algorithms.statistics.models.regression

Module: algorithms.statistics.models.regression

Inheritance diagram for nipy.algorithms.statistics.models.regression:

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This module implements some standard regression models: OLS and WLS models, as well as an AR(p) regression model.

Models are specified with a design matrix and are fit using their ‘fit’ method.

Subclasses that have more complicated covariance matrices should write over the ‘whiten’ method as the fit method prewhitens the response by calling ‘whiten’.

General reference for regression models:

‘Introduction to Linear Regression Analysis’, Douglas C. Montgomery,

Elizabeth A. Peck, G. Geoffrey Vining. Wiley, 2006.

Classes

AREstimator

class nipy.algorithms.statistics.models.regression.AREstimator(model, p=1)

Bases: object

A class to estimate AR(p) coefficients from residuals

__init__(model, p=1)

Bias-correcting AR estimation class

Parameters

model : OSLModel instance

A models.regression.OLSmodel instance, where model has attribute design

p : int, optional

Order of AR(p) noise

ARModel

class nipy.algorithms.statistics.models.regression.ARModel(design, rho)

Bases: nipy.algorithms.statistics.models.regression.OLSModel

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.

The linear autoregressive process of order p–AR(p)–is defined as:

TODO

Examples

>>> from nipy.algorithms.statistics.api import Term, Formula
>>> data = np.rec.fromarrays(([1,3,4,5,8,10,9], range(1,8)),
...                          names=('Y', 'X'))
>>> f = Formula([Term("X"), 1])
>>> dmtx = f.design(data, return_float=True)
>>> model = ARModel(dmtx, 2)

We go through the model.iterative_fit procedure long-hand:

>>> for i in range(6):
...     results = model.fit(data['Y'])
...     print("AR coefficients:", model.rho)
...     rho, sigma = yule_walker(data["Y"] - results.predicted,
...                              order=2,
...                              df=model.df_resid)
...     model = ARModel(model.design, rho) 
...
AR coefficients: [ 0.  0.]
AR coefficients: [-0.61530877 -1.01542645]
AR coefficients: [-0.72660832 -1.06201457]
AR coefficients: [-0.7220361  -1.05365352]
AR coefficients: [-0.72229201 -1.05408193]
AR coefficients: [-0.722278   -1.05405838]
>>> results.theta 
array([ 1.59564228, -0.58562172])
>>> results.t() 
array([ 38.0890515 ,  -3.45429252])
>>> print(results.Tcontrast([0,1]))  
<T contrast: effect=-0.58562172384377043, sd=0.16953449108110835,
t=-3.4542925165805847, df_den=5>
>>> print(results.Fcontrast(np.identity(2)))  
<F contrast: F=4216.810299725842, df_den=5, df_num=2>

Reinitialize the model, and do the automated iterative fit

>>> model.rho = np.array([0,0])
>>> model.iterative_fit(data['Y'], niter=3)
>>> print(model.rho)  
[-0.7220361  -1.05365352]

__init__(design, rho)

Initialize AR model instance

Parameters

design : ndarray

2D array with design matrix

rho : int 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).

iterative_fit(Y, niter=3)

Perform an iterative two-stage procedure to estimate AR(p) parameters and regression coefficients simultaneously.

Parameters

Y : ndarray

data to which to fit model

niter : optional, int

the number of iterations (default 3)

Returns

None

whiten(X)

Whiten a series of columns according to AR(p) covariance structure

Parameters

X : array-like of shape (n_features)

array to whiten

Returns

wX : ndarray

X whitened with order self.order AR

fit(Y)

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

has_intercept()

Check if column of 1s is in column space of design

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) / n where 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).

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.

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

R2

23. Green. “Econometric Analysis,” 5th ed., Pearson, 2003.

predict(design=None)

After a model has been fit, results are (assumed to be) stored in self.results, which itself should have a predict method.

rank()

Compute rank of design matrix

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.

GLSModel

class nipy.algorithms.statistics.models.regression.GLSModel(design, sigma)

Bases: nipy.algorithms.statistics.models.regression.OLSModel

Generalized least squares model with a general covariance structure

__init__(design, sigma)

Parameters

design : array-like

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

fit(Y)

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

has_intercept()

Check if column of 1s is in column space of design

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) / n where 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).

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.

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

R3

23. Green. “Econometric Analysis,” 5th ed., Pearson, 2003.

predict(design=None)

After a model has been fit, results are (assumed to be) stored in self.results, which itself should have a predict method.

rank()

Compute rank of design matrix

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.

whiten(Y)

Whiten design matrix

Parameters

X : array

design matrix

Returns

wX : array

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.

