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doc: follow up for normalizing histogram
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‎galleries/examples/statistics/histogram_normalization.py renamed to ‎galleries/tutorials/histogram_normalization.py

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@@ -1,6 +1,8 @@
11
"""
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.. redirect-from:: /gallery/statistics/histogram_features
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.. _histogram_normalization:
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46
===================================
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Histogram bins, density, and weight
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===================================
@@ -34,65 +36,90 @@
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# changing the style of the histogram bars just to make it
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# very clear where the boundaries of the bins are:
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style = {'facecolor': 'none', 'edgecolor': 'C0', 'linewidth': 3}
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style = {'facecolor': 'none', 'edgecolor': 'C0', 'linewidth': 3, 'alpha': .5}
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41+
fig, ax = plt.subplots(figsize=(6, 3))
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fig, ax = plt.subplots(layout='constrained', figsize=(8, 4))
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39-
fig, ax = plt.subplots()
45+
# count the number of values in xdata between each value in xbins
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ax.hist(xdata, bins=xbins, **style)
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42-
# plot the xdata locations on the x axis:
43-
ax.plot(xdata, 0*xdata, 'd')
44-
ax.set_ylabel('Number per bin')
45-
ax.set_xlabel('x bins (dx=1.0)')
48+
# plot the xdata events:
49+
ax.eventplot(xdata, orientation='vertical', color='C1', alpha=.5)
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51+
ax.set(xlabel='Number per bin', ylabel='x bins (dx=1.0)', title='histogram')
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# %%
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# Modifying bins
49-
# ==============
54+
# Choose bins
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# ===========
5056
#
5157
# Changing the bin size changes the shape of this sparse histogram, so its a
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# good idea to choose bins with some care with respect to your data. Here we
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# make the bins half as wide.
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# good idea to choose bins with some care with respect to your data. The `.Axes.hist`
59+
# *bins* parameter accepts either the number of bins or a list of bin edges.
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#
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#
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# Set *bins* using fixed edges
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# ----------------------------
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#
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# Here the bins are set to the list of edges [1, 1.5, 2, 2.5, 3, 3.5, 4].
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# This is half as wide as the previous example.
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xbins = np.arange(1, 4.5, 0.5)
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57-
fig, ax = plt.subplots()
70+
fig, ax = plt.subplots(layout='constrained', figsize=(8, 4))
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5872
ax.hist(xdata, bins=xbins, **style)
59-
ax.plot(xdata, 0*xdata, 'd')
60-
ax.set_ylabel('Number per bin')
61-
ax.set_xlabel('x bins (dx=0.5)')
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ax.eventplot(xdata, orientation='vertical', color='C1', alpha=.5)
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ax.set(ylabel='cpunt', xlabel='x bins (dx=0.5)',
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title='fixed bin edges: bins=np.arange(1, 4.5, .5)',)
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# %%
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#
81+
# Set *bins* using number of bins
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# -------------------------------
83+
#
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# We can also let numpy (via Matplotlib) choose the bins automatically, or
