Construction of some special classes of stable processes that - PowerPoint PPT Presentation

Motivation and application examples Overview on α -stable variables and processes Construction of some special classes of stable processes that generalises spatial or temporal Gaussian processes. Nourddine Azzaoui works with G.W. Peters, L. Clavier, M. Egan and A. Guillin STM 2016: International Workshop on Spatial and Temporal Modeling from Statistical, Machine Learning and Engineering perspectives 20 July 2016 Azzaoui et al ... Special classes of stable Processes...

Motivation and application examples Overview on α -stable variables and processes Motivation and application examples The question is to find Z t ( s ) at any time t or/and any location s in the field D . Parametric solution: reduce the knowledge of the process to some su ffi cient functionals: = ) Gaussian Processes (GP): reduce the processes to some su ffi cient functionals (covariances and mean functionals) = ) why not heavy tailed models (stables processes) with covariation, spectral densities... Let { s 1 , . . . , s n } measuring sensors placed in a Estimates these functionals from observations of spatial field D . the process. n o Z t ( s ) , s 2 D ⇢ R d , d = 1 , 2 , 3 a time Other concurrent solutions exist: non parametric varying spatial random variable. techniques, Bayesian methods... Azzaoui et al ... Special classes of stable Processes...

Motivation and application examples Overview on α -stable variables and processes Motivation and application examples Consider a channel or a filter, Z s ( t ) = e ⇤ h ( t ) = e ( t � τ ) h ( τ ) d τ Or equivalently by Fourier transform, = ) N Z e ιω t h ( t ) dt a k δ t � τ k e i θ k , X H ( ω ) = h ( t ) = k = 1 But real world is random and ( h ( t ) , t � 0 ) is considered as a stochastic process = ) A harmonizable process Z e ιω t d ξ ( t ) H ( ω ) = ( ξ t ) (resp d ξ ( . ) ) is heavy tailed process (resp. random measure) Azzaoui et al ... Special classes of stable Processes...

Motivation and application examples Overview on α -stable variables and processes Why α -stables? Theoretical interest Practical modelings It is an extension of gaussian distributions and processes (case α = 2) The convolution stability: a combination of i.i.d stable variables is a stable one The central limit theorem: α -stable distributions are the only possible limit distribution for normalized sum of random variables. It is a parametric family having only 4 parameters (tail index α , scale, location and skewness Heavier tail with the decrease of α . parameters) α -stables take into account extreme values usually seen as outliers for Gaussians. α -stable are better models the high variability phenomena (infinite variance, impulsive signals...). Azzaoui et al ... Special classes of stable Processes...

Motivation and application examples Overview on α -stable variables and processes The aim of the talk Let us consider a stochastic integral: Z X t = f ( t , λ ) d ξ ( λ ) , where ξ is an α -stable stochastic process, We focus here on the symmetric case ( S α S process) How to characterize this process with a spectral bi-measure? = ) spectral representation We Focus on the particular case of harmonisable processes ( f ( t , λ ) = e ι t λ ) In this case how the bimeasure is linked to the dependance structure of the process. Given observations how to estimate this bi-measure. What is the physical interpretation of the spectral measure in this case. Azzaoui et al ... Special classes of stable Processes...

Motivation and application examples Overview on α -stable variables and processes α -stable variables 1 A random variable X is said stable (or have a stable distribution) if and only if for any positive real A and B their exist a unique positive C and real D s.t: AX 1 + BX 2 = d CX + D X 1 and X 2 i.i.d copies of X (in the symmetric case D=0) 2 It was shown that in this cas there exist a unique 0  α  2 such that C is given by A α + B α = C α Hence the prefix α the characteristic function of Symmetric α -stable variables (S α S) is given by: 3 E [ e ı θ X ] = e � σ α | θ | α where 0 < α  2 and σ > 0 . φ X ( θ ) = I Unfortunately the form of the characteristic function suggest that the density function of these 4 distribution is impossible to calculate except for three special cases ( α = 1 , 2 , or 1 2 ) Azzaoui et al ... Special classes of stable Processes...

Motivation and application examples Overview on α -stable variables and processes α -stable random vectors A random vector X = ( X 1 , . . . , X d ) is α -stable S α S if for every A and B positives, their exist C > 0 such 1 that: AX ( 1 ) + BX ( 2 ) d = CX , where X ( 1 ) and X ( 2 ) are i.i.d. copies of X and A α + B α = C α equivalently we can show that the vector X is symmetric α -stable if and only if every linear combinaison 2 d X Y = b k X k is a an S α S univariate variable. k = 1 ( d ) = ( X 1 , . . . , X d ) is given by: The Characteristic function of an S α S real vector X 3 Z | θ 1 s 1 + · · · + θ d s d | α d Γ φ X ( θ 1 , . . . , θ d ) = exp { � X ( d ) ( s 1 , . . . , s d ) } Sd X ( d ) is a unique positive finite measure on the unit sphere of R d where Γ 4 Complexe random variables and vectors: X = X 1 + ı. X 2 est α -stable if and only i ff the vector ( X 1 , X 2 ) is α -stable on R 2 . More generally a vector ( X 1 , . . . , X d ) with X j = X 1 j + ı. X 2 j , is α -stable if and only if ( X 1 1 , X 2 1 , . . . , X 1 d , X 2 d ) is α -stable vector on R 2 d . 5 A complexe S α S, X = X 1 + ı. X 2 is said isotropic (rotationally invariant) if for any φ 2 [ 0 , 2 π [ d = e ı φ . X . X Azzaoui et al ... Special classes of stable Processes...