OLSModel

class nipy.algorithms.statistics.models.regression.OLSModel(design)

Bases: nipy.algorithms.statistics.models.model.LikelihoodModel

A simple ordinary least squares model.

Parameters

design : array-like

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

Examples

>>> from nipy.algorithms.statistics.api import Term, Formula
>>> data = np.rec.fromarrays(([1,3,4,5,2,3,4], range(1,8)),
...                          names=('Y', 'X'))
>>> f = Formula([Term("X"), 1])
>>> dmtx = f.design(data, return_float=True)
>>> model = OLSModel(dmtx)
>>> results = model.fit(data['Y'])
>>> results.theta
array([ 0.25      ,  2.14285714])
>>> results.t()
array([ 0.98019606,  1.87867287])
>>> print(results.Tcontrast([0,1]))  
<T contrast: effect=2.14285714286, sd=1.14062281591, t=1.87867287326,
df_den=5>
>>> print(results.Fcontrast(np.eye(2)))  
<F contrast: F=19.4607843137, df_den=5, df_num=2>

Attributes

design	(ndarray) This is the design, or X, matrix.
wdesign	(ndarray) This is the whitened design matrix. design == wdesign by default for the OLSModel, though models that inherit from the OLSModel will whiten the design.
calc_beta	(ndarray) This is the Moore-Penrose pseudoinverse of the whitened design matrix.
normalized_cov_beta	(ndarray) np.dot(calc_beta, calc_beta.T)
df_resid	(scalar) Degrees of freedom of the residuals. Number of observations less the rank of the design.
df_model	(scalar) Degrees of freedome of the model. The rank of the design.

Methods

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

__init__(design)

Parameters

design : array-like

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

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.

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

R4

23. Green. “Econometric Analysis,” 5th ed., Pearson, 2003.

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.

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) / n where 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).

whiten(X)

Whiten design matrix

Parameters

X : array

design matrix

Returns

wX : array

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.

has_intercept()

Check if column of 1s is in column space of design

rank()

Compute rank of design matrix

fit(Y)

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

predict(design=None)

After a model has been fit, results are (assumed to be) stored in self.results, which itself should have a predict method.

RegressionResults

class nipy.algorithms.statistics.models.regression.RegressionResults(theta, Y, model, wY, wresid, cov=None, dispersion=1.0, nuisance=None)

Bases: nipy.algorithms.statistics.models.model.LikelihoodModelResults

This class summarizes the fit of a linear regression model.

It handles the output of contrasts, estimates of covariance, etc.

__init__(theta, Y, model, wY, wresid, cov=None, dispersion=1.0, nuisance=None)

See LikelihoodModelResults constructor.

The only difference is that the whitened Y and residual values are stored for a regression model.

resid()

Residuals from the fit.

norm_resid()

Residuals, normalized to have unit length.

Notes

Is this supposed to return “stanardized residuals,” residuals standardized to have mean zero and approximately unit variance?

d_i = e_i / sqrt(MS_E)

Where MS_E = SSE / (n - k)

See: Montgomery and Peck 3.2.1 p. 68

Davidson and MacKinnon 15.2 p 662

predicted()

Return linear predictor values from a design matrix.

R2_adj()

Return the R^2 value for each row of the response Y.

Notes

Changed to the textbook definition of R^2.

See: Davidson and MacKinnon p 74

R2()

Return the adjusted R^2 value for each row of the response Y.

Notes

Changed to the textbook definition of R^2.

See: Davidson and MacKinnon p 74

SST()

Total sum of squares. If not from an OLS model this is “pseudo”-SST.

SSE()

Error sum of squares. If not from an OLS model this is “pseudo”-SSE.

SSR()

Regression sum of squares

MSR()

Mean square (regression)

MSE()

Mean square (error)

MST()

Mean square (total)

F_overall()

Overall goodness of fit F test, comparing model to a model with just an intercept. If not an OLS model this is a pseudo-F.

AIC()

Akaike Information Criterion

BIC()

Schwarz’s Bayesian Information Criterion

Fcontrast(matrix, dispersion=None, invcov=None)

Compute an Fcontrast for a contrast matrix matrix.

Here, matrix M is assumed to be non-singular. More precisely

\[M pX pX' M'\]

is assumed invertible. Here, \(pX\) is the generalized inverse of the design matrix of the model. There can be problems in non-OLS models where the rank of the covariance of the noise is not full.

