Sample manipulation

This example will describe the main statistical functionalities on data through the Sample object. The Sample is an output variable of interest.

from __future__ import print_function
import openturns as ot
ot.Log.Show(ot.Log.NONE)

A typical example

A recurring issue in uncertainty quantification is to perform analysis on an output variable of interest Y obtained through a model f and input parameters X. Here we shall consider the input parameters as two independent standard normal distributions . We therefore use an IndependentCopula to describe the link between the two marginals.

# input parameters
inputDist = ot.ComposedDistribution([ot.Normal()] * 2, ot.IndependentCopula(2))
inputDist.setDescription(['X1', 'X2'])

We create a vector from the 2D-distribution created before :

inputVector = ot.RandomVector(inputDist)

Suppose our model f is known and reads as :

We define our model f with a SymbolicFunction

f = ot.SymbolicFunction(["x1", "x2"], ["x1^2+x2", "x2^2+x1"])

Our output vector is Y=f(X), the image of the inputVector by the model

outputVector = ot.CompositeRandomVector(f, inputVector)

We can now get a sample out of Y, that is realizations (here 1000) of the random outputVector

size = 1000
sample = outputVector.getSample(size)

The sample may be seen as a matrix of size . We print the 5 first samples (out of 1000) :

sample[:5]

	y0	y1
0	-0.5815072	0.7240122
1	3.26726	-0.5563772
2	-0.3683326	-0.08640049
3	-1.139952	1.854578
4	5.692328	-1.219674

Basic operations on samples

We have access to basic information about a sample such as

• minimum and maximum per component

sample.getMin(), sample.getMax()

Out:

(class=Point name=Unnamed dimension=2 values=[-2.56587,-2.84726], class=Point name=Unnamed dimension=2 values=[9.93535,12.1777])

• the range per component (max-min)

sample.computeRange()

[12.5012,15.025]

More elaborate functionalities are also available :

• get the median per component

sample.computeMedian()

[0.680688,0.874763]

• compute the covariance

sample.computeCovariance()

[[ 2.59234 -0.0758625 ]
[ -0.0758625 3.30636 ]]

• get the empirical 0.95 quantile per component

sample.computeQuantilePerComponent(0.95)

[3.67518,4.13131]

• get the value of the empirical CDF at a point

point = [1.1, 2.2]
sample.computeEmpiricalCDF(point)

Out:

0.518

Estimate the statistical moments

Oftentimes, we need to estimate the first moments of the output data. We can then estimate statistical moments from the output sample :

• estimate the moment of order 1 : mean

sample.computeMean()

[0.903872,1.15217]

• estimate the standard deviation (returns the Cholesky factor)

sample.computeStandardDeviation()

[[ 1.61007 0 ]
[ -0.0471174 1.81773 ]]

• estimate the standard deviation for each component

sample.computeStandardDeviationPerComponent()

[1.61007,1.81834]

• estimate the moment of order 2 : variance

sample.computeVariance()

[2.59234,3.30636]

• estimate the moment of order 3 : skewness

sample.computeSkewness()

[1.28241,1.80582]

• estimate the moment of order 4 : kurtosis

sample.computeKurtosis()

[6.40216,9.59074]

Test the correlation

Some statistical test for correlation are available :

• get the sample Pearson correlation matrix :

sample.computePearsonCorrelation()

[[ 1 -0.0259123 ]
[ -0.0259123 1 ]]

• get the sample Kendall correlation matrix :

sample.computeKendallTau()

[[ 1 0.0183584 ]
[ 0.0183584 1 ]]

• get the sample Spearman correlation matrix :

sample.computeSpearmanCorrelation()

[[ 1 0.0200394 ]
[ 0.0200394 1 ]]