Motivation and application examples Overview on α -stable variables and processes α -stable processes, S α S random measures A stochastic process ξ = ( ξ t , t 2 R ) is symmetric if and only if its finite dimensional distributions are 1 S α S vectors. An S α S random measure is a random set function d ξ : B ( R ) 7� ! R ( or C ) such that, for any Borel sets 2 A 1 , . . . , A n , the vector ( d ξ ( A 1 ) , . . . , d ξ ( A n )) is an S α S random vector A random measure d ξ is said independently scattered if for any disjoint Borel sets A 1 , . . . , A n the 3 variables d ξ ( A 1 ) , . . . , d ξ ( A n ) are independents. Azzaoui et al ... Special classes of stable Processes...

Motivation and application examples Overview on α -stable variables and processes Dependence structure: the covariation Let X = ( X 1 , X 2 ) jointly S α S vector with corresponding measure on the sphere Γ , the covariation of X 1 on X 2 is defined by : Z s 1 . ( s 2 ) < α � 1 > d Γ ( s 1 , s 2 ) [ X 1 , X 2 ] α = S 2 where s < β > = sign ( s ) . | s | β In case where X = ( X 1 , X 2 ) is complex i.e. X 1 = X 1 2 and X 2 = X 2 1 + ı X 1 1 + ı X 2 2 , then the covariation of X 1 on X 2 is : Z [ X 1 , X 2 ] α = ( s 1 1 + ı s 1 2 ) . ( s 2 1 + ı s 2 2 ) < α � 1 > d Γ X ( s 1 1 , s 1 2 , s 2 1 , s 2 2 ) S 4 and the notation z < β > = | z | β � 1 z . A useful result: For any S α S vector X on R d with spectral measure Γ X then, " d ! < α � 1 > d # d ! d Z X X X a i s i X b i s i a i X i , b i X i = . d Γ X ( s 1 , . . . , s d ) S d i = 1 i = 1 i = 1 i = 1 α Azzaoui et al ... Special classes of stable Processes...

Motivation and application examples Overview on α -stable variables and processes Properties of the covariation 1 Linearity with respect to the first component i.e. for any S α S vector ( X 1 , X 2 , Y ) we have, [ X 1 + X 2 , Y ] α = [ X 1 , Y ] α + [ X 2 , Y ] α . if X and Y are independent jointly S α S variables, then [ X , Y ] α = 0. the inverse is not true in general 2 3 The covariation is additive with respect to its second component, [ X , Y 1 + Y 2 ] α = [ X , Y 1 ] α + [ X , Y 2 ] α if Y 1 and Y 2 are independents. for any real or complexe a and b , [ a . X , b . Y ] α = ab < α � 1 > [ X , Y ] α . 4 Let X = ( X t ) t an S α S process and denote l ( X ) the space of finite linear combinations of X . The 5 application, ! R + l ( X ) � k . k α : 1 k Y k α , ([ Y , Y ] α ) Y 7� ! α is a norm covariation norm . In this case ( l ( X ) , k . k α ) is a Banach space 6 In ( l ( X ) , k . k α ) , the covariation is continuous moreover we have: | [ Z 1 , Z 2 ] α � [ Z 1 , Z 3 ] α |  2 k Z 1 k α . k Z 2 � Z 3 k α � 1 . α Azzaoui et al ... Special classes of stable Processes...

Motivation and application examples Overview on α -stable variables and processes Second order processes - the stationary case Let r ( t ) be the covariance function of a second order stationary process X t , Z 1 e ı t λ F ( d � ) r(t) is positive definite r ( t ) = Bochner’s Th. �1 n n ⇐ ⇒ X X c i c j r ( t i � t j ) � 0 F is a positive measure i = 1 j = 1 Z 1 e ı t λ d ⇠ ( � ) Cramer-Kolmogorov X t = �1 ⇐ ⇒ ⇠ have orthogonal increments Azzaoui et al ... Special classes of stable Processes...

Motivation and application examples Overview on α -stable variables and processes Second ordre - non stationary case now the covariance is bivariate r ( s , t ) and F ( A , B ) = cov ( ⇠ ( A ) , ⇠ ( B )) the cov. is bilinear ⇒ = positive definite F positive definite bi-measure Z 1 X t = f ( t , � ) d ⇠ ( � ) Cramer-Rao �1 ⇐ ⇒ Z Z f ( s , λ ) f ( t , λ 0 ) F ( d λ , d λ 0 ) r ( s , t ) = The bimeasure F is positive definite in the sense, n n X X c i c j F ( A i , A j ) � 0 , i = 1 j = 1 Azzaoui et al ... Special classes of stable Processes...