from statsmodels.compat.python import lzip, string_types
import numpy as np
from scipy import stats
from statsmodels.regression.linear_model import OLS
from statsmodels.tools.tools import add_constant
from statsmodels.tools.decorators import (resettable_cache,
cache_readonly,
cache_writable)
from . import utils
__all__ = ['qqplot', 'qqplot_2samples', 'qqline', 'ProbPlot']
[docs]class ProbPlot(object):
"""
Class for convenient construction of Q-Q, P-P, and probability plots.
Can take arguments specifying the parameters for dist or fit them
automatically. (See fit under kwargs.)
Parameters
----------
data : array-like
1d data array
dist : A scipy.stats or statsmodels distribution
Compare x against dist. The default is
scipy.stats.distributions.norm (a standard normal).
distargs : tuple
A tuple of arguments passed to dist to specify it fully
so dist.ppf may be called.
loc : float
Location parameter for dist
a : float
Offset for the plotting position of an expected order
statistic, for example. The plotting positions are given
by (i - a)/(nobs - 2*a + 1) for i in range(0,nobs+1)
scale : float
Scale parameter for dist
fit : boolean
If fit is false, loc, scale, and distargs are passed to the
distribution. If fit is True then the parameters for dist
are fit automatically using dist.fit. The quantiles are formed
from the standardized data, after subtracting the fitted loc
and dividing by the fitted scale.
See Also
--------
scipy.stats.probplot
Notes
-----
1) Depends on matplotlib.
2) If `fit` is True then the parameters are fit using the
distribution's `fit()` method.
3) The call signatures for the `qqplot`, `ppplot`, and `probplot`
methods are similar, so examples 1 through 4 apply to all
three methods.
4) The three plotting methods are summarized below:
ppplot : Probability-Probability plot
Compares the sample and theoretical probabilities (percentiles).
qqplot : Quantile-Quantile plot
Compares the sample and theoretical quantiles
probplot : Probability plot
Same as a Q-Q plot, however probabilities are shown in the scale of
the theoretical distribution (x-axis) and the y-axis contains
unscaled quantiles of the sample data.
Examples
--------
>>> import statsmodels.api as sm
>>> from matplotlib import pyplot as plt
>>> # example 1
>>> data = sm.datasets.longley.load()
>>> data.exog = sm.add_constant(data.exog)
>>> model = sm.OLS(data.endog, data.exog)
>>> mod_fit = model.fit()
>>> res = mod_fit.resid # residuals
>>> probplot = sm.ProbPlot(res)
>>> probplot.qqplot()
>>> plt.show()
qqplot of the residuals against quantiles of t-distribution with 4
degrees of freedom:
>>> # example 2
>>> import scipy.stats as stats
>>> probplot = sm.ProbPlot(res, stats.t, distargs=(4,))
>>> fig = probplot.qqplot()
>>> plt.show()
qqplot against same as above, but with mean 3 and std 10:
>>> # example 3
>>> probplot = sm.ProbPlot(res, stats.t, distargs=(4,), loc=3, scale=10)
>>> fig = probplot.qqplot()
>>> plt.show()
Automatically determine parameters for t distribution including the
loc and scale:
>>> # example 4
>>> probplot = sm.ProbPlot(res, stats.t, fit=True)
>>> fig = probplot.qqplot(line='45')
>>> plt.show()
A second `ProbPlot` object can be used to compare two seperate sample
sets by using the `other` kwarg in the `qqplot` and `ppplot` methods.
>>> # example 5
>>> import numpy as np
>>> x = np.random.normal(loc=8.25, scale=2.75, size=37)
>>> y = np.random.normal(loc=8.75, scale=3.25, size=37)
>>> pp_x = sm.ProbPlot(x, fit=True)
>>> pp_y = sm.ProbPlot(y, fit=True)
>>> fig = pp_x.qqplot(line='45', other=pp_y)
>>> plt.show()
The following plot displays some options, follow the link to see the
code.
