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weatherwax_Box_N_Jenkins

weatherwax_Box_N_Jenkins - Notes on the Book Time Series...

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Notes on the Book: TimeSeries Analysis: Forecastingand Control by George E. P. Boxand Gwilym M. Jenkins John L. Weatherwax June 20, 2008 * [email protected] 1
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Chapter 2 (The Autocorrelation Function and the Spec- trum) Notes on the Text Notes on positive definiteness and the autocovariance matrix The book defined the autocovariance matrix Γ n of a stochastic process as Γ n = γ 0 γ 1 γ 2 · · · γ n 1 γ 1 γ 0 γ 1 · · · γ n 2 γ 2 γ 1 γ 0 · · · γ n 3 . . . . . . . . . · · · . . . γ n 1 γ n 2 γ n 3 · · · γ 0 . (1) Then holding the definition for a second, if we consider the derived time series L t given by L t = l 1 z t + l 2 z t 1 + · · · + l n z t n +1 , we can compute the variance of this series using the definition var[ L t ] = E [( L t ¯ L ) 2 ]. We first evaluate the mean of L t ¯ L = E [ l 1 z t + l 2 z t 1 + · · · + l n z t n +1 ] = ( l 1 + l 2 + · · · + l n ) μ, since z t is assumed stationary so that E [ z t ] = μ for all t . We then have that L t ¯ L = l 1 ( z t μ ) + l 2 ( z t 1 μ ) + l 3 ( z t 2 μ ) + · · · + l n ( z t n +1 μ ) , so that when we square this expression we get ( L t ¯ L ) 2 = n summationdisplay i =1 n summationdisplay j =1 l i l j ( z t ( i 1) μ )( z t ( j 1) μ ) . Taking the expectation of both sides to compute the variance and using E [( z t ( i 1) μ )( z t ( j 1) μ )] = γ | i j | , gives var[ L t ] = n summationdisplay i =1 n summationdisplay j =1 l i l j γ | i j | . As the expression on the right-hand-side is the same as the quadratic form bracketleftbig l 1 l 2 l 3 · · · l n bracketrightbig γ 0 γ 1 γ 2 · · · γ n 1 γ 1 γ 0 γ 1 · · · γ n 2 γ 2 γ 1 γ 0 · · · γ n 3 . . . . . . . . . · · · . . . γ n 1 γ n 2 γ n 3 · · · γ 0 l 1 l 2 l 3 . . . l n . (2) 2
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Thus since var[ L t ] > 0 (from its definition) for all possible values for l 1 ,l 2 ,l 3 , · · · l n 1 we have shown that the inner product given by Equation 2 is positive for all nonzero vectors with components l 1 ,l 2 ,l 3 , · · · l n 1 we have shown that the autocovariance matrix Γ n is positive definite. Since the autocorrelation matrix, P n , is a scaled version of Γ n it too is positive definite. Given the fact that P n is positive definite we can use standard properties of positive definite matrices to derive properties of the correlations ρ k . Given a matrix Q of size n × n , we define the principal minors of Q to be determinants of smaller square matrices obtained from the matrix Q . The smaller submatrices are selected from Q by selecting a set of indices from 1 to n representing the rows (and columns) we want to downsample from. Thus if you view the indices selected as the indices of rows from the original matrix Q , the columns we select must equal the indices of the rows we select. As an example, if the matrix Q is 6 × 6 we could construct one of the principal minors from the first, third, and sixth rows. If we denote the
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