Time Series - STAT - EPFL

More documents

Recommendations

Info

Second-order theory of stationary random processes slide 126 Reminder: Basic definitions Definition 28 (a) A random function is a set of random variables {Y (t)} such that the time-index t can take any real value. (b) The trend of {Y t } is the non-random function µ(t) = E{Y (t)}, and its autocovariance function is γ(t,s) = cov{Y (t),Y (s)} = E[{Y (t) − µ(t)}{Y (s) − µ(s)}], t,s ∈ R. The mean and covariance functions constitute the second-order properties of {Y (t)}. They determine the entire distribution of the random function if the joint distribution of any finite collection of random variables {Y (t 1 ),... ,Y (t k )} is multivariate normal. (c) The random function is stationary if µ(t) = µ and γ(t,s) = γ(|t − s|): the trend is constant and the covariance between Y (t) and Y (s) depends only on their time separation t − s. In this case we can define the autocorrelation function ρ(t) = γ(t,0)/γ(0,0). (d) A random sequence is a collection of random variables {Y t } in which the time index takes only integer values: t ∈ Z. We use a subscript notation to make this clear. (e) A white noise sequence is a random sequence consisting of mutually independent random variables each with mean zero and variance σ 2 . Time Series Spring 2010 – slide 127 Spectrum Definition 29 (a) The autocovariance generating function of a stationary random sequence {Y t } with autocovariances γ k = cov(Y t ,Y t+k ) is (b) The spectrum of {Y t } is G(z) = ∞∑ k=−∞ f(ω) = G(e −2πiω ) = γ k z k , z ∈ C. ∞∑ k=−∞ γ k e −2πikω , ω ∈ R, where i 2 = −1; this may also be written as the real-valued function f(ω) = γ 0 + 2 ∞∑ γ k cos(2πkω), ω ∈ R. (5) k=1 The normalised spectrum is f ∗ (ω) = f(ω)/γ 0 . The spectrum provides a convenient summary of the second-order properties of the process in a single function, and also shows the effect of linear operations on the series very simply. Time Series Spring 2010 – slide 128 127
Examples Example 30 Find the spectrum and normalised spectrum for white noise. Example 31 Show that the spectrum of the AR(1) process Y t = αY t−1 + ε t with |α| < 1 is and that the normalised spectrum is f(ω) = f ∗ (ω) = σ 2 1 − 2α cos(2πω) + α 2, 1 − α 2 1 − 2α cos(2πω) + α 2. Time Series Spring 2010 – slide 129 Normalized spectra for AR(1) models Y −4 −2 0 1 2 3 Y −4 −2 0 1 2 3 0 50 alpha=−0.5 100 150 200 Time alpha=0.5 0 50 100 150 200 Time f*(w) 0.5 1.0 1.5 2.0 2.5 3.0 f*(w) 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.1 0.2 0.3 0.4 0.5 w 0.0 0.1 0.2 0.3 0.4 0.5 w alpha=0.9 Y −4 −2 0 2 4 f*(w) 0 5 10 15 0 50 100 150 200 0.0 0.1 0.2 0.3 0.4 0.5 Time w Time Series Spring 2010 – slide 130 128
Page 1: Time Series Anthony Davison c○201
Page 5 and 6: Example Example 33 Show that the co
Page 7 and 8: Moving average process Definition 3
Page 9: Comments □ Why use ARMA processes

Time Series - STAT - EPFL

Create successful ePaper yourself

Delete template?

Save as template?