'''Partial Regression plot and residual plots to find misspecification
Author: Josef Perktold
License: BSD-3
Created: 2011-01-23
update
2011-06-05 : start to convert example to usable functions
2011-10-27 : docstrings
'''
from statsmodels.compat.python import lrange, string_types, lzip, range
import numpy as np
import pandas as pd
from patsy import dmatrix
from statsmodels.regression.linear_model import OLS, GLS, WLS
from statsmodels.genmod.generalized_linear_model import GLM
from statsmodels.genmod.generalized_estimating_equations import GEE
from statsmodels.sandbox.regression.predstd import wls_prediction_std
from statsmodels.graphics import utils
from statsmodels.nonparametric.smoothers_lowess import lowess
from statsmodels.tools.tools import maybe_unwrap_results
from statsmodels.base import model
from ._regressionplots_doc import (
_plot_added_variable_doc,
_plot_partial_residuals_doc,
_plot_ceres_residuals_doc)
__all__ = ['plot_fit', 'plot_regress_exog', 'plot_partregress', 'plot_ccpr',
'plot_regress_exog', 'plot_partregress_grid', 'plot_ccpr_grid',
'add_lowess', 'abline_plot', 'influence_plot',
'plot_leverage_resid2', 'added_variable_resids',
'partial_resids', 'ceres_resids', 'plot_added_variable',
'plot_partial_residuals', 'plot_ceres_residuals']
#TODO: consider moving to influence module
def _high_leverage(results):
#TODO: replace 1 with k_constant
return 2. * (results.df_model + 1)/results.nobs
def add_lowess(ax, lines_idx=0, frac=.2, **lowess_kwargs):
"""
Add Lowess line to a plot.
Parameters
----------
ax : matplotlib Axes instance
The Axes to which to add the plot
lines_idx : int
This is the line on the existing plot to which you want to add
a smoothed lowess line.
frac : float
The fraction of the points to use when doing the lowess fit.
lowess_kwargs
Additional keyword arguments are passes to lowess.
Returns
-------
fig : matplotlib Figure instance
The figure that holds the instance.
"""
y0 = ax.get_lines()[lines_idx]._y
x0 = ax.get_lines()[lines_idx]._x
lres = lowess(y0, x0, frac=frac, **lowess_kwargs)
ax.plot(lres[:, 0], lres[:, 1], 'r', lw=1.5)
return ax.figure
[docs]def plot_fit(results, exog_idx, y_true=None, ax=None, **kwargs):
"""Plot fit against one regressor.
This creates one graph with the scatterplot of observed values compared to
fitted values.
Parameters
----------
results : result instance
result instance with resid, model.endog and model.exog as attributes
x_var : int or str
Name or index of regressor in exog matrix.
y_true : array_like
(optional) If this is not None, then the array is added to the plot
ax : Matplotlib AxesSubplot instance, optional
If given, this subplot is used to plot in instead of a new figure being
created.
kwargs
The keyword arguments are passed to the plot command for the fitted
values points.
Returns
-------
fig : Matplotlib figure instance
If `ax` is None, the created figure. Otherwise the figure to which
`ax` is connected.
Examples
--------
Load the Statewide Crime data set and perform linear regression with
`poverty` and `hs_grad` as variables and `murder` as the response
>>> import statsmodels.api as sm
>>> import matplotlib.pyplot as plt
>>> data = sm.datasets.statecrime.load_pandas().data
>>> murder = data['murder']
>>> X = data[['poverty', 'hs_grad']]
>>> X["constant"] = 1
>>> y = murder
>>> model = sm.OLS(y, X)
>>> results = model.fit()
Create a plot just for the variable 'Poverty':
>>> fig, ax = plt.subplots()
>>> fig = sm.graphics.plot_fit(results, 0, ax=ax)
>>> ax.set_ylabel("Murder Rate")
>>> ax.set_xlabel("Poverty Level")
>>> ax.set_title("Linear Regression")
>>> plt.show()
.. plot:: plots/graphics_plot_fit_ex.py
"""
fig, ax = utils.create_mpl_ax(ax)
exog_name, exog_idx = utils.maybe_name_or_idx(exog_idx, results.model)
results = maybe_unwrap_results(results)
#maybe add option for wendog, wexog
y = results.model.endog
x1 = results.model.exog[:, exog_idx]
x1_argsort = np.argsort(x1)
y = y[x1_argsort]
x1 = x1[x1_argsort]
ax.plot(x1, y, 'bo', label=results.model.endog_names)
if not y_true is None:
ax.plot(x1, y_true[x1_argsort], 'b-', label='True values')
title = 'Fitted values versus %s' % exog_name
prstd, iv_l, iv_u = wls_prediction_std(results)
ax.plot(x1, results.fittedvalues[x1_argsort], 'D', color='r',
label='fitted', **kwargs)
ax.vlines(x1, iv_l[x1_argsort], iv_u[x1_argsort], linewidth=1, color='k',
alpha=.7)
#ax.fill_between(x1, iv_l[x1_argsort], iv_u[x1_argsort], alpha=0.1,
# color='k')
ax.set_title(title)
ax.set_xlabel(exog_name)
ax.set_ylabel(results.model.endog_names)
ax.legend(loc='best', numpoints=1)
return fig
[docs]def plot_regress_exog(results, exog_idx, fig=None):
"""Plot regression results against one regressor.
