Kolmogorov-Smirnov : understand the p-valueΒΆ

In this example, we illustrate the calculation of the Kolmogorov-Smirnov p-value.

  • We generate a sample from a gaussian distribution.

  • We create a Uniform distribution with known parameters.

  • The Kolmogorov-Smirnov statistics is computed and plot on the empirical cumulated distribution function.

  • We plot the p-value as the area under the part of the curve exceeding the observed statistics.

import openturns as ot
import openturns.viewer as viewer
from matplotlib import pylab as plt


We generate a sample from a standard gaussian distribution.

dist = ot.Normal()
samplesize = 10
sample = dist.getSample(samplesize)
testdistribution = ot.Normal()
result = ot.FittingTest.Kolmogorov(sample, testdistribution, 0.01)
pvalue = result.getPValue()
KSstat = result.getStatistic()

Compute exact Kolmogorov PDF.

Create a function which returns the CDF given the KS distance.

def pKolmogorovPy(x):
    y = ot.DistFunc.pKolmogorov(samplesize, x[0])
    return [y]
pKolmogorov = ot.PythonFunction(1, 1, pKolmogorovPy)

Create a function which returns the KS PDF given the KS distance: use the gradient method.

def kolmogorovPDF(x):
    return pKolmogorov.gradient(x)[0, 0]
def dKolmogorov(x):
    Compute Kolmogorov PDF for given x.
    x : a Sample, the points where the PDF must be evaluated
    Numerical Derivatives in Scilab, Michael Baudin, May 2009
    n = x.getSize()
    y = ot.Sample(n, 1)
    for i in range(n):
        y[i, 0] = kolmogorovPDF(x[i])
    return y
def linearSample(xmin, xmax, npoints):
    """Returns a sample created from a regular grid
    from xmin to xmax with npoints points."""
    step = (xmax - xmin) / (npoints - 1)
    rg = ot.RegularGrid(xmin, step, npoints)
    vertices = rg.getVertices()
    return vertices
n = 1000  # Number of points in the plot
s = linearSample(0.001, 0.999, n)
y = dKolmogorov(s)

Create a regular grid starting from the observed KS statistics.

nplot = 100
x = linearSample(KSstat, 0.6, nplot)

Compute the bounds to fill: the lower vertical bound is 0 and the upper vertical bound is the KS PDF.

vLow = [0.0] * nplot
vUp = [pKolmogorov.gradient(x[i])[0, 0] for i in range(nplot)]
boundsPoly = ot.Polygon.FillBetween(x.asPoint(), vLow, vUp)
boundsPoly.setLegend("pvalue = %.4f" % (pvalue))
curve = ot.Curve(s, y)
curve.setLegend("Exact distribution")
curveStat = ot.Curve([KSstat, KSstat], [0.0, kolmogorovPDF([KSstat])])
curveStat.setLegend("KS-statistics = %.4f" % (KSstat))
graph = ot.Graph(
    "Kolmogorov-Smirnov distribution (known parameters)",
    "upper right",
graph.setLegends(["Empirical distribution"])
graph.setTitle("Kolmogorov-Smirnov distribution (known parameters)")
view = viewer.View(graph)
Kolmogorov-Smirnov distribution (known parameters)

We observe that the p-value is the area of the curve which corresponds to the KS distances greater than the observed KS statistics.