Gaussian distributions and processes have long been accepted as useful tools for stochastic

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1 Chapter 3 Alpha-Stable Random Variables and Processes Gaussian distributions and processes have long been accepted as useful tools for stochastic modeling. In this section, we introduce a statistical model based on the class of symmetric -stable (SS) distributions which iswell-suited for describing signals that are impulsive in nature. A review of the state of the art on stable processes from a statistical point of view is provided by a collection of papers edited by Cambanis, Samorodnitsky and Taqqu [11]. Several statisticians including Cambanis, Zolotarev, Weron, et al. have published extensively on the theory and applications of stable processes. They studied the properties of stable processes [6, 8, 9, 33, 36, 43, 46, 48, 65, 9, 98], their spectral representation [7, 7, 47], as well as prediction and linear ltering problems [9, 1, 35, 37, 6,94]. Textbooks in the area were written by Samorodnitsky and Taqqu [64] and by Janicki and Weron [3]. An extensive review of stable processes from a signal processing point of view can be found in a tutorial paper by Shao and Nikias [69] aswell as in a monogram written by the same authors [5]. 3.1 The Class of Real SS Distributions The symmetric -stable (SS) distribution is best dened by its characteristic function '(!) = exp(! ; j!j ) (3:1) where is the characteristic exponent restricted to the values <, (;1 < < 1) is the location parameter, and ( > ) is the dispersion of the distribution. For values of in the interval (1 ], the location parameter corresponds to the mean of the SS 6

2 alpha=.5... alpha=1 alpha=1.5 alpha=.4 f(x) x Figure 3.1: Standard SS densities. distribution, while for < 1, corresponds to its median. The dispersion parameter determines the spread of the distribution around its location parameter, similar to the variance of the Gaussian distribution. The characteristic exponent is the most important parameter of the SS distribution and it determines the shape of the distribution. A stable distribution is called standard if = and = 1. Clearly, if a random variable X is stable with parameters, then (X ; )= 1= is standard with characteristic exponent. The standard SS density functions for a few values of the characteristic exponent are shown in Figure 3.1. By letting take the values 1 and, we gettwo important special cases of SS distributions, namely, thecauchy ( = 1), and the Gaussian ( = ): Cauchy Gaussian f 1 ( x)= 1 +(x ; ) (3:) f ( x)= p 1 (x ; ) exp [; ]: (3:3) 4 4 7

3 alpha=.5... alpha=1 alpha=1.5 alpha=. f(x) x Figure 3.: A close-up view of the tails of the densities in Figure 3.1. Unfortunately, no closed form expressions exist for general SS distributions other than the Cauchy and the Gaussian. However, power series expansions can be derived for f ( x). In the following, we shall assume that all SS distributions are centered at the origin, i.e., the location parameter =. This is equivalent to the zero-mean assumption for Gaussian distributions. Then, the standard SS density function is given by [69] f (x) = 8 >< >: 1 x P 1k=1 (;1) k;1 ;(k +1)x ;k sin( k k! ) for <<1 1 for =1 (x +1) P 1 1k= (;1) k ;( k+1 k! 1 p exp [; x 4 )xk for 1 << ] for =: (3:4) Although the SS density behaves approximately like a Gaussian density near the origin, its tails decay at a lower rate than the Gaussian density tails. While the Gaussian density has exponential tails, the stable densities have algebraic tails (cf. Figure 3.). The smaller the characteristic exponent is, the heavier the tails of the SS density. This implies that random variables following SS distributions with small characteristic exponents are highly impulsive. It is this heavy-tail characteristic that makes the SS densities appropriate for modeling signals and noise or interference which are impulsive in 8