This plots four graphs in a 2 by 2 figure: 'endog versus exog',
'residuals versus exog', 'fitted versus exog' and
'fitted plus residual versus exog'
Parameters
----------
results : result instance
result instance with resid, model.endog and model.exog as attributes
exog_idx : int
index of regressor in exog matrix
fig : Matplotlib figure instance, optional
If given, this figure is simply returned. Otherwise a new figure is
created.
Returns
-------
fig : matplotlib figure instance
"""
fig = utils.create_mpl_fig(fig)
exog_name, exog_idx = utils.maybe_name_or_idx(exog_idx, results.model)
results = maybe_unwrap_results(results)
#maybe add option for wendog, wexog
y_name = results.model.endog_names
x1 = results.model.exog[:, exog_idx]
prstd, iv_l, iv_u = wls_prediction_std(results)
ax = fig.add_subplot(2, 2, 1)
ax.plot(x1, results.model.endog, 'o', color='b', alpha=0.9, label=y_name)
ax.plot(x1, results.fittedvalues, 'D', color='r', label='fitted',
alpha=.5)
ax.vlines(x1, iv_l, iv_u, linewidth=1, color='k', alpha=.7)
ax.set_title('Y and Fitted vs. X', fontsize='large')
ax.set_xlabel(exog_name)
ax.set_ylabel(y_name)
ax.legend(loc='best')
ax = fig.add_subplot(2, 2, 2)
ax.plot(x1, results.resid, 'o')
ax.axhline(y=0, color='black')
ax.set_title('Residuals versus %s' % exog_name, fontsize='large')
ax.set_xlabel(exog_name)
ax.set_ylabel("resid")
ax = fig.add_subplot(2, 2, 3)
exog_noti = np.ones(results.model.exog.shape[1], bool)
exog_noti[exog_idx] = False
exog_others = results.model.exog[:, exog_noti]
from pandas import Series
fig = plot_partregress(results.model.data.orig_endog,
Series(x1, name=exog_name,
index=results.model.data.row_labels),
exog_others, obs_labels=False, ax=ax)
ax.set_title('Partial regression plot', fontsize='large')
#ax.set_ylabel("Fitted values")
#ax.set_xlabel(exog_name)
ax = fig.add_subplot(2, 2, 4)
fig = plot_ccpr(results, exog_idx, ax=ax)
ax.set_title('CCPR Plot', fontsize='large')
#ax.set_xlabel(exog_name)
#ax.set_ylabel("Fitted values + resids")
fig.suptitle('Regression Plots for %s' % exog_name, fontsize="large")
fig.tight_layout()
fig.subplots_adjust(top=.90)
return fig
def _partial_regression(endog, exog_i, exog_others):
"""Partial regression.
regress endog on exog_i conditional on exog_others
uses OLS
Parameters
----------
endog : array_like
exog : array_like
exog_others : array_like
Returns
-------
res1c : OLS results instance
(res1a, res1b) : tuple of OLS results instances
results from regression of endog on exog_others and of exog_i on
exog_others
"""
#FIXME: This function doesn't appear to be used.
res1a = OLS(endog, exog_others).fit()
res1b = OLS(exog_i, exog_others).fit()
res1c = OLS(res1a.resid, res1b.resid).fit()
return res1c, (res1a, res1b)
[docs]def plot_partregress(endog, exog_i, exog_others, data=None,
title_kwargs={}, obs_labels=True, label_kwargs={},
ax=None, ret_coords=False, **kwargs):
"""Plot partial regression for a single regressor.
