.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_data_analysis/distribution_fitting/plot_smoothing_mixture.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_distribution_fitting_plot_smoothing_mixture.py: Bandwidth sensitivity in kernel smoothing ========================================= .. GENERATED FROM PYTHON SOURCE LINES 6-33 Introduction ------------ We consider the distribution .. math:: f_1(x) = w_1 f_A(x) + w_2 f_B(x) for any :math:`x\in\mathbb{R}` where :math:`f_A` is the density of the Normal distribution :math:`\mathcal{N}(0,1)`, :math:`f_B` is the density of the Normal distribution :math:`\mathcal{N}(3/2,(1/3)^2)` and the weights are :math:`w_1 = \frac{3}{4}` and :math:`w_2 = \frac{1}{4}`. This is a mixture of two Normal distributions: 1/4th of the observations have the :math:`\mathcal{N}(0,1)` distribution and 3/4th of the observations have the :math:`\mathcal{N}(3/2,(1/3)^2)` distribution. This example is considered in (Wand, Jones, 1994), page 14, Figure 2.3. We consider a sample generated from independent realizations of :math:`f_1` and want to approximate the distribution from kernel smoothing. More precisely, we want to observe the sensitivity of the resulting density to the bandwidth. References ---------- * "Kernel Smoothing", M.P.Wand, M.C.Jones. Chapman and Hall / CRC (1994). .. GENERATED FROM PYTHON SOURCE LINES 36-40 Generate the mixture by merging two samples ------------------------------------------- In this section, we show that a mixture of two Normal distributions is nothing more than the merged sample of two independent Normal distributions. In order to generate a sample with size :math:`n`, we sample :math:`\lfloor w_1 n\rfloor` points from the first Normal distribution :math:`f_A` and :math:`\lfloor w_2 n\rfloor` points from the second Normal distribution :math:`f_B`. Then we merge the two samples. .. GENERATED FROM PYTHON SOURCE LINES 42-47 .. code-block:: default import openturns as ot import openturns.viewer as otv import pylab as pl import numpy as np .. GENERATED FROM PYTHON SOURCE LINES 48-49 We choose a rather large sample size: :math:`n=1000`. .. GENERATED FROM PYTHON SOURCE LINES 51-53 .. code-block:: default n = 1000 .. GENERATED FROM PYTHON SOURCE LINES 54-55 Then we define the two Normal distributions and their parameters. .. GENERATED FROM PYTHON SOURCE LINES 57-62 .. code-block:: default w1 = 0.75 w2 = 1.0 - w1 distribution1 = ot.Normal(0.0, 1.0) distribution2 = ot.Normal(1.5, 1.0 / 3.0) .. GENERATED FROM PYTHON SOURCE LINES 63-64 We generate two independent sub-samples from the two Normal distributions. .. GENERATED FROM PYTHON SOURCE LINES 66-69 .. code-block:: default sample1 = distribution1.getSample(int(w1 * n)) sample2 = distribution2.getSample(int(w2 * n)) .. GENERATED FROM PYTHON SOURCE LINES 70-71 Then we merge the sub-samples into a larger one with the `add` method of the `Sample` class. .. GENERATED FROM PYTHON SOURCE LINES 73-77 .. code-block:: default sample = ot.Sample(sample1) sample.add(sample2) sample.getSize() .. rst-class:: sphx-glr-script-out Out: .. code-block:: none 1000 .. GENERATED FROM PYTHON SOURCE LINES 78-79 In order to see the result, we build a kernel smoothing approximation on the sample. In order to keep it simple, let us use the default bandwidth selection rule. .. GENERATED FROM PYTHON SOURCE LINES 81-84 .. code-block:: default factory = ot.KernelSmoothing() fit = factory.build(sample) .. GENERATED FROM PYTHON SOURCE LINES 85-88 .. code-block:: default graph = fit.drawPDF() view = otv.View(graph) .. image-sg:: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_smoothing_mixture_001.png :alt: plot smoothing mixture :srcset: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_smoothing_mixture_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 89-90 We see that the distribution of the merged sample has two modes. However, these modes are not clearly distinct. To distinguish them, we could increase the sample size. However, it might be interesting to see if the bandwidth selection rule can be better chosen: this is the purpose of the next section. .. GENERATED FROM PYTHON SOURCE LINES 92-96 Simulation based on a mixture ----------------------------- Since the distribution that we approximate is a mixture, it will be more convenient to create it from the `Mixture` class. It takes as input argument a list of distributions and a list of weights. .. GENERATED FROM PYTHON SOURCE LINES 98-100 .. code-block:: default distribution = ot.Mixture([distribution1, distribution2], [w1, w2]) .. GENERATED FROM PYTHON SOURCE LINES 101-102 Then we generate a sample from it. .. GENERATED FROM PYTHON SOURCE LINES 104-106 .. code-block:: default sample = distribution.getSample(n) .. GENERATED FROM PYTHON SOURCE LINES 107-110 .. code-block:: default factory = ot.KernelSmoothing() fit = factory.build(sample) .. GENERATED FROM PYTHON SOURCE LINES 111-113 .. code-block:: default factory.getBandwidth() .. raw:: html

