22 - Introduction to Time Series Analysis Lecture 22 1...

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Introduction to Time Series Analysis. Lecture 22. 1. Review: The smoothed periodogram. 2. Examples. 3. Parametric spectral density estimation. 1
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Review: Periodogram The periodogram is defined as I ( ν ) = | X ( ν ) | 2 = X 2 c ( ν ) + X 2 s ( ν ) . X c ( ν ) = 1 n n summationdisplay t =1 cos(2 πtν ) x t , X s ( ν ) = 1 n n summationdisplay t =1 sin(2 πtν j ) x t . Under general conditions, X c ( ν j ) , X s ( ν j ) are asymptotically independent and N (0 ,f ( ν j ) / 2) . Thus, E I ν ( n ) ) f ( ν ) , but Var ( I ν ( n ) )) f ( ν ) 2 . 2
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Review: Smoothed spectral estimators ˆ f ( ν ) = summationdisplay | j |≤ L n W n ( j ) I ν ( n ) j/n ) , where the spectral window function satisfies L n → ∞ , L n /n 0 , W n ( j ) 0 , W n ( j ) = W n ( j ) , W n ( j ) = 1 , and W 2 n ( j ) 0 . Then ˆ f ( ν ) f ( ν ) (in the mean square sense), and asymptotically ˆ f ( ν k ) f ( ν k ) χ 2 d d , where d = 2 / W 2 n ( j ) . 3
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Introduction to Time Series Analysis. Lecture 22. 1. Review: The smoothed periodogram. 2. Examples. 3. Parametric spectral density estimation. 4
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Example: Southern Oscillation Index Figure 4.4 in the text shows the periodogram of the SOI time series. The SOI is the scaled, standardized, mean-adjusted, difference between monthly average air pressure at sea level in Tahiti and Darwin: SOI = 10 P Tahiti P Darwin σ ¯ x. For the time series in the text, n = 453 months. The periodogram has a large peak at ν = 0 . 084 cycles/sample. This corresponds to 0 . 084 cycles per month, or a period of 1 / 0 . 084 = 11 . 9 months. There are smaller peaks at ν 0 . 02 : I (0 . 02) 1 . 0 . The frequency ν = 0 . 02 corresponds to a period of 50 months, or 4 . 2 years. 5
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Example: Southern Oscillation Index Consider the hypothesized El Ni ˜ no effect, at a period of around four years. The approximate 95% confidence interval at this frequency, ν = 1 / (4 × 12) , is 2 I ( ν ) χ 2 2 (0 . 025) f ( ν ) 2 I ( ν ) χ 2 2 (0 . 975) 2 × 0 . 64 7 . 3778 f ( ν ) 2 × 0 . 64 0 . 0506 0 . 17 f ( ν ) 25 . 5 . The lower extreme of this confidence interval is around the noise baseline, so it is difficult to conclude much about the hypothesized El Ni ˜ no effect.
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