algorithms.statistics.models.regression

Module: algorithms.statistics.models.regression

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

Inheritance diagram of nipy.algorithms.statistics.models.regression

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

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)
whiten(Y)

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.

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)
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

[R112]
  1. Green. “Econometric Analysis,” 5th ed., Pearson, 2003.
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(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.

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.

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.

MSE()

Mean square (error)

MSR()

Mean square (regression)

MST()

Mean square (total)

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

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

SSE()

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

SSR()

Regression sum of squares

SST()

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

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.

resid()

Residuals from the fit.

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)
whiten(X)

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

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 [Worsley200234].

This routine implements the bias correction described in appendix A.1 of [Worsley200234].

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

[Worsley200234](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 [Worsley200256].

This routine implements the bias correction described in appendix A.1 of [Worsley200256].

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

[Worsley200256](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