See the contrast module to see how to specify contrasts. In particular, the matrices from these contrasts will always be non-singular in the sense above.

Parameters

matrix : 1D array-like

contrast matrix

dispersion : None or float, optional

If None, use self.dispersion

invcov : None or array, optional

Known inverse of variance covariance matrix. If None, calculate this matrix.

Returns

f_res : FContrastResults instance

with attributes F, df_den, df_num

Notes

For F contrasts, we now specify an effect and covariance

Tcontrast(matrix, store=('t', 'effect', 'sd'), dispersion=None)

Compute a Tcontrast for a row vector matrix

To get the t-statistic for a single column, use the ‘t’ method.

Parameters

matrix : 1D array-like

contrast matrix

store : sequence, optional

components of t to store in results output object. Defaults to all components (‘t’, ‘effect’, ‘sd’).

dispersion : None or float, optional

Returns

res : TContrastResults object

conf_int(alpha=0.05, cols=None, dispersion=None)

The confidence interval of the specified theta estimates.

Parameters

alpha : float, optional

The alpha level for the confidence interval. ie., alpha = .05 returns a 95% confidence interval.

cols : tuple, optional

cols specifies which confidence intervals to return

dispersion : None or scalar

scale factor for the variance / covariance (see class docstring and vcov method docstring)

Returns

cis : ndarray

cis is shape (len(cols), 2) where each row contains [lower, upper] for the given entry in cols

Notes

Confidence intervals are two-tailed. TODO: tails : string, optional

tails can be “two”, “upper”, or “lower”

Examples

>>> from numpy.random import standard_normal as stan
>>> from nipy.algorithms.statistics.models.regression import OLSModel
>>> x = np.hstack((stan((30,1)),stan((30,1)),stan((30,1))))
>>> beta=np.array([3.25, 1.5, 7.0])
>>> y = np.dot(x,beta) + stan((30))
>>> model = OLSModel(x).fit(y)
>>> confidence_intervals = model.conf_int(cols=(1,2))

logL()

The maximized log-likelihood

t(column=None)

Return the (Wald) t-statistic for a given parameter estimate.

Use Tcontrast for more complicated (Wald) t-statistics.

vcov(matrix=None, column=None, dispersion=None, other=None)

Variance/covariance matrix of linear contrast

Parameters

matrix: (dim, self.theta.shape[0]) array, optional

numerical contrast specification, where dim refers to the ‘dimension’ of the contrast i.e. 1 for t contrasts, 1 or more for F contrasts.

column: int, optional

alternative way of specifying contrasts (column index)

dispersion: float or (n_voxels,) array, optional

value(s) for the dispersion parameters

other: (dim, self.theta.shape[0]) array, optional

alternative contrast specification (?)

Returns

cov: (dim, dim) or (n_voxels, dim, dim) array

the estimated covariance matrix/matrices

Returns the variance/covariance matrix of a linear contrast of the

estimates of theta, multiplied by dispersion which will often be an

estimate of dispersion, like, sigma^2.

The covariance of interest is either specified as a (set of) column(s)

or a matrix.

WLSModel

class nipy.algorithms.statistics.models.regression.WLSModel(design, weights=1)

Bases: nipy.algorithms.statistics.models.regression.OLSModel

A regression model with diagonal but non-identity covariance structure.

The weights are presumed to be (proportional to the) inverse of the variance of the observations.

Examples

>>> from nipy.algorithms.statistics.api import Term, Formula
>>> data = np.rec.fromarrays(([1,3,4,5,2,3,4], range(1,8)),
...                          names=('Y', 'X'))
>>> f = Formula([Term("X"), 1])
>>> dmtx = f.design(data, return_float=True)
>>> model = WLSModel(dmtx, weights=range(1,8))
>>> results = model.fit(data['Y'])
>>> results.theta
array([ 0.0952381 ,  2.91666667])
>>> results.t()
array([ 0.35684428,  2.0652652 ])
>>> print(results.Tcontrast([0,1]))  
<T contrast: effect=2.91666666667, sd=1.41224801095, t=2.06526519708,
df_den=5>
>>> print(results.Fcontrast(np.identity(2)))  
<F contrast: F=26.9986072423, df_den=5, df_num=2>

__init__(design, weights=1)

Parameters

design : array-like

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

whiten(X)

Whitener for WLS model, multiplies by sqrt(self.weights)

fit(Y)

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

has_intercept()

Check if column of 1s is in column space of design

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) / n where 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).

