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17 - Introduction to Time Series Analysis Lecture 17 1...

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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. Time-invariant 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)e-2ih 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 e-2i 2 i=0 i Wt-i = (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 (e-2i )(e2i ) (e-2i ) (e2i ) (e-2i ) (e-2i ) 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 e-2i - zj |e-2i 2 p - pj | 5 Rational spectra 2 q q j=1 p j=1 2 f () = w e-2i - zj |e-2i 2 2. 2 p - pj | As varies from 0 to 1/2, e-2i moves clockwise around the unit circle from 1 to e-i = -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 Xt-1 + 2 Xt-2 + 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 |e-2i 2 2 w , 2 |e-2i - p |2 - p1 | 2 and this gets very peaked when e-2i passes near 1.054e-2i0.16165 . 8 Example: AR(2) Spectral density of AR(2): Xt = Xt-1 - 0.9 Xt-2 + 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 Xt-12 + Wt . (B) = 1 , 12 1 - 1 B 2 f () = w 1 (1 - 1 e-2i12 )(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 Xt-12 + 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 = e-i 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 e-2i passes near |1 |-1/12 1, e-i/6 , e-i/3 , e-i/2 , e-i2/3 , e-i5/6 , -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 e-2i is at one of the 12th roots of 1; the AR(1) poles give a peak near e-2i = 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. Time-invariant linear filters 5. Frequency response 15 Time-invariant 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 Wt-j 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,t-j is independent of t (at,t-j = j ), then we say that the filter is time-invariant. 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 Time-invariant linear filters: Examples 1. Yt = X-t is linear, but not time-invariant. 1 2. Yt = 3 (Xt-1 + Xt + Xt+1 ) is linear, time-invariant, 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, time-invariant, causal filter. 17 Time-invariant linear filters The operation j Xt-j j=- is called the convolution of X with . 18 Time-invariant 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 Xt-j = 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. Time-invariant linear filters 5. Frequency response 20 Frequency response of a time-invariant 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 time-invariant 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, frequency-by-frequency, 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 Xt-j k=- j=- k Xt+h-k j j=- k=- k E [Xt+h-k Xt-j ] = j=- j k=- k x (h + j - k) = j=- j l=- h+j-l 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)e-2ih = h=- j=- j l=- h+j-l x (l)e-2ih = j=- j e2ij l=- x (l)e-2il h=- h+j-l e-2i(h+j-l) = (e2ij )fx () h=- h e-2ih = (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 (e-2i ) (e-2i ) 2 q 2 2 = w 2 p q j=1 p j=1 e-2i - zj |e-2i 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 Xt-j . j=-k This is a time invariant linear filter (but it is not causal). Its transfer function is the Dirichlet kernel 1 e-2ij (e-2i ) = 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 low-pass 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 (e-2i ) = 1 - e-2i , so |(e-2i )|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 high-pass 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. Time-invariant linear filters 5. Frequency response 31 ...
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