sources. Assume that each model Mi, i¼1,…,L, satisfies the identifiability condition (C.B) in the above subsection. Then, the criterion function B↦inf ℓ ¼ 1;…;L inf θℓ A Θℓ KLm ðcℓθℓ ; cY Þ is nonnegative and achieves its minimum value zero iff B ¼ A  1 (up to scale and permutation indeterminacies). Hence, we propose to apply the method described in the above subsection for each model, and then choose the solution that minimizes the criterion over all considered models, i.e., ℓn b ≔arg inf inf KL d B m ðcθ ; cY Þ B

θ A Θ ℓn

where

ℓ d ℓn ≔arg min inf inf KL m ðcθ ; cY Þ: ℓ ¼ 1;…;L B

θ A Θℓ

6. Simulation results

n ¼ 1; …; N. We obtain then the following algorithm. Algorithm 1. A copula based BSS algorithm for dependent component sources. Data : the observed signals xðnÞ; n ¼ 1; …; N b ðnÞ; n ¼ 1; …; N Result : the estimated sources y Initialization : y0 ðnÞ ¼ B0 xðnÞ, B0 ¼ I p . Given ε 4 0 and μ 40 suitably choosen Do  Update B and y: cm ðcθ ;cY Þ dKL 0 Bk þ 1 ¼ Bk  μ  dB B ¼ Bk

yk þ 1 ðnÞ ¼ Bk þ 1 xðnÞ; n ¼ 1; …; N Until J Bk þ 1  Bk J o ε b ðnÞ ¼ yk þ 1 ðnÞ; n ¼ 1; …; N y

5.2. When the model is known and the parameter is unknown Denote by fcθ ðÞ; θ A Θ  Rd g the semiparametric copula density model of the source components. Since the parameter θ is unknown, we propose to adapt the above criterion and to estimate the demixing matrix by b ≔arg inf B inf θ A Θ KL d B m ðcθ ; cY Þ, which can be computed using the gradient (on both B and θ) of the criterion d function ðB; θÞ↦KL m ðcθ ; cY Þ. Note that the criterion function

B↦inf θ A Θ KLm ðcθ ; cY Þ is nonnegative and achieves its mini-

mum value zero iff B ¼ A  1 (up to scale and permutation

In this section, we give simulation results for the proposed method. We dealt with instantaneous mixtures (2 mixtures of 2 sources) of five kinds of sample sources: uniform i.i.d. with independent components (Fig. 1a); i.i.d. sources with independent components where the marginals are drawn from the 4-ASK (Amplitude Shift Keying) alphabets (  3, 1,1,3 with equal probabilities 0.25) at which was added a centered Gaussian random variable with variance equal to 0.25 (Fig. 1b); i.i.d. (with uniform marginals) vector sources with dependent components generated from Fairlie–Gumbel–Morgenstern (FGM)copula with θ0 ¼ 0:8 (Fig. 2); i.i.d. (with uniform marginals in Fig. 3a and binary phase-shift keying (BPSK)-marginals in Fig. 3b) vector sources with dependent components generated from Ali–Mikhail–Haq (AMH)-copula with θ0 ¼ 0:6. The accuracy of source estimation is evaluated through the signal-noise-ratio (SNR), defined by SNRi ≔ 2 2 N b 10 log10 ∑N In the n ¼ 1 si ðnÞ =∑n ¼ 1 ðy i ðnÞ si ðnÞÞ ; i ¼ 1; 2. standard case of independent source components, the proposed method described in Section 4 is compared with the MI one proposed by [14], under the same conditions, see Fig. 1. The used mixing matrix is A ≔½1 0:5; 0:5 1. The number of samples is N ¼2000, and all simulations are repeated 100 times. The gradient descent parameter is taken μ ¼0.1 in all cases. We observe from Fig. 1 that the proposed method gives good results for the standard case of independent component sources. Moreover, we see, from Figs. 2 and 3, that our proposed method is able to separate, with good performance, mixtures of dependent source components.

A. Keziou et al. / Signal Processing 104 (2014) 319–324

55

50

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40 35

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SNR (dB)

SNR (dB)

40

30 25

30 25 20

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15

0

10

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KLm_C MI

10

KLm_C MI

10 5

323

60

5

70

0

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Iterations

30

40

50

Iterations

Fig. 1. SNRs vs iterations with independent sources. (a) Uniform sources. (b) ASK sources.

50

0.7

45

0.6

40 0.5 criterion value

SNR (dB)

35 30 25

0.4

0.3

20 0.2 15 10 5

0.1

SNR_y1 SNR_y2 0

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55

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SNR (dB)

SNR (dB)

Fig. 2. Separation of dependent sources from FGM copula. (a) SNRs vs iterations. (b) Criterion vs iterations.

30 25

30 25

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15

15 SNR_y1 SNR_y2

10 5

0

50

100 Iterations

150

SNR_y1 SNR_y2

10 200

5

0

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100 Iterations

Fig. 3. SNRs vs iterations with dependent sources from AMH copula. (a) Uniform sources. (b) BPSK sources.

150

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7. Conclusion We have proposed a new BSS approach by minimizing the empirical KLm-divergence between copula densities. The approach is able to separate instantaneous linear mixtures of both independent and dependent source components. In Section 6, the accuracy and the consistency of the obtained algorithms are illustrated by simulation, for 2  2 mixture-source. Other simulation examples give similar results for the case of 3  3 mixture-source. It should be mentioned that our proposed algorithms based on copula densities, rather than the classical ones based on probability densities, are more time consuming, since we estimate both copulas density of the vector and the marginal distribution function of each component. The present approach can be extended to deal with convolutive mixtures, and it will be interesting also to investigate a theoretical study of the identifiability assumptions as well as the convergence of the proposed algorithms. These developments are beyond the scope of the present paper, and will be addressed in future communications.

Acknowledgments Thanks to Egide PHC-Volubilis program for funding the project 27093ND - MA/12/270. The authors wish to thank the reviewers for their constructive comments and criticisms leading to improvement of this paper. References [1] S. Bouzebda, A. Keziou, New estimates and tests of independence in some copula models via ϕ-divergences, Kybernetika 46 (1) (2010) 178–201.

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