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.

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

R5

23. Green. “Econometric Analysis,” 5th ed., Pearson, 2003.

predict(design=None)

After a model has been fit, results are (assumed to be) stored in self.results, which itself should have a predict method.

rank()

Compute rank of design matrix

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.

Functions

nipy.algorithms.statistics.models.regression.ar_bias_correct(results, order, invM=None)

Apply bias correction in calculating AR(p) coefficients from results

There is a slight bias in the rho estimates on residuals due to the correlations induced in the residuals by fitting a linear model. See [Worsley20025].

This routine implements the bias correction described in appendix A.1 of [Worsley20025].

Parameters

results : ndarray or results object

If ndarray, assume these are residuals, from a simple model. If a results object, with attribute resid, then use these for the residuals. See Notes for more detail

order : int

Order p of AR(p) model

invM : None or array

Known bias correcting matrix for covariance. If None, calculate from results.model

Returns

rho : array

Bias-corrected AR(p) coefficients

Notes

If results has attributes resid and scale, then assume scale has come from a fit of a potentially customized model, and we use that for the sum of squared residuals. In this case we also need results.df_resid. Otherwise we assume this is a simple Gaussian model, like OLS, and take the simple sum of squares of the residuals.

References

Worsley20025(1,2,3)

K.J. Worsley, C.H. Liao, J. Aston, V. Petre, G.H. Duncan, F. Morales, A.C. Evans (2002) A General Statistical Analysis for fMRI Data. Neuroimage 15:1:15

nipy.algorithms.statistics.models.regression.ar_bias_corrector(design, calc_beta, order=1)

Return bias correcting matrix for design and AR order order

There is a slight bias in the rho estimates on residuals due to the correlations induced in the residuals by fitting a linear model. See [Worsley20026].

This routine implements the bias correction described in appendix A.1 of [Worsley20026].

Parameters

design : array

Design matrix

calc_beta : array

Moore-Penrose pseudoinverse of the (maybe) whitened design matrix. This is the matrix that, when applied to the (maybe whitened) data, produces the betas.

order : int, optional

Order p of AR(p) process

Returns

invM : array

Matrix to bias correct estimated covariance matrix in calculating the AR coefficients

References

Worsley20026(1,2,3)

K.J. Worsley, C.H. Liao, J. Aston, V. Petre, G.H. Duncan, F. Morales, A.C. Evans (2002) A General Statistical Analysis for fMRI Data. Neuroimage 15:1:15

nipy.algorithms.statistics.models.regression.isestimable(C, D)

True if (Q, P) contrast C is estimable for (N, P) design D

From an Q x P contrast matrix C and an N x P design matrix D, checks if the contrast C is estimable by looking at the rank of vstack([C,D]) and verifying it is the same as the rank of D.

Parameters

C : (Q, P) array-like

contrast matrix. If C has is 1 dimensional assume shape (1, P)

D: (N, P) array-like

design matrix

Returns

tf : bool

True if the contrast C is estimable on design D

Examples

>>> D = np.array([[1, 1, 1, 0, 0, 0],
...               [0, 0, 0, 1, 1, 1],
...               [1, 1, 1, 1, 1, 1]]).T
>>> isestimable([1, 0, 0], D)
False
>>> isestimable([1, -1, 0], D)
True

nipy.algorithms.statistics.models.regression.yule_walker(X, order=1, method='unbiased', df=None, inv=False)

Estimate AR(p) parameters from a sequence X using Yule-Walker equation.

unbiased or maximum-likelihood estimator (mle)

See, for example:

http://en.wikipedia.org/wiki/Autoregressive_moving_average_model

Parameters

X : ndarray of shape(n)

order : int, optional

Order of AR process.

method : str, optional

Method can be “unbiased” or “mle” and this determines denominator in estimate of autocorrelation function (ACF) at lag k. If “mle”, the denominator is n=X.shape[0], if “unbiased” the denominator is n-k.

df : int, optional

Specifies the degrees of freedom. If df is supplied, then it is assumed the X has df degrees of freedom rather than n.

inv : bool, optional

Whether to return the inverse of the R matrix (see code)

Returns

rho : (order,) ndarray

sigma : int

standard deviation of the residuals after fit

R_inv : ndarray

If inv is True, also return the inverse of the R matrix

Notes

See also http://en.wikipedia.org/wiki/AR_model#Calculation_of_the_AR_parameters