Proof of Wiener-like linear regression of isotropic complex symmetric alpha-stable random variables

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Proof of Wiener-like linear regression of isotropic

complex symmetric alpha-stable random variables

Roland Badeau, Antoine Liutkus

To cite this version:

Roland Badeau, Antoine Liutkus. Proof of Wiener-like linear regression of isotropic complex symmetric

alpha-stable random variables. 2014. �hal-01069612�

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PROOF OF WIENER-LIKE LINEAR REGRESSION

OF ISOTROPIC COMPLEX SYMMETRIC ALPHA-STABLE RANDOM VARIABLES

Roland Badeau

T´el´ecom ParisTech, CNRS LTCI

Paris, France

roland.badeau@telecom-paristech.fr

Antoine Liutkus

INRIA Nancy - Grand Est

Villers-l`es-Nancy, France

antoine.liutkus@inria.fr

This document features supplementary materials to the reference paper [1]. It provides the proof of equation (8) in [1]. This proof concerns a particular regression property of complex isotropic symmetric α-stable random variables (SαS, see [2]). In [1], this property is shown paramount in building efficient filters for separating SαS processes. Such processes exhibit very large dynamic ranges while being locally stationary, and have been shown appropriate for audio modeling.

Proposition 1 (Wiener-like linear regression of isotropic complex SαS random variables). Let α∈]0, 2]. Let s1and s2be two

independent isotropic complex SαS random variables of scale parameters σ1and σ2, respectively. Let x= s1+ s2. Then the

conditional expectation of s1given x is expressed as follows:

E[s1|x] = σ α 1 σα 1 + σ2α x. (1)

Proof. Let sr1 = Re(s1), si1 = Im(s1), s2r = Re(s2), si2 = Im(s2), xr = Re(x), and xi = Im(x), where Re(.) and Im(.) denote the real and imaginary parts of complex numbers, respectively. For j∈ {1, 2}, the characteristic function of the isotropic complex SαS random variable sj is φsj(θ

r x, θix) = e−σ α j(|θrx| 2 +|θi x| 2 )α2

[2]. Therefore the characteristic function of the random vector(sr 1, xr, xi) ∈ R3is φ(sr 1,xr,xi)(θ r 1, θrx, θix) = E[ei(θ r 1sr1+θxrxr+θixxi)] = E[ei(θ1rsr1+θxr(sr1+sr2)+θix(si1+si2))]

= E[ei((θr1+θxr)sr1+θixs1i)]E[ei(θxrs2r+θixsi2)]

= φs1(θ r 1+ θrx, θx) φsi 2(θ r x, θx)i = e− σα 1(|θr1+θxr| 2 +|θi x| 2 )α2+σα 2(|θrx| 2 +|θi x| 2 )α2 . Using [2, eq. (5.1.7) p. 226], the conditional characteristic function of sr

1given xrand xi ∈ R is expressed as

φ(sr 1|xr,xi)(θ r 1) = R R2φ(sr 1,xr,xi)(θ r 1, θxr, θxi)e−i(θ r xx r_+θi xx i₎ dθr xdθix R R2φ(sr₁,xr_,xi)(0, θr_x, θ_xi)e−i(θxrxr+θixxi)dθr_xdθ_xi = R R2e −σα 1(|θr1+θxr| 2 +|θi x| 2 )α2+σα 2(|θxr| 2 +|θi x| 2 )α2 e−i(θr xxr+θixxi)dθr xdθix R R2e−(σ α 1+σα2)(θrx|2+|θix|2) α 2 e−i(θr xxr+θixxi)dθ_xrdθi_x . (2)