Parameters
----------
endog : ndarray or string
endogenous or response variable. If string is given, you can use a
arbitrary translations as with a formula.
exog_i : ndarray or string
exogenous, explanatory variable. If string is given, you can use a
arbitrary translations as with a formula.
exog_others : ndarray or list of strings
other exogenous, explanatory variables. If a list of strings is given,
each item is a term in formula. You can use a arbitrary translations
as with a formula. The effect of these variables will be removed by
OLS regression.
data : DataFrame, dict, or recarray
Some kind of data structure with names if the other variables are
given as strings.
title_kwargs : dict
Keyword arguments to pass on for the title. The key to control the
fonts is fontdict.
obs_labels : bool or array-like
Whether or not to annotate the plot points with their observation
labels. If obs_labels is a boolean, the point labels will try to do
the right thing. First it will try to use the index of data, then
fall back to the index of exog_i. Alternatively, you may give an
array-like object corresponding to the obseveration numbers.
labels_kwargs : dict
Keyword arguments that control annotate for the observation labels.
ax : Matplotlib AxesSubplot instance, optional
If given, this subplot is used to plot in instead of a new figure being
created.
ret_coords : bool
If True will return the coordinates of the points in the plot. You
can use this to add your own annotations.
kwargs
The keyword arguments passed to plot for the points.
Returns
-------
fig : Matplotlib figure instance
If `ax` is None, the created figure. Otherwise the figure to which
`ax` is connected.
coords : list, optional
If ret_coords is True, return a tuple of arrays (x_coords, y_coords).
Notes
-----
The slope of the fitted line is the that of `exog_i` in the full
multiple regression. The individual points can be used to assess the
influence of points on the estimated coefficient.
See Also
--------
plot_partregress_grid : Plot partial regression for a set of regressors.
"""
#NOTE: there is no interaction between possible missing data and
#obs_labels yet, so this will need to be tweaked a bit for this case
fig, ax = utils.create_mpl_ax(ax)
# strings, use patsy to transform to data
if isinstance(endog, string_types):
endog = dmatrix(endog + "-1", data)
if isinstance(exog_others, string_types):
RHS = dmatrix(exog_others, data)
elif isinstance(exog_others, list):
RHS = "+".join(exog_others)
RHS = dmatrix(RHS, data)
else:
RHS = exog_others
RHS_isemtpy = False
if isinstance(RHS, np.ndarray) and RHS.size==0:
RHS_isemtpy = True
elif isinstance(RHS, pd.DataFrame) and RHS.empty:
RHS_isemtpy = True
if isinstance(exog_i, string_types):
exog_i = dmatrix(exog_i + "-1", data)
# all arrays or pandas-like
if RHS_isemtpy:
ax.plot(endog, exog_i, 'o', **kwargs)
fitted_line = OLS(endog, exog_i).fit()
x_axis_endog_name = 'x' if isinstance(exog_i, np.ndarray) else exog_i.name
y_axis_endog_name = 'y' if isinstance(endog, np.ndarray) else endog.design_info.column_names[0]
else:
res_yaxis = OLS(endog, RHS).fit()
res_xaxis = OLS(exog_i, RHS).fit()
xaxis_resid = res_xaxis.resid
yaxis_resid = res_yaxis.resid
x_axis_endog_name = res_xaxis.model.endog_names
y_axis_endog_name = res_yaxis.model.endog_names
ax.plot(xaxis_resid, yaxis_resid, 'o', **kwargs)
fitted_line = OLS(yaxis_resid, xaxis_resid).fit()
fig = abline_plot(0, fitted_line.params[0], color='k', ax=ax)
if x_axis_endog_name == 'y': # for no names regression will just get a y
x_axis_endog_name = 'x' # this is misleading, so use x
ax.set_xlabel("e(%s | X)" % x_axis_endog_name)
ax.set_ylabel("e(%s | X)" % y_axis_endog_name)
ax.set_title('Partial Regression Plot', **title_kwargs)
#NOTE: if we want to get super fancy, we could annotate if a point is
#clicked using this widget
#http://stackoverflow.com/questions/4652439/
#is-there-a-matplotlib-equivalent-of-matlabs-datacursormode/
#4674445#4674445
if obs_labels is True:
if data is not None:
obs_labels = data.index
elif hasattr(exog_i, "index"):
obs_labels = exog_i.index
else:
obs_labels = res_xaxis.model.data.row_labels
#NOTE: row_labels can be None.
#Maybe we should fix this to never be the case.
if obs_labels is None:
obs_labels = lrange(len(exog_i))
if obs_labels is not False: # could be array-like
if len(obs_labels) != len(exog_i):
raise ValueError("obs_labels does not match length of exog_i")
label_kwargs.update(dict(ha="center", va="bottom"))
ax = utils.annotate_axes(lrange(len(obs_labels)), obs_labels,
lzip(res_xaxis.resid, res_yaxis.resid),
[(0, 5)] * len(obs_labels), "x-large", ax=ax,
**label_kwargs)
if ret_coords:
return fig, (res_xaxis.resid, res_yaxis.resid)
else:
return fig
def plot_partregress_grid(results, exog_idx=None, grid=None, fig=None):
"""Plot partial regression for a set of regressors.