.. plot:: plots/graphics_gofplots_qqplot.py
"""
[docs] def __init__(self, data, dist=stats.norm, fit=False,
distargs=(), a=0, loc=0, scale=1):
self.data = data
self.a = a
self.nobs = data.shape[0]
self.distargs = distargs
self.fit = fit
if isinstance(dist, string_types):
dist = getattr(stats, dist)
self.fit_params = dist.fit(data)
if fit:
self.loc = self.fit_params[-2]
self.scale = self.fit_params[-1]
if len(self.fit_params) > 2:
self.dist = dist(*self.fit_params[:-2],
**dict(loc = 0, scale = 1))
else:
self.dist = dist(loc=0, scale=1)
elif distargs or loc == 0 or scale == 1:
self.dist = dist(*distargs, **dict(loc=loc, scale=scale))
self.loc = loc
self.scale = scale
else:
self.dist = dist
self.loc = loc
self.scale = scale
# propertes
self._cache = resettable_cache()
@cache_readonly
[docs] def theoretical_percentiles(self):
return plotting_pos(self.nobs, self.a)
@cache_readonly
[docs] def theoretical_quantiles(self):
try:
return self.dist.ppf(self.theoretical_percentiles)
except TypeError:
msg = '%s requires more parameters to ' \
'compute ppf'.format(self.dist.name,)
raise TypeError(msg)
except:
msg = 'failed to compute the ppf of {0}'.format(self.dist.name,)
raise
@cache_readonly
[docs] def sorted_data(self):
sorted_data = np.array(self.data, copy=True)
sorted_data.sort()
return sorted_data
@cache_readonly
[docs] def sample_quantiles(self):
if self.fit and self.loc != 0 and self.scale != 1:
return (self.sorted_data-self.loc)/self.scale
else:
return self.sorted_data
@cache_readonly
[docs] def sample_percentiles(self):
quantiles = \
(self.sorted_data - self.fit_params[-2])/self.fit_params[-1]
return self.dist.cdf(quantiles)
[docs] def ppplot(self, xlabel=None, ylabel=None, line=None, other=None,
ax=None, **plotkwargs):
"""
P-P plot of the percentiles (probabilities) of x versus the
probabilities (percetiles) of a distribution.
Parameters
----------
xlabel, ylabel : str or None, optional
User-provided lables for the x-axis and y-axis. If None (default),
other values are used depending on the status of the kwarg `other`.
line : str {'45', 's', 'r', q'} or None, optional
Options for the reference line to which the data is compared:
- '45' - 45-degree line
- 's' - standardized line, the expected order statistics are scaled
by the standard deviation of the given sample and have the mean
added to them
- 'r' - A regression line is fit
- 'q' - A line is fit through the quartiles.
- None - by default no reference line is added to the plot.
other : `ProbPlot` instance, array-like, or None, optional
If provided, the sample quantiles of this `ProbPlot` instance are
plotted against the sample quantiles of the `other` `ProbPlot`
instance. If an array-like object is provided, it will be turned
into a `ProbPlot` instance using default parameters. If not provided
(default), the theoretical quantiles are used.
ax : Matplotlib AxesSubplot instance, optional
If given, this subplot is used to plot in instead of a new figure
being created.
**plotkwargs : additional matplotlib arguments to be passed to the
`plot` command.
Returns
-------
fig : Matplotlib figure instance
If `ax` is None, the created figure. Otherwise the figure to which
`ax` is connected.
