.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_meta_modeling/general_purpose_metamodels/plot_overfitting_model_selection.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_meta_modeling_general_purpose_metamodels_plot_overfitting_model_selection.py: Over-fitting and model selection ================================ .. GENERATED FROM PYTHON SOURCE LINES 8-21 Introduction ------------ In this notebook, we present the problem of over-fitting a model to data. We consider noisy observations of the sine function. We estimate the coefficients of the univariate polynomial based on linear least squares and show that, when the degree of the polynomial becomes too large, the overall prediction quality decreases. This shows why and how model selection can come into play in order to select the degree of the polynomial: there is is a trade-off between fitting the data and preserving the quality of future predictions. In this example, we use cross validation as a model selection method. .. GENERATED FROM PYTHON SOURCE LINES 23-28 References ---------- * Bishop Christopher M., 1995, Neural networks for pattern recognition. Figure 1.4, page 7 .. GENERATED FROM PYTHON SOURCE LINES 30-34 Compute the data ---------------- In this section, we generate noisy observations from the sine function. .. GENERATED FROM PYTHON SOURCE LINES 36-40 .. code-block:: Python import openturns as ot import pylab as pl import openturns.viewer as otv .. GENERATED FROM PYTHON SOURCE LINES 41-43 .. code-block:: Python ot.RandomGenerator.SetSeed(0) .. GENERATED FROM PYTHON SOURCE LINES 44-45 We define the function that we are going to approximate. .. GENERATED FROM PYTHON SOURCE LINES 47-49 .. code-block:: Python g = ot.SymbolicFunction(["x"], ["sin(2*pi_*x)"]) .. GENERATED FROM PYTHON SOURCE LINES 50-59 .. code-block:: Python graph = ot.Graph("Polynomial curve fitting", "x", "y", True, "upper right") # The "unknown" function curve = g.draw(0, 1) curve.setColors(["green"]) curve.setLegends(['"Unknown" function']) graph.add(curve) view = otv.View(graph) .. image-sg:: /auto_meta_modeling/general_purpose_metamodels/images/sphx_glr_plot_overfitting_model_selection_001.png :alt: Polynomial curve fitting :srcset: /auto_meta_modeling/general_purpose_metamodels/images/sphx_glr_plot_overfitting_model_selection_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 60-61 This seems a nice, smooth function to approximate with polynomials. .. GENERATED FROM PYTHON SOURCE LINES 64-73 .. code-block:: Python def linearSample(xmin, xmax, npoints): """Returns a sample created from a regular grid from xmin to xmax with npoints points.""" step = (xmax - xmin) / (npoints - 1) rg = ot.RegularGrid(xmin, step, npoints) vertices = rg.getVertices() return vertices .. GENERATED FROM PYTHON SOURCE LINES 74-75 We consider 10 observation points in the interval [0,1]. .. GENERATED FROM PYTHON SOURCE LINES 77-80 .. code-block:: Python n_train = 10 x_train = linearSample(0, 1, n_train) .. GENERATED FROM PYTHON SOURCE LINES 81-82 Assume that the observations are noisy and that the noise follows a Normal distribution with zero mean and small standard deviation. .. GENERATED FROM PYTHON SOURCE LINES 84-87 .. code-block:: Python noise = ot.Normal(0, 0.1) noiseSample = noise.getSample(n_train) .. GENERATED FROM PYTHON SOURCE LINES 88-90 The following computes the observation as the sum of the function value and of the noise. The couple (`x_train`,`y_train`) is the training set: it is used to compute the coefficients of the polynomial model. .. GENERATED FROM PYTHON SOURCE LINES 92-94 .. code-block:: Python y_train = g(x_train) + noiseSample .. GENERATED FROM PYTHON SOURCE LINES 95-107 .. code-block:: Python graph = ot.Graph("Polynomial curve fitting", "x", "y", True, "upper right") # The "unknown" function curve = g.draw(0, 1) curve.setColors(["green"]) graph.add(curve) # Training set cloud = ot.Cloud(x_train, y_train) cloud.setPointStyle("circle") cloud.setLegend("Observations") graph.add(cloud) view = otv.View(graph) .. image-sg:: /auto_meta_modeling/general_purpose_metamodels/images/sphx_glr_plot_overfitting_model_selection_002.png :alt: Polynomial curve fitting :srcset: /auto_meta_modeling/general_purpose_metamodels/images/sphx_glr_plot_overfitting_model_selection_002.