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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 integrala 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)e2ih for  < < . 7 Spectral density: Some facts
1. We have h= (h)e2ih < . 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 e2ih 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)e2ih + (0) +
h=1 (h)e2ih , f () =
h= (h)e2i(h) + (0) +
h=1 1 (h)e2i(h) , (h)e2ih =
h=1 (h)e2ih + (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= e2i(jh) (j) d
1/2 =
j= (j)
1/2 e2i(jh) d (j) ei(jh)  ei(jh) 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)e2ih 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 Xt1 + Wt , we have seen that (h) = w 1 /(1  2 ). Thus, 1 h f () =
h= (h)e2ih 2 w = 1  2 1 1 e2ih
h= h 2 w = 1  2 1 2 w = 1  2 1 1+
h=1 h e2ih + e2ih 1 1 e2i 1 e2i 1+ + 1  1 e2i 1  1 e2i 2 w 1  1 e2i 1 e2i = (1  2 ) (1  1 e2i )(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 Xt1 + Wt , (h) = w 1 /(1  2 ). 1 h f () = 2 w . 121 cos(2)+2 1 If 1 > 0 (positive autocorrelation), spectrum is dominated by low frequency componentssmooth in the time domain. If 1 < 0 (negative autocorrelation), spectrum is dominated by high frequency componentsrough in the time domain. 13 Example: AR(1)
Spectral density of AR(1): Xt = +0.9 Xt1 + 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 Xt1 + 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.
 Fall '08
 Staff

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