Modelling nonstationary signals using stochastic and nonstochastic - PowerPoint PPT Presentation

Cyclostationary signals FOT approach Stochastic approach Acknowledgement References-selected Modelling nonstationary signals using stochastic and nonstochastic approach Jacek Leśkow Institute of Mathematics Cracow University of Technology Cracow, Poland Będlewo, November 2016 Jacek Leśkow Modelling nonstationary signals

Cyclostationary signals FOT approach Stochastic approach Acknowledgement References-selected Plan of the talk Cyclostationary signals 1 Examples FOT approach 2 Basics Relative measure Joint relative measurability FOT: weak convergence FOT: CLT FOT resampling Stochastic approach 3 APC processes, inference APC, limit theorems Mixing conditions Resampling for APC Validity Jacek Leśkow Modelling nonstationary signals Acknowledgement 4

Cyclostationary signals FOT approach Stochastic approach Examples Acknowledgement References-selected GRF signals for a jogger Jacek Leśkow Modelling nonstationary signals

Cyclostationary signals FOT approach Stochastic approach Examples Acknowledgement References-selected Motivating example no 2. Engine signal (Lafon, Antoni, Sidahmed, Polac, Journal of Sound and Vibration (2011)) Jacek Leśkow Modelling nonstationary signals

Cyclostationary signals FOT approach Stochastic approach Examples Acknowledgement References-selected Motivating example no 3. Wheel bearing signal - normally operating and inner race default Jacek Leśkow Modelling nonstationary signals

Cyclostationary signals FOT approach Stochastic approach Examples Acknowledgement References-selected Motivating example no 4. Jacek Leśkow Modelling nonstationary signals

Cyclostationary signals FOT approach Stochastic approach Examples Acknowledgement References-selected Research on cyclostationarity Over 5000 papers in international journals published in recent 10 years Wide ranging applicability: biology, medicine, climate, finances, vibroacoustics, structural health monitoring,... Many challenges for inferential models: identification, estimation, bootstrap, time/frequency signature,... Jacek Leśkow Modelling nonstationary signals

Cyclostationary signals FOT approach Stochastic approach Examples Acknowledgement References-selected Main task of this presentation to briefly present two most popular approaches: nonstochastic (FOT) and stochastic (bootstrap) to show their impact on inference for cyclostationary signals Jacek Leśkow Modelling nonstationary signals

Basics Cyclostationary signals Relative measure FOT approach Joint relative measurability Stochastic approach FOT: weak convergence Acknowledgement FOT: CLT References-selected FOT resampling Basics of the FOT approach In the FOT ( Fraction-of-Time ) approach we look at signals { x ( t ) , t ∈ R or t ∈ Z } as deterministic functions . The probability is generated by level-crossings of the function x ( t ) . We define empirical FOT distribution F T , x , t 0 ( ξ ) = 1 T µ { u ∈ [ t 0 , t 0 + T ]; x ( u ) ≤ ξ } , µ - Lebesque measure. FOT theoretical distribution F x , t 0 ( ξ ) = lim T F T , x , t 0 ( ξ ) . For a large class of functions (relatively measurable functions) the above limit exists and does not depend on t 0 . Jacek Leśkow Modelling nonstationary signals

Basics Cyclostationary signals Relative measure FOT approach Joint relative measurability Stochastic approach FOT: weak convergence Acknowledgement FOT: CLT References-selected FOT resampling Basics of the FOT approach Having empirical and theoretical FOT distributions we define R ξ k dF x ( ξ ) . � Theoretical moments m k = � R ξ k dF T , x ( ξ ) Empirical moments ˆ m k , T = Covariances via joint FOT distributions Joint FOT distribution of two functions. Take x ( t ) and y ( t ) . Define 1 F x , y ( ξ 1 , ξ 2 ) = lim T µ { u ∈ [ − T / 2 , T / 2 ]; x ( u ) ≤ ξ 1 , y ( u ) ≤ ξ 2 } T Jacek Leśkow Modelling nonstationary signals

Basics Cyclostationary signals Relative measure FOT approach Joint relative measurability Stochastic approach FOT: weak convergence Acknowledgement FOT: CLT References-selected FOT resampling Basics of the FOT approach Having a joint FOT of x ( t ) and x ( t + τ ) we define empirical autocovariance R T t 0 , x as � R x ( τ, T ) = R 2 ξ 1 ξ 2 dF T , x , x ( · + τ ) ( ξ 1 , ξ 2 ) . Jacek Leśkow Modelling nonstationary signals

Basics Cyclostationary signals Relative measure FOT approach Joint relative measurability Stochastic approach FOT: weak convergence Acknowledgement FOT: CLT References-selected FOT resampling Relative measure as a probability measure Given a set A ∈ B R - the σ -field of the Borel subsets and µ the Lebesgue measure on the real line R , the relative measure of A is defined as 1 µ R ( A ) def = lim T µ ( A ∩ [ t 0 − T / 2 , t 0 + T / 2 ]) (1) T →∞ provided that the limit exists. In such a case, the limit does not depend on t 0 and the set A is said to be relatively measurable (RM). Relatively measurable sets do NOT form a σ -algebra. It is relatively easy to create a non relatively measurable set. Jacek Leśkow Modelling nonstationary signals