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 108-133 Compute the coefficients of the polynomial decomposition -------------------------------------------------------- Let :math:`y \in \mathbb{R}^n` be a vector of observations. The polynomial model is .. math:: P(x) = \beta_0 + \beta_1 x + ... + \beta_p x^p, for any :math:`x\in\mathbb{R}`, where :math:`p` is the polynomial degree and :math:`\beta\in\mathbb{R}^{p+1}` is the vector of the coefficients of the model. Let :math:`n` be the training sample size and let :math:`x_1,...,x_n \in \mathbb{R}` be the abscissas of the training set. The design matrix :math:`X \in \mathbb{R}^{n \times (p+1)}` is .. math:: x_{i,j} = x^j_i, for :math:`i=1,...,n` and :math:`j=0,...,p`. The least squares solution is: .. math:: \beta^\star = \textrm{argmin}_{\beta \in \mathbb{R}^{p+1}} \| X\beta - y\|_2^2. .. GENERATED FROM PYTHON SOURCE LINES 135-140 In order to approximate the function with polynomials up to degree 4, we create a list of strings containing the associated monomials. We do not include a constant in the polynomial basis, as this constant term is automatically included in the model by the `LinearLeastSquares` class. We perform the loop from 1 up to `total_degree` (but the `range` function takes `total_degree + 1` as its second input argument). .. GENERATED FROM PYTHON SOURCE LINES 142-146 .. code-block:: Python total_degree = 4 polynomialCollection = ["x^%d" % (degree) for degree in range(1, total_degree + 1)] polynomialCollection .. rst-class:: sphx-glr-script-out .. code-block:: none ['x^1', 'x^2', 'x^3', 'x^4'] .. GENERATED FROM PYTHON SOURCE LINES 147-148 Given the list of strings, we create a symbolic function which computes the values of the monomials. .. GENERATED FROM PYTHON SOURCE LINES 150-153 .. code-block:: Python basis = ot.SymbolicFunction(["x"], polynomialCollection) basis .. raw:: html
[x]->[x^1,x^2,x^3,x^4]


.. GENERATED FROM PYTHON SOURCE LINES 154-157 .. code-block:: Python designMatrix = basis(x_train) designMatrix .. raw:: html
y0y1y2y3
00000
10.11111110.012345680.0013717420.0001524158
20.22222220.049382720.010973940.002438653
30.33333330.11111110.037037040.01234568
40.44444440.19753090.08779150.03901844
50.55555560.3086420.17146780.09525987
60.66666670.44444440.29629630.1975309
70.77777780.60493830.47050750.3659503
80.88888890.79012350.7023320.6242951
91111


.. GENERATED FROM PYTHON SOURCE LINES 158-161 .. code-block:: Python myLeastSquares = ot.LinearLeastSquares(designMatrix, y_train) myLeastSquares.run() .. GENERATED FROM PYTHON SOURCE LINES 162-164 .. code-block:: Python responseSurface = myLeastSquares.getMetaModel() .. GENERATED FROM PYTHON SOURCE LINES 165-166 The couple (`x_test`,`y_test`) is the test set: it is used to assess the quality of the polynomial model with points that were not used for training. .. GENERATED FROM PYTHON SOURCE LINES 168-172 .. code-block:: Python n_test = 50 x_test = linearSample(0, 1, n_test) y_test = responseSurface(basis(x_test)) .. GENERATED FROM PYTHON SOURCE LINES 173-189 .. code-block:: Python graph = ot.Graph("Polynomial curve fitting", "x", "y", True, "upper right") # The "unknown" function curve = g.draw(0, 1) curve.setColors(["green"]) graph.add(curve) # Training set cloud = ot.Cloud(x_train, y_train) cloud.setPointStyle("circle") graph.add(cloud) # Predictions curve = ot.Curve(x_test, y_test) curve.setLegend("Polynomial Degree = %d" % (total_degree)) curve.setColor("red") graph.add(curve) view = otv.View(graph) .. image-sg:: /auto_meta_modeling/general_purpose_metamodels/images/sphx_glr_plot_overfitting_model_selection_003.png :alt: Polynomial curve fitting :srcset: /auto_meta_modeling/general_purpose_metamodels/images/sphx_glr_plot_overfitting_model_selection_003.