Time Series - STAT - EPFL

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Random walk Example 6 Let T = {0,1,2,...}, let {ε t } be white noise, let Y 0 = 0, and define Y t = Y t−1 + ε t , t = 1,2,... . Show that this is not a stationary time series. Y 0 10 20 30 40 0 100 200 300 400 500 Time Time Series Autumn 2008 – slide 28 Autoregression Example 7 Let T = Z, let {ε t } be white noise, and define Y t = αY t−1 + ε t . This is an autoregressive process of order 1, AR(1), model. Find a necessary condition for it to be stationary. The graph below shows examples with α = ±0.9. Y(t) −4 −2 0 2 4 Y(t) −6 −4 −2 0 2 4 6 0 50 100 150 200 0 50 100 150 200 Time Time Time Series Autumn 2008 – slide 29 14
Periodic series Example 8 Let {ε t } be white noise with unit variance, and define ( ) 2πt Y t = cos + 5ε t , t = 1,... ,500. 50 This is a periodic signal obscured by noise. Compute its mean and autocorrelation function. −2 0 1 2 0 100 200 300 400 500 Time −20 0 10 0 100 200 300 400 500 Time Time Series Autumn 2008 – slide 30 Summary Today we □ saw some examples of time series □ introduced some basic ideas: – use of stochastic process as model for time series – mean, covariance and correlation functions – stationary and strictly stationary series – white noise – simple examples: moving average, random walk, autoregression, periodic series Next week: □ simple approaches to removing systematic variation (trend and seasonality) Time Series Autumn 2008 – slide 31 15
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Time Series - STAT - EPFL

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