We know that E[sr1|xr, xi] is defined if and only if φ(sr

1|xr,xi)(θ

r

1) is differentiable at θr1 = 0, and that in that case dφ_(sr

1 |xr ,xi)

dθr

1 (0) = iE[s

r

1|xr, xi]. Firstly, it is easy to prove that the first order derivative of φ(sr

1|xr,xi) is well-defined and

has the following expression:

dφ_(sr 1 |xr ,xi) dθr 1 (θ r 1) = R R2−ασ α 1(θr1+θxr)(|θ r 1+θrx| 2_+|θi x| 2₎( α₂−1)_e− σα_{1 (|θ}r_{1 +θ}r_{x |}2 +|θi_{x |}2 )α2 +σα_{2 (|θ}r x |2 +|θix |2 ) α 2 e−i(θr_{x x}r +θi_{x x}i )_dθr xdθ i x R R2e −(σα_{1 +}σα_{2 )(}θr_{x |}2 +|θix |2 ) α 2 e−i(θr_{x x}r +θi_{x x}i )_dθr xdθxi . (3)

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Indeed, differentiating under theR

sign in the numerator of the last member of (2) is allowed, because the term following theR sign in the numerator of (3) can be upper bounded by an integrable function independently of θ1r∈ R:

−ασα 1(θr1+ θxr)(|θr1+ θrx|2+ |θix|2)( α 2−1)e− σα 1(|θr1+θxr| 2_+|θi x| 2₎α₂_+σα 2(|θrx| 2_+|θi x| 2₎α₂ e−i(θrxx r_+θi xx i₎ ≤ ασα 1(|θ1r+ θrx|2+ |θxi|2) α−1 2 e− σ1α(|θ1r+θrx| 2_+|θi x| 2₎α₂_+σα 2(|θrx| 2_+|θi x| 2₎α₂ ≤ g(|θr 1+ θxr|2+ |θ i x|2) h(θ r x, θ i x) ≤ kgk∞h(θr x, θ i x),

where the nonnegative functions g∈ L∞_{(R) and h ∈ L}1_(R2_{) are defined according to the value of α:} • if 1 ≤ α ≤ 2, g(t) = ασα 1|t| α−1 2 e−σ α 1|t| α 2 and h(θr x, θix) = e−σ α 2(|θxr| 2 +|θi x| 2 )α2 ; • if 0 < α < 1, g(t) = ασα 1e−σ α 1|t| α 2 and h(θr x, θix) = e−σα2 (|θrx |2 +|θix |2 ) α 2 |θi x|1−α .

Then applying equation (3) to θr

1= 0 yields dφ_(sr 1 |xr ,xi) dθr 1 (0) = R R2−ασ α 1θrx(|θ r x|2+ |θ i x|2)( α 2−1)e−(σ α 1+σα2)(|θxr| 2_+|θi x| 2₎α₂ e−i(θxrx r_+θi xx i₎ dθrxdθ i x R R2e−(σ α 1+σα2)(θrx|2+|θix|2) α 2 e−i(θr xxr+θixxi)dθ_xrdθi_x = σα1 σα 1+σα2 R R2 de−(σα1 +σα2 )(|θx |r2 +|θix |2 ) α 2 dθr x e −i(θr xx r_+θi xx i₎ dθr xdθix R R2e−(σ α 1+σα2)(θrx|2+|θix|2) α 2 e−i(θr xxr+θxixi)dθr_xdθi_x = − σα1 σα 1+σα2 R R2e−(σ α 1+σα2)(|θxr| 2_+|θi x| 2₎α₂ _de−i(θr_{x x}r +θi_{x x}i ) dθr x dθ r xdθix R R2e−(σ α 1+σ2α)(θxr|2+|θix|2) α 2 e−i(θr xxr+θixxi)dθr_xdθ_xi = i σ1α σα 1+σ2αx r_.

This proves that E[sr

1|xr, xi] = σ1α

σα 1+σ2αx

r

. In exactly the same way, it is proved that E[si

1|xr, xi] = σ1α σα 1+σα2 x i_{, which finally} proves equation (1). 1. REFERENCES

[1] A. Liutkus and R. Badeau, Generalized Wiener filtering with fractional power spectrograms, submitted to IEEE Interna-tional Conference on Acoustics, Speech, and Signal Processing (ICASSP), Brisbane, Australia, April 2015.

[2] G. Samoradnitsky and M. Taqqu, Stable non-Gaussian random processes: stochastic models with infinite variance, vol. 1, CRC Press, 1994.