import openturns as ot import openturns.viewer as viewer from matplotlib import pylab as plt ot.Log.Show(ot.Log.NONE)
Sample independence test¶
In this paragraph we perform tests to assess whether two 1-d samples are independent or not.
The following tests are available :
the ChiSquared test: it tests if both scalar samples (discrete ones only) are independent. If is the number of values of the sample in the modality , , and the ChiSquared test evaluates the decision variable:
which tends towards the distribution. The hypothesis of independence is rejected if is too high (depending on the p-value threshold).
the Pearson test: it tests if there exists a linear relation between two scalar samples which form a gaussian vector (which is equivalent to have a linear correlation coefficient not equal to zero). If both samples are and , and and , the Pearson test evaluates the decision variable:
The variable tends towards a , under the hypothesis of normality of both samples. The hypothesis of a linear coefficient equal to 0 is rejected (which is equivalent to the independence of the samples) if D is too high (depending on the p-value threshold).
the Spearman test: it tests if there exists a monotonous relation between two scalar samples. If both samples are and ,, the Spearman test evaluates the decision variable:
where and . is such that tends towards the standard normal distribution.
The continuous case¶
We create two different continuous samples :
sample1 = ot.Normal().getSample(100) sample2 = ot.Normal().getSample(100)
We first use the Pearson test and store the result :
resultPearson = ot.HypothesisTest.Pearson(sample1, sample2, 0.10)
We can then display the result of the test as a yes/no answer with the getBinaryQualityMeasure. We can retrieve the p-value and the threshold with the getPValue and getThreshold methods.
print('Component is normal?', resultPearson.getBinaryQualityMeasure(), 'p-value=%.6g' % resultPearson.getPValue(), 'threshold=%.6g' % resultPearson.getThreshold())
Component is normal? False p-value=0.0451584 threshold=0.1
We can also use the Spearman test :
resultSpearman = ot.HypothesisTest.Spearman(sample1, sample2, 0.10) print('Component is normal?', resultSpearman.getBinaryQualityMeasure(), 'p-value=%.6g' % resultSpearman.getPValue(), 'threshold=%.6g' % resultSpearman.getThreshold())
Component is normal? False p-value=0.0603411 threshold=0.1
The discrete case¶
Testing is also possible for discrete distribution. Let us create discrete two different samples :
sample1 = ot.Poisson(0.2).getSample(100) sample2 = ot.Poisson(0.2).getSample(100)
We use the Chi2 test to check independence and store the result :
resultChi2 = ot.HypothesisTest.ChiSquared(sample1, sample2, 0.10)
and display the results :
print('Component is normal?', resultChi2.getBinaryQualityMeasure(), 'p-value=%.6g' % resultChi2.getPValue(), 'threshold=%.6g' % resultChi2.getThreshold())
Component is normal? True p-value=0.20552 threshold=0.1
Test samples independence using regression¶
Independence testing with regression is also an option in OpenTURNS. It consists in detecting a linear relation between two scalar samples.
We generate a sample of dimension 3 with component 0 correlated to component 2 :
marginals = [ot.Normal()] * 3 S = ot.CorrelationMatrix(3) S[0, 2] = 0.9 copula = ot.NormalCopula(S) distribution = ot.ComposedDistribution(marginals, copula) sample = distribution.getSample(30)
Next, we split it in two samples : firstSample of dimension=2, secondSample of dimension=1.
firstSample = sample[:, :2] secondSample = sample[:, 2]
We test independence of each component of firstSample against the secondSample :
test_results = ot.LinearModelTest.FullRegression(firstSample, secondSample) for i in range(len(test_results)): print('Component', i, 'is independent?', test_results[i].getBinaryQualityMeasure(), 'p-value=%.6g' % test_results[i].getPValue(), 'threshold=%.6g' % test_results[i].getThreshold())
Component 0 is independent? True p-value=0.646138 threshold=0.05 Component 1 is independent? False p-value=1.30057e-10 threshold=0.05 Component 2 is independent? True p-value=0.342379 threshold=0.05
Total running time of the script: ( 0 minutes 0.006 seconds)