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Modelling nonstationary signals using stochastic and nonstochastic - - PowerPoint PPT Presentation

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Cyclostationary signals FOT approach Stochastic approach Acknowledgement References-selected Modelling nonstationary signals using stochastic and nonstochastic approach Jacek Lekow Institute of Mathematics Cracow University of Technology

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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

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Cyclostationary signals FOT approach Stochastic approach Acknowledgement References-selected

Plan of the talk

1

Cyclostationary signals Examples

2

FOT approach Basics Relative measure Joint relative measurability FOT: weak convergence FOT: CLT FOT resampling

3

Stochastic approach APC processes, inference APC, limit theorems Mixing conditions Resampling for APC Validity

4

Acknowledgement

Jacek Leśkow Modelling nonstationary signals

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Cyclostationary signals FOT approach Stochastic approach Acknowledgement References-selected Examples

GRF signals for a jogger

Jacek Leśkow Modelling nonstationary signals

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Cyclostationary signals FOT approach Stochastic approach Acknowledgement References-selected Examples

Motivating example no 2. Engine signal (Lafon, Antoni, Sidahmed, Polac, Journal of Sound and Vibration (2011))

Jacek Leśkow Modelling nonstationary signals

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Cyclostationary signals FOT approach Stochastic approach Acknowledgement References-selected Examples

Motivating example no 3. Wheel bearing signal - normally operating and inner race default

Jacek Leśkow Modelling nonstationary signals

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Cyclostationary signals FOT approach Stochastic approach Acknowledgement References-selected Examples

Motivating example no 4.

Jacek Leśkow Modelling nonstationary signals

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Cyclostationary signals FOT approach Stochastic approach Acknowledgement References-selected Examples

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

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Cyclostationary signals FOT approach Stochastic approach Acknowledgement References-selected Examples

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

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Cyclostationary signals FOT approach Stochastic approach Acknowledgement References-selected Basics Relative measure Joint relative measurability FOT: weak convergence FOT: CLT 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 FT,x,t0(ξ) = 1

T µ{u ∈ [t0, t0 + T]; x(u) ≤ ξ}, µ - Lebesque

measure. FOT theoretical distribution Fx,t0(ξ) = limT FT,x,t0(ξ).

For a large class of functions (relatively measurable functions) the above limit exists and does not depend on t0 .

Jacek Leśkow Modelling nonstationary signals

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Cyclostationary signals FOT approach Stochastic approach Acknowledgement References-selected Basics Relative measure Joint relative measurability FOT: weak convergence FOT: CLT FOT resampling

Basics of the FOT approach

Having empirical and theoretical FOT distributions we define Theoretical moments mk =

  • R ξkdFx(ξ).

Empirical moments ˆ mk,T =

  • R ξkdFT,x(ξ)

Covariances via joint FOT distributions Joint FOT distribution of two functions. Take x(t) and y(t). Define Fx,y(ξ1, ξ2) = lim

T

1 T µ{u ∈ [−T/2, T/2]; x(u) ≤ ξ1, y(u) ≤ ξ2}

Jacek Leśkow Modelling nonstationary signals

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Cyclostationary signals FOT approach Stochastic approach Acknowledgement References-selected Basics Relative measure Joint relative measurability FOT: weak convergence FOT: CLT FOT resampling

Basics of the FOT approach

Having a joint FOT of x(t) and x(t + τ) we define empirical autocovariance RT

t0,x as

Rx(τ, T) =

  • R2 ξ1ξ2dFT,x,x(·+τ)(ξ1, ξ2).

Jacek Leśkow Modelling nonstationary signals

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Cyclostationary signals FOT approach Stochastic approach Acknowledgement References-selected Basics Relative measure Joint relative measurability FOT: weak convergence FOT: CLT FOT resampling

Relative measure as a probability measure

Given a set A ∈ BR - 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 µR(A) def = lim

T→∞

1 T µ(A ∩ [t0 − T/2, t0 + T/2]) (1) provided that the limit exists. In such a case, the limit does not depend on t0 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

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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 − Ξ0, where Ξ0 is at most a countable set of points. Each RM function ϕ(t) generates a function Fϕ(ξ) def = µR({t ∈ R : ϕ(t) ≤ ξ}) = lim

T→∞

1 T µ({t ∈ [−T/2, T/2] : ϕ(t) ≤ ξ}) = lim

T→∞

1 T T/2

−T/2

1(−∞,ξ](ϕ(t)) dt in all points ξ where the limit exists.

