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.

__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
initialize(design)[source]#
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

The parameter above is what is sometimes referred to as a nuisance parameter. That is, the likelihood is considered as a function of , but to evaluate it, a value of is needed.

If is not provided, then its maximum likelihood estimate:

is plugged in. This likelihood is now a function of only 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.