Unformatted text preview: Introduction to Time Series Analysis. Lecture 17.
1. Review: Spectral distribution function, spectral density. 2. Rational spectra. Poles and zeros. 3. Examples. 4. Timeinvariant linear filters 5. Frequency response 1 Review: Spectral density and spectral distribution function If a time series {Xt } has autocovariance satisfying h= (h) < , then we define its spectral density as f () =
h= (h)e2ih for  < < . We have
1/2 1/2 (h) =
1/2 e2ih f () d =
1/2 e2ih dF (), where dF () = f ()d. f measures how the variance of Xt is distributed across the spectrum.
2 Review: Spectral density of a linear process
If Xt is a linear process, it can be written Xt = Then
2 f () = w e2i 2 i=0 i Wti = (B)Wt . . That is, the spectral density f () of a linear process measures the modulus of the (MA()) polynomial at the point e2i on the unit circle. 3 Spectral density of a linear process
For an ARMA(p,q), (B) = (B)/(B), so f () =
2 w (e2i )(e2i ) (e2i ) (e2i ) (e2i ) (e2i )
2 2 = w . This is known as a rational spectrum. 4 Rational spectra
Consider the factorization of and as (z) = q (z  z1 )(z  z2 ) (z  zq ) (z) = p (z  p1 )(z  p2 ) (z  pp ), where z1 , . . . , zq and p1 , . . . , pp are called the zeros and poles. f () =
2 w q p
2 q q 2i j=1 (e p 2i j=1 (e q j=1 p j=1  zj )  pj )
2 2. 2 2 = w e2i  zj e2i 2 p  pj  5 Rational spectra
2 q q j=1 p j=1 2 f () = w e2i  zj e2i 2 2. 2 p  pj  As varies from 0 to 1/2, e2i moves clockwise around the unit circle from 1 to ei = 1. And the value of f () goes up as this point moves closer to (further from) the poles pj (zeros zj ). 6 Example: ARMA
Recall AR(1): (z) = 1  1 z. The pole is at 1/1 . If 1 > 0, the pole is to the right of 1, so the spectral density decreases as moves away from 0. If 1 < 0, the pole is to the left of 1, so the spectral density is at its maximum when = 0.5. Recall MA(1): (z) = 1 + 1 z. The zero is at 1/1 . If 1 > 0, the zero is to the left of 1, so the spectral density decreases as moves towards 1. If 1 < 0, the zero is to the right of 1, so the spectral density is at its minimum when = 0. 7 Example: AR(2)
Consider Xt = 1 Xt1 + 2 Xt2 + Wt . Example 4.6 in the text considers 2 this model with 1 = 1, 2 = 0.9, and w = 1. In this case, the poles are at p1 , p2 0.5555 i0.8958 1.054ei1.01567 1.054e2i0.16165 . Thus, we have f () = 2 e2i 2
2 w , 2 e2i  p 2  p1  2 and this gets very peaked when e2i passes near 1.054e2i0.16165 . 8 Example: AR(2)
Spectral density of AR(2): Xt = Xt1  0.9 Xt2 + Wt 140 120 100 80 f() 60 40 20 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 9 Example: Seasonal ARMA
Consider Xt = 1 Xt12 + Wt . (B) = 1 , 12 1  1 B 2 f () = w 1 (1  1 e2i12 )(1  1 e2i12 ) 1 2 = w 2. 1  21 cos(24) + 1 Notice that f () is periodic with period 1/12. 10 Example: Seasonal ARMA
Spectral density of AR(1)12: Xt = +0.2 Xt12 + Wt 1.6 1.4 1.2 1 f() 0.8 0.6 0.4 0.2 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 11 Example: Seasonal ARMA
Another view: 1  1 z 12 = 0 with z = rei , ei12 = ei arg(1 ) . r = 1 1/12 , For 1 > 0, the twelve poles are at 1 1/12 eik/6 for k = 0, 1, . . . , 5, 6. So the spectral density gets peaked as e2i passes near 1 1/12 1, ei/6 , ei/3 , ei/2 , ei2/3 , ei5/6 , 1 . 12 Example: Multiplicative seasonal ARMA
Consider (1  1 B 12 )(1  1 B)Xt = Wt .
2 f () = w 1 2 )(1  2 cos(2) + 2 ) . (1  21 cos(24) + 1 1 1 This is a scaled product of the AR(1) spectrum and the (periodic) AR(1)12 spectrum. The AR(1)12 poles give peaks when e2i is at one of the 12th roots of 1; the AR(1) poles give a peak near e2i = 1. 13 Example: Multiplicative seasonal ARMA
Spectral density of AR(1)AR(1) : (1+0.5 B)(1+0.2 B ) X = W
12 t 12 t 7 6 5 4 f() 3 2 1 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 14 Introduction to Time Series Analysis. Lecture 17.
