Time Series - STAT - EPFL

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Stationarity If S is a set, then we use u + S to denote the set {u + s : s ∈ S}, and Y S to denote the set of random variables {Y s : s ∈ S}. Definition 3 A stochastic process {Y t } t∈T is said to be (a) strictly stationary if for any finite subset S ⊂ T and any u such that u + S ⊂ T , the joint distributions of Y S+u and Y S are the same; (b) second-order stationary (or weakly stationary) if the mean µ t is constant and the covariance function γ(s,t) depends only on t − s. □ When T = Z = {0, ±1, ±2,...} and the process is stationary, say, where h is called the lag. γ(t,t + h) = γ(0,h) = γ(0, −h) ≡ γ |h| = γ h , h ∈ Z, □ Similarly, we can write ρ(t,t + h) ≡ ρ |h| = ρ h , say, for h ∈ Z. □ Thus in the stationary case the covariance and correlation functions are symmetric around h = 0. Time Series Autumn 2008 – slide 24 Stationarity □ In practice strict stationarity is impossible to verify, and many computations require only second-order stationarity. □ Hereafter ‘stationary’ will mean second-order stationary, when used without comment. □ We can also define third- and higher-order stationarity by extending (b) to higher moments. □ In practice we often preprocess the data, by removing trend/seasonality, and model the processed series using a stationary stochastic process. □ However treating variation as random or as trend depends on the purpose of analysis. Consider the figure below, or the temperature data ... Y(t) 0.0 0.5 1.0 1.5 2.0 2.5 Y(t) 0.0 0.5 1.0 1.5 2.0 2.5 0 20 40 60 80 100 t 0 2 4 6 8 10 t Time Series Autumn 2008 – slide 25 12
White noise Definition 4 A stochastic process {Y t } is called white noise if its elements are all uncorrelated, with mean E(Y t ) = 0 and variance var(Y t ) = σ 2 . If in addition the Y t are normally (Gaussianly) distributed, then we have Gaussian white noise, iid Y t ∼ N(0,σ 2 ). The term ‘white’ comes from an analogy with white light, and indicates that all frequencies are equally present ... −3 −1 1 3 0 100 200 300 400 500 Time −3 −1 1 3 0 100 200 300 400 500 Time Time Series Autumn 2008 – slide 26 Moving average □ The panels on the previous page showed Gaussian white noise {ε t } above, and a smoothed version Y t = 1 3 (ε t + ε t−1 + ε t−2 ). □ Averaging reduces the variance, and introduces correlation in {Y t }. Example 5 Compute the autocorrelation function of the above moving average and show that it is stationary. Discuss the figure below. Y_{t+1} −1.5 −0.5 0.5 1.5 Y_{t+2} −1.5 −0.5 0.5 1.5 Y_{t+3} −1.5 −0.5 0.5 1.5 −1.5 −0.5 0.5 1.5 −1.5 −0.5 0.5 1.5 −1.5 −0.5 0.5 1.5 Y_t Y_t Y_t Time Series Autumn 2008 – slide 27 13
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Page 15: Periodic series Example 8 Let {ε t

Time Series - STAT - EPFL

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