.. 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
=========================================
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).
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.
.. code-block:: default
import openturns as ot
import openturns.viewer as otv
import pylab as pl
import numpy as np
We choose a rather large sample size: :math:`n=1000`.
.. code-block:: default
n = 1000
Then we define the two Normal distributions and their parameters.
.. 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)
We generate two independent sub-samples from the two Normal distributions.
.. code-block:: default
sample1 = distribution1.getSample(int(w1 * n))
sample2 = distribution2.getSample(int(w2 * n))
Then we merge the sub-samples into a larger one with the `add` method of the `Sample` class.
.. code-block:: default
sample = ot.Sample(sample1)
sample.add(sample2)
sample.getSize()
.. rst-class:: sphx-glr-script-out
Out:
.. code-block:: none
1000
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.
.. code-block:: default
factory = ot.KernelSmoothing()
fit = factory.build(sample)
.. code-block:: default
graph = fit.drawPDF()
view = otv.View(graph)
.. image:: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_smoothing_mixture_001.png
:alt: plot smoothing mixture
:class: sphx-glr-single-img
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.
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.
.. code-block:: default
distribution = ot.Mixture([distribution1, distribution2], [w1, w2])
Then we generate a sample from it.
.. code-block:: default
sample = distribution.getSample(n)
.. code-block:: default
factory = ot.KernelSmoothing()
fit = factory.build(sample)
.. code-block:: default
factory.getBandwidth()
.. raw:: html
[0.208514]
We see that the default bandwidth is close to 0.17.
.. 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:: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_smoothing_mixture_002.png
:alt: plot smoothing mixture
:class: sphx-glr-single-img
We see that the result of the kernel smoothing approximation of the mixture is similar to the previous one and could be improved.
Sensitivity to the bandwidth
----------------------------
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.
.. 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:: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_smoothing_mixture_003.png
:alt: plot smoothing mixture
:class: sphx-glr-multi-img
*
.. image:: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_smoothing_mixture_004.png
:alt: plot smoothing mixture
:class: sphx-glr-multi-img
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.
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.
.. code-block:: default
h1 = factory.computeSilvermanBandwidth(sample)[0]
h1
.. rst-class:: sphx-glr-script-out
Out:
.. code-block:: none
0.3445636453391276
.. code-block:: default
h2 = factory.computePluginBandwidth(sample)[0]
h2
.. rst-class:: sphx-glr-script-out
Out:
.. code-block:: none
0.2021709523195656
.. code-block:: default
h3 = factory.computeMixedBandwidth(sample)[0]
h3
.. rst-class:: sphx-glr-script-out
Out:
.. code-block:: none
0.20851397168332242
.. code-block:: default
factory.getBandwidth()[0]
.. rst-class:: sphx-glr-script-out
Out:
.. code-block:: none
0.18
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.
.. 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:: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_smoothing_mixture_005.png
:alt: plot smoothing mixture
:class: sphx-glr-multi-img
*
.. image:: /auto_data_analysis/distribution_fitting/images/sphx_glr_plot_smoothing_mixture_006.png
:alt: plot smoothing mixture
:class: sphx-glr-multi-img
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 0.834 seconds)
.. _sphx_glr_download_auto_data_analysis_distribution_fitting_plot_smoothing_mixture.py:
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