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 ).[2] 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.[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)
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
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
linearity of the Fourier transform, Gaussianity in Fourier domain implies Gaussianity in time domain and vice versa. What makes the Whittle likelihood only approximately accurate is related to the
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]
^
abcHannan, 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
^Pawitan, Y. (1998), "Whittle likelihood", in Kotz, S.; Read, C. B.; Banks, D. L. (eds.), Encyclopedia of Statistical Sciences, vol. Update Volume 2, New York: Wiley & Sons, pp. 708–710,
doi:
10.1002/0471667196.ess0753,
ISBN978-0471667193
^Countreras-Cristán, A.; Gutiérrez-Peña, E.; Walker, S. G. (2006). "A Note on Whittle's Likelihood". Communications in Statistics – Simulation and Computation. 35 (4): 857–875.
doi:
10.1080/03610910600880203.
S2CID119395974.