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.[1] It is commonly used in
Context
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 ).
Definition
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)
where is the th Fourier frequency. This approximate model immediately leads to the (logarithmic) likelihood function
where denotes the absolute value with .[3][4][6]
Special case of a known noise spectrum
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.
Accuracy of approximation
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. The efficiency of the Whittle approximation always depends on the particular circumstances.[7] [8]
Note that due to
Applications
Parameter estimation
Whittle's likelihood is commonly used to estimate signal parameters for signals that are buried in non-white noise. The
Signal detection
Signal detection is commonly performed with the matched filter, which is based on the Whittle likelihood for the case of a known noise power spectral density.[10][11] The matched filter effectively does a
The matched filter may be generalized to an analogous procedure based on a
Spectrum estimation
The Whittle likelihood is also applicable for estimation of the
See also
- Coloured noise
- Discrete Fourier transform
- Likelihood function
- Matched filter
- Power spectral density
- Statistical signal processing
- Weighted least squares
References
- ^ Whittle, P. (1951). Hypothesis testing in times series analysis. Uppsala: Almqvist & Wiksells Boktryckeri AB.
- NYU Stern.
- ^ ISBN 978-0-387-94989-5
See also: Calder, M.; Davis, R. A. (1996), "An introduction to Whittle (1953) "The analysis of multiple stationary time series"", Technical report 1996/41, Department of Statistics, Colorado State University - ^ a b c Hannan, E. J. (1994), "The Whittle likelihood and frequency estimation", in Kelly, F. P. (ed.), Probability, statistics and optimization; a tribute to Peter Whittle, Chichester: Wiley
- ISBN 978-0471667193
- ^ S2CID 46673503.
- .
- S2CID 119395974.
- S2CID 19004097.
- S2CID 5128742.
- ^ Wainstein, L. A.; Zubakov, V. D. (1962). Extraction of signals from noise. Englewood Cliffs, NJ: Prentice-Hall.
- ^ .
- S2CID 17906077.
- S2CID 11508218.