Whittle likelihood

In statistics, Whittle likelihood is an approximation to the likelihood function of a stationary Gaussian time series. It is named after the mathematician and statistician Peter Whittle, who introduced it in his PhD thesis in 1951. It is commonly used in time series analysis and signal processing for parameter estimation and signal detection. In a stationary Gaussian time series model, the likelihood function is (as usual in Gaussian models) a function of the associated mean and covariance parameters. With a large number () of observations, the () covariance matrix may become very large, making computations very costly in practice. However, due to stationarity, the covariance matrix has a rather simple structure, and by using an approximation, computations may be simplified considerably (from to ). The idea effectively boils down to assuming a heteroscedastic zero-mean Gaussian model in Fourier domain; the model formulation is based on the time series' discrete Fourier transform and its power spectral density. Let be a stationary Gaussian time series with (one-sided) power spectral density , where is even and samples are taken at constant sampling intervals . Let be the (complex-valued) discrete Fourier transform (DFT) of the time series. Then for the Whittle likelihood one effectively assumes independent zero-mean Gaussian distributions for all with variances for the real and imaginary parts given by where is the th Fourier frequency. This approximate model immediately leads to the (logarithmic) likelihood function where denotes the absolute value with . In case the noise spectrum is assumed a-priori known, and noise properties are not to be inferred from the data, the likelihood function may be simplified further by ignoring constant terms, leading to the sum-of-squares expression This expression also is the basis for the common matched filter. The Whittle likelihood in general is only an approximation, it is only exact if the spectrum is constant, i.e., in the trivial case of white noise.