# 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.

### Weighted least squares

Batch Least Squareslinear least squaresweighted
Weighted least squares

### Stationary process

stationarynon-stationarystationarity
Whittle likelihood

### Spectral density

frequency spectrumpower spectrumspectrum
Whittle likelihood