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

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

System Message: WARNING/2 (\ell(\beta,\sigma,Y)= -\frac{n}{2}\log(2\pi\sigma^2) - \|Y-X\beta\|^2/(2\sigma^2))

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The parameter

System Message: WARNING/2 (\sigma)

above is what is sometimes referred to as a nuisance parameter. That is, the likelihood is considered as a function of

System Message: WARNING/2 (\beta)

, but to evaluate it, a value of

System Message: WARNING/2 (\sigma)

is needed.

System Message: WARNING/2 (\sigma)

is not provided, then its maximum likelihood estimate:

System Message: WARNING/2 (\hat{\sigma}(\beta) = \frac{\text{SSE}(\beta)}{n})

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

System Message: WARNING/2 (\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.