Basics Cyclostationary signals Relative measure FOT approach Joint relative measurability Stochastic approach FOT: weak convergence Acknowledgement FOT: CLT References-selected FOT resampling Definition Relative Measurability of Functions. A Lebesgue measurable function ϕ ( t ) is said to be relatively measurable if and only if the set { t ∈ R : ϕ ( t ) ≤ ξ } is RM for every ξ ∈ R − Ξ 0 , where Ξ 0 is at most a countable set of points. Each RM function ϕ ( t ) generates a function F ϕ ( ξ ) def = µ R ( { t ∈ R : ϕ ( t ) ≤ ξ } ) = 1 T µ ( { t ∈ [ − T / 2 , T / 2 ] : ϕ ( t ) ≤ ξ } ) = lim T →∞ � T / 2 1 lim 1 ( −∞ ,ξ ] ( ϕ ( t )) dt T T →∞ − T / 2 in all points ξ where the limit exists. Jacek Leśkow Modelling nonstationary signals

Basics Cyclostationary signals Relative measure FOT approach Joint relative measurability Stochastic approach FOT: weak convergence Acknowledgement FOT: CLT References-selected FOT resampling Examples of relatively measurable functions (i) Almost periodic functions x AP ( t ) in the sense of Besicovitch (uniform limits of Fourier polynomials) (ii) Asymptotic almost periodic functions y AAP ( t ) = x AP ( t ) + η 0 ( t ) with η 0 ∈ L 1 ( R ) . (AP signals + finite energy signals). (iii) Pseudorandom function x ( t ) = cos ( 2 π P ([ t ])) , where P ( t ) is the l th order polynomial with l ≥ 2 and at least one its coefficient is irrational. (iv) Pseudorandom function y ( t ) = cos ( π [ P ([ t ])]) , where P ( · ) as in (iii). Functions (iii) and (iv) are popular in communication signals modelling and known as PAM - Pulse Amplitude Modulations. PAM in (iii) and (iv) are not AP. Jacek Leśkow Modelling nonstationary signals

Basics Cyclostationary signals Relative measure FOT approach Joint relative measurability Stochastic approach FOT: weak convergence Acknowledgement FOT: CLT References-selected FOT resampling The Lebesgue measurable functions ϕ 1 ( t ) , . . . , ϕ n ( t ) , t ∈ R , are said to be jointly RM if the limit F ϕ 1 ··· ϕ n ( ξ 1 , . . . , ξ n ) def = µ R ( { t ∈ R : ϕ 1 ( t ) ≤ ξ 1 } ∩ · · · ∩ { t ∈ R : ϕ n ( t ) ≤ ξ n } ) = 1 lim T µ ( { t ∈ [ − T / 2 , T / 2 ] : ϕ 1 ( t ) ≤ ξ 1 , . . . , ϕ n ( t ) ≤ ξ n } ) T →∞ exists for all ( ξ 1 , . . . , ξ n ) ∈ R n − Ξ 0 , where Ξ 0 is at most a countable set of ( n − 1 ) -dimensional manifolds of R n . Jacek Leśkow Modelling nonstationary signals

Basics Cyclostationary signals Relative measure FOT approach Joint relative measurability Stochastic approach FOT: weak convergence Acknowledgement FOT: CLT References-selected FOT resampling Applications of RM functions Let us take x 1 ( t ) = cos ( π [ P ([ t / T p ])]) , where √ √ P ( t ) = p 2 t 2 + p 1 T + p 0 , where p 2 = 2 , p 1 = 3 , p 0 = 0 and T p = 10 T s , with T s denoting the sampling period. Clearly x 1 is PAM. The sample correlogram � T / 2 = 1 R x 1 ( τ, T ) def x 1 ( t + τ ) x 1 ( t ) dt T − T / 2 approaches the limit function R x 1 ( τ ) = ( 1 − | τ | ) T p when | τ | ≤ T p and R x 1 ( τ ) = 0 otherwise. Jacek Leśkow Modelling nonstationary signals

Basics Cyclostationary signals Relative measure FOT approach Joint relative measurability Stochastic approach FOT: weak convergence Acknowledgement FOT: CLT References-selected FOT resampling Application of RM function - picture Jacek Leśkow Modelling nonstationary signals

Basics Cyclostationary signals Relative measure FOT approach Joint relative measurability Stochastic approach FOT: weak convergence Acknowledgement FOT: CLT References-selected FOT resampling Application of nonRM function Consider now x 2 ( t ) that is RM but its lagged product x 2 ( t ) x 2 ( t + τ ) is not RM. Details of the construction of such x 2 are in Leśkow, Napolitano (2006) and use the idea of the nonRM set constructed before. For such x 2 calculate the sample correlogram � T / 2 = 1 R x 2 ( τ, T ) def x 2 ( t + τ ) x 2 ( t ) dt T − T / 2 However, lim T →∞ R x 2 ( τ, T ) does not exist ! Just see the picture next frame. Jacek Leśkow Modelling nonstationary signals