4 nature. Figure 3.3 depicts some representative time series of SS deviates for comparison purposes. Clearly, several outliers can be observed in the data series when the characteristic exponent takes values other than. As decreases both the occurrence rate and the strength of the outliers increase, resulting to very impulsive processes. SS densities obey two important properties which further justify their role in data modeling: The stability property, which states that the random variables X 1 ::: X n are independent and symmetrically stable with the same characteristic exponent if and only if for any constants a 1 ::: a n, the linear combination P n i=1 a ix i is also SS The generalized central limit theorem, which states that the family of stable distributions contains all limiting distributions of sums of i.i.d. random variables. An important dierence between the Gaussian and the other distributions of the SS family is that only moments of order less than exist for the non-gaussian SS family members. The fractional lower order moments (FLOM's) of a SS random variable with zero location parameter and dispersion are given by: EjXj p = C(p ) p for <p< (3:5) where C(p )= p+1 ;( p+1 );(; p ) p ;(; p ) (3:6) and ;() is the Gamma function. 3. Bivariate Isotropic Stable Distributions Multivariate stable distributions, much like the univariate stable distributions, are characterized by the stability property and the generalized central limit theorem. However, they are much more dicult to describe because they form a nonparametric set [69]. An exception is the family of multidimensional isotropic stable distributions. Here, we concentrate on the two dimensional (bivariate) case which is appropriate for modeling signals and noise in the bearing estimation problem. The characteristic function of a bivariate isotropic -stable distribution has the form '(! 1! ) = exp( ( 1! 1 +! ) ; j!j ) (3:7) 9

5 5 (a) 1 (b) (c) (d) (e) x (f) Figure 3.3: SS time series. (a): =: (Gaussian), (b): =1:95, (c): =1:5, (d): =1: (Cauchy), (e): =:85, (f): =:45. 3

6 q where! =(! 1! ) and j!j =! + 1!. Again here, the parameters and are the characteristic exponent and the dispersion, respectively. The parameters 1 are the location parameters. The distribution is isotropic with respect to the point ( 1 ). Note that the two marginal distributions of the isotropic stable distribution are SS with parameters ( 1 ) and ( ). In the following, we will assume that ( 1 )=( ). The bivariate isotropic Cauchy and Gaussian distributions are special cases for = 1 and =, respectively. As in the case of the univariate SS density function, when 6= 1or 6=, no closed form expressions exist for the density function of the bivariate stable random variable. By q using the polar coordinate r = jxj = x + 1 x, the density function can be written as f (x 1 x )= (r), and can be expressed in a power series expansion form: (r) = 8 >< >: P 1 1k=1 k (;1) k;1 = for <<1 (;(k=+1)) sin( k)( r ) ;k; k! 1= for =1 (r + ) 3= P 1 1k= (;1) k = 1 4 ;( k+ k+1 (k!) )( r 1= ) k for 1 << exp [; r 4 ] for =: (3:8) The density function (r) described aboveisalsoaheavy-tailed function. An expression for the FLOM's, similar to the one for the single-dimensional case, can be found in [98]. If X is a random vector in R n having the isotropic stable distribution with dispersion, then EjXj p = C n (p ) p for <p< (3:9) where C n (p )= p+1 ;( p+n );(; p ) ;( n );(; p ) : (3:1) 3.3 Symmetric Alpha-Stable Processes A collection of random variables fx(t) t T g where T is an arbitrary index set, is said to be a SS stochastic process if for all combinations of distinct indices t 1 ::: t n T, the random variables X(t 1 ) ::: X(t n )arejointly SS with the same characteristic exponent. The stochastic process fx(t) t T g is stationary if the random vectors (X(t 1 ) ::: X(t n )) and (X(t 1 +s) ::: X(t n +s)) are identically distributed for each choice of s t 1 ::: t n T. The family of stable processes has many members with mutually 31