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# specify a number of bins to choose automatically:
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67-
fig, ax = plt.subplot_mosaic([['auto', 'n4']],
68-
sharex=True, sharey=True, layout='constrained')
87+
fig, ax = plt.subplot_mosaic([['auto'], ['n4']],
88+
sharex=True, sharey=True,
89+
layout='constrained', figsize=(8, 4))
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7091
ax['auto'].hist(xdata, **style)
71-
ax['auto'].plot(xdata, 0*xdata, 'd')
72-
ax['auto'].set_ylabel('Number per bin')
73-
ax['auto'].set_xlabel('x bins (auto)')
92+
ax['auto'].eventplot(xdata, orientation='vertical', color='C1', alpha=.5)
93+
94+
ax['auto'].set(ylabel='count', xlabel='x bins',
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title='dynamically computed bin edges: bins="auto"')
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ax['n4'].hist(xdata, bins=4, **style)
76-
ax['n4'].plot(xdata, 0*xdata, 'd')
77-
ax['n4'].set_xlabel('x bins ("bins=4")')
98+
ax['n4'].eventplot(xdata, orientation='vertical', color='C1', alpha=.5)
99+
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ax['n4'].set(ylabel='count', xlabel='x bins',
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title='fixed number of bins: bins=4',)
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# %%
80-
# Normalizing histograms: density and weight
81-
# ==========================================
104+
# Normalize histogram
105+
# ===================
82106
#
83107
# Counts-per-bin is the default length of each bar in the histogram. However,
84108
# we can also normalize the bar lengths as a probability density function using
85109
# the ``density`` parameter:
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87-
fig, ax = plt.subplots()
111+
fig, ax = plt.subplots(layout='constrained', figsize=(8, 4))
112+
88113
ax.hist(xdata, bins=xbins, density=True, **style)
89-
ax.set_ylabel('Probability density [$V^{-1}$])')
90-
ax.set_xlabel('x bins (dx=0.5 $V$)')
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115+
ax.set(ylabel='Probability density [$V^{-1}$])',
116+
xlabel='x bins (dx=0.5 $V$)',
117+
title='normalizing histogram using density')
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92119
# %%
93120
# This normalization can be a little hard to interpret when just exploring the
94121
# data. The value attached to each bar is divided by the total number of data
95-
# points *and* the width of the bin, and thus the values _integrate_ to one
122+
# points *and* the width of the bin, and thus the values *integrate* to one
96123
# when integrating across the full range of data.
97124
# e.g. ::
98125
#
@@ -117,127 +144,154 @@
117144
pdf = 1 / (np.sqrt(2 * np.pi)) * np.exp(-xpdf**2 / 2)
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119146
# %%
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# *density* parameter
148+
# -------------------
149+
#
120150
# If we don't use ``density=True``, we need to scale the expected probability
121151
# distribution function by both the length of the data and the width of the
122152
# bins:
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124-
fig, ax = plt.subplot_mosaic([['False', 'True']], layout='constrained')
125154
dx = 0.1
126155
xbins = np.arange(-4, 4, dx)
127-
ax['False'].hist(xdata, bins=xbins, density=False, histtype='step', label='Counts')
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157+
fig, ax = plt.subplot_mosaic([['False', 'True']], layout='constrained',
158+
figsize=(8, 4))
159+
160+
161+
ax['False'].hist(xdata, bins=xbins, density=False, histtype='step', label='Counts')
129162
# scale and plot the expected pdf:
130-
ax['False'].plot(xpdf, pdf * len(xdata) * dx, label=r'$N\,f_X(x)\,\delta x$')
131-
ax['False'].set_ylabel('Count per bin')
132-
ax['False'].set_xlabel('x bins [V]')
133-
ax['False'].legend()
163+
ax['False'].plot(xpdf, pdf * len(xdata) * dx, label=r'$N\,f_X(x)\,\delta x$', alpha=.5)
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135166
ax['True'].hist(xdata, bins=xbins, density=True, histtype='step', label='density')
136-
ax['True'].plot(xpdf, pdf, label='$f_X(x)$')
137-
ax['True'].set_ylabel('Probability density [$V^{-1}$]')
138-
ax['True'].set_xlabel('x bins [$V$]')
167+
ax['True'].plot(xpdf, pdf, label='$f_X(x)$', alpha=.5)
168+
169+
170+
ax['False'].set(ylabel='Count per bin', xlabel='x bins [V]',
171+
title="normalization using scaling, density=False")
172+
ax['False'].legend()
173+
ax['True'].set(ylabel='Probability density [$V^{-1}$]', xlabel='x bins [$V$]',
174+
title="density=True")
139175
ax['True'].legend()