Parameters
----------
results : results instance
A regression model results instance
exog_idx : None, list of ints, list of strings
(column) indices of the exog used in the plot, default is all.
grid : None or tuple of int (nrows, ncols)
If grid is given, then it is used for the arrangement of the subplots.
If grid is None, then ncol is one, if there are only 2 subplots, and
the number of columns is two otherwise.
fig : Matplotlib figure instance, optional
If given, this figure is simply returned. Otherwise a new figure is
created.
Returns
-------
fig : Matplotlib figure instance
If `fig` is None, the created figure. Otherwise `fig` itself.
Notes
-----
A subplot is created for each explanatory variable given by exog_idx.
The partial regression plot shows the relationship between the response
and the given explanatory variable after removing the effect of all other
explanatory variables in exog.
See Also
--------
plot_partregress : Plot partial regression for a single regressor.
plot_ccpr
References
----------
See http://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/partregr.htm
"""
import pandas
fig = utils.create_mpl_fig(fig)
exog_name, exog_idx = utils.maybe_name_or_idx(exog_idx, results.model)
#maybe add option for using wendog, wexog instead
y = pandas.Series(results.model.endog, name=results.model.endog_names)
exog = results.model.exog
k_vars = exog.shape[1]
#this function doesn't make sense if k_vars=1
if not grid is None:
nrows, ncols = grid
else:
if len(exog_idx) > 2:
nrows = int(np.ceil(len(exog_idx)/2.))
ncols = 2
title_kwargs = {"fontdict" : {"fontsize" : 'small'}}
else:
nrows = len(exog_idx)
ncols = 1
title_kwargs = {}
# for indexing purposes
other_names = np.array(results.model.exog_names)
for i, idx in enumerate(exog_idx):
others = lrange(k_vars)
others.pop(idx)
exog_others = pandas.DataFrame(exog[:, others],
columns=other_names[others])
ax = fig.add_subplot(nrows, ncols, i+1)
plot_partregress(y, pandas.Series(exog[:, idx],
name=other_names[idx]),
exog_others, ax=ax, title_kwargs=title_kwargs,
obs_labels=False)
ax.set_title("")
fig.suptitle("Partial Regression Plot", fontsize="large")
fig.tight_layout()
fig.subplots_adjust(top=.95)
return fig
[docs]def plot_ccpr(results, exog_idx, ax=None):
"""Plot CCPR against one regressor.
Generates a CCPR (component and component-plus-residual) plot.
Parameters
----------
results : result instance
A regression results instance.
exog_idx : int or string
Exogenous, explanatory variable. If string is given, it should
be the variable name that you want to use, and you can use arbitrary
translations as with a formula.
ax : Matplotlib AxesSubplot instance, optional
If given, it is used to plot in instead of a new figure being
created.
Returns
-------
fig : Matplotlib figure instance
If `ax` is None, the created figure. Otherwise the figure to which
`ax` is connected.
See Also
--------
plot_ccpr_grid : Creates CCPR plot for multiple regressors in a plot grid.
Notes
-----
The CCPR plot provides a way to judge the effect of one regressor on the
response variable by taking into account the effects of the other
independent variables. The partial residuals plot is defined as
Residuals + B_i*X_i versus X_i. The component adds the B_i*X_i versus
X_i to show where the fitted line would lie. Care should be taken if X_i
is highly correlated with any of the other independent variables. If this
is the case, the variance evident in the plot will be an underestimate of
the true variance.
References
----------
http://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/ccpr.htm
"""
fig, ax = utils.create_mpl_ax(ax)
exog_name, exog_idx = utils.maybe_name_or_idx(exog_idx, results.model)
results = maybe_unwrap_results(results)
x1 = results.model.exog[:, exog_idx]
#namestr = ' for %s' % self.name if self.name else ''
x1beta = x1*results.params[exog_idx]
ax.plot(x1, x1beta + results.resid, 'o')
from statsmodels.tools.tools import add_constant
mod = OLS(x1beta, add_constant(x1)).fit()
params = mod.params
fig = abline_plot(*params, **dict(ax=ax))
#ax.plot(x1, x1beta, '-')
ax.set_title('Component and component plus residual plot')
ax.set_ylabel("Residual + %s*beta_%d" % (exog_name, exog_idx))
ax.set_xlabel("%s" % exog_name)
return fig
def plot_ccpr_grid(results, exog_idx=None, grid=None, fig=None):
"""Generate CCPR plots against a set of regressors, plot in a grid.
Generates a grid of CCPR (component and component-plus-residual) plots.
