.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_data_analysis/sample_analysis/plot_linear_regression.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_data_analysis_sample_analysis_plot_linear_regression.py: Build and validate a linear model ================================= .. GENERATED FROM PYTHON SOURCE LINES 7-32 In this example, we build a linear regression model and validate it numerically and graphically. The linear model 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 :math:`\mathcal{N}(0.0, 1.0)`. 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 can also be validated 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 34-38 .. code-block:: Python import openturns as ot import openturns.viewer as otv .. GENERATED FROM PYTHON SOURCE LINES 39-40 Generate `X, Y` samples .. GENERATED FROM PYTHON SOURCE LINES 40-44 .. code-block:: Python 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 45-46 Generate a particular scalar sampleX .. GENERATED FROM PYTHON SOURCE LINES 46-48 .. code-block:: Python particularXSample = ot.Triangular(1.0, 5.0, 10.0).getSample(N) .. GENERATED FROM PYTHON SOURCE LINES 49-50 Create the linear model from `Y, X` samples .. GENERATED FROM PYTHON SOURCE LINES 50-62 .. code-block:: Python 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 .. code-block:: none coefficients of the linear regression model = [0.592409,2.98159] confidence intervals of the coefficients = [0.435545, 0.749274] [2.95382, 3.00935] .. GENERATED FROM PYTHON SOURCE LINES 63-64 Validate the model with a visual test .. GENERATED FROM PYTHON SOURCE LINES 64-67 .. code-block:: Python graph = ot.VisualTest.DrawLinearModel(Xsample, Ysample, result) view = otv.View(graph) .. image-sg:: /auto_data_analysis/sample_analysis/images/sphx_glr_plot_linear_regression_001.svg :alt: Linear model visual test :srcset: /auto_data_analysis/sample_analysis/images/sphx_glr_plot_linear_regression_001.svg :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 68-69 Draw the graph of the residual values .. GENERATED FROM PYTHON SOURCE LINES 69-72 .. code-block:: Python graph = ot.VisualTest.DrawLinearModelResidual(Xsample, Ysample, result) view = otv.View(graph) .. image-sg:: /auto_data_analysis/sample_analysis/images/sphx_glr_plot_linear_regression_002.svg :alt: residual(i) versus residual(i-1) :srcset: /auto_data_analysis/sample_analysis/images/sphx_glr_plot_linear_regression_002.svg :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 73-74 Check the nullity of the regression linear model coefficients .. GENERATED FROM PYTHON SOURCE LINES 74-81 .. code-block:: Python 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 .. code-block:: none Test Success ? False p-value of the LinearModelFisher Test = 0.0 p-value threshold = 0.1 .. GENERATED FROM PYTHON SOURCE LINES 82-83 Check, under the hypothesis of a Gaussian sample, if the mean of the residuals is equal to zero .. GENERATED FROM PYTHON SOURCE LINES 83-94 .. code-block:: Python 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()) .. rst-class:: sphx-glr-script-out .. code-block:: none Test Success ? True p-value of the LinearModelResidualMean Test = 0.9999999999995843 p-value threshold = 0.1 .. GENERATED FROM PYTHON SOURCE LINES 95-96 .. code-block:: Python otv.View.ShowAll() .. _sphx_glr_download_auto_data_analysis_sample_analysis_plot_linear_regression.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_linear_regression.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_linear_regression.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_linear_regression.zip `