.. 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_meta_modeling_general_purpose_metamodels_plot_overfitting_model_selection.py: Over-fitting and model selection ================================ 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. References ---------- * Bishop Christopher M., 1995, Neural networks for pattern recognition. Figure 1.4, page 7 Compute the data ---------------- In this section, we generate noisy observations from the sine function. .. code-block:: default import openturns as ot import pylab as pl import openturns.viewer as otv .. code-block:: default ot.RandomGenerator.SetSeed(0) We define the function that we are going to approximate. .. code-block:: default g = ot.SymbolicFunction(["x"], ["sin(2*pi_*x)"]) .. code-block:: default graph = ot.Graph("Polynomial curve fitting", "x", "y", True, "topright") # The "unknown" function curve = g.draw(0, 1) curve.setColors(["green"]) curve.setLegends(['"Unknown" function']) graph.add(curve) view = otv.View(graph) .. image:: /auto_meta_modeling/general_purpose_metamodels/images/sphx_glr_plot_overfitting_model_selection_001.png :alt: Polynomial curve fitting :class: sphx-glr-single-img This seems a nice, smooth function to approximate with polynomials. .. code-block:: default 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 We consider 10 observation points in the interval [0,1]. .. code-block:: default n_train = 10 x_train = linearSample(0, 1, n_train) Assume that the observations are noisy and that the noise follows a Normal distribution with zero mean and small standard deviation. .. code-block:: default noise = ot.Normal(0, 0.1) noiseSample = noise.getSample(n_train) 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. .. code-block:: default y_train = g(x_train) + noiseSample .. code-block:: default graph = ot.Graph("Polynomial curve fitting", "x", "y", True, "topright") # 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:: /auto_meta_modeling/general_purpose_metamodels/images/sphx_glr_plot_overfitting_model_selection_002.png :alt: Polynomial curve fitting :class: sphx-glr-single-img 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. 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). .. code-block:: default total_degree = 4 polynomialCollection = ["x^%d" % (degree) for degree in range(1, total_degree + 1)] polynomialCollection .. rst-class:: sphx-glr-script-out Out: .. code-block:: none ['x^1', 'x^2', 'x^3', 'x^4'] Given the list of strings, we create a symbolic function which computes the values of the monomials. .. code-block:: default basis = ot.SymbolicFunction(["x"], polynomialCollection) basis .. raw:: html

[x]->[x^1,x^2,x^3,x^4]



.. code-block:: default 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


.. code-block:: default myLeastSquares = ot.LinearLeastSquares(designMatrix, y_train) myLeastSquares.run() .. code-block:: default responseSurface = myLeastSquares.getMetaModel() 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. .. code-block:: default n_test = 50 x_test = linearSample(0, 1, n_test) y_test = responseSurface(basis(x_test)) .. code-block:: default graph = ot.Graph("Polynomial curve fitting", "x", "y", True, "topright") # 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:: /auto_meta_modeling/general_purpose_metamodels/images/sphx_glr_plot_overfitting_model_selection_003.png :alt: Polynomial curve fitting :class: sphx-glr-single-img For each observation in the training set, the error is the vertical distance between the model and the observation. .. code-block:: default graph = ot.Graph( "Least squares minimizes the sum of the squares of the vertical bars", "x", "y", True, "topright", ) # 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:: /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 :class: sphx-glr-single-img 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. We gather the previous computation in two different functions. The `myPolynomialDataFitting` function computes the least squares solution and `myPolynomialCurveFittingGraph` plots the results. .. code-block:: default 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 .. code-block:: default 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, "topright") # 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 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 ?). .. code-block:: default 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:: /auto_meta_modeling/general_purpose_metamodels/images/sphx_glr_plot_overfitting_model_selection_005.png :alt: Polynomial curve fitting :class: sphx-glr-single-img 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. 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. .. code-block:: default sqrt = ot.SymbolicFunction(["x"], ["sqrt(x)"]) 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. .. code-block:: default total_degree = 4 responseSurface, basis = myPolynomialDataFitting(total_degree, x_train, y_train) Then we create a test set, with the same method as before. .. code-block:: default def createDataset(n): x = linearSample(0, 1, n) noiseSample = noise.getSample(n) y = g(x) + noiseSample return x, y .. code-block:: default n_test = 100 x_test, y_test = createDataset(n_test) On this test set, we evaluate the polynomial. .. code-block:: default ypredicted_test = responseSurface(basis(x_test)) The vector of residuals is the vector of the differences between the observations and the predictions. .. code-block:: default residuals = y_test.asPoint() - ypredicted_test.asPoint() 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). .. code-block:: default RMSE = sqrt([residuals.normSquare() / n_test])[0] RMSE .. rst-class:: sphx-glr-script-out Out: .. code-block:: none 0.1446476675291092 The following function gathers the RMSE computation to make the experiment easier. .. code-block:: default def computeRMSE(responseSurface, basis, x, y): ypredicted = responseSurface(basis(x)) residuals = y.asPoint() - ypredicted.asPoint() RMSE = sqrt([residuals.normSquare() / n_test])[0] return RMSE .. code-block:: default 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) .. code-block:: default degreeSample = ot.Sample([[i] for i in range(maximum_degree)]) graph = ot.Graph("Root mean square error", "Degree", "RMSE", True, "topright") # 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:: /auto_meta_modeling/general_purpose_metamodels/images/sphx_glr_plot_overfitting_model_selection_006.png :alt: Root mean square error :class: sphx-glr-single-img 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. Increasing the training set --------------------------- We wonder what happens when the training dataset size is increased. .. code-block:: default 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:: /auto_meta_modeling/general_purpose_metamodels/images/sphx_glr_plot_overfitting_model_selection_007.png :alt: Polynomial curve fitting :class: sphx-glr-single-img 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. .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 0.903 seconds) .. _sphx_glr_download_auto_meta_modeling_general_purpose_metamodels_plot_overfitting_model_selection.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_overfitting_model_selection.py ` .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_overfitting_model_selection.ipynb ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_