Jacek Leśkow Modelling nonstationary signals

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Cyclostationary signals FOT approach Stochastic approach Acknowledgement References-selected Basics Relative measure Joint relative measurability FOT: weak convergence FOT: CLT FOT resampling

Examples of relatively measurable functions

(i) Almost periodic functions xAP(t) in the sense of Besicovitch (uniform limits of Fourier polynomials) (ii) Asymptotic almost periodic functions yAAP(t) = xAP(t) + η0(t) with η0 ∈ L1(R) . (AP signals + finite energy signals). (iii) Pseudorandom function x(t) = cos(2πP([t])), where P(t) is the lth 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

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Cyclostationary signals FOT approach Stochastic approach Acknowledgement References-selected Basics Relative measure Joint relative measurability FOT: weak convergence FOT: CLT FOT resampling

The Lebesgue measurable functions ϕ1(t), . . . , ϕn(t), t ∈ R, are said to be jointly RM if the limit Fϕ1···ϕn(ξ1, . . . , ξn) def = µR({t ∈ R : ϕ1(t) ≤ ξ1} ∩ · · · ∩ {t ∈ R : ϕn(t) ≤ ξn}) = lim

T→∞

1 T µ({t ∈ [−T/2, T/2] : ϕ1(t) ≤ ξ1, . . . , ϕn(t) ≤ ξn}) exists for all (ξ1, . . . , ξn) ∈ Rn − Ξ0, where Ξ0 is at most a countable set of (n − 1)-dimensional manifolds of Rn.

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Applications of RM functions

Let us take x1(t) = cos(π[P([t/Tp])]), where P(t) = p2t2 + p1T + p0, where p2 = √ 2, p1 = √ 3, p0 = 0 and Tp = 10Ts, with Ts denoting the sampling period. Clearly x1 is

  • PAM. The sample correlogram

Rx1(τ, T) def = 1 T T/2

−T/2

x1(t + τ)x1(t)dt approaches the limit function Rx1(τ) = (1 − |τ| Tp ) when |τ| ≤ Tp and Rx1(τ) = 0 otherwise.

Jacek Leśkow Modelling nonstationary signals

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Cyclostationary signals FOT approach Stochastic approach Acknowledgement References-selected Basics Relative measure Joint relative measurability FOT: weak convergence FOT: CLT FOT resampling

Application of RM function - picture

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Application of nonRM function

Consider now x2(t) that is RM but its lagged product x2(t)x2(t + τ) is not RM. Details of the construction of such x2 are in Leśkow, Napolitano (2006) and use the idea of the nonRM set constructed before. For such x2 calculate the sample correlogram Rx2(τ, T) def = 1 T T/2

−T/2

x2(t + τ)x2(t)dt However, limT→∞ Rx2(τ, T) does not exist ! Just see the picture next frame.

Jacek Leśkow Modelling nonstationary signals

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Cyclostationary signals FOT approach Stochastic approach Acknowledgement References-selected Basics Relative measure Joint relative measurability FOT: weak convergence FOT: CLT FOT resampling

Application of non RM function - picture

Jacek Leśkow Modelling nonstationary signals

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Conclusions Relatively measurable functions and FOT approach provide an useful approach for signal processing and secure communication Mathematical approach is based on measure theory where measures are induced by level crossings Questions Is it possible to introduce statistical inference in the FOT approach ? Central Limit Theorem ? Resampling ?

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Cyclostationary signals FOT approach Stochastic approach Acknowledgement References-selected Basics Relative measure Joint relative measurability FOT: weak convergence FOT: CLT FOT resampling

Weak convergence

Let {xn(t)}n∈N be a sequence of RM functions and let Φn(u) def = lim

T→∞

1 T T/2

−T/2

ejuxn(t)dt be the (temporal) characteristic function of xn(t).

