.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_data_analysis/manage_data_and_samples/plot_linear_regression.py" .. LINE NUMBERS ARE GIVEN BELOW. .. 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 ================================= .. GENERATED FROM PYTHON SOURCE LINES 6-28 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). .. GENERATED FROM PYTHON SOURCE LINES 30-35 .. code-block:: default import openturns as ot import openturns.viewer as viewer from matplotlib import pylab as plt ot.Log.Show(ot.Log.NONE) .. GENERATED FROM PYTHON SOURCE LINES 36-37 Generate X,Y samples .. GENERATED FROM PYTHON SOURCE LINES 37-41 .. 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) .. GENERATED FROM PYTHON SOURCE LINES 42-43 Generate a particular scalar sampleX .. GENERATED FROM PYTHON SOURCE LINES 43-45 .. code-block:: default particularXSample = ot.Triangular(1.0, 5.0, 10.0).getSample(N) .. GENERATED FROM PYTHON SOURCE LINES 46-47 Create the linear model from Y,X samples .. GENERATED FROM PYTHON SOURCE LINES 47-57 .. 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] .. GENERATED FROM PYTHON SOURCE LINES 58-59 Validate the model with a visual test .. GENERATED FROM PYTHON SOURCE LINES 59-62 .. code-block:: default graph = ot.VisualTest.DrawLinearModel(Xsample, Ysample, result) view = viewer.View(graph) .. image-sg:: /auto_data_analysis/manage_data_and_samples/images/sphx_glr_plot_linear_regression_001.png :alt: Linear model visual test :srcset: /auto_data_analysis/manage_data_and_samples/images/sphx_glr_plot_linear_regression_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 63-64 Draw the graph of the residual values .. GENERATED FROM PYTHON SOURCE LINES 64-67 .. code-block:: default graph = ot.VisualTest.DrawLinearModelResidual(Xsample, Ysample, result) view = viewer.View(graph) .. image-sg:: /auto_data_analysis/manage_data_and_samples/images/sphx_glr_plot_linear_regression_002.png :alt: residual(i) versus residual(i-1) :srcset: /auto_data_analysis/manage_data_and_samples/images/sphx_glr_plot_linear_regression_002.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 68-69 Check the nullity of the regression linear model coefficients .. GENERATED FROM PYTHON SOURCE LINES 69-76 .. 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 .. GENERATED FROM PYTHON SOURCE LINES 77-78 Check, under the hypothesis of a gaussian sample, if the mean of the residual is equal to zero .. GENERATED FROM PYTHON SOURCE LINES 78-85 .. 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 = 0.9999999999997742 p-value threshold = 0.1 .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 0.136 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 `_