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141177
# %%
142-
# One advantage of using the density is therefore that the shape and amplitude
143-
# of the histogram does not depend on the size of the bins. Consider an
144-
# extreme case where the bins do not have the same width. In this example, the
145-
# bins below ``x=-1.25`` are six times wider than the rest of the bins. By
178+
# Preserving distribution shape
179+
# -----------------------------
180+
# One advantage of using the density is that the shape and amplitude of the histogram
181+
# does not depend on the size of the bins.
182+
#
183+
# Irregularly spaced bins
184+
# ^^^^^^^^^^^^^^^^^^^^^^^
185+
# Consider an extreme case where the bins do not have the same width. In this example,
186+
# the bins below ``x=-1.25`` are six times wider than the rest of the bins. By
146187
# normalizing by density, we preserve the shape of the distribution, whereas if
147188
# we do not, then the wider bins have much higher counts than the thinner bins:
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149-
fig, ax = plt.subplot_mosaic([['False', 'True']], layout='constrained')
150190
dx = 0.1
151191
xbins = np.hstack([np.arange(-4, -1.25, 6*dx), np.arange(-1.25, 4, dx)])
192+
193+
fig, ax = plt.subplot_mosaic([['False', 'True']],
194+
layout='constrained', figsize=(8, 4))
195+
196+
152197
ax['False'].hist(xdata, bins=xbins, density=False, histtype='step', label='Counts')
153-
ax['False'].plot(xpdf, pdf * len(xdata) * dx, label=r'$N\,f_X(x)\,\delta x_0$')
154-
ax['False'].set_ylabel('Count per bin')
155-
ax['False'].set_xlabel('x bins [V]')
156-
ax['False'].legend()
198+
ax['False'].plot(xpdf, pdf * len(xdata) * dx, label=r'$N\,f_X(x)\,\delta x_0$', alpha=.5)
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158200
ax['True'].hist(xdata, bins=xbins, density=True, histtype='step', label='density')
159-
ax['True'].plot(xpdf, pdf, label='$f_X(x)$')
160-
ax['True'].set_ylabel('Probability density [$V^{-1}$]')
161-
ax['True'].set_xlabel('x bins [$V$]')
201+
ax['True'].plot(xpdf, pdf, label='$f_X(x)$', alpha=.5)
202+
203+
204+
ax['False'].set(ylabel='Count per bin', xlabel='x bins [V]',
205+
title="irregularly spaced bins, density=False")
206+
ax['False'].legend()
207+
208+
ax['True'].set(ylabel='Probability density [$V^{-1}$]', xlabel='x bins [$V$]',
209+
title="irregularly spaced bins, density=True",)
162210
ax['True'].legend()
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164212
# %%
213+
# Histograms with different bin widths
214+
# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
165215
# Similarly, if we want to compare histograms with different bin widths, we may
166216
# want to use ``density=True``:
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168-
fig, ax = plt.subplot_mosaic([['False', 'True']], layout='constrained')
218+
fig, ax = plt.subplot_mosaic([['False', 'True']],
219+
layout='constrained', figsize=(8, 4))
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170221
# expected PDF
171222
ax['True'].plot(xpdf, pdf, '--', label='$f_X(x)$', color='k')
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173224
for nn, dx in enumerate([0.1, 0.4, 1.2]):
174225
xbins = np.arange(-4, 4, dx)
175226
# expected histogram:
176-
ax['False'].plot(xpdf, pdf*1000*dx, '--', color=f'C{nn}')
227+
ax['False'].plot(xpdf, pdf*1000*dx, '--', color=f'C{nn}', alpha=.5)
177228
ax['False'].hist(xdata, bins=xbins, density=False, histtype='step')
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179-
ax['True'].hist(xdata, bins=xbins, density=True, histtype='step', label=dx)
230+
ax['True'].hist(xdata, bins=xbins, density=True,
231+
histtype='step', label=dx, alpha=style['alpha'])
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181233
# Labels:
182-
ax['False'].set_xlabel('x bins [$V$]')
183-
ax['False'].set_ylabel('Count per bin')
184-
ax['True'].set_ylabel('Probability density [$V^{-1}$]')
185-
ax['True'].set_xlabel('x bins [$V$]')
234+
ax['False'].set(ylabel='Count per bin', xlabel='x bins [$V$]',
235+
title="density=False")
236+
ax['True'].set(ylabel='Probability density [$V^{-1}$]', xlabel='x bins [$V$]',
237+
title='density=True')
186238
ax['True'].legend(fontsize='small', title='bin width:')
