def norm_resid(self):
"""
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
"""
return self.resid * pos_recipr(np.sqrt(self.dispersion))
def norm_resid(self):
"""
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
"""
return self.resid * pos_recipr(np.sqrt(self.dispersion))
def 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
[Worsley2002]_.
This routine implements the bias correction described in appendix A.1 of
[Worsley2002]_.
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
----------
.. [Worsley2002] 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
"""
if invM is None:
# We need a model from ``results`` if invM is not specified
model = results.model
invM = ar_bias_corrector(model.design, model.calc_beta, order)
if hasattr(results, 'resid'):
resid = results.resid
else:
resid = results
in_shape = resid.shape
n_features = in_shape[0]
# Allows results residuals to have shapes other than 2D. This allows us to
# use this routine for image data as well as more standard 2D model data
resid = resid.reshape((n_features, - 1))
# glm.Model fit methods fill in a ``scale`` estimate. For simpler
# models, there is no scale estimate written into the results.
# However, the same calculation resolves (with Gaussian family)
# to ``np.sum(resid**2) / results.df_resid``.
# See ``estimate_scale`` from glm.Model
if hasattr(results, 'scale'):
sum_sq = results.scale.reshape(resid.shape[1:]) * results.df_resid
else: # No scale in results
sum_sq = np.sum(resid ** 2, axis=0)
cov = np.zeros((order + 1,) + sum_sq.shape)
cov[0] = sum_sq
for i in range(1, order + 1):
cov[i] = np.sum(resid[i:] * resid[0:- i], axis=0)
# cov is shape (order + 1, V) where V = np.product(in_shape[1:])
cov = np.dot(invM, cov)
output = cov[1:] * pos_recipr(cov[0])
return np.squeeze(output.reshape((order,) + in_shape[1:]))