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.. GENERATED FROM PYTHON SOURCE LINES 114-115 We see that the default bandwidth is close to 0.17. .. GENERATED FROM PYTHON SOURCE LINES 117-125 .. code-block:: default graph = distribution.drawPDF() curve = fit.drawPDF() graph.add(curve) graph.setColors(["dodgerblue3", "darkorange1"]) graph.setLegends(["Mixture", "Kernel smoothing"]) graph.setLegendPosition("topleft") view = otv.View(graph) .. image-sg:: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_smoothing_mixture_002.png :alt: plot smoothing mixture :srcset: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_smoothing_mixture_002.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 126-127 We see that the result of the kernel smoothing approximation of the mixture is similar to the previous one and could be improved. .. GENERATED FROM PYTHON SOURCE LINES 129-131 Sensitivity to the bandwidth ---------------------------- .. GENERATED FROM PYTHON SOURCE LINES 133-134 In this section, we observe the sensitivity of the kernel smoothing to the bandwidth. We consider the three following bandwidths: the small bandwidth 0.05, the large bandwidth 0.54 and 0.18 which is in-between. For each bandwidth, we use the second optional argument of the `build` method in order to select a specific bandwidth value. .. GENERATED FROM PYTHON SOURCE LINES 136-157 .. code-block:: default hArray = [0.05, 0.54, 0.18] nLen = len(hArray) fig = pl.figure(figsize=(10, 8)) for i in range(nLen): ax = fig.add_subplot(2, 2, i + 1) fit = factory.build(sample, [hArray[i]]) graph = fit.drawPDF() graph.setColors(["dodgerblue3"]) graph.setLegends(["h=%.4f" % (hArray[i])]) exact = distribution.drawPDF() curve = exact.getDrawable(0) curve.setColor("darkorange1") curve.setLegend("Mixture") curve.setLineStyle("dashed") graph.add(curve) graph.setLegendPosition("topleft") graph.setXTitle("X") view = otv.View(graph, figure=fig, axes=[ax]) pl.ylim(top=0.5) # Common y-range view = otv.View(graph) .. rst-class:: sphx-glr-horizontal * .. image-sg:: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_smoothing_mixture_003.png :alt: plot smoothing mixture :srcset: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_smoothing_mixture_003.png :class: sphx-glr-multi-img * .. image-sg:: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_smoothing_mixture_004.png :alt: plot smoothing mixture :srcset: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_smoothing_mixture_004.png :class: sphx-glr-multi-img .. GENERATED FROM PYTHON SOURCE LINES 158-159 We see that when the bandwidth is too small, the resulting kernel smoothing has many more modes than the distribution it is supposed to approximate. When the bandwidth is too large, the approximated distribution is too smooth and has only one mode instead of the expected two modes which are in the mixture distribution. When the bandwidth is equal to 0.18, the two modes are correctly represented. .. GENERATED FROM PYTHON SOURCE LINES 161-165 Sensitivity to the bandwidth rule --------------------------------- The library provides three different rules to compute the bandwidth. In this section, we compare the results that we can get with them. .. GENERATED FROM PYTHON SOURCE LINES 167-170 .. code-block:: default h1 = factory.computeSilvermanBandwidth(sample)[0] h1 .. rst-class:: sphx-glr-script-out Out: .. code-block:: none 0.3445636453391276 .. GENERATED FROM PYTHON SOURCE LINES 171-174 .. code-block:: default h2 = factory.computePluginBandwidth(sample)[0] h2 .. rst-class:: sphx-glr-script-out Out: .. code-block:: none 0.2021709523195656 .. GENERATED FROM PYTHON SOURCE LINES 175-178 .. code-block:: default h3 = factory.computeMixedBandwidth(sample)[0] h3 .. rst-class:: sphx-glr-script-out Out: .. code-block:: none 0.20851397168332242 .. GENERATED FROM PYTHON SOURCE LINES 179-181 .. code-block:: default factory.getBandwidth()[0] .. rst-class:: sphx-glr-script-out Out: .. code-block:: none 0.18 .. GENERATED FROM PYTHON SOURCE LINES 182-185 We see that the default rule is the "Mixed" rule. This is because the sample is in dimension 1 and the sample size is quite large. For a small sample in 1 dimension, the "Plugin" rule would have been used. The following script compares the results produced by the three rules. .. GENERATED FROM PYTHON SOURCE LINES 187-213 .. code-block:: default hArray = [h1, h2, h3] legends = ["Silverman", "Plugin", "Mixed"] nLen = len(hArray) fig = pl.figure(figsize=(10, 8)) for i in range(nLen): ax = fig.add_subplot(2, 2, i + 1) fit = factory.build(sample, [hArray[i]]) graph = fit.drawPDF() graph.setColors(["dodgerblue3"]) graph.setLegends(["h=%.4f, %s" % (hArray[i], legends[i])]) exact = distribution.drawPDF() curve = exact.getDrawable(0) curve.setColor("darkorange1") curve.setLegend("Mixture") curve.setLineStyle("dashed") graph.add(curve) graph.setLegendPosition("topleft") graph.setXTitle("X") if i > 0: graph.setYTitle("") view = otv.View(graph, figure=fig, axes=[ax]) pl.ylim(top=0.5) # Common y-range view = otv.View(graph) otv.View.ShowAll() .. rst-class:: sphx-glr-horizontal * .. image-sg:: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_smoothing_mixture_005.png :alt: plot smoothing mixture :srcset: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_smoothing_mixture_005.png :class: sphx-glr-multi-img * .. image-sg:: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_smoothing_mixture_006.png :alt: plot smoothing mixture :srcset: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_smoothing_mixture_006.png :class: sphx-glr-multi-img .. GENERATED FROM PYTHON SOURCE LINES 214-215 We see that the bandwidth produced by Silverman's rule is too large, leading to an oversmoothed distribution. The results produced by the Plugin and Mixed rules are comparable in this case. .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 0.729 seconds) .. _sphx_glr_download_auto_data_analysis_distribution_fitting_plot_smoothing_mixture.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_smoothing_mixture.py ` .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_smoothing_mixture.ipynb ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_