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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 ω plus a random error term. We will consider the following model: X t = μ + α cos( ωt ) + β sin( ωt ) + Z t , t = 1 , . . ., N (7.1) Here Z i is a purely random process, and ( μ, α, β ) 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 ) = Aθ where x T = ( x 1 , . . ., x n ) , θ T = ( μ, α, β ) and A N × 3 = 1 cos( ω ) sin( ω ) 1 cos(2 ω ) sin(2 ω ) . . . . . . . . . 1 cos( Nω ) sin( Nω ) . The least squares estimator is given by ˆ θ = ( A T A )- 1 A T x where A T A = N ∑ N k =1 cos( kω ) ∑ N k =1 sin( kω ) ∑ N k =1 cos( kω ) ∑ N k =1 cos 2 ( kω ) ∑ N k =1 cos( kω ) sin( kω ) ∑ N k =1 sin( kω ) ∑ N k =1 cos( kω ) sin( kω ) ∑ N k =1 sin 2 ( kω ) . They become very simple if ω is restricted to one of the values ω p = 2 πp/N, p = 1 , . . ., N/ 2....
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- Spring '09
- Normal Distribution, Cos, spectral density, periodogram