"""
if other is not None:
check_other = isinstance(other, ProbPlot)
if not check_other:
other = ProbPlot(other)
fig, ax = _do_plot(other.sample_percentiles,
self.sample_percentiles,
self.dist, ax=ax, line=line,
**plotkwargs)
if xlabel is None:
xlabel = 'Probabilities of 2nd Sample'
if ylabel is None:
ylabel = 'Probabilities of 1st Sample'
else:
fig, ax = _do_plot(self.theoretical_percentiles,
self.sample_percentiles,
self.dist, ax=ax, line=line,
**plotkwargs)
if xlabel is None:
xlabel = "Theoretical Probabilities"
if ylabel is None:
ylabel = "Sample Probabilities"
ax.set_xlabel(xlabel)
ax.set_ylabel(ylabel)
ax.set_xlim([0.0, 1.0])
ax.set_ylim([0.0, 1.0])
return fig
[docs] def qqplot(self, xlabel=None, ylabel=None, line=None, other=None,
ax=None, **plotkwargs):
"""
Q-Q plot of the quantiles of x versus the quantiles/ppf of a
distribution or the quantiles of another `ProbPlot` instance.
Parameters
----------
xlabel, ylabel : str or None, optional
User-provided lables for the x-axis and y-axis. If None (default),
other values are used depending on the status of the kwarg `other`.
line : str {'45', 's', 'r', q'} or None, optional
Options for the reference line to which the data is compared:
- '45' - 45-degree line
- 's' - standardized line, the expected order statistics are scaled
by the standard deviation of the given sample and have the mean
added to them
- 'r' - A regression line is fit
- 'q' - A line is fit through the quartiles.
- None - by default no reference line is added to the plot.
other : `ProbPlot` instance, array-like, or None, optional
If provided, the sample quantiles of this `ProbPlot` instance are
plotted against the sample quantiles of the `other` `ProbPlot`
instance. If an array-like object is provided, it will be turned
into a `ProbPlot` instance using default parameters. If not
provided (default), the theoretical quantiles are used.
ax : Matplotlib AxesSubplot instance, optional
If given, this subplot is used to plot in instead of a new figure
being created.
**plotkwargs : additional matplotlib arguments to be passed to the
`plot` command.
Returns
-------
fig : Matplotlib figure instance
If `ax` is None, the created figure. Otherwise the figure to which
`ax` is connected.
"""
if other is not None:
check_other = isinstance(other, ProbPlot)
if not check_other:
other = ProbPlot(other)
fig, ax = _do_plot(other.sample_quantiles,
self.sample_quantiles,
self.dist, ax=ax, line=line,
**plotkwargs)
if xlabel is None:
xlabel = 'Quantiles of 2nd Sample'
if ylabel is None:
ylabel = 'Quantiles of 1st Sample'
else:
fig, ax = _do_plot(self.theoretical_quantiles,
self.sample_quantiles,
self.dist, ax=ax, line=line,
**plotkwargs)
if xlabel is None:
xlabel = "Theoretical Quantiles"
if ylabel is None:
ylabel = "Sample Quantiles"
ax.set_xlabel(xlabel)
ax.set_ylabel(ylabel)
return fig
[docs] def probplot(self, xlabel=None, ylabel=None, line=None,
exceed=False, ax=None, **plotkwargs):
"""
Probability plot of the unscaled quantiles of x versus the
probabilities of a distibution (not to be confused with a P-P plot).
The x-axis is scaled linearly with the quantiles, but the probabilities
are used to label the axis.
Parameters
----------
xlabel, ylabel : str or None, optional
User-provided lables for the x-axis and y-axis. If None (default),
other values are used depending on the status of the kwarg `other`.
line : str {'45', 's', 'r', q'} or None, optional
Options for the reference line to which the data is compared:
- '45' - 45-degree line
- 's' - standardized line, the expected order statistics are scaled
by the standard deviation of the given sample and have the mean
added to them
- 'r' - A regression line is fit
- 'q' - A line is fit through the quartiles.
- None - by default no reference line is added to the plot.
exceed : boolean, optional
- If False (default) the raw sample quantiles are plotted against
the theoretical quantiles, show the probability that a sample
will not exceed a given value
- If True, the theoretical quantiles are flipped such that the
figure displays the probability that a sample will exceed a
given value.
ax : Matplotlib AxesSubplot instance, optional
If given, this subplot is used to plot in instead of a new figure
being created.
