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Kernel Density Estimation with SciPy
# -*- coding: utf-8 -*-
# <nbformat>2</nbformat>
# <markdowncell>
# Kernel Density Estimation with SciPy
# ====================================
# <codecell>
import numpy as np
from scipy import stats
import matplotlib.pyplot as plt
# <markdowncell>
# Univariate estimation
# ---------------------
#
# We start with a minimal amount of data in order to see how `gaussian_kde` works,
# and what the different options for bandwidth selection do.
# The data sampled from the PDF is show as blue dashes at the bottom of the figure,
# ( this is called a rug plot).
# <codecell>
x1 = np.array([-7, -5, 1, 4, 5], dtype=np.float)
kde1 = stats.gaussian_kde(x1)
kde2 = stats.gaussian_kde(x1, bw_method='silverman')
fig = plt.figure()
ax = fig.add_subplot(111)
ax.plot(x1, np.zeros(x1.shape), 'b+', ms=20) # rug plot
x_eval = np.linspace(-10, 10, num=200)
ax.plot(x_eval, kde1(x_eval), 'k-', label="Scott's Rule")
ax.plot(x_eval, kde1(x_eval), 'r-', label="Silverman's Rule")
# <markdowncell>
# We see that there is very little difference between Scott's Rule and Silverman's Rule,
# and that the bandwidth selection with a limited amount of data is probably a bit too wide.
# We can define our own bandwidth function to get a less smoothed out result.
# <codecell>
def my_kde_bandwidth(obj, fac=1./5):
"""We use Scott's Rule, multiplied by a constant factor."""
return np.power(obj.n, -1./(obj.d+4)) * fac
fig = plt.figure()
ax = fig.add_subplot(111)
ax.plot(x1, np.zeros(x1.shape), 'b+', ms=20) # rug plot
kde3 = stats.gaussian_kde(x1, bw_method=my_kde_bandwidth)
ax.plot(x_eval, kde3(x_eval), 'g-', label="With smaller BW")
# <markdowncell>
# We see that if we set bandwidth to be very narrow, the obtained estimate for the probability
# density function (PDF) is simply the sum of Gaussians around each data point.
# <markdowncell>
# We now take a more realistic example, and look at the difference between the two available
# bandwidth selection rules. Those rules are known to work well for (close to) normal
# distributions, but even for uni-modal distributions that are quite strongly non-normal
# they work reasonably well. As a non-normal distribution we take a Student's T distribution
# with 5 degrees of freedom.
# <codecell>
np.random.seed(12456)
x1 = np.random.normal(size=200) # random data, normal distribution
xs = np.linspace(x1.min()-1, x1.max()+1, 200)
kde1 = stats.gaussian_kde(x1)
kde2 = stats.gaussian_kde(x1, bw_method='silverman')
fig = plt.figure(figsize=(8, 6))
ax1 = fig.add_subplot(211)
ax1.plot(x1, np.zeros(x1.shape), 'b+', ms=12) # rug plot
ax1.plot(xs, kde1(xs), 'k-', label="Scott's Rule")
ax1.plot(xs, kde2(xs), 'b-', label="Silverman's Rule")
ax1.plot(xs, stats.norm.pdf(xs), 'r--', label="True PDF")
ax1.set_xlabel('x')
ax1.set_ylabel('Density')
ax1.set_title("Normal (top) and Student's T$_{df=5}$ (bottom) distributions")
ax1.legend(loc=1)
x2 = stats.t.rvs(5, size=200) # random data, T distribution
xs = np.linspace(x2.min() - 1, x2.max() + 1, 200)
kde3 = stats.gaussian_kde(x2)
kde4 = stats.gaussian_kde(x2, bw_method='silverman')
ax2 = fig.add_subplot(212)
ax2.plot(x2, np.zeros(x2.shape), 'b+', ms=12) # rug plot
ax2.plot(xs, kde3(xs), 'k-', label="Scott's Rule")
ax2.plot(xs, kde4(xs), 'b-', label="Silverman's Rule")
ax2.plot(xs, stats.t.pdf(xs, 5), 'r--', label="True PDF")
ax2.set_xlabel('x')
ax2.set_ylabel('Density')
