Whittle likelihood

In statistics, Whittle likelihood is an approximation to the likelihood function of a stationary Gaussian time series.wikipedia
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Peter Whittle (mathematician)

Peter WhittleWhittle, Peter
It is named after the mathematician and statistician Peter Whittle, who introduced it in his PhD thesis in 1951.
* Whittle likelihood

Matched filter

matched filteringmatched-filteringNorth filters
This expression also is the basis for the common matched filter.
The matched filter may be derived in a variety of ways, but as a special case of a least squares procedure it may also be interpreted as a maximum likelihood method in the context of a (coloured) Gaussian noise model and the associated Whittle likelihood.

Spectral density estimation

spectral estimationfrequency estimationspectral analysis
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.
Whittle likelihood

Colors of noise

blue noisecolored noisenoise
Coloured noise
Whittle likelihood

Statistics

statisticalstatistical analysisstatistician
In statistics, Whittle likelihood is an approximation to the likelihood function of a stationary Gaussian time series.

Likelihood function

likelihoodlog-likelihoodlikelihood ratio
In statistics, Whittle likelihood is an approximation to the likelihood function of a stationary Gaussian time series. 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.

Time series

time series analysistime-seriestime-series analysis
In statistics, Whittle likelihood is an approximation to the likelihood function of a stationary Gaussian time series. It is commonly utilized in time series analysis and signal processing for parameter estimation and signal detection.

Gaussian process

GaussianGaussian random processGaussian distribution
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.

Heteroscedasticity

heteroscedasticheteroskedasticheteroskedasticity
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.

Frequency domain

frequency-domainfrequencyspectral analysis
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.

Discrete Fourier transform

DFTcircular convolution theoremFourier transform
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.

Normal distribution

normally distributednormalGaussian
Then for the Whittle likelihood one effectively assumes independent zero-mean Gaussian distributions for all \tilde{X}_j with variances for the real and imaginary parts given by

White noise

whitestaticnoise
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.

Efficiency (statistics)

efficientefficiencyinefficient
The efficiency of the Whittle approximation always depends on the particular circumstances.

Linearity

linearlinearlycomplex linear
Note that due to linearity of the Fourier transform, Gaussianity in Fourier domain implies Gaussianity in time domain and vice versa.

Nyquist–Shannon sampling theorem

sampling theoremsampling theoryNyquist
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).

Spectral leakage

leakagemain lobe
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).

Window function

window functionswindowingwindowed
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).

Maximum likelihood estimation

maximum likelihoodmaximum likelihood estimatormaximum likelihood estimate
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.

Likelihood-ratio test

likelihood ratio testlikelihood ratiolikelihood-ratio
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.

Student's t-distribution

Student's ''t''-distributiont''-distributiont-distribution
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.

Signal processing

signal analysissignalsignal processor
It is commonly utilized in time series analysis and signal processing for parameter estimation and signal detection.

Spectral density

frequency spectrumpower spectrumspectrum
Whittle likelihood