Motivation and application examples Overview on α -stable variables and processes Bimeausres, spectral measures and other characerization of heavy tail processes Nourddine Azzaoui, collaboration with Laurent Clavier, Arnaud Guillin and Gareth Peters workshop on Complex systems Modeling and Estimation Challenges in big data (CSM 2014) The Institute of Statistical mathematics (ISM) Azzaoui et al... Bi-measures, spectral representation...
Motivation and application examples Overview on α -stable variables and processes Motivation and application examples Consider a channel or a filter, Azzaoui et al... Bi-measures, spectral representation...
Motivation and application examples Overview on α -stable variables and processes Motivation and application examples Consider a channel or a filter, � s ( t ) = e ∗ h ( t ) = e ( t − τ ) h ( τ ) d τ Azzaoui et al... Bi-measures, spectral representation...
Motivation and application examples Overview on α -stable variables and processes Motivation and application examples Consider a channel or a filter, � s ( t ) = e ∗ h ( t ) = e ( t − τ ) h ( τ ) d τ Or equivalently by Fourier transform, Azzaoui et al... Bi-measures, spectral representation...
Motivation and application examples Overview on α -stable variables and processes Motivation and application examples Consider a channel or a filter, � s ( t ) = e ∗ h ( t ) = e ( t − τ ) h ( τ ) d τ Or equivalently by Fourier transform, = ⇒ � e ιω t h ( t ) dt H ( ω ) = Azzaoui et al... Bi-measures, spectral representation...
Motivation and application examples Overview on α -stable variables and processes Motivation and application examples Consider a channel or a filter, � s ( t ) = e ∗ h ( t ) = e ( t − τ ) h ( τ ) d τ Or equivalently by Fourier transform, = ⇒ � e ιω t h ( t ) dt H ( ω ) = Azzaoui et al... Bi-measures, spectral representation...
Motivation and application examples Overview on α -stable variables and processes Motivation and application examples Consider a channel or a filter, � s ( t ) = e ∗ h ( t ) = e ( t − τ ) h ( τ ) d τ Or equivalently by Fourier transform, = ⇒ N � � a k δ t − τ k e i θ k , e ιω t h ( t ) dt h ( t ) = H ( ω ) = k = 1 But real world is random and ( h ( t ) , t ≥ 0 ) is considered as a stochastic process = ⇒ A harmonizable process � e ιω t d ξ ( t ) H ( ω ) = ( ξ t ) (resp d ξ ( . ) ) is heavy tailed process (resp. random measure) Azzaoui et al... Bi-measures, spectral representation...
Motivation and application examples Overview on α -stable variables and processes Why α -stables? Theoretical interest 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 parameters) Azzaoui et al... Bi-measures, spectral representation...
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... Bi-measures, spectral representation...
Motivation and application examples Overview on α -stable variables and processes The aim of the talk Let us consider a stochastic integral: � X t = f ( t , λ ) d ξ ( λ ) , where ξ is an α -stable stochastic process, Azzaoui et al... Bi-measures, spectral representation...
Motivation and application examples Overview on α -stable variables and processes The aim of the talk Let us consider a stochastic integral: � 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) Azzaoui et al... Bi-measures, spectral representation...
Motivation and application examples Overview on α -stable variables and processes The aim of the talk Let us consider a stochastic integral: � 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? Azzaoui et al... Bi-measures, spectral representation...
Motivation and application examples Overview on α -stable variables and processes The aim of the talk Let us consider a stochastic integral: � 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 Azzaoui et al... Bi-measures, spectral representation...
Motivation and application examples Overview on α -stable variables and processes The aim of the talk Let us consider a stochastic integral: � 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 Given a characteristic bi-measure, how to generate the process from it. Azzaoui et al... Bi-measures, spectral representation...
Motivation and application examples Overview on α -stable variables and processes The aim of the talk Let us consider a stochastic integral: � 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 Given a characteristic bi-measure, how to generate the process from it. = ⇒ Lepage series expansions Azzaoui et al... Bi-measures, spectral representation...
Motivation and application examples Overview on α -stable variables and processes The aim of the talk Let us consider a stochastic integral: � 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 Given a characteristic bi-measure, how to generate the process from it. = ⇒ Lepage series expansions We Focus on the particular case of harmonisable processes ( f ( t , λ ) = e ι t λ ) Azzaoui et al... Bi-measures, spectral representation...
Motivation and application examples Overview on α -stable variables and processes The aim of the talk Let us consider a stochastic integral: � 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 Given a characteristic bi-measure, how to generate the process from it. = ⇒ Lepage series expansions 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. Azzaoui et al... Bi-measures, spectral representation...
Motivation and application examples Overview on α -stable variables and processes The aim of the talk Let us consider a stochastic integral: � 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 Given a characteristic bi-measure, how to generate the process from it. = ⇒ Lepage series expansions 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. Azzaoui et al... Bi-measures, spectral representation...
Motivation and application examples Overview on α -stable variables and processes The aim of the talk Let us consider a stochastic integral: � 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 Given a characteristic bi-measure, how to generate the process from it. = ⇒ Lepage series expansions 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. Prove that it is a natural model for the communication channel. Azzaoui et al... Bi-measures, spectral representation...
Recommend
More recommend
