15 - Introduction to Time Series Analysis. Lecture 15....

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Unformatted text preview: Introduction to Time Series Analysis. Lecture 15. Spectral Analysis 1. Spectral density: Facts and examples. 2. Spectral distribution function. 3. Wold's decomposition. 1 Spectral Analysis Idea: decompose a stationary time series {Xt } into a combination of sinusoids, with random (and uncorrelated) coefficients. Just as in Fourier analysis, where we decompose (deterministic) functions into combinations of sinusoids. This is referred to as `spectral analysis' or analysis in the `frequency domain,' in contrast to the time domain approach we have considered so far. The frequency domain approach considers regression on sinusoids; the time domain approach considers regression on past values of the time series. 2 A periodic time series Consider Xt = A sin(2t) + B cos(2t) = C sin(2t + ), where A, B are uncorrelated, mean zero, variance 2 = 1, and C 2 = A2 + B 2 , tan = B/A. Then t = E[Xt ] = 0 (t, t + h) = cos(2h). So {Xt } is stationary. 3 An aside: Some trigonometric identities sin , cos sin2 + cos2 = 1, tan = sin(a + b) = sin a cos b + cos a sin b, cos(a + b) = cos a cos b - sin a sin b. 4 A periodic time series For Xt = A sin(2t) + B cos(2t), with uncorrelated A, B (mean 0, variance 2 ), (h) = 2 cos(2h). The autocovariance of the sum of two uncorrelated time series is the sum of their autocovariances. Thus, the autocovariance of a sum of random sinusoids is a sum of sinusoids with the corresponding frequencies: k Xt = j=1 k (Aj sin(2j t) + Bj cos(2j t)) , 2 j cos(2j h), j=1 (h) = 2 where Aj , Bj are uncorrelated, mean zero, and Var(Aj ) = Var(Bj ) = j . 5 A periodic time series k k Xt = j=1 (Aj sin(2j t) + Bj cos(2j t)) , (h) = j=1 2 j cos(2j h). Thus, we can represent (h) using a Fourier series. The coefficients are the variances of the sinusoidal components. The spectral density is the continuous analog: the Fourier transform of . (The analogous spectral representation of a stationary process Xt involves a stochastic integral--a sum of discrete components at a finite number of frequencies is a special case. We won't consider this representation in this course.) 6 Spectral density If a time series {Xt } has autocovariance satisfying h=- |(h)| < , then we define its spectral density as f () = h=- (h)e-2ih for - < < . 7 Spectral density: Some facts 1. We have h=- (h)e-2ih < . This is because |ei | = | cos + i sin | = (cos2 + sin2 )1/2 = 1, and because of the absolute summability of . 2. f is periodic, with period 1. This is true since e-2ih is a periodic function of with period 1. Thus, we can restrict the domain of f to -1/2 1/2. (The text does this.) 8 Spectral density: Some facts 3. f is even (that is, f () = f (-)). To see this, write -1 f () = h=- -1 (h)e-2ih + (0) + h=1 (h)e-2ih , f (-) = h=- (h)e-2i(-h) + (0) + h=1 -1 (h)e-2i(-h) , (-h)e-2ih = h=1 (-h)e-2ih + (0) + h=- = f (). 4. f () 0. 9 Spectral density: Some facts 1/2 5. (h) = -1/2 1/2 e2ih f () d. 1/2 e2ih f () d = -1/2 -1/2 j=- e-2i(j-h) (j) d 1/2 = j=- (j) -1/2 e-2i(j-h) d (j) ei(j-h) - e-i(j-h) 2i(j - h) (j) sin((j - h)) = (h). (j - h) = (h) + j=h = (h) + j=h 10 Example: White noise 2 For white noise {Wt }, we have seen that (0) = w and (h) = 0 for h = 0. Thus, f () = h=- (h)e-2ih 2 = (0) = w . That is, the spectral density is constant across all frequencies: each frequency in the spectrum contributes equally to the variance. This is the origin of the name white noise: it is like white light, which is a uniform mixture of all frequencies in the visible spectrum. 11 Example: AR(1) 2 For Xt = 1 Xt-1 + Wt , we have seen that (h) = w 1 /(1 - 2 ). Thus, 1 |h| f () = h=- (h)e-2ih 2 w = 1 - 2 1 1 e-2ih h=- |h| 2 w = 1 - 2 1 2 w = 1 - 2 1 1+ h=1 h e-2ih + e2ih 1 1 e-2i 1 e2i 1+ + 1 - 1 e-2i 1 - 1 e2i 2 w 1 - 1 e-2i 1 e2i = (1 - 2 ) (1 - 1 e-2i )(1 - 1 e2i ) 1 2 w = 2. 1 - 21 cos(2) + 1 12 Examples 2 White noise: {Wt }, (0) = w and (h) = 0 for h = 0. 2 f () = (0) = w . 2 AR(1): Xt = 1 Xt-1 + Wt , (h) = w 1 /(1 - 2 ). 1 |h| f () = 2 w . 1-21 cos(2)+2 1 If 1 > 0 (positive autocorrelation), spectrum is dominated by low frequency components--smooth in the time domain. If 1 < 0 (negative autocorrelation), spectrum is dominated by high frequency components--rough in the time domain. 13 Example: AR(1) Spectral density of AR(1): Xt = +0.9 Xt-1 + Wt 100 90 80 70 60 f() 50 40 30 20 10 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 14 Example: AR(1) Spectral density of AR(1): Xt = -0.9 Xt-1 + Wt 100 90 80 70 60 f() 50 40 30 20 10 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 15 ...
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This note was uploaded on 05/14/2011 for the course STAT 153 taught by Professor Staff during the Fall '08 term at University of California, Berkeley.

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