# <markdowncell>
# We now take a look at a bimodal distribution with one wider and one narrower Gaussian feature.
# We expect that this will be a more difficult density to approximate, due to the
# different bandwidths required to accurately resolve each feature.
# <codecell>
from functools import partial
loc1, scale1, size1 = (-2, 1, 175)
loc2, scale2, size2 = (2, 0.2, 50)
x2 = np.concatenate([np.random.normal(loc=loc1, scale=scale1, size=size1),
np.random.normal(loc=loc2, scale=scale2, size=size2)])
x_eval = np.linspace(x2.min() - 1, x2.max() + 1, 500)
kde = stats.gaussian_kde(x2)
kde2 = stats.gaussian_kde(x2, bw_method='silverman')
kde3 = stats.gaussian_kde(x2, bw_method=partial(my_kde_bandwidth, fac=0.2))
kde4 = stats.gaussian_kde(x2, bw_method=partial(my_kde_bandwidth, fac=0.5))
pdf = stats.norm.pdf
bimodal_pdf = pdf(x_eval, loc=loc1, scale=scale1) * float(size1) / x2.size + \
pdf(x_eval, loc=loc2, scale=scale2) * float(size2) / x2.size
fig = plt.figure(figsize=(8, 6))
ax = fig.add_subplot(111)
ax.plot(x2, np.zeros(x2.shape), 'b+', ms=12)
ax.plot(x_eval, kde(x_eval), 'k-', label="Scott's Rule")
ax.plot(x_eval, kde2(x_eval), 'b-', label="Silverman's Rule")
ax.plot(x_eval, kde3(x_eval), 'g-', label="Scott * 0.2")
ax.plot(x_eval, kde4(x_eval), 'c-', label="Scott * 0.5")
ax.plot(x_eval, bimodal_pdf, 'r--', label="Actual PDF")
ax.set_xlim([x_eval.min(), x_eval.max()])
ax.legend(loc=2)
ax.set_xlabel('x')
ax.set_ylabel('Density')
# <markdowncell>
# As expected, the KDE is not as close to the true PDF as we would like due to the different
# characteristic size of the two features of the bimodal distribution. By halving the default
# bandwidth (`Scott * 0.5`) we can do somewhat better, while using a factor 5 smaller bandwidth
# than the default doesn't smooth enough. What we really need though in this case is
# a non-uniform (adaptive) bandwidth.
# <markdowncell>
# Multivariate estimation
# -----------------------
#
# With `gaussian_kde` we can perform multivariate as well as univariate estimation.
# We demonstrate the bivariate case. First we generate some random data with a model in
# which the two variates are correlated.
# <codecell>
def measure(n):
"""Measurement model, return two coupled measurements."""
m1 = np.random.normal(size=n)
m2 = np.random.normal(scale=0.5, size=n)
return m1+m2, m1-m2
m1, m2 = measure(2000)
xmin = m1.min()
xmax = m1.max()
ymin = m2.min()
ymax = m2.max()
# <markdowncell>
# The we apply the KDE to the data:
# <codecell>
X, Y = np.mgrid[xmin:xmax:100j, ymin:ymax:100j]
positions = np.vstack([X.ravel(), Y.ravel()])
values = np.vstack([m1, m2])
kernel = stats.gaussian_kde(values)
Z = np.reshape(kernel.evaluate(positions).T, X.shape)
# <markdowncell>
# Finally we plot the estimated bivariate distribution as a colormap, and plot the individual
# data points on top.
# <codecell>
fig = plt.figure(figsize=(8, 6))
ax = fig.add_subplot(111)
ax.imshow(np.rot90(Z), cmap=plt.cm.gist_earth_r,
extent=[xmin, xmax, ymin, ymax])
ax.plot(m1, m2, 'k.', markersize=2)
ax.set_xlim([xmin, xmax])
ax.set_ylim([ymin, ymax])
plt.show()
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