Parameters
----------
results : result instance
uses exog and params of the result instance
exog_idx : None or list of int
(column) indices of the exog used in the plot
grid : None or tuple of int (nrows, ncols)
If grid is given, then it is used for the arrangement of the subplots.
If grid is None, then ncol is one, if there are only 2 subplots, and
the number of columns is two otherwise.
fig : Matplotlib figure instance, optional
If given, this figure is simply returned. Otherwise a new figure is
created.
Returns
-------
fig : Matplotlib figure instance
If `ax` is None, the created figure. Otherwise the figure to which
`ax` is connected.
Notes
-----
Partial residual plots are formed as::
Res + Betahat(i)*Xi versus Xi
and CCPR adds::
Betahat(i)*Xi versus Xi
See Also
--------
plot_ccpr : Creates CCPR plot for a single regressor.
References
----------
See http://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/ccpr.htm
"""
fig = utils.create_mpl_fig(fig)
exog_name, exog_idx = utils.maybe_name_or_idx(exog_idx, results.model)
if grid is not None:
nrows, ncols = grid
else:
if len(exog_idx) > 2:
nrows = int(np.ceil(len(exog_idx)/2.))
ncols = 2
else:
nrows = len(exog_idx)
ncols = 1
seen_constant = 0
for i, idx in enumerate(exog_idx):
if results.model.exog[:, idx].var() == 0:
seen_constant = 1
continue
ax = fig.add_subplot(nrows, ncols, i+1-seen_constant)
fig = plot_ccpr(results, exog_idx=idx, ax=ax)
ax.set_title("")
fig.suptitle("Component-Component Plus Residual Plot", fontsize="large")
fig.tight_layout()
fig.subplots_adjust(top=.95)
return fig
[docs]def abline_plot(intercept=None, slope=None, horiz=None, vert=None,
model_results=None, ax=None, **kwargs):
"""
Plots a line given an intercept and slope.
intercept : float
The intercept of the line
slope : float
The slope of the line
horiz : float or array-like
Data for horizontal lines on the y-axis
vert : array-like
Data for verterical lines on the x-axis
model_results : statsmodels results instance
Any object that has a two-value `params` attribute. Assumed that it
is (intercept, slope)
ax : axes, optional
Matplotlib axes instance
kwargs
Options passed to matplotlib.pyplot.plt
Returns
-------
fig : Figure
The figure given by `ax.figure` or a new instance.
Examples
--------
>>> import numpy as np
>>> import statsmodels.api as sm
>>> np.random.seed(12345)
>>> X = sm.add_constant(np.random.normal(0, 20, size=30))
>>> y = np.dot(X, [25, 3.5]) + np.random.normal(0, 30, size=30)
>>> mod = sm.OLS(y,X).fit()
>>> fig = sm.graphics.abline_plot(model_results=mod)
>>> ax = fig.axes[0]
>>> ax.scatter(X[:,1], y)
>>> ax.margins(.1)
>>> import matplotlib.pyplot as plt
>>> plt.show()
"""
if ax is not None: # get axis limits first thing, don't change these
x = ax.get_xlim()
else:
x = None
fig, ax = utils.create_mpl_ax(ax)
if model_results:
intercept, slope = model_results.params
if x is None:
x = [model_results.model.exog[:, 1].min(),
model_results.model.exog[:, 1].max()]
else:
if not (intercept is not None and slope is not None):
raise ValueError("specify slope and intercepty or model_results")
if x is None:
x = ax.get_xlim()
data_y = [x[0]*slope+intercept, x[1]*slope+intercept]
ax.set_xlim(x)
#ax.set_ylim(y)
from matplotlib.lines import Line2D
class ABLine2D(Line2D):
def __init__(self, *args, **kwargs):
super(ABLine2D, self).__init__(*args, **kwargs)
self.id_xlim_callback = None
self.id_ylim_callback = None
def remove(self):
ax = self.axes
if self.id_xlim_callback:
ax.callbacks.disconnect(self.id_xlim_callback)
if self.id_ylim_callback:
ax.callbacks.disconnect(self.id_ylim_callback)
super(ABLine2D, self).remove()
def update_datalim(self, ax):
ax.set_autoscale_on(False)
children = ax.get_children()
ablines = [child for child in children if child is self]
abline = ablines[0]
x = ax.get_xlim()
y = [x[0] * slope + intercept, x[1] * slope + intercept]
abline.set_data(x, y)
ax.figure.canvas.draw()
# TODO: how to intercept something like a margins call and adjust?
line = ABLine2D(x, data_y, **kwargs)
ax.add_line(line)
line.id_xlim_callback = ax.callbacks.connect('xlim_changed', line.update_datalim)
line.id_ylim_callback = ax.callbacks.connect('ylim_changed', line.update_datalim)
if horiz:
ax.hline(horiz)
if vert:
ax.vline(vert)
return fig
[docs]def influence_plot(results, external=True, alpha=.05, criterion="cooks",
size=48, plot_alpha=.75, ax=None, **kwargs):
"""
Plot of influence in regression. Plots studentized resids vs. leverage.