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 190-191 For each observation in the training set, the error is the vertical distance between the model and the observation. .. GENERATED FROM PYTHON SOURCE LINES 193-219 .. code-block:: Python graph = ot.Graph( "Least squares minimizes the sum of the squares of the vertical bars", "x", "y", True, "upper right", ) # Training set observations cloud = ot.Cloud(x_train, y_train) cloud.setPointStyle("circle") graph.add(cloud) # Predictions curve = ot.Curve(x_test, y_test) curve.setLegend("Polynomial Degree = %d" % (total_degree)) curve.setColor("red") graph.add(curve) # Errors ypredicted_train = responseSurface(basis(x_train)) for i in range(n_train): curve = ot.Curve([x_train[i], x_train[i]], [y_train[i], ypredicted_train[i]]) curve.setColor("green") curve.setLineWidth(2) graph.add(curve) view = otv.View(graph) .. image-sg:: /auto_meta_modeling/general_purpose_metamodels/images/sphx_glr_plot_overfitting_model_selection_004.png :alt: Least squares minimizes the sum of the squares of the vertical bars :srcset: /auto_meta_modeling/general_purpose_metamodels/images/sphx_glr_plot_overfitting_model_selection_004.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 220-221 The least squares method minimizes the sum of the squared errors i.e. the sum of the squares of the lengths of the vertical segments. .. GENERATED FROM PYTHON SOURCE LINES 223-225 We gather the previous computation in two different functions. The `myPolynomialDataFitting` function computes the least squares solution and `myPolynomialCurveFittingGraph` plots the results. .. GENERATED FROM PYTHON SOURCE LINES 228-242 .. code-block:: Python def myPolynomialDataFitting(total_degree, x_train, y_train): """Computes the polynomial curve fitting with given total degree. This is for learning purposes only: please consider a serious metamodel for real applications, e.g. polynomial chaos or kriging.""" polynomialCollection = ["x^%d" % (degree) for degree in range(1, total_degree + 1)] basis = ot.SymbolicFunction(["x"], polynomialCollection) designMatrix = basis(x_train) myLeastSquares = ot.LinearLeastSquares(designMatrix, y_train) myLeastSquares.run() responseSurface = myLeastSquares.getMetaModel() return responseSurface, basis .. GENERATED FROM PYTHON SOURCE LINES 243-270 .. code-block:: Python def myPolynomialCurveFittingGraph(total_degree, x_train, y_train): """Returns the graphics for a polynomial curve fitting with given total degree""" responseSurface, basis = myPolynomialDataFitting(total_degree, x_train, y_train) # Graphics n_test = 100 x_test = linearSample(0, 1, n_test) ypredicted_test = responseSurface(basis(x_test)) # Graphics graph = ot.Graph("Polynomial curve fitting", "x", "y", True, "upper right") # The "unknown" function curve = g.draw(0, 1) curve.setColors(["green"]) graph.add(curve) # Training set cloud = ot.Cloud(x_train, y_train) cloud.setPointStyle("circle") cloud.setLegend("N=%d" % (x_train.getSize())) graph.add(cloud) # Predictions curve = ot.Curve(x_test, ypredicted_test) curve.setLegend("Polynomial Degree = %d" % (total_degree)) curve.setColor("red") graph.add(curve) return graph .. GENERATED FROM PYTHON SOURCE LINES 271-272 In order to see the effect of the polynomial degree, we compare the polynomial fit with degrees equal to 0 (constant), 1 (linear), 3 (cubic) and 9 (enneagonic ?). .. GENERATED FROM PYTHON SOURCE LINES 274-293 .. code-block:: Python