algorithms.statistics.models.glm¶
Module: algorithms.statistics.models.glm¶
Inheritance diagram for nipy.algorithms.statistics.models.glm:
General linear models¶
Model¶
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class
nipy.algorithms.statistics.models.glm.Model(design, family=<nipy.algorithms.statistics.models.family.family.Gaussian object>)[source]¶ Bases:
nipy.algorithms.statistics.models.regression.WLSModel,object-
__init__(design, family=<nipy.algorithms.statistics.models.family.family.Gaussian object>)[source]¶ - Parameters
design : array-like
This is your design matrix. Data are assumed to be column ordered with observations in rows.
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niter= 10¶
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deviance(Y=None, results=None, scale=1.0)[source]¶ Return (unnormalized) log-likelihood for GLM.
Note that self.scale is interpreted as a variance in old_model, so we divide the residuals by its sqrt.
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fit(Y)[source]¶ Fit model to data Y
Full fit of the model including estimate of covariance matrix, (whitened) residuals and scale.
- Parameters
Y : array-like
The dependent variable for the Least Squares problem.
- Returns
fit : RegressionResults
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has_intercept()¶ Check if column of 1s is in column space of design
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information(beta, nuisance=None)¶ Returns the information matrix at (beta, Y, nuisance).
See logL for details.
- Parameters
beta : ndarray
The parameter estimates. Must be of length df_model.
nuisance : dict
A dict with key ‘sigma’, which is an estimate of sigma. If None, defaults to its maximum likelihood estimate (with beta fixed) as
sum((Y - X*beta)**2) / nwhere n=Y.shape[0], X=self.design.- Returns
info : array
The information matrix, the negative of the inverse of the Hessian of the of the log-likelihood function evaluated at (theta, Y, nuisance).
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initialize(design)¶ Initialize (possibly re-initialize) a Model instance.
For instance, the design matrix of a linear model may change and some things must be recomputed.
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logL(beta, Y, nuisance=None)¶ 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.
- Parameters
beta : ndarray
The parameter estimates. Must be of length df_model.
Y : ndarray
The dependent variable
nuisance : dict, 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
loglf : float
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
- R1
Green. “Econometric Analysis,” 5th ed., Pearson, 2003.
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predict(design=None)[source]¶ After a model has been fit, results are (assumed to be) stored in self.results, which itself should have a predict method.
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rank()¶ Compute rank of design matrix
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score(beta, Y, nuisance=None)¶ Gradient of the loglikelihood function at (beta, Y, nuisance).
The graient of the loglikelihood function at (beta, Y, nuisance) is the score function.
See
logL()for details.- Parameters
beta : ndarray
The parameter estimates. Must be of length df_model.
Y : ndarray
The dependent variable.
nuisance : dict, 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
The gradient of the loglikelihood function.
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whiten(X)¶ Whitener for WLS model, multiplies by sqrt(self.weights)
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