Jacek Leśkow Modelling nonstationary signals

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Theorem Weak Convergence With Respect to Relative Measure. If lim

n→∞ Φn(u) = Φ(u)

for any u with Φ(u) ∈ L1(R), then there exists a function x(t) with a distribution function Fx(ξ), a temporal characteristic function Φ(u), and a continuous probability density function fx(ξ) = 1 2π

  • R

Φ(u) ejuξdu

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Cyclostationary signals FOT approach Stochastic approach Acknowledgement References-selected Basics Relative measure Joint relative measurability FOT: weak convergence FOT: CLT FOT resampling

FOT CLT

Let us consider a sequence of real-valued functions {ϕk(t)}k∈N such that the following assumptions are fulfilled. A1) For every k the function ϕk(t) is zero mean lim

T→∞

1 T T/2

−T/2

ϕk(t)dt = 0. A2) The functions ϕk(t) have temporal mean-square values lim

T→∞

1 T T/2

−T/2

ϕ2

k(t)dt = σ2 k < ∞

such that lim

n→∞

1 n

n

  • k=1

σ2

k = σ2 < ∞ .

Jacek Leśkow Modelling nonstationary signals

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Cyclostationary signals FOT approach Stochastic approach Acknowledgement References-selected Basics Relative measure Joint relative measurability FOT: weak convergence FOT: CLT FOT resampling

FOT CLT

A3) The functions ϕk(t) are such that ak

def

= lim sup

T→∞

1 T T/2

−T/2

|ϕk(t)|3dt < ∞ with

n

  • k=1

ak = o(n3/2) as n → ∞ Put also xn(t) def = 1 √n

n

  • k=1

ϕk(t)

Jacek Leśkow Modelling nonstationary signals

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Cyclostationary signals FOT approach Stochastic approach Acknowledgement References-selected Basics Relative measure Joint relative measurability FOT: weak convergence FOT: CLT FOT resampling

FOT CLT

Theorem Central Limit Theorem. Let {ϕk(t)}k∈N be a sequence of jointly RM independent functions satisfying assumptions A1, A2, and A3. We have lim

n→∞ µR

  • {t ∈ R : a < xn(t) ≤ b}
  • = µR
  • {t ∈ R : a < x(t) ≤ b}
  • = (1/2πσ2)(1/2)

b

a

e− ξ2

2σ2 dξ .

Moreover, x(t) has a Gaussian FOT distribution function.

Jacek Leśkow Modelling nonstationary signals

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Cyclostationary signals FOT approach Stochastic approach Acknowledgement References-selected Basics Relative measure Joint relative measurability FOT: weak convergence FOT: CLT FOT resampling

FOT resampling

Convenient framework for considering resampling in FOT context is provided by subsampling (Politis (1999), Dehay, Dudek, Leśkow (2014)). Recall the basics for continuous time signal x(t). Let b > 0, h > 0 and t ≥ 0. Define the blocks Eb,th = {u ∈ R : th ≤ u ≤ th + b} as subintervals of the length b, whose location depends on t and have overlap h. The picture below shows the construction of overlapping blocks Eb,th of the length b with the overlap parameter h.

Jacek Leśkow Modelling nonstationary signals

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FOT resampling

In the definition of Eb,th the parameter h is the overlap factor, when t varies. The minimal overlap is obtained for h = b. For the continuous time argument we get the maximal overlap as h → 0. Define now the subsampling version ˆ mT,b(t) of the estimator m∗

T(t) that estimates the mean m with the use of blocks Eb,th via

the following formula: ˆ mT,b(t) def = 1 b

  • Eb,th

m∗

T(u)du

Let also FT,m,b be the empirical FOT distribution of √ b( ˆ mT,b(t) − m∗

Z(t)).

Consistency of the subsampling is equivalent to proving that FT,m,b

T→∞

− → N(0, σ2)

Jacek Leśkow Modelling nonstationary signals

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Application of FOT for financial data

Typical situation for FOT application comes with the following

  • signal. Here x(t) corresponds to the signal of daily prices of stocks
  • KGHM copper conglomerate of Poland. The task is to calculate

short-term (20-days) prediction intervals. The key formula here is FT+δ(ξ) ∈ [ T T + δFT(ξ), T T + δ(FT(ξ) + δ T )] The symbol FT+δ(ξ) is the empirical FOT distribution for x on the interval [0, T + δ] (predictive FOT) while FT(ξ) is the empirical FOT for the data on x available on [0, T]. We obtain two α quantiles approximations:

Jacek Leśkow Modelling nonstationary signals

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FOT and VaR

VaR prediction algorithm. Step 1. Given the observation of the continuous function x calculate the FOT FT. Step 2. Calculate ˆ qT+δ

α

(1) = inf{ξ : T T + δFT(ξ) + δ T + δ ≥ α} and ˆ qT+δ

α

(2) = inf{ξ : T T + δFT(ξ) ≥ α}.

