Sample manipulation

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

import openturns as ot

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.JointDistribution([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 , 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	0.5519547	0.6413422
1	2.069226	-1.048988
2	-2.211924	5.340953
3	0.7773735	1.66219
4	4.325861	-1.994259

Basic operations on samples

We have access to basic information about a sample such as

• minimum and maximum per component

sample.getMin(), sample.getMax()

(class=Point name=Unnamed dimension=2 values=[-3.24513,-2.98342], class=Point name=Unnamed dimension=2 values=[10.6987,10.5037])

• the range per component (max-min)

sample.computeRange()

class=Point name=Unnamed dimension=2 values=[13.9438,13.4871]

More elaborate functionalities are also available :

• get the median per component

sample.computeMedian()

class=Point name=Unnamed dimension=2 values=[0.631033,0.715266]

• compute the covariance

sample.computeCovariance()

[[ 2.77895 0.0913307 ]
[ 0.0913307 3.30989 ]]

• get the empirical 0.95 quantile per component

sample.computeQuantilePerComponent(0.95)

class=Point name=Unnamed dimension=2 values=[3.96521,4.45879]

• get the value of the empirical CDF at a point

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

0.569

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()

class=Point name=Unnamed dimension=2 values=[0.928558,1.01799]

• estimate the standard deviation for each component

sample.computeStandardDeviation()

class=Point name=Unnamed dimension=2 values=[1.66702,1.81931]

• estimate the moment of order 2 : variance

sample.computeVariance()

class=Point name=Unnamed dimension=2 values=[2.77895,3.30989]

• estimate the moment of order 3 : skewness

sample.computeSkewness()

class=Point name=Unnamed dimension=2 values=[1.40965,1.73437]

• estimate the moment of order 4 : kurtosis

sample.computeKurtosis()

class=Point name=Unnamed dimension=2 values=[6.84373,7.96431]

Test the correlation

Some statistical test for correlation are available :

• get the sample linear correlation matrix :

sample.computeLinearCorrelation()

[[ 1 0.0301141 ]
[ 0.0301141 1 ]]

• get the sample Kendall correlation matrix :

sample.computeKendallTau()

[[ 1 -0.010026 ]
[ -0.010026 1 ]]

• get the sample Spearman correlation matrix :

sample.computeSpearmanCorrelation()

[[ 1 0.00483728 ]
[ 0.00483728 1 ]]