# The Kolmogorov-Smirnov p-value¶

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


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)
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):

:

def dKolmogorov(x,samplesize):
"""
Compute Kolmogorov PDF for given x.
x : a Sample, the points where the PDF must be evaluated
samplesize : the size of the sample
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,samplesize)

:

def drawInTheBounds(vLow,vUp,n_test):
'''
Draw the area within the bounds.
'''
palette = ot.Drawable.BuildDefaultPalette(2)
myPaletteColor = palette
polyData = [[vLow[i], vLow[i+1], vUp[i+1], vUp[i]] for i in range(n_test-1)]
polygonList = [ot.Polygon(polyData[i], myPaletteColor, myPaletteColor) for i in range(n_test-1)]
boundsPoly = ot.PolygonArray(polygonList)
return boundsPoly


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 zero and the upper vertical bound is the KS PDF.

:

vLow = [[x[i,0],0.] for i in range(nplot)]
vUp = [[x[i,0],pKolmogorov.gradient(x[i])[0,0]] for i in range(nplot)]

:

boundsPoly = drawInTheBounds(vLow,vUp,nplot)
boundsPoly.setLegend("pvalue = %.4f" % (pvalue))
curve = ot.Curve(s,y)
curve.setLegend("Exact distribution")
curveStat = ot.Curve([KSstat,KSstat],[0.,kolmogorovPDF([KSstat])])
curveStat.setColor("red")
curveStat.setLegend("KS-statistics = %.4f" % (KSstat))
graph = ot.Graph('Kolmogorov-Smirnov distribution (known parameters)', 'KS-Statistics', 'PDF', True, 'topright')
graph.setLegends(["Empirical distribution"])

: 