# 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

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.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 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 1.002228 1.122468 2.982256 -1.643145 -0.2918633 2.278239 -0.3874231 0.009052058 1.351702 -1.126908

## Basic operations on samples¶

• minimum and maximum per component

sample.getMin(), sample.getMax()

(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()

class=Point name=Unnamed dimension=2 values=[12.5012,15.025]

More elaborate functionalities are also available :

• get the median per component

sample.computeMedian()

class=Point name=Unnamed dimension=2 values=[0.68633,0.879481]

• compute the covariance

sample.computeCovariance()


[[ 2.56005 -0.0561621 ]
[ -0.0561621 3.30845 ]]

• get the empirical 0.95 quantile per component

sample.computeQuantilePerComponent(0.95)

class=Point name=Unnamed dimension=2 values=[3.63824,4.13131]

• get the value of the empirical CDF at a point

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

0.517


## 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=[0.903865,1.15424]

• estimate the standard deviation for each component

sample.computeStandardDeviation()

class=Point name=Unnamed dimension=2 values=[1.60001,1.81891]

• estimate the moment of order 2 : variance

sample.computeVariance()

class=Point name=Unnamed dimension=2 values=[2.56005,3.30845]

• estimate the moment of order 3 : skewness

sample.computeSkewness()

class=Point name=Unnamed dimension=2 values=[1.28143,1.80235]

• estimate the moment of order 4 : kurtosis

sample.computeKurtosis()

class=Point name=Unnamed dimension=2 values=[6.47685,9.56975]

## Test the correlation¶

Some statistical test for correlation are available :

• get the sample linear correlation matrix :

sample.computeLinearCorrelation()


[[ 1 -0.0192978 ]
[ -0.0192978 1 ]]

• get the sample Kendall correlation matrix :

sample.computeKendallTau()


[[ 1 0.0250531 ]
[ 0.0250531 1 ]]

• get the sample Spearman correlation matrix :

sample.computeSpearmanCorrelation()


[[ 1 0.0291728 ]
[ 0.0291728 1 ]]