Instatistics,Whittle likelihood is an approximation to thelikelihood function of a stationary Gaussiantime series. It is named after the mathematician and statisticianPeter Whittle, who introduced it in his PhD thesis in 1951.^[1]It is commonly used intime series analysis andsignal processing for parameter estimation and signal detection.

In astationary Gaussian time series model, thelikelihood function is (as usual in Gaussian models) a function of the associated mean and covariance parameters. With a large number (${\displaystyle N}$) of observations, the (${\displaystyle N\times N}$) 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${\displaystyle O(N^2)}$ to${\displaystyle O(N\log (N))}$).^[2] The idea effectively boils down to assuming aheteroscedastic zero-mean Gaussian model inFourier domain; the model formulation is based on the time series'discrete Fourier transform and itspower spectral density.^[3][4][5]

Let${\displaystyle X_1,\dots ,X_N}$ be a stationary Gaussian time series with (one-sided) power spectral density${\displaystyle S_1(f)}$, where${\displaystyle N}$ is even and samples are taken at constant sampling intervals${\displaystyle {\mathrm{\Delta }}_t}$.Let${\displaystyle {\tilde{X}}_1,\dots ,{\tilde{X}}_{N/2+1}}$ be the (complex-valued)discrete Fourier transform (DFT) of the time series. Then for the Whittle likelihood one effectively assumes independent zero-meanGaussian distributions for all${\displaystyle {\tilde{X}}_j}$ with variances for the real and imaginary parts given by

${\displaystyle \mathrm{Var}\left(\mathrm{Re}({\tilde{X}}_j)\right)=\mathrm{Var}\left(\mathrm{Im}({\tilde{X}}_j)\right)=S_1(f_j)}$

where${\displaystyle f_j=\frac{j}{N{\mathrm{\Delta }}_t}}$ is the${\displaystyle j}$th Fourier frequency. This approximate model immediately leads to the (logarithmic) likelihood function

${\displaystyle \log \left(P(x_1,\dots ,x_N)\right)\propto -\sum _j\left(\log \left(S_1(f_j)\right)+\frac{|{\tilde{x}}_j{|}^2}{\frac{N}{2{\mathrm{\Delta }}_t}{S}_1(f_j)}\right)}$

where${\displaystyle |\cdot |}$ denotes the absolute value with${\displaystyle |{\tilde{x}}_j{|}^2={\left(\mathrm{Re}({\tilde{x}}_j)\right)}^2+{\left(\mathrm{Im}({\tilde{x}}_j)\right)}^2}$.^[3][4][6]

In case the noise spectrum is assumed a-prioriknown, 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

${\displaystyle \log \left(P(x_1,\dots ,x_N)\right)\propto -\sum _j\frac{|{\tilde{x}}_j{|}^2}{\frac{N}{2{\mathrm{\Delta }}_t}{S}_1(f_j)}}$

This expression also is the basis for the commonmatched filter.

The Whittle likelihood in general is only an approximation, it is only exact if the spectrum is constant, i.e., in the trivial case ofwhite noise.Theefficiency of the Whittle approximation always depends on the particular circumstances.^[7][8]

Note that due tolinearity 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 thesampling theorem—the effect of Fourier-transforming only afinite number of data points, which also manifests itself asspectral 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 (${\displaystyle x_1}$ and${\displaystyle x_N}$), which are effectively treated as "neighbouring" samples (like${\displaystyle x_1}$ and${\displaystyle x_2}$).

Whittle's likelihood is commonly used to estimate signal parameters for signals that are buried in non-white noise. Thenoise spectrum then may be assumed known,^[9]or it may be inferred along with the signal parameters.^[4][6]

Signal detection is commonly performed with thematched filter, which is based on the Whittle likelihood for the case of aknown noise power spectral density.^[10][11]The matched filter effectively does amaximum-likelihood fit of the signal to the noisy data and uses the resultinglikelihood ratio as the detection statistic.^[12]

The matched filter may be generalized to an analogous procedure based on aStudent-t distribution by also considering uncertainty (e.g.estimation uncertainty) in the noise spectrum. On the technical side, theEM algorithm may be utilized here, effectively leading to repeated or iterative matched-filtering.^[12]

The Whittle likelihood is also applicable for estimation of thenoise spectrum, either alone or in conjunction with signal parameters.^[13][14]

• Coloured noise – Power spectrum of a noise signal
• Discrete Fourier transform – Function in discrete mathematics
• Likelihood function – Function related to statistics and probability theory
• Matched filter – Filters used in signal processing that are optimal in some sense
• Power spectral density – Relative importance of certain frequencies in a composite signalPages displaying short descriptions of redirect targets
• Statistical signal processing – Field of electrical engineeringPages displaying short descriptions of redirect targets
• Weighted least squares – Method for model fitting in statistics

• Time series
• Time series models
• Frequency-domain analysis
• Statistical inference
• Statistical models
• Statistical signal processing
• Signal estimation
• Normal distribution

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