# Kolmogorov-Smirnov : understand the 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
import openturns.viewer as viewer
from matplotlib import pylab as plt
ot.Log.Show(ot.Log.NONE)
```

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
```

Out:

```0.39539868588509486
```
```KSstat = result.getStatistic()
KSstat
```

Out:

```0.2685817728820096
```

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"]) 