"""
Bandwidth sensitivity in kernel smoothing
=========================================
"""
# %%
#
# Introduction
# ------------
#
# When we have a sample, we may estimate the probability density function of the
# underlying distribution using kernel smoothing.
# One of the parameters of this method is the bandwidth, which can be either
# set by the user, or estimated from the data.
# This is especially true when the density is multimodal.
# In this example, we consider a bimodal distribution and see how the bandwidth
# can change the estimated probability density function.
#
# 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.
# %%
import openturns as ot
import openturns.viewer as otv
# %%
# We choose a rather large sample size: :math:`n=1000`.
# %%
n = 1000
# %%
# Then we define the two Normal distributions and their parameters.
# %%
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.
# %%
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.
# %%
sample = ot.Sample(sample1)
sample.add(sample2)
sample.getSize()
# %%
# 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.
# %%
factory = ot.KernelSmoothing()
fit = factory.build(sample)
# %%
graph = fit.drawPDF()
view = otv.View(graph)
# %%
# 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.
# %%
distribution = ot.Mixture([distribution1, distribution2], [w1, w2])
# %%
# Then we generate a sample from it.
# %%
sample = distribution.getSample(n)
# %%
factory = ot.KernelSmoothing()
fit = factory.build(sample)
# %%
factory.getBandwidth()
# %%
# We see that the default bandwidth is close to 0.17.
# %%
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)
# %%
# 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.
# %%
hArray = [0.05, 0.54, 0.18]
nLen = len(hArray)
grid = ot.GridLayout(1, len(hArray))
index = 0
for i in range(nLen):
fit = factory.build(sample, [hArray[i]])
graph = fit.drawPDF()
exact = distribution.drawPDF()
curve = exact.getDrawable(0)
curve.setLegend("Mixture")
curve.setLineStyle("dashed")
graph.add(curve)
graph.setXTitle("X")
graph.setTitle("h=%.4f" % (hArray[i]))
graph.setLegends([""])
graph.setColors(ot.Drawable.BuildDefaultPalette(2))
bounding_box = graph.getBoundingBox()
upper_bound = bounding_box.getUpperBound()
upper_bound[1] = 0.5 # Common y-range
graph.setBoundingBox(bounding_box)
grid.setGraph(0, index, graph)
index += 1
view = otv.View(grid, figure_kw={"figsize": (10.0, 4.0)})
# %%
# 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.
# %%
h1 = factory.computeSilvermanBandwidth(sample)[0]
h1
# %%
h2 = factory.computePluginBandwidth(sample)[0]
h2
# %%
h3 = factory.computeMixedBandwidth(sample)[0]
h3
# %%
factory.getBandwidth()[0]
# %%
# 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.
# %%
hArray = [h1, h2, h3]
legends = ["Silverman", "Plugin", "Mixed"]
nLen = len(hArray)
grid = ot.GridLayout(1, len(hArray))
index = 0
for i in range(nLen):
fit = factory.build(sample, [hArray[i]])
graph = fit.drawPDF()
exact = distribution.drawPDF()
curve = exact.getDrawable(0)
curve.setLegend("Mixture")
curve.setLineStyle("dashed")
graph.add(curve)
graph.setLegends([""])
graph.setTitle("h=%.4f, %s" % (hArray[i], legends[i]))
graph.setXTitle("X")
graph.setColors(ot.Drawable.BuildDefaultPalette(2))
if i > 0:
graph.setYTitle("")
bounding_box = graph.getBoundingBox()
upper_bound = bounding_box.getUpperBound()
upper_bound[1] = 0.5 # Common y-range
graph.setBoundingBox(bounding_box)
grid.setGraph(0, index, graph)
index += 1
view = otv.View(grid, figure_kw={"figsize": (10.0, 4.0)})
otv.View.ShowAll()
# %%
# 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.