Parameters
----------
results : results instance
A fitted model.
external : bool
Whether to use externally or internally studentized residuals. It is
recommended to leave external as True.
alpha : float
The alpha value to identify large studentized residuals. Large means
abs(resid_studentized) > t.ppf(1-alpha/2, dof=results.df_resid)
criterion : str {'DFFITS', 'Cooks'}
Which criterion to base the size of the points on. Options are
DFFITS or Cook's D.
size : float
The range of `criterion` is mapped to 10**2 - size**2 in points.
plot_alpha : float
The `alpha` of the plotted points.
ax : matplotlib Axes instance
An instance of a matplotlib Axes.
Returns
-------
fig : matplotlib figure
The matplotlib figure that contains the Axes.
Notes
-----
Row labels for the observations in which the leverage, measured by the
diagonal of the hat matrix, is high or the residuals are large, as the
combination of large residuals and a high influence value indicates an
influence point. The value of large residuals can be controlled using the
`alpha` parameter. Large leverage points are identified as
hat_i > 2 * (df_model + 1)/nobs.
"""
fig, ax = utils.create_mpl_ax(ax)
infl = results.get_influence()
if criterion.lower().startswith('coo'):
psize = infl.cooks_distance[0]
elif criterion.lower().startswith('dff'):
psize = np.abs(infl.dffits[0])
else:
raise ValueError("Criterion %s not understood" % criterion)
# scale the variables
#TODO: what is the correct scaling and the assumption here?
#we want plots to be comparable across different plots
#so we would need to use the expected distribution of criterion probably
old_range = np.ptp(psize)
new_range = size**2 - 8**2
psize = (psize - psize.min()) * new_range/old_range + 8**2
leverage = infl.hat_matrix_diag
if external:
resids = infl.resid_studentized_external
else:
resids = infl.resid_studentized_internal
from scipy import stats
cutoff = stats.t.ppf(1.-alpha/2, results.df_resid)
large_resid = np.abs(resids) > cutoff
large_leverage = leverage > _high_leverage(results)
large_points = np.logical_or(large_resid, large_leverage)
ax.scatter(leverage, resids, s=psize, alpha=plot_alpha)
# add point labels
labels = results.model.data.row_labels
if labels is None:
labels = lrange(len(resids))
ax = utils.annotate_axes(np.where(large_points)[0], labels,
lzip(leverage, resids),
lzip(-(psize/2)**.5, (psize/2)**.5), "x-large",
ax)
#TODO: make configurable or let people do it ex-post?
font = {"fontsize" : 16, "color" : "black"}
ax.set_ylabel("Studentized Residuals", **font)
ax.set_xlabel("H Leverage", **font)
ax.set_title("Influence Plot", **font)
return fig
[docs]def plot_leverage_resid2(results, alpha=.05, ax=None,
**kwargs):
"""
Plots leverage statistics vs. normalized residuals squared
Parameters
----------
results : results instance
A regression results instance
alpha : float
Specifies the cut-off for large-standardized residuals. Residuals
are assumed to be distributed N(0, 1) with alpha=alpha.
ax : Axes instance
Matplotlib Axes instance
Returns
-------
fig : matplotlib Figure
A matplotlib figure instance.
"""
from scipy.stats import zscore, norm
fig, ax = utils.create_mpl_ax(ax)
infl = results.get_influence()
leverage = infl.hat_matrix_diag
resid = zscore(results.resid)
ax.plot(resid**2, leverage, 'o', **kwargs)
ax.set_xlabel("Normalized residuals**2")
ax.set_ylabel("Leverage")
ax.set_title("Leverage vs. Normalized residuals squared")
large_leverage = leverage > _high_leverage(results)
#norm or t here if standardized?