7 exclusive properties. In the following, we present three important types of stable processes that are commonly used. 1) Sub-Gaussian Processes: A stable process fx(t) t T g is said to be an -sub- Gaussian process (-SG(R)) if for distinct indices t 1 ::: t n T,(X(t 1 ) ::: X(t n )) has characteristic function given by '(!) =exp(;[ 1 nx l m=1! l! m R(t l t m )] = ) (3:11) where R(t s) is a positive denite function,! =[! 1 :::! n ] T and takes values in (1 ]. When =,X(t) is a Gaussian process with zero mean and covariance function R(t s). A sub-gaussian process is stationary if and only if R(t s) =R(t ; s) =R(s ; t). Sub-Gaussian processes share many common features with Gaussian processes. In fact, sub-gaussian processes are variance mixtures of Gaussian processes [1]. Specically, if X(t) is-sg(r), then X(t)=S 1= Y (t) (3:1) where S is a positive -stable random variable and independent ofy (t) which is a Gaussian process with zero-mean and covariance function R(t s). An important distinction between Gaussian and sub-gaussian processes is that, while linear spaces of Gaussian random variables may contain non-degenerate independent elements, sub-gaussian random variables cannot be independent [1]. A sub-gaussian process X(t) =S 1= Y (t) is stationary if and only if the Gaussian process Y (t) is stationary. ) Linear Stable Processes: LetfU(n) n= 1 :::g be a family of i.i.d. SS random variables. Then, X(n) = +1X i=;1 a i U(n ; i) (3:13) denes a stationary SS random process if P +1 ;1 ja i j ; < 1 for some < < when <<1, or if P +1 ;1 ja i j < 1 when 1. These processes are called linear stable processes or stable processes with moving-average representation. Examples of linear stable processes include nite-order autoregressive (AR), moving-average (MA) and autoregressive moving-average (ARMA) processes [69]. 3

8 3) Harmonizable Stable Processes: A complex valued SS process X(t) is called harmonizable if it can be expressed as: X(t) = Z 1 ;1 e t! d(!) ;1 <t<1 (3:14) where d(!) is a SS process with independent increments satisfying fejd(!)j p g =p = C(p )(!) d! for all <p< (3:15) and C(p ) is a constant depending on p and and (!) is a nonnegative function called the spectral density of X(t) [7]. The spectral density (!) describes fully the distribution of the process X(t). In another sharp contrast with the Gaussian case, the class of SS harmonizable processes is disjoint from the class of linear processes. In addition, sub- Gaussian processes are neither linear nor harmonizable [1]. In modeling the signals and/or noise for the parameter estimation problem, we need a complex model for the noise samples. We also need to dene quantities which describe correlations between random variables. In the following section, we present the family of isotropic complex SS distributions and describe their correlation properties by means of the covariation quantity. 3.4 Complex SS Random Variables and Covariations A complex random variable (r.v.) X = X 1 + X is symmetric -stable (SS)ifX 1 and X are jointly SS and its characteristic function is written as '(!) = E exp[ <(!X )] = E exp[ (! 1 X 1 +! X )] Z = exp ; j! 1 x 1 +! x j d; X1 X (x 1 x ) (3.16) S where! =! 1 +!, <[] is the real part operator, and ; X1 X is a symmetric measure on the unit sphere S, called the spectral measure of the random variable X. A complex random variable X = X 1 + X is isotropic if and only if (X 1 X ) has a uniform spectral measure. In this case, the characteristic function of X can be written as '(!) =E exp( <[!X ]) = exp(;j!j ) (3:17) 33

9 where ( > ) is the dispersion of the distribution. In the theory of second-order processes, the concept of covariance plays an important role in problems of linear prediction, ltering and smoothing. Since SS processes do not possess nite pth order moments for p, covariances do not exist on the space of SS random variables. Instead, a quantity calledcovariation plays an analogous role for statistical signal processing problems involving SS processes to the role played by covariance in the case of second-order processes. Several complex r.v.'s are jointly SS if their real and imaginary parts are jointly SS. When X = X 1 + X and Y = Y 1 + Y are jointly SS with 1 <, the covariation of X and Y is dened by [X Y ] = Z S 4 (x 1 + x )(y 1 + y ) <;1> d; X1 X Y 1 Y (x 1 x y 1 y ) (3:18) where we use throughout the convention Y <> = jy j ;1 Y : (3:19) It can be shown that for every 1 p<, the covariation can be expressed as a function of moments [9] [X Y ] = EXY <p;1> where Y is the dispersion of the r.v. Y given by EjY j p Y (3:) p= Y = EjY jp C(p ) for <p< (3:1) with Obviously, from (3.) it holds that C(p )= p+1 ;( p+ );(; p ) ;( 1 );(; p ) : (3:) [X X] = X : (3:3) Also, the covariation coecient of X and Y is dened by X Y = [X Y ] [Y Y ] (3:4) 34