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188240
# %%
241+
# Assign weights
242+
# ==============
243+
#
189244
# Sometimes people want to normalize so that the sum of counts is one. This is
190245
# analogous to a `probability mass function
191246
# <https://en.wikipedia.org/wiki/Probability_mass_function>`_ for a discrete
192-
# variable where the sum of probabilities for all the values equals one. Using
193-
# ``hist``, we can get this normalization if we set the *weights* to 1/N.
247+
# variable where the sum of probabilities for all the values equals one.
248+
#
249+
# *weights* parameter
250+
# -------------------
251+
# Using ``hist``, we can get this normalization if we set the *weights* to 1/N.
194252
# Note that the amplitude of this normalized histogram still depends on
195-
# width and/or number of the bins:
253+
# width and/or number of bins:
196254

197-
fig, ax = plt.subplots(layout='constrained', figsize=(3.5, 3))
255+
fig, ax = plt.subplots(layout='constrained', figsize=(8, 4))
198256

199257
for nn, dx in enumerate([0.1, 0.4, 1.2]):
200258
xbins = np.arange(-4, 4, dx)
201259
ax.hist(xdata, bins=xbins, weights=1/len(xdata) * np.ones(len(xdata)),
202260
histtype='step', label=f'{dx}')
203-
ax.set_xlabel('x bins [$V$]')
204-
ax.set_ylabel('Bin count / N')
261+
262+
ax.set(ylabel='Bin count / N', xlabel='x bins [$V$]',
263+
title="histogram normalization using weights")
205264
ax.legend(fontsize='small', title='bin width:')
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207266
# %%
267+
# Populations of different sizes
268+
# ------------------------------
208269
# The value of normalizing histograms is comparing two distributions that have
209-
# different sized populations. Here we compare the distribution of ``xdata``
270+
# different sized populations. Here we compare the distribution of ``xdata``
210271
# with a population of 1000, and ``xdata2`` with 100 members.
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212273
xdata2 = rng.normal(size=100)
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214-
fig, ax = plt.subplot_mosaic([['no_norm', 'density', 'weight']],
215-
layout='constrained', figsize=(8, 4))
275+
fig, ax = plt.subplot_mosaic([['no_norm'], ['density'], ['weight']],
276+
layout='constrained', figsize=(8,2))
216277

217278
xbins = np.arange(-4, 4, 0.25)
218279

219-
ax['no_norm'].hist(xdata, bins=xbins, histtype='step')
220-
ax['no_norm'].hist(xdata2, bins=xbins, histtype='step')
221-
ax['no_norm'].set_ylabel('Counts')
222-
ax['no_norm'].set_xlabel('x bins [$V$]')
223-
ax['no_norm'].set_title('No normalization')
224-
225-
ax['density'].hist(xdata, bins=xbins, histtype='step', density=True)
226-
ax['density'].hist(xdata2, bins=xbins, histtype='step', density=True)
227-
ax['density'].set_ylabel('Probability density [$V^{-1}$]')
228-
ax['density'].set_title('Density=True')
229-
ax['density'].set_xlabel('x bins [$V$]')
230-
231-
ax['weight'].hist(xdata, bins=xbins, histtype='step',
232-
weights=1 / len(xdata) * np.ones(len(xdata)),
233-
label='N=1000')
234-
ax['weight'].hist(xdata2, bins=xbins, histtype='step',
235-
weights=1 / len(xdata2) * np.ones(len(xdata2)),
236-
label='N=100')
237-
ax['weight'].set_xlabel('x bins [$V$]')
238-
ax['weight'