Note

Go to the end to download the full example code

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 X=(X_1, X_2). 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 :

f(X) = \begin{pmatrix} x_1^2 + x_2 \\ x_1 + x_2^2 \end{pmatrix}

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 1000 \times 2. We print the 5 first samples (out of 1000) :

sample[:5]
y0y1
01.0022281.122468
12.982256-1.643145
2-0.29186332.278239
3-0.38742310.009052058
41.351702-1.126908


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=[-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()

[0.68633,0.879481]



  • 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)

[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()

[0.903865,1.15424]



  • estimate the standard deviation for each component

sample.computeStandardDeviation()

[1.60001,1.81891]



  • estimate the moment of order 2 : variance

sample.computeVariance()

[2.56005,3.30845]



  • estimate the moment of order 3 : skewness

sample.computeSkewness()

[1.28143,1.80235]



  • estimate the moment of order 4 : kurtosis

sample.computeKurtosis()

[6.47685,9.56975]



Test the correlation

Some statistical test for correlation are available :

  • get the sample Pearson correlation matrix :

sample.computePearsonCorrelation()

[[ 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 ]]