**plotkwargs : additional matplotlib arguments to be passed to the
`plot` command.
Returns
-------
fig : Matplotlib figure instance
If `ax` is None, the created figure. Otherwise the figure to which
`ax` is connected.
"""
if exceed:
fig, ax = _do_plot(self.theoretical_quantiles[::-1],
self.sorted_data,
self.dist, ax=ax, line=line,
**plotkwargs)
if xlabel is None:
xlabel = 'Probability of Exceedance (%)'
else:
fig, ax = _do_plot(self.theoretical_quantiles,
self.sorted_data,
self.dist, ax=ax, line=line,
**plotkwargs)
if xlabel is None:
xlabel = 'Non-exceedance Probability (%)'
if ylabel is None:
ylabel = "Sample Quantiles"
ax.set_xlabel(xlabel)
ax.set_ylabel(ylabel)
_fmt_probplot_axis(ax, self.dist, self.nobs)
return fig
[docs]def qqplot(data, dist=stats.norm, distargs=(), a=0, loc=0, scale=1, fit=False,
line=None, ax=None):
"""
Q-Q plot of the quantiles of x versus the quantiles/ppf of a distribution.
Can take arguments specifying the parameters for dist or fit them
automatically. (See fit under Parameters.)
Parameters
----------
data : array-like
1d data array
dist : A scipy.stats or statsmodels distribution
Compare x against dist. The default
is scipy.stats.distributions.norm (a standard normal).
distargs : tuple
A tuple of arguments passed to dist to specify it fully
so dist.ppf may be called.
loc : float
Location parameter for dist
a : float
Offset for the plotting position of an expected order statistic, for
example. The plotting positions are given by (i - a)/(nobs - 2*a + 1)
for i in range(0,nobs+1)
scale : float
Scale parameter for dist
fit : boolean
If fit is false, loc, scale, and distargs are passed to the
distribution. If fit is True then the parameters for dist
are fit automatically using dist.fit. The quantiles are formed
from the standardized data, after subtracting the fitted loc
and dividing by the fitted scale.
line : str {'45', 's', 'r', q'} or None
Options for the reference line to which the data is compared:
- '45' - 45-degree line
- 's' - standardized line, the expected order statistics are scaled
by the standard deviation of the given sample and have the mean
added to them
- 'r' - A regression line is fit
- 'q' - A line is fit through the quartiles.
- None - by default no reference line 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.
Returns
-------
fig : Matplotlib figure instance
If `ax` is None, the created figure. Otherwise the figure to which
`ax` is connected.
See Also
--------
scipy.stats.probplot
Examples
--------
>>> import statsmodels.api as sm
>>> from matplotlib import pyplot as plt
>>> data = sm.datasets.longley.load()
>>> data.exog = sm.add_constant(data.exog)
>>> mod_fit = sm.OLS(data.endog, data.exog).fit()
>>> res = mod_fit.resid # residuals
>>> fig = sm.qqplot(res)
>>> plt.show()
qqplot of the residuals against quantiles of t-distribution with 4 degrees
of freedom:
>>> import scipy.stats as stats
>>> fig = sm.qqplot(res, stats.t, distargs=(4,))
>>> plt.show()
qqplot against same as above, but with mean 3 and std 10:
>>> fig = sm.qqplot(res, stats.t, distargs=(4,), loc=3, scale=10)
>>> plt.show()
Automatically determine parameters for t distribution including the
loc and scale:
>>> fig = sm.qqplot(res, stats.t, fit=True, line='45')
>>> plt.show()
The following plot displays some options, follow the link to see the code.
.. plot:: plots/graphics_gofplots_qqplot.py
Notes
-----
Depends on matplotlib. If `fit` is True then the parameters are fit using
the distribution's fit() method.
