Time Series - STAT - EPFL

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Comments □ The form of the spectrum suggests that f(ω) = f(−ω), f(ω) = f(ω + k), k ∈ Z, so f need only be defined in 0 < ω < 1 2 —recall the definition of the periodogram. □ The variance of the average Y of data Y 1 ,... ,Y n from a stationary process satisfies { lim nvar(Y ) = lim n→∞ n→∞ } n−1 ∑ ∞∑ γ 0 + 2 (1 − h/n)γ h = γ 0 + 2 γ h = f(0) h=1 for large n. Thus the spectral density at the origin, if finite, equals the (rescaled) variance of Y . □ We easily see that ∫ 1/2 −1/2 f(ω)dω = γ 0. □ The spectrum is the discrete Fourier transform of the autocovariance sequence. □ The inverse Fourier transform gives the autocovariance function by γ k = ∫ 1/2 −1/2 e 2πikω f(ω)dω = 2 ∫ 1/2 □ The spectral density need not always exist; cf. Example 33. 0 h=1 cos(2πkω)f(ω)dω, k ∈ Z. (6) Time Series Spring 2010 – slide 131 Spectral distribution Theorem 32 (a) A set of numbers {γ h } h∈Z is the autocovariance function of a stationary random sequence iff there exists a unique bounded function F defined on [− 1 2 , 1 2 ] such that F(−1/2) = 0, F is right-continuous and non-decreasing, with symmetric increments about zero, and ∫ γ h = e 2πihu dF(u), h ∈ Z. (−1/2,1/2] The function F is called the spectral distribution function of γ h , and its derivative, if it exists, is called the spectral density function, or spectrum. (b) If ∑ h |γ h| < ∞, then f exists. (c) A function f(ω) defined on [−1/2,1/2] is the spectrum of a stationary process if and only if f(ω) = f(−ω), f(ω) ≥ 0, and ∫ 1/2 −1/2 f(ω)dω < ∞. □ The symmetric increments property means that if 0 ≤ a < b ≤ 1 2 , then F(b) − F(a) = F(−a) − F(−b). □ The interpretation of F is that F(ω 2 ) − F(ω 1 ) measures the variation accounted for by fluctuations in frequency in the interval (ω 1 ,ω 2 ). □ Part (c) of the theorem suggests how we may construct covariance functions with desired properties, by choosing an appropriate spectrum—note that f may be any scaled density function. Time Series Spring 2010 – slide 132 129
Example Example 33 Show that the covariance function of the stationary random sequence given by Y t = U 1 cos(2πω 0 t) + U 2 sin(2πω 0 t), U 1 ,U 2 iid ∼ N(0,σ 2 ), may be written as γ h = ∫ 1/2 −1/2 ⎧ ⎪⎨ 0, ω < −ω 0 , e 2πiωh dF(ω), F(ω) = σ ⎪⎩ 2 /2, −ω 0 ≤ ω < ω 0 , σ 2 , ω 0 ≤ ω. Time Series Spring 2010 – slide 133 Linear filters Definition 34 A linear filter is a transformation of the random sequence {U t } of the form Y t = ∞∑ j=−∞ □ If {U t } is stationary and a j U t−j . (7) – only a finite number of the a j are non-zero, then {Y t } is stationary; – infinitely many of the a j are non-zero, then the properties of {Y t } depend on their values. □ The relation between the spectra of the sequences is given by the following theorem: Theorem 35 The spectra of two stationary random sequences {U t } and {Y t } satisfying (7) are related by f Y (ω) = |a(ω)| 2 f U (ω), where a(ω) = ∑ ∞ j=−∞ a je −2πijω is the transfer function of the linear filter. Time Series Spring 2010 – slide 134 Effect of filtering Example 36 Find the spectrum of a three-point moving average of an AR(1) process. Spectrum Squared modulus of transfer function Filtered spectrum f(w) 0 1 2 3 4 |a(w)|^2 0.0 0.2 0.4 0.6 0.8 1.0 f(w)|a(w)|^2 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 w w w Time Series Spring 2010 – slide 135 130
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Page 9: Comments □ Why use ARMA processes

Time Series - STAT - EPFL

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