Note
This page is a reference documentation. It only explains the class signature, and not how to use it. Please refer to the user guide for the big picture.
nilearn.glm.ARModel#
- class nilearn.glm.ARModel(design, rho)[source]#
A regression model with an AR(p) covariance structure.
In terms of a LikelihoodModel, the parameters are beta, the usual regression parameters, and sigma, a scalar nuisance parameter that shows up as multiplier in front of the AR(p) covariance.
Notes
This class is experimental. It may change in any future release of Nilearn.
- __init__(design, rho)[source]#
Initialize AR model instance
- Parameters
- designndarray
2D array with design matrix.
- rhoint or array-like
If int, gives order of model, and initializes rho to zeros. If ndarray, gives initial estimate of rho. Be careful as
ARModel(X, 1) != ARModel(X, 1.0).
- whiten(X)[source]#
Whiten a series of columns according to AR(p) covariance structure
- Parameters
- Xarray-like of shape (n_features)
Array to whiten.
- Returns
- whitened_Xndarray
X whitened with order self.order AR.
- fit(Y)[source]#
Fit model to data Y
Full fit of the model including estimate of covariance matrix, (whitened) residuals and scale.
- Parameters
- Yarray-like
The dependent variable for the Least Squares problem.
- Returns
- fitRegressionResults
- logL(beta, Y, nuisance=None)[source]#
Returns the value of the loglikelihood function at beta.
Given the whitened design matrix, the loglikelihood is evaluated at the parameter vector, beta, for the dependent variable, Y and the nuisance parameter, sigma 1.
- Parameters
- betandarray
The parameter estimates. Must be of length df_model.
- Yndarray
The dependent variable
- nuisancedict, optional
A dict with key ‘sigma’, which is an optional estimate of sigma. If None, defaults to its maximum likelihood estimate (with beta fixed) as
sum((Y - X*beta)**2) / n, where n=Y.shape[0], X=self.design.
- Returns
- loglffloat
The value of the loglikelihood function.
Notes
The log-Likelihood Function is defined as
\ell(\beta,\sigma,Y)= -\frac{n}{2}\log(2\pi\sigma^2) - \|Y-X\beta\|^2/(2\sigma^2)
The parameter \sigma above is what is sometimes referred to as a nuisance parameter. That is, the likelihood is considered as a function of \beta, but to evaluate it, a value of \sigma is needed.
If \sigma is not provided, then its maximum likelihood estimate:
\hat{\sigma}(\beta) = \frac{\text{SSE}(\beta)}{n}
is plugged in. This likelihood is now a function of only \beta and is technically referred to as a profile-likelihood.
References
- 1
William H. Greene. Econometric Analysis. Pearson Education, fifth edition, 2003. ISBN 0-13-066189-9. URL: http://pages.stern.nyu.edu/~wgreene/Text/econometricanalysis.htm.