"""
probplot = ProbPlot(data, dist=dist, distargs=distargs,
fit=fit, a=a, loc=loc, scale=scale)
fig = probplot.qqplot(ax=ax, line=line)
return fig
[docs]def qqplot_2samples(data1, data2, xlabel=None, ylabel=None, line=None, ax=None):
"""
Q-Q Plot of two samples' quantiles.
Can take either two `ProbPlot` instances or two array-like objects. In the
case of the latter, both inputs will be converted to `ProbPlot` instances
using only the default values - so use `ProbPlot` instances if
finer-grained control of the quantile computations is required.
Parameters
----------
data1, data2 : array-like (1d) or `ProbPlot` instances
xlabel, ylabel : str or None
User-provided labels for the x-axis and y-axis. If None (default),
other values are used.
line : str {'45', 's', 'r', q'} or None
Options for the reference line to which the data is compared:
- '45' - 45-degree line
- 's' - standardized line, the expected order statistics are scaled
by the standard deviation of the given sample and have the mean
added to them
- 'r' - A regression line is fit
- 'q' - A line is fit through the quartiles.
- None - by default no reference line 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.
Returns
-------
fig : Matplotlib figure instance
If `ax` is None, the created figure. Otherwise the figure to which
`ax` is connected.
See Also
--------
scipy.stats.probplot
Examples
--------
>>> x = np.random.normal(loc=8.5, scale=2.5, size=37)
>>> y = np.random.normal(loc=8.0, scale=3.0, size=37)
>>> pp_x = sm.ProbPlot(x)
>>> pp_y = sm.ProbPlot(y)
>>> qqplot_2samples(data1, data2, xlabel=None, ylabel=None, line=None, ax=None):
Notes
-----
1) Depends on matplotlib.
2) If `data1` and `data2` are not `ProbPlot` instances, instances will be
created using the default parameters. Therefore, it is recommended to use
`ProbPlot` instance if fine-grained control is needed in the computation
of the quantiles.
"""
check_data1 = isinstance(data1, ProbPlot)
check_data2 = isinstance(data2, ProbPlot)
if not check_data1 and not check_data2:
data1 = ProbPlot(data1)
data2 = ProbPlot(data2)
fig = data1.qqplot(xlabel=xlabel, ylabel=ylabel,
line=line, other=data2, ax=ax)
return fig
[docs]def qqline(ax, line, x=None, y=None, dist=None, fmt='r-'):
"""
Plot a reference line for a qqplot.
Parameters
----------
ax : matplotlib axes instance
The axes on which to plot the line
line : str {'45','r','s','q'}
Options for the reference line to which the data is compared.:
- '45' - 45-degree line
- 's' - standardized line, the expected order statistics are scaled by
the standard deviation of the given sample and have the mean
added to them
- 'r' - A regression line is fit
- 'q' - A line is fit through the quartiles.
- None - By default no reference line is added to the plot.
x : array
X data for plot. Not needed if line is '45'.
y : array
Y data for plot. Not needed if line is '45'.
dist : scipy.stats.distribution
A scipy.stats distribution, needed if line is 'q'.
Notes
-----
There is no return value. The line is plotted on the given `ax`.
"""
if line == '45':
end_pts = lzip(ax.get_xlim(), ax.get_ylim())
end_pts[0] = min(end_pts[0])
end_pts[1] = max(end_pts[1])
ax.plot(end_pts, end_pts, fmt)
ax.set_xlim(end_pts)
ax.set_ylim(end_pts)
return # does this have any side effects?
if x is None and y is None:
raise ValueError("If line is not 45, x and y cannot be None.")