Jacek Leśkow Modelling nonstationary signals

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FOT and VaR

The picture looks as follows:

Jacek Leśkow Modelling nonstationary signals

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Inference for APC

Stochastic process {X (t) ; t ∈ R} is almost periodically correlated (APC), when µX (t) = E (Xt) and the autocovariance function BX (t, τ) =

  • λ∈Λ

a (λ, τ) eiλt. When X is mean-zero, the estimator of the parameter a (λ, τ) is given by: ˆ aT(λ, τ) = 1 T T−|τ|

|τ|

X(s + τ)X(s)S(λs) ds for −T/2 ≤ τ ≤ T/2, and ˆ aT(λ, τ) = 0 otherwise, where S(θ) = (cos(θ), sin(−θ)).

Jacek Leśkow Modelling nonstationary signals

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Given various types of mixing conditions ˆ aT (λ, τ) is asymptotically

  • normal. However, variance-covariance matrix is very complicated.

Consider a particular result: Theorem - Dehay, Dudek, Leśkow (2014))

(A1) supt E

  • |X(t)|4+δ

< ∞ for some δ > 0, the function v → cov

  • X(u + v + τ)X(u + v), X(v + τ)X(v)
  • is almost periodic for each u

and τ. The process X is α-mixing and the mixing coefficient satisfies ∞ αX (t)δ/(4+δ) dt < ∞, (A2) For each λ ∈ Λ

  • λ′∈Λ\{λ}
  • a(λ′, τ)

λ′ − λ

  • < ∞.

We have √ T{ aT (λ, τ) − a(λ, τ)}

L

− → N2(0, V (λ, τ)), where

Jacek Leśkow Modelling nonstationary signals

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V (λ, τ) = lim

T→∞ T Var

  • aT (λ, τ)
  • = 1

2

  • R
  • Bc(2λ, u + τ, τ, u)S1(λu) + Bs(2λ, u + τ, τ, u)S2(λu)

+Bc(0, u + τ, τ, u)S3(λu)

  • du,

with Bc

  • λ, u, v, w
  • =

lim

T→∞

1 T T cov

  • X(s)X(s + u), X(s + v)X(s + w)
  • cos(λs) ds

Bs

  • λ, u, v, w
  • =

lim

T→∞

1 T T cov

  • X(s)X(s + u), X(s + v)X(s + w)
  • sin(λs) ds

S1(θ) = cos(θ) sin(θ) sin(θ) − cos(θ)

  • ,

S2(θ) = − sin(θ) cos(θ) cos(θ) sin(θ)

  • ,

S3(θ) =

  • cos(θ)

sin(θ) − sin(θ) cos(θ)

  • Jacek Leśkow

Modelling nonstationary signals

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Difficulty with mixing

A stochastic process {Xt : t ∈ R} fulfills α-mixing condition if for τ ∈ R and αX(τ) def = sup

t∈R

sup A ∈ FX(−∞, t) B ∈ FX(t + τ, ∞) |P(A ∩ B) − P(A)P(B)|, we have that αX(τ) → 0 for τ → ∞. In practice, this condition can be verified for gaussian

  • structures. Many signals are not gaussian but heavy tailed.

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Possible remedy: weak dependence

Assume (E, · ) - normed space, u ∈ N∗. Assume that h : E u − → R belongs to the class L = {h : E u → R, h ∞≤ 1, Lip(h) < ∞}, where Lip(h) = supx=y

|h(x)−h(y)| x−y1

and x 1= u

i=1 xi .

A sequence {Xt}t∈Z of random variables taking values in E = Rd (d ∈ N∗ = N \ {0}) is (ǫ, L, Ψ)−weakly dependent if there exists Ψ : L × L × N∗ × N∗ → R and a sequence {ǫr}r∈N (ǫr → 0) such that for any (f , g) ∈ L × L, and (u, v, r) ∈ N∗2 × N |Cov(f (Xi1, ..., Xiu), g(Xj1, ..., Xjv ))| ≤ Ψ(f , g, u, v)ǫr whenever i1 < i2 < ... < iu ≤ r + iu ≤ j1 < j2 < ... < jv.

