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'] 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 """ 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 def theoretical_percentiles(self): return plotting_pos(self.nobs, self.a) @cache_readonly 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 def sorted_data(self): sorted_data = np.array(self.data, copy=True) sorted_data.sort() return sorted_data @cache_readonly 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 def sample_percentiles(self): quantiles = \ (self.sorted_data - self.fit_params[-2])/self.fit_params[-1] return self.dist.cdf(quantiles) 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 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 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 def qqplot(data, dist=stats.norm, distargs=(), a=0, loc=0, scale=1, fit=False, line=None, ax=None, **plotkwargs): """ 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. **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. 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, **plotkwargs) return fig 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(pp_x, pp_y) 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 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 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")