fig = pl.figure(figsize=(12, 9)) _ = fig.suptitle("Polynomial curve fitting") ax_1 = fig.add_subplot(2, 2, 1) _ = ot.viewer.View( myPolynomialCurveFittingGraph(0, x_train, y_train), figure=fig, axes=[ax_1] ) ax_2 = fig.add_subplot(2, 2, 2) _ = ot.viewer.View( myPolynomialCurveFittingGraph(1, x_train, y_train), figure=fig, axes=[ax_2] ) ax_3 = fig.add_subplot(2, 2, 3) _ = ot.viewer.View( myPolynomialCurveFittingGraph(3, x_train, y_train), figure=fig, axes=[ax_3] ) ax_4 = fig.add_subplot(2, 2, 4) _ = ot.viewer.View( myPolynomialCurveFittingGraph(9, x_train, y_train), figure=fig, axes=[ax_4] ) .. image-sg:: /auto_meta_modeling/general_purpose_metamodels/images/sphx_glr_plot_overfitting_model_selection_005.png :alt: Polynomial curve fitting :srcset: /auto_meta_modeling/general_purpose_metamodels/images/sphx_glr_plot_overfitting_model_selection_005.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 294-306 When the polynomial degree is low, the fit is satisfying. The polynomial is close to the observations, although there is still some residual error. However, when the polynomial degree is high, it produces large oscillations which significantly deviate from the true function. This is *overfitting*. This is a pity, given the fact that the polynomial *exactly* interpolates the observations: the residuals are zeroed. If the locations of the x abscissas could be changed, then the oscillations could be made smaller. This is the method used in gaussian quadrature, where the nodes of interpolation are made closer on the left and right bounds. In our situation, we make the asssumption that these abscissas cannot be changed: the most obvious choice is to limit the degree of the polynomial. Another possibility is to include a regularization into the least squares solution. .. GENERATED FROM PYTHON SOURCE LINES 308-312 Root mean squared error ----------------------- In order to assess the quality of the polynomial fit, we create a second dataset, the *test set* and compare the value of the polynomial with the test observations. .. GENERATED FROM PYTHON SOURCE LINES 314-316 .. code-block:: Python sqrt = ot.SymbolicFunction(["x"], ["sqrt(x)"]) .. GENERATED FROM PYTHON SOURCE LINES 317-320 In order to see how close the model is to the observations, we compute the root mean square error. First, we create a degree 4 polynomial which fits the data. .. GENERATED FROM PYTHON SOURCE LINES 322-326 .. code-block:: Python total_degree = 4 responseSurface, basis = myPolynomialDataFitting(total_degree, x_train, y_train) .. GENERATED FROM PYTHON SOURCE LINES 327-328 Then we create a test set, with the same method as before. .. GENERATED FROM PYTHON SOURCE LINES 331-338 .. code-block:: Python def createDataset(n): x = linearSample(0, 1, n) noiseSample = noise.getSample(n) y = g(x) + noiseSample return x, y .. GENERATED FROM PYTHON SOURCE LINES 339-342 .. code-block:: Python n_test = 100 x_test, y_test = createDataset(n_test) .. GENERATED FROM PYTHON SOURCE LINES 343-344 On this test set, we evaluate the polynomial. .. GENERATED FROM PYTHON SOURCE LINES 346-348 .. code-block:: Python ypredicted_test = responseSurface(basis(x_test)) .. GENERATED FROM PYTHON SOURCE LINES 349-350 The vector of residuals is the vector of the differences between the observations and the predictions. .. GENERATED FROM PYTHON SOURCE LINES 352-354 .. code-block:: Python residuals = y_test.asPoint() - ypredicted_test.asPoint() .. GENERATED FROM PYTHON SOURCE LINES 355-358 The `normSquare` method computes the square of the Euclidian norm (i.e. the 2-norm). We divide this by the test sample size (so as to compare the error for different sample sizes) and compute the square root of the result (so that the result has the same unit as y). .. GENERATED FROM PYTHON SOURCE LINES 360-364 .. code-block:: Python RMSE = sqrt([residuals.normSquare() / n_test])[0] RMSE .. rst-class:: sphx-glr-script-out .. code-block:: none 0.14464766752910935 .. GENERATED FROM PYTHON SOURCE LINES 365-366 The following function gathers the RMSE computation to make the experiment easier. .. GENERATED FROM PYTHON SOURCE LINES 369-376 .. code-block:: Python def computeRMSE(responseSurface, basis, x, y): ypredicted = responseSurface(basis(x)) residuals = y.asPoint() - ypredicted.asPoint() RMSE = sqrt([residuals.normSquare() / n_test])[0] return RMSE .. GENERATED FROM PYTHON SOURCE LINES 377-385 .. code-block:: Python maximum_degree = 10 RMSE_train = ot.Sample(maximum_degree, 1) RMSE_test = ot.Sample(maximum_degree, 1) for total_degree in range(maximum_degree): responseSurface, basis = myPolynomialDataFitting(total_degree, x_train, y_train) RMSE_train[total_degree, 0] = computeRMSE(responseSurface, basis, x_train, y_train) RMSE_test[total_degree, 0] = computeRMSE(responseSurface, basis, x_test, y_test) .. GENERATED FROM PYTHON SOURCE LINES 386-402 .. code-block:: Python degreeSample = ot.Sample([[i] for i in range(maximum_degree)]) graph = ot.Graph("Root mean square error", "Degree", "RMSE", True, "upper right") # Train cloud = ot.Curve(degreeSample, RMSE_train) cloud.setColor("blue") cloud.setLegend("Train") cloud.setPointStyle("circle") graph.add(cloud) # Test cloud = ot.Curve(degreeSample, RMSE_test) cloud.setColor("red") cloud.setLegend("Test") cloud.setPointStyle("circle") graph.add(cloud) view = otv.View(graph) .. image-sg:: /auto_meta_modeling/general_purpose_metamodels/images/sphx_glr_plot_overfitting_model_selection_006.png :alt: Root mean square error :srcset: /auto_meta_modeling/general_purpose_metamodels/images/sphx_glr_plot_overfitting_model_selection_006.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 403-411 We see that the RMSE on the train set continuously decreases, reaching zero when the polynomial degree is so that the number of coefficients is equal to the train dataset sample size. In this extreme situation, the least squares solution is equivalent to solving a linear system of equations: this leads to a zero residual. On the test set however, the RMSE decreases, reaches a flat region, then increases dramatically when the degree is equal to 9. Hence, limiting the polynomial degree limits overfitting. .. GENERATED FROM PYTHON SOURCE LINES 413-417 Increasing the training set --------------------------- We wonder what happens when the training dataset size is increased. .. GENERATED FROM PYTHON SOURCE LINES 419-442 .. code-block:: Python total_degree = 9 fig = pl.figure(figsize=(12, 9)) _ = fig.suptitle("Polynomial curve fitting") # ax_1 = fig.add_subplot(2, 2, 1) n_train = 11 x_train, y_train = createDataset(n_train) _ = ot.viewer.View( myPolynomialCurveFittingGraph(total_degree, x_train, y_train), figure=fig, axes=[ax_1], ) # n_train = 100 x_train, y_train = createDataset(n_train) ax_2 = fig.add_subplot(2, 2, 2) _ = ot.viewer.View( myPolynomialCurveFittingGraph(total_degree, x_train, y_train), figure=fig, axes=[ax_2], ) pl.show() .. image-sg:: /auto_meta_modeling/general_purpose_metamodels/images/sphx_glr_plot_overfitting_model_selection_007.png :alt: Polynomial curve fitting :srcset: /auto_meta_modeling/general_purpose_metamodels/images/sphx_glr_plot_overfitting_model_selection_007.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 443-444 We see that the polynomial oscillates with a dataset with size 11, but does not with the larger dataset: increasing the training dataset mitigates the oscillations. .. _sphx_glr_download_auto_meta_modeling_general_purpose_metamodels_plot_overfitting_model_selection.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_overfitting_model_selection.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_overfitting_model_selection.py `