cutoff = norm.ppf(1.-alpha/2)
large_resid = np.abs(resid) > cutoff
labels = results.model.data.row_labels
if labels is None:
labels = lrange(int(results.nobs))
index = np.where(np.logical_or(large_leverage, large_resid))[0]
ax = utils.annotate_axes(index, labels, lzip(resid**2, leverage),
[(0, 5)]*int(results.nobs), "large",
ax=ax, ha="center", va="bottom")
ax.margins(.075, .075)
return fig
def plot_added_variable(results, focus_exog, resid_type=None,
use_glm_weights=True, fit_kwargs=None, ax=None):
# Docstring attached below
model = results.model
fig, ax = utils.create_mpl_ax(ax)
endog_resid, focus_exog_resid =\
added_variable_resids(results, focus_exog,
resid_type=resid_type,
use_glm_weights=use_glm_weights,
fit_kwargs=fit_kwargs)
ax.plot(focus_exog_resid, endog_resid, 'o', alpha=0.6)
ax.set_title('Added variable plot', fontsize='large')
if type(focus_exog) is str:
xname = focus_exog
else:
xname = model.exog_names[focus_exog]
ax.set_xlabel(xname, size=15)
ax.set_ylabel(model.endog_names + " residuals", size=15)
return fig
plot_added_variable.__doc__ = _plot_added_variable_doc % {
'extra_params_doc' : "results: object\n\tResults for a fitted regression model"}
def plot_partial_residuals(results, focus_exog, ax=None):
# Docstring attached below
model = results.model
focus_exog, focus_col = utils.maybe_name_or_idx(focus_exog, model)
pr = partial_resids(results, focus_exog)
focus_exog_vals = results.model.exog[:, focus_col]
fig, ax = utils.create_mpl_ax(ax)
ax.plot(focus_exog_vals, pr, 'o', alpha=0.6)
ax.set_title('Partial residuals plot', fontsize='large')
if type(focus_exog) is str:
xname = focus_exog
else:
xname = model.exog_names[focus_exog]
ax.set_xlabel(xname, size=15)
ax.set_ylabel("Component plus residual", size=15)
return fig
plot_partial_residuals.__doc__ = _plot_partial_residuals_doc % {
'extra_params_doc' : "results: object\n\tResults for a fitted regression model"}
def plot_ceres_residuals(results, focus_exog, frac=0.66, cond_means=None,
ax=None):
# Docstring attached below
model = results.model
focus_exog, focus_col = utils.maybe_name_or_idx(focus_exog, model)
presid = ceres_resids(results, focus_exog, frac=frac,
cond_means=cond_means)
focus_exog_vals = model.exog[:, focus_col]
fig, ax = utils.create_mpl_ax(ax)
ax.plot(focus_exog_vals, presid, 'o', alpha=0.6)
ax.set_title('CERES residuals plot', fontsize='large')
ax.set_xlabel(focus_exog, size=15)
ax.set_ylabel("Component plus residual", size=15)
return fig
plot_ceres_residuals.__doc__ = _plot_ceres_residuals_doc % {
'extra_params_doc' : "results: object\n\tResults for a fitted regression model"}
def ceres_resids(results, focus_exog, frac=0.66, cond_means=None):
"""
Calculate the CERES residuals (Conditional Expectation Partial
Residuals) for a fitted model.
Parameters
----------
results : model results instance
The fitted model for which the CERES residuals are calculated.
focus_exog : int
The column of results.model.exog used as the 'focus variable'.
frac : float, optional
Lowess smoothing parameter for estimating the conditional
means. Not used if `cond_means` is provided.
cond_means : array-like, optional
If provided, the columns of this array are the conditional
means E[exog | focus exog], where exog ranges over some
or all of the columns of exog other than focus exog. If
this is an empty nx0 array, the conditional means are
treated as being zero. If None, the conditional means are
estimated.
Returns
-------
An array containing the CERES residuals.
Notes
-----
If `cond_means` is not provided, it is obtained by smoothing each
column of exog (except the focus column) against the focus column.
Currently only supports GLM, GEE, and OLS models.
"""
model = results.model
if not isinstance(model, (GLM, GEE, OLS)):
raise ValueError("ceres residuals not available for %s" %
model.__class__.__name__)
focus_exog, focus_col = utils.maybe_name_or_idx(focus_exog, model)
# Indices of non-focus columns
ix_nf = range(len(results.params))
ix_nf = list(ix_nf)
ix_nf.pop(focus_col)
nnf = len(ix_nf)
# Estimate the conditional means if not provided.
if cond_means is None:
# Below we calculate E[x | focus] where x is each column other
# than the focus column. We don't want the intercept when we do
# this so we remove it here.
pexog = model.exog[:, ix_nf]
pexog -= pexog.mean(0)
u, s, vt = np.linalg.svd(pexog, 0)
ii = np.flatnonzero(s > 1e-6)
pexog = u[:, ii]
fcol = model.exog[:, focus_col]
cond_means = np.empty((len(fcol), pexog.shape[1]))
for j in range(pexog.shape[1]):
