Estimate correlation coefficients

In this example we are going to estimate the correlation between an output sample Y and the corresponding inputs using various estimators:

• Pearson coefficients

• Spearman coefficients

• PCC: Partial Correlation Coefficients

• PRCC: Partial Rank Correlation Coefficient

• SRC: Standard Regression Coefficients

• SRRC: Standard Rank Regression Coefficient

from openturns.usecases import ishigami_function
import openturns as ot
import openturns.viewer as viewer
from matplotlib import pylab as plt

ot.Log.Show(ot.Log.NONE)

To illustrate the usage of the method mentioned above, we define a set of X and Y data using the Ishigami model. This classical model is defined in a data class:

im = ishigami_function.IshigamiModel()

Create X/Y data We get the input variables description :

input_names = im.inputDistribution.getDescription()

size = 100
inputDesign = ot.SobolIndicesExperiment(im.inputDistribution, size, True).generate()
outputDesign = im.model(inputDesign)

Create a CorrelationAnalysis object to compute various estimates of the correlation between the inputs and the output.

corr_analysis = ot.CorrelationAnalysis(inputDesign, outputDesign)

PCC coefficients

pcc_indices = corr_analysis.computePCC()
print(pcc_indices)

[0.457032,0.00109669,-0.0520883]

graph = ot.SobolIndicesAlgorithm.DrawCorrelationCoefficients(
    pcc_indices, input_names, "PCC coefficients"
)
view = viewer.View(graph)

PRCC coefficients

prcc_indices = corr_analysis.computePRCC()
print(prcc_indices)

[0.468439,-0.0114603,-0.0337681]

graph = ot.SobolIndicesAlgorithm.DrawCorrelationCoefficients(
    prcc_indices, input_names, "PRCC coefficients"
)
view = viewer.View(graph)

SRC coefficients

src_indices = corr_analysis.computeSRC()
print(src_indices)

[0.456421,0.000974935,-0.0463861]

graph = ot.SobolIndicesAlgorithm.DrawCorrelationCoefficients(
    src_indices, input_names, "SRC coefficients"
)
view = viewer.View(graph)

Normalized squared SRC coefficients (coefficients are made to sum to 1) :

squared_src_indices = corr_analysis.computeSquaredSRC(True)
print(squared_src_indices)

[0.989772,4.51602e-06,0.010223]

graph = ot.SobolIndicesAlgorithm.DrawCorrelationCoefficients(
    squared_src_indices, input_names, "Squared SRC coefficients"
)
view = viewer.View(graph)

SRRC coefficients

srrc_indices = corr_analysis.computeSRRC()
print(srrc_indices)

[0.468235,-0.0101317,-0.0298819]

graph = ot.SobolIndicesAlgorithm.DrawCorrelationCoefficients(
    srrc_indices, input_names, "SRRC coefficients"
)
view = viewer.View(graph)

Pearson coefficients

We compute here the Pearson coefficients.

pearson_correlation = corr_analysis.computeLinearCorrelation()
print(pearson_correlation)

[0.457802,0.00787278,-0.059923]

graph = ot.SobolIndicesAlgorithm.DrawCorrelationCoefficients(
    pearson_correlation, input_names, "Pearson correlation coefficients"
)
view = viewer.View(graph)

Spearman coefficients

We compute here the Spearman coefficients.

spearman_correlation = corr_analysis.computeSpearmanCorrelation()
print(spearman_correlation)

[0.469077,-0.0030321,-0.044356]

graph = ot.SobolIndicesAlgorithm.DrawCorrelationCoefficients(
    spearman_correlation, input_names, "Spearman correlation coefficients"
)
view = viewer.View(graph)
plt.show()