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.

Notes

This class is experimental. It may change in any future release of Nilearn.

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]#

Parameters

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

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))

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.20 (TeX Live 2019/Debian) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2020-02-02> patch level 2 L3 programming layer <2020-02-14> (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2019/12/20 v1.4l Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?' option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) ! LaTeX Error: File `anyfontsize.sty' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.8 \usepackage {bm}^^M No pages of output. Transcript written on math.log.

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.

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