elif line == 'r':
# could use ax.lines[0].get_xdata(), get_ydata(),
# but don't know axes are 'clean'
y = OLS(y, add_constant(x)).fit().fittedvalues
ax.plot(x,y,fmt)
elif line == 's':
m,b = y.std(), y.mean()
ref_line = x*m + b
ax.plot(x, ref_line, fmt)
elif line == 'q':
_check_for_ppf(dist)
q25 = stats.scoreatpercentile(y, 25)
q75 = stats.scoreatpercentile(y, 75)
theoretical_quartiles = dist.ppf([0.25, 0.75])
m = (q75 - q25) / np.diff(theoretical_quartiles)
b = q25 - m*theoretical_quartiles[0]
ax.plot(x, m*x + b, fmt)
#about 10x faster than plotting_position in sandbox and mstats
[docs]def plotting_pos(nobs, a):
"""
Generates sequence of plotting positions
Parameters
----------
nobs : int
Number of probability points to plot
a : float
Offset for the plotting position of an expected order statistic, for
example.
Returns
-------
plotting_positions : array
The plotting positions
Notes
-----
The plotting positions are given by (i - a)/(nobs - 2*a + 1) for i in
range(0,nobs+1)
See also
--------
scipy.stats.mstats.plotting_positions
"""
return (np.arange(1.,nobs+1) - a)/(nobs- 2*a + 1)
def _fmt_probplot_axis(ax, dist, nobs):
"""
Formats a theoretical quantile axis to display the corresponding
probabilities on the quantiles' scale.
Parameteters
------------
ax : Matplotlib AxesSubplot instance, optional
The axis to be formatted
nobs : scalar
Numbero of observations in the sample
dist : scipy.stats.distribution
A scipy.stats distribution sufficiently specified to impletment its
ppf() method.
Returns
-------
There is no return value. This operates on `ax` in place
"""
_check_for_ppf(dist)
if nobs < 50:
axis_probs = np.array([1,2,5,10,20,30,40,50,60,
70,80,90,95,98,99,])/100.0
elif nobs < 500:
axis_probs = np.array([0.1,0.2,0.5,1,2,5,10,20,30,40,50,60,70,
80,90,95,98,99,99.5,99.8,99.9])/100.0
else:
axis_probs = np.array([0.01,0.02,0.05,0.1,0.2,0.5,1,2,5,10,
20,30,40,50,60,70,80,90,95,98,99,99.5,
99.8,99.9,99.95,99.98,99.99])/100.0
axis_qntls = dist.ppf(axis_probs)
ax.set_xticks(axis_qntls)
ax.set_xticklabels(axis_probs*100, rotation=45,
rotation_mode='anchor',
horizontalalignment='right',
verticalalignment='center')
ax.set_xlim([axis_qntls.min(), axis_qntls.max()])
def _do_plot(x, y, dist=None, line=False, ax=None, fmt='bo', **kwargs):
"""
Boiler plate plotting function for the `ppplot`, `qqplot`, and
`probplot` methods of the `ProbPlot` class
Parameteters
------------
x, y : array-like
Data to be plotted
dist : scipy.stats.distribution
A scipy.stats distribution, needed if `line` is 'q'.
line : str {'45', 's', 'r', q'} or None
Options for the reference line to which the data is compared.
ax : Matplotlib AxesSubplot instance, optional
If given, this subplot is used to plot in instead of a new figure being
created.
fmt : str, optional
matplotlib-compatible formatting string for the data markers
kwargs : keywords
These are passed to matplotlib.plot
Returns
-------
fig : Matplotlib Figure instance
ax : Matplotlib AxesSubplot instance (see Parameters)
"""
fig, ax = utils.create_mpl_ax(ax)
ax.set_xmargin(0.02)
ax.plot(x, y, fmt, **kwargs)
if line:
if line not in ['r','q','45','s']:
msg = "%s option for line not understood" % line
raise ValueError(msg)
qqline(ax, line, x=x, y=y, dist=dist)
return fig, ax
def _check_for_ppf(dist):
if not hasattr(dist, 'ppf'):
raise ValueError("distribution must have a ppf method")