The scipy.stats.qqplot and scipy.stats.probplot functions plot expected values
versus actual data values for visualization of fit to a distribution. First a
one-D array of expected percentiles is generated for a sample of size N; then
that is passed to dist.ppf, the per cent point function for the chosen
distribution, to return an array of expected values. The visualized data
points are pairs of expected and actual values, and a linear regression is done
on these to produce the line data points in this distribution should lie on.
Where x is the input data array and dist the chosen distribution we have:
> osr = np.sort(x)
> osm_uniform = _calc_uniform_order_statistic_medians(len(x))
> osm = dist.ppf(osm_uniform)
> slope, intercept, r, prob, sterrest = stats.linregress(osm, osr)
My question concerns the plot display.
> ax.plot(osm, osr, 'bo', osm, slope*osm + intercept, 'r-')
The x-axis of the resulting plot is labeled quantiles, but the xticks and
xticklabels produced produced by qqplot and problplot do not seem correct for
the their intended interpretations. First the numbers on the x-axis do not
represent quantiles; the intervals between them do not in general contain equal
numbers of points. For a normal distribution with sigma=1, they represent
standard deviations. Changing the label on the x-axis does not seem like a
very good solution, because the interpretation of the values on the x-axis will
be different for different distributions. Rather the right solution seems to
be to actually show quantiles on the x-axis. The numbers on the x-axis can stay
as they are, representing quantile indexes, but they need to be spaced so as to
show the actual division points that carve the population up into groups of
the same size. This can be done in something like the following way.
> import numpy as np
> xt = np.arange(-3,3,dtype=int)
> # Find the 5 quantiles to divide the data into sixths
> percentiles = [x*.167 + .502 for x in xt]
> percentiles = np.array(percentiles + [.999])
> vals = dist.ppf(percentiles)
> ax.set_xticks(vals)
> xt = np.array(list(xt)+[3])
> ax.set_xticklabels(xt)
> ax.set_xlabel('Quantile')
> plt.show()
I’ve attached two images to show the difference between the current
visualization and the suggested one.
Mark Gawron
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