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Resampling for APC

As in the FOT case, we define the blocks Eb,th = {u ∈ R : th ≤ u ≤ th + b} as subintervals of the length b, whose location depends on t and have overlap h. The picture below shows the construction of overlapping blocks Eb,th of the length b with the overlap parameter h.

Jacek Leśkow Modelling nonstationary signals

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Estimator

We consider the following estimator of the parameter a(λ, τ). ˆ aT(λ, τ) = 1 T T−|τ|

|τ|

X(s + τ)X(s)S(λs) ds for −T/2 ≤ τ ≤ T/2, and ˆ aT(λ, τ) = 0 otherwise, where S(θ) = (cos(θ), sin(−θ)).

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The subsampling version ˜ ab,th(λ, τ) of the estimator ˆ aT(λ, τ) generated by the data Yb,th is defined as ˜ ab,th = 1 b b−|τ|

|τ|

X(th + u + τ)X(th + u) S(λu)du for |τ| ≤ b/2 and ˜ ab,th = ˜ ab,th(λ, τ) = 0 otherwise. Unfortunately, lim

b→∞ E{˜

ab,th} = a(λ, τ) S1(λth) for all fixed λ, τ, t and h. Thus, we need to modify the subsampling version ˜ ab,th to get an asymptotically unbiased estimator of a(λ, τ). The modified subsampling version ˆ ab,th = ˆ ab,th(λ, τ) of the estimator ˆ aT(λ, τ) is defined by ˆ ab,th = ˜ ab,th S1(−λth) for |τ| ≤ b/2 and ˆ ab,th = ˆ ab,th(λ, τ) = 0 otherwise.

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Asymptotic validity theorem

Let APC process X fulfill the following assumptions: (A1) supt E

  • |X(t)|4+δ

< ∞ for some δ > 0, the fourth moment is almost periodic in the following sense : the function t → cov

  • X(t)X(t + τ), X(t + u)X(t + u + τ)
  • is almost

periodic for each u. Moreover the process X is α-mixing with αX(·)1+(4/δ) integrable on [0, ∞) . (A2) For each λ ∈ Λ the following separability property is fulfilled

  • λ′∈Λ\{λ}
  • a(λ′, τ)

λ′ − λ

  • < ∞.

(A3) b/T → 0 while both T, b → ∞. (There is an optimal choice for b = C · T 1/3)

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Asymptotic validity theorem

If the conditions (A1), (A2) and (A3) are fulfilled then the subsampling estimator ˆ ab,th is asymptotically valid that is the sequence of processes LT,b,h(x) = 1 q

q−1

  • j=0

1 √ b

  • ˆ

ab,jh − ˆ aT(λ, τ)

  • ≤ x
  • where q = qT = ⌊ T−b

h ⌋ + 1 is the number of intervals Eb,jh

contained in [0, T] and JT(x) = P √ T(ˆ aT(λ, τ) − a(λ, τ)) ≤ x

  • are asymptotically equivalent.

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Subsampling in practice

Typical cyclostationary signal analysis. Underlying residual signal

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Subsampling - wheel bearing signal

Wheel bearing data - frequency signature. Healthy structure (left) and damaged structure (right).

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Acknowledgements Financial support of the Collaborative Linkage Grant of NATO and the support of Polish National Center for Science NCN grant no: UMO – 2013/10/M/ST1/00096 are gratefully acknowledged.

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Cyclostationary signals FOT approach Stochastic approach Acknowledgement References-selected

References - selected

Dehay, D., Leśkow, J. and Napolitano, A. (2013), ’Central Limit Theorem in the Functional Approach’, IEEE Transactions

  • n Signal Processing, Vol 61, No. 16, pp. 4025 - 4037.

Dehay, D., Dudek, A. and Leśkow, J. (2014), Subsampling for continuous-time almost periodically correlated processes, Journal of Statistical Planning and Inference, vol 150, pp. 142

  • 158.

Leśkow, J., Napolitano, A. (2006), Foundations of the functional approach for signal analysis, Signal Processing, Vol 86, pp. 3796-3825.

Jacek Leśkow Modelling nonstationary signals