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.OLSModel¶

class nilearn.glm.OLSModel(design)[source]¶

A simple ordinary least squares model.

Parameters:

designarray-like: This is your design matrix. Data are assumed to be column ordered with observations in rows.

Attributes:

designndarray: This is the design, or X, matrix.
whitened_designndarray: This is the whitened design matrix. design == whitened_design by default for the OLSModel, though models that inherit from the OLSModel will whiten the design.
calc_betandarray: This is the Moore-Penrose pseudoinverse of the whitened design matrix.
normalized_cov_betandarray: np.dot(calc_beta, calc_beta.T)
df_residualsscalar: Degrees of freedom of the residuals. Number of observations less the rank of the design.
df_modelscalar: Degrees of freedome of the model. The rank of the design.

Methods

model.__init___(design)
model.logL(b=self.beta, Y)

__init__(design)[source]¶

initialize(design)[source]¶: Construct instance.

logL(beta, Y, nuisance=None)[source]¶

Return 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 Greene[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

whiten(X)[source]¶

Whiten design matrix.

Parameters:

Xarray: design matrix

Returns:

whitened_Xarray: This matrix is the matrix whose pseudoinverse is ultimately used in estimating the coefficients. For OLSModel, it is does nothing. For WLSmodel, ARmodel, it pre-applies a square root of the covariance matrix to X.

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