Whittle likelihood

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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

time series analysis and signal processing
for parameter estimation and signal detection.

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 ).

power spectral density.[3][4][5]

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)

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 .[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

sampling theorem—the effect of Fourier-transforming only a finite number of data points, which also manifests itself as spectral leakage in related problems (and which may be ameliorated using the same methods, namely, windowing
). In the present case, the implicit periodicity assumption implies correlation between the first and last samples ( and ), which are effectively treated as "neighbouring" samples (like and ).

Applications

Parameter estimation

Whittle's likelihood is commonly used to estimate signal parameters for signals that are buried in non-white noise. The

noise spectrum then may be assumed known,[9]
or it may be inferred along with the signal parameters.[4][6]

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

maximum-likelihood fit of the signal to the noisy data and uses the resulting likelihood ratio as the detection statistic.[12]

The matched filter may be generalized to an analogous procedure based on a

Student-t distribution by also considering uncertainty (e.g. estimation uncertainty) in the noise spectrum. On the technical side, this entails repeated or iterative matched-filtering.[12]

Spectrum estimation

The Whittle likelihood is also applicable for estimation of the

noise spectrum, either alone or in conjunction with signal parameters.[13][14]

See also

References

  1. ^ Whittle, P. (1951). Hypothesis testing in times series analysis. Uppsala: Almqvist & Wiksells Boktryckeri AB.
  2. NYU Stern
    .
  3. ^
    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
  4. ^ 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
  5. ^ .
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  10. ^ Wainstein, L. A.; Zubakov, V. D. (1962). Extraction of signals from noise. Englewood Cliffs, NJ: Prentice-Hall.
  11. ^ .
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