# Get the fitted values for column i given the other
# columns (skip the intercept).
y0 = pexog[:, j]
cf = lowess(y0, fcol, frac=frac, return_sorted=False)
cond_means[:, j] = cf
new_exog = np.concatenate((model.exog[:, ix_nf], cond_means), axis=1)
# Refit the model using the adjusted exog values
klass = model.__class__
init_kwargs = model._get_init_kwds()
new_model = klass(model.endog, new_exog, **init_kwargs)
new_result = new_model.fit()
# The partial residual, with respect to l(x2) (notation of Cook 1998)
presid = model.endog - new_result.fittedvalues
if isinstance(model, (GLM, GEE)):
presid *= model.family.link.deriv(new_result.fittedvalues)
if new_exog.shape[1] > nnf:
presid += np.dot(new_exog[:, nnf:], new_result.params[nnf:])
return presid
def partial_resids(results, focus_exog):
"""
Returns partial residuals for a fitted model with respect to a
'focus predictor'.
Parameters
----------
results : results instance
A fitted regression model.
focus col : int
The column index of model.exog with respect to which the
partial residuals are calculated.
Returns
-------
An array of partial residuals.
References
----------
RD Cook and R Croos-Dabrera (1998). Partial residual plots in
generalized linear models. Journal of the American Statistical
Association, 93:442.
"""
# TODO: could be a method of results
# TODO: see Cook et al (1998) for a more general definition
# The calculation follows equation (8) from Cook's paper.
model = results.model
resid = model.endog - results.predict()
if isinstance(model, (GLM, GEE)):
resid *= model.family.link.deriv(results.fittedvalues)
elif isinstance(model, (OLS, GLS, WLS)):
pass # No need to do anything
else:
raise ValueError("Partial residuals for '%s' not implemented."
% type(model))
if type(focus_exog) is str:
focus_col = model.exog_names.index(focus_exog)
else:
focus_col = focus_exog
focus_val = results.params[focus_col] * model.exog[:, focus_col]
return focus_val + resid
def added_variable_resids(results, focus_exog, resid_type=None,
use_glm_weights=True, fit_kwargs=None):
"""
Residualize the endog variable and a 'focus' exog variable in a
regression model with respect to the other exog variables.
Parameters
----------
results : regression results instance
A fitted model including the focus exog and all other
predictors of interest.
focus_exog : integer or string
The column of results.model.exog or a variable name that is
to be residualized against the other predictors.
resid_type : string
The type of residuals to use for the dependent variable. If
None, uses `resid_deviance` for GLM/GEE and `resid` otherwise.
use_glm_weights : bool
Only used if the model is a GLM or GEE. If True, the
residuals for the focus predictor are computed using WLS, with
the weights obtained from the IRLS calculations for fitting
the GLM. If False, unweighted regression is used.
fit_kwargs : dict, optional
Keyword arguments to be passed to fit when refitting the
model.
Returns
-------
endog_resid : array-like
The residuals for the original exog
focus_exog_resid : array-like
The residuals for the focus predictor
Notes
-----
The 'focus variable' residuals are always obtained using linear
regression.
Currently only GLM, GEE, and OLS models are supported.
"""
model = results.model
if not isinstance(model, (GEE, GLM, OLS)):
raise ValueError("model type %s not supported for added variable residuals" %
model.__class__.__name__)
exog = model.exog
endog = model.endog
focus_exog, focus_col = utils.maybe_name_or_idx(focus_exog, model)
focus_exog_vals = exog[:, focus_col]
# Default residuals
if resid_type is None:
if isinstance(model, (GEE, GLM)):
resid_type = "resid_deviance"
else:
resid_type = "resid"
ii = range(exog.shape[1])
ii = list(ii)
ii.pop(focus_col)
reduced_exog = exog[:, ii]
start_params = results.params[ii]
klass = model.__class__
kwargs = model._get_init_kwds()
new_model = klass(endog, reduced_exog, **kwargs)
args = {"start_params": start_params}
if fit_kwargs is not None:
args.update(fit_kwargs)
new_result = new_model.fit(**args)
if not new_result.converged:
raise ValueError("fit did not converge when calculating added variable residuals")
try:
endog_resid = getattr(new_result, resid_type)
except AttributeError:
raise ValueError("'%s' residual type not available" % resid_type)
import statsmodels.regression.linear_model as lm
if isinstance(model, (GLM, GEE)) and use_glm_weights:
weights = model.family.weights(results.fittedvalues)
if hasattr(model, "data_weights"):
weights = weights * model.data_weights
lm_results = lm.WLS(focus_exog_vals, reduced_exog, weights).fit()
else:
lm_results = lm.OLS(focus_exog_vals, reduced_exog).fit()
focus_exog_resid = lm_results.resid
return endog_resid, focus_exog_resid