10 and by using (3.), it can be expressed as X Y = EXY <p;1> EjY j p for 1 p<: (3:5) The covariation of complex jointly SS r.v.'s is not generally symmetric and has the following properties [6]: P1 If X 1, X and Y are jointly SS, then [ax 1 + bx Y] = a[x 1 Y] + b[x Y] (3:6) for any complex constants a and b. P If Y 1 and Y are independent and X 1, X and Y are jointly SS, then [ax 1 by 1 + cy ] = ab <;1> [X 1 Y 1 ] + ac <;1> [X 1 Y ] (3:7) for any complex constants a, b and c. P3 If X and Y are independent SS, then [X Y ] = P4 Let fu i i =1 ::: ng be independent complex SS r.v.'s with dispersions i. For any complex numbers fa i b i i =1 ::: ng form X = a 1 U a n U n Y = b 1 U b n U n : Then [X X] = 1 ja 1 j + + n ja n j [Y Y ] = 1 jb 1 j + + n jb n j [X Y ] = 1 a 1 b <;1> n a n b <;1> n : (3:8) 35

11 3.5 Generation of Complex Isotropic SS Random Variables The generation of complex isotropic SS deviates of characteristic exponent is based on the following proposition found in [64]: Proposition 3.1 Acomplex SS ( <) random variable X = X 1 + X is isotropic if and only if there exist two i.i.d. zero-mean Gaussian variables G 1 and G and a real stable random variable A of characteristic exponent =, dispersion cos (=4) and skewness =1(we write A S = (cos (=4) 1)), independent of (G 1 G ) such that (X 1 X ) d = (A 1= G 1 A 1= G ). We say that the vector (X 1 X )issub-gaussian with underlying vector (G 1 G ). It can be shown that the real and imaginary parts of X are always dependent, unless G 1 and G are degenerate. Hence, every complex isotropic SS random variable with < can be expressed as X = A 1= (G 1 + G ) (3:9) and its generation involves the generation of a real, totally skewed stable random variable. The problem of generating a real stable deviate is studied in [14] and [3]. Here, we present the result for easy reference. To generate a real standard stable random variable A S (1 )ofcharacteristic exponent, skewness and unit dispersion = 1, the following representations can be deduced: sin (U ; U ) cos(u ; (U ; U )) 1; S( 1) = D for 6= 1 (3:3) (cos U) 1= W and S(1 1) = # "( + U) tan U ; ln( W cos U + U ) (3:31) where W is standard exponential with PrfW > wg = e ;w w >, and U is uniform on (; ). Also, D = [cos(arctan( tan(=)))] ;1=,andU = ; [k()=] with k() =1;j1 ; j. Then, a stable variate, A 1, of dispersion can be obtained from A by A 1 = 1= A. 36

12 To conclude, the following proposition gives the relationship between the dispersion of the complex r.v. X = X 1 + X and the variance of the complex Gaussian r.v. G = G 1 + G. Proposition 3. The dispersion of the complex r.v. X = X 1 + X generated according to Proposition 3.1 is given by =(=), where is the variance of the underlying complex Gaussian random variable. Proof The Laplace transform of the r.v. A S = (cos (=4) 1) is given by [64]: Efexp(;sA)g = exp(;s = ) s>: (3:3) Also, since G = G 1 + G N C ( ), its characteristic function is given by ' G (!) =exp(; j!j =4) (3:33) where! =! 1 +!. Then, the characteristic function of X can be expressed as ' X (!) = Efexp( <[!X ])g = Efexp( (! 1 A 1= G 1 +! A 1= G ))g = EfEfexp( (! 1 A 1= G 1 +! A 1= G ))jagg = Efexp(; (! 1 +! )A=4)g [by use of 3.33] = exp((=) j!j ) [by use of 3.3]: (3.34) Comparing (3.34) with (3.7) we conclude that =(=). 37

Stable Process. 2. Multivariate Stable Distributions. July, 2006

Stable Process. 2. Multivariate Stable Distributions. July, 2006 Stable Process 2. Multivariate Stable Distributions July, 2006 1. Stable random vectors. 2. Characteristic functions. 3. Strictly stable and symmetric stable random vectors. 4. Sub-Gaussian random vectors.