.. only:: html .. note:: :class: sphx-glr-download-link-note Click :ref:`here ` to download the full example code .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_data_analysis_manage_data_and_samples_plot_linear_regression.py: Build and validate a linear model ================================= In this example we are going to build a linear regression model and validate it numerically and graphically. The linear model between links a scalar variable :math:`Y` and to an n-dimensional one :math:`\underline{X} = (X_i)_{i \leq n}`, as follows: .. math:: \tilde{Y} = a_0 + \sum_{i=1}^n a_i X_i + \varepsilon where :math:`\varepsilon` is the residual, supposed to follow the Normal(0.0, 1.0) distribution. The linear model may be validated graphically if :math:`\underline{X}` is of dimension 1, by drawing on the same graph the cloud :math:`(X_i, Y_i)`. The linear model also be validate numerically with several tests: - LinearModelFisher: tests the nullity of the regression linear model coefficients (Fisher distribution used), - LinearModelResidualMean: tests, under the hypothesis of a gaussian sample, if the mean of the residual is equal to zero. It is based on the Student test (equality of mean for two gaussian samples). The hypothesis on the residuals (centered gaussian distribution) may be validated: - graphically if :math:`\underline{X}` is of dimension 1, by drawing the residual couples (:math:`\varepsilon_i, \varepsilon_{i+1}`), where the residual :math:`\varepsilon_i` is evaluated on the samples :math:`(X, Y)`. - numerically with the LinearModelResidualMean Test which tests, under the hypothesis of a gaussian sample, if the mean of the residual is equal to zero. It is based on the Student test (equality of mean for two gaussian samples). .. code-block:: default from __future__ import print_function import openturns as ot import openturns.viewer as viewer from matplotlib import pylab as plt ot.Log.Show(ot.Log.NONE) Generate X,Y samples .. code-block:: default N = 1000 Xsample = ot.Triangular(1.0, 5.0, 10.0).getSample(N) Ysample = Xsample * 3.0 + ot.Normal(0.5, 1.0).getSample(N) Generate a particular scalar sampleX .. code-block:: default particularXSample = ot.Triangular(1.0, 5.0, 10.0).getSample(N) Create the linear model from Y,X samples .. code-block:: default result = ot.LinearModelAlgorithm(Xsample, Ysample).getResult() # Get the coefficients ai print("coefficients of the linear regression model = ", result.getCoefficients()) # Get the confidence intervals of the ai coefficients print("confidence intervals of the coefficients = ", ot.LinearModelAnalysis(result).getCoefficientsConfidenceInterval(0.9)) .. rst-class:: sphx-glr-script-out Out: .. code-block:: none coefficients of the linear regression model = [0.620986,2.98488] confidence intervals of the coefficients = [0.464408, 0.777565] [2.95727, 3.0125] Validate the model with a visual test .. code-block:: default graph = ot.VisualTest.DrawLinearModel(Xsample, Ysample, result) view = viewer.View(graph) .. image:: /auto_data_analysis/manage_data_and_samples/images/sphx_glr_plot_linear_regression_001.png :alt: Linear model visual test :class: sphx-glr-single-img Draw the graph of the residual values .. code-block:: default graph = ot.VisualTest.DrawLinearModelResidual(Xsample, Ysample, result) view = viewer.View(graph) .. image:: /auto_data_analysis/manage_data_and_samples/images/sphx_glr_plot_linear_regression_002.png :alt: residual(i) versus residual(i-1) :class: sphx-glr-single-img Check the nullity of the regression linear model coefficients .. code-block:: default resultLinearModelFisher = ot.LinearModelTest.LinearModelFisher(Xsample, Ysample, result, 0.10) print("Test Success ? ", resultLinearModelFisher.getBinaryQualityMeasure()) print("p-value of the LinearModelFisher Test = ", resultLinearModelFisher.getPValue()) print("p-value threshold = ", resultLinearModelFisher.getThreshold()) .. rst-class:: sphx-glr-script-out Out: .. code-block:: none Test Success ? False p-value of the LinearModelFisher Test = 0.0 p-value threshold = 0.1 Check, under the hypothesis of a gaussian sample, if the mean of the residual is equal to zero .. code-block:: default resultLinearModelResidualMean = ot.LinearModelTest.LinearModelResidualMean(Xsample, Ysample, result, 0.10) print("Test Success ? ", resultLinearModelResidualMean.getBinaryQualityMeasure()) print("p-value of the LinearModelResidualMean Test = ", resultLinearModelResidualMean.getPValue()) print("p-value threshold = ", resultLinearModelResidualMean.getThreshold()) plt.show() .. rst-class:: sphx-glr-script-out Out: .. code-block:: none Test Success ? True p-value of the LinearModelResidualMean Test = 0.9999999999998087 p-value threshold = 0.1 .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 0.162 seconds) .. _sphx_glr_download_auto_data_analysis_manage_data_and_samples_plot_linear_regression.py: .. only :: html .. container:: sphx-glr-footer :class: sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_linear_regression.py ` .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_linear_regression.ipynb ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_