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Unformatted text preview: 7 Spectral Analysis 7.1 A Simple Sinusoidal Model Suppose we suspect that a given time series, with observations made at unit time intervals, contains a sinusoidal component of frequency plus a random error term. We will consider the following model: X t = + cos( t ) + sin( t ) + Z t , t = 1 , . . ., N (7.1) Here Z i is a purely random process, and ( , , ) are to be estimated from data x 1 , . . ., x N . To make the algebra easy, we only consider the case of N even. We represent (7.1) in matrix notation by E ( X ) = A where x T = ( x 1 , . . ., x n ) , T = ( , , ) and A N 3 = 1 cos( ) sin( ) 1 cos(2 ) sin(2 ) . . . . . . . . . 1 cos( N ) sin( N ) . The least squares estimator is given by = ( A T A )- 1 A T x where A T A = N N k =1 cos( k ) N k =1 sin( k ) N k =1 cos( k ) N k =1 cos 2 ( k ) N k =1 cos( k ) sin( k ) N k =1 sin( k ) N k =1 cos( k ) sin( k ) N k =1 sin 2 ( k ) . They become very simple if is restricted to one of the values p = 2 p/N, p = 1 , . . ., N/ 2....
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- Spring '09