pastas.objective_functions.GaussianLikelihoodAr1#
- class pastas.objective_functions.GaussianLikelihoodAr1#
Gaussian likelihood function for homoscedastic, autocorrelated residuals.
Notes
The Gaussian log-likelihood function with AR1 autocorrelated residuals [Smith et al., 2015] is defined as:
\[\log(L) = -\frac{N-1}{2}\log(2\pi\sigma^2) - \frac{\sum_{t=1}^N(\epsilon_t - \phi \epsilon_{t-\Delta t})^2} {2\sigma^2}\]where \(N\) is the number of observations, \(\sigma^2\) is the variance of the residuals, \(\epsilon_t\) is the residual at time \(t\). \(\Delta t\) is the time step between the observations. \(\phi\) is the autoregressive parameter. The parameters \(\phi\) and \(\sigma^2\) need to be estimated.
The current implementation is valid for equidistant time series only.
Methods#
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Get the initial parameters for the log-likelihood function. |
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Compute the log-likelihood. |