Kolmogorov-Smirnov : understand the p-value

In this example, we illustrate the calculation of the Kolmogorov-Smirnov (KS) 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 pyplot 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()
pvalue

0.5520956737074482

KSstat = result.getStatistic()
KSstat

0.23684644362352725

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
    Reference
    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.setColor("red")
curveStat.setLegend("KS-statistics = %.4f" % (KSstat))
graph = ot.Graph(
    "Kolmogorov-Smirnov distribution (known parameters)",
    "KS-Statistics",
    "PDF",
    True,
    "upper right",
)
graph.setLegends(["Empirical distribution"])
graph.add(curve)
graph.add(curveStat)
graph.add(boundsPoly)
graph.setTitle("Kolmogorov-Smirnov distribution (known parameters)")
view = viewer.View(graph)
plt.show()

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