1. Review: Spectral distribution function, spectral density. 2. Rational spectra. Poles and zeros. 3. Examples. 4. Timeinvariant linear filters 5. Frequency response 15 Timeinvariant linear filters
A filter is an operator; given a time series {Xt }, it maps to a time series {Yt }. We can think of a linear process Xt = j=0 j Wtj as the output of a causal linear filter with a white noise input. A time series {Yt } is the output of a linear filter A = {at,j : t, j Z} with input {Xt } if Yt =
j= at,j Xj . If at,tj is independent of t (at,tj = j ), then we say that the filter is timeinvariant. If j = 0 for j < 0, we say the filter is causal. We'll see that the name `filter' arises from the frequency domain viewpoint.
16 Timeinvariant linear filters: Examples
1. Yt = Xt is linear, but not timeinvariant.
1 2. Yt = 3 (Xt1 + Xt + Xt+1 ) is linear, timeinvariant, but not causal: 1 if j 1, j = 3 0 otherwise. 3. For polynomials (B), (B) with roots outside the unit circle, (B) = (B)/(B) is a linear, timeinvariant, causal filter. 17 Timeinvariant linear filters
The operation j Xtj
j= is called the convolution of X with . 18 Timeinvariant linear filters
The sequence is also called the impulse response, since the output {Yt } of the linear filter in response to a unit impulse, 1 if t = 0, Xt = 0 otherwise, is Yt = (B)Xt =
j= j Xtj = t . 19 Introduction to Time Series Analysis. Lecture 17.
1. Review: Spectral distribution function, spectral density. 2. Rational spectra. Poles and zeros. 3. Examples. 4. Timeinvariant linear filters 5. Frequency response 20 Frequency response of a timeinvariant linear filter
Suppose that {Xt } has spectral density fx () and is stable, that is, j= j  < . Then Yt = (B)Xt has spectral density fy () = e
2i 2 fx (). The function (e2i ) (the polynomial (z) evaluated on the unit circle) is known as the frequency response or transfer function of the linear filter. The squared modulus, (e2i )2 is known as the power transfer function of the filter. 21 Frequency response of a timeinvariant linear filter
For stable , Yt = (B)Xt has spectral density fy () = e2i
2 fx (). We have seen that a linear process, Yt = (B)Wt , is a special case, since 2 fy () = (e2i )2 w = (e2i )2 fw (). When we pass a time series {Xt } through a linear filter, the spectral density is multiplied, frequencybyfrequency, by the squared modulus of the frequency response (e2i )2 . This is a version of the equality Var(aX) = a2 Var(X), but the equality is true for the component of the variance at every frequency. This is also the origin of the name `filter.'
22 Frequency response of a filter: Details
Why is fy () = e2i 2 fx ()? First, y (h) = E = j Xtj
k= j= k Xt+hk j
j= k= k E [Xt+hk Xtj ] =
j= j
k= k x (h + j  k) =
j= j
l= h+jl x (l). It is easy to check that j  < and j= h= x (h) < imply that h= y (h) < . Thus, the spectral density of y is defined.
23 Frequency response of a filter: Details fy () =
h= (h)e2ih =
h= j= j
l= h+jl x (l)e2ih =
j= j e2ij
l= x (l)e2il
h= h+jl e2i(h+jl) = (e2ij )fx ()
h= h e2ih = (e2ij ) fx (). 2 24 Frequency response: Examples
For a linear process Yt = (B)Wt , fy () = e2i
2 2 w . For an ARMA model, (B) = (B)/(B), so {Yt } has the rational spectrum
2 fy () = w (e2i ) (e2i )
2 q 2 2 = w 2 p q j=1 p j=1 e2i  zj e2i 2 2,  pj  where pj and zj are the poles and zeros of the rational function z (z)/(z). 25 Frequency response: Examples
Consider the moving average 1 Yt = 2k + 1
k Xtj .
j=k This is a time invariant linear filter (but it is not causal). Its transfer function is the Dirichlet kernel 1 e2ij (e2i ) = Dk (2) = 2k + 1 j=k 1 if = 0, = sin(2(k+1/2)) otherwise.
(2k+1) sin() 26 k Example: Moving average
Transfer function of moving average (k=5) 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 27 Example: Moving average
Squared modulus of transfer function of moving average (k=5) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 This is a lowpass filter: It preserves low frequencies and diminishes high frequencies. It is often used to estimate a monotonic trend component of a series.
28 Example: Differencing
Consider the first difference Yt = (1  B)Xt . This is a time invariant, causal, linear filter. Its transfer function is (e2i ) = 1  e2i , so (e2i )2 = 2(1  cos(2)). 29 Example: Differencing
Transfer function of first difference 4 3.5 3 2.5 2 1.5 1 0.5 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 This is a highpass filter: It preserves high frequencies and diminishes low frequencies. It is often used to eliminate a trend component of a series. 30 Introduction to Time Series Analysis. Lecture 17.
1. Review: Spectral distribution function, spectral density. 2. Rational spectra. Poles and zeros. 3. Examples. 4. Timeinvariant linear filters 5. Frequency response 31 ...
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 Fall '08
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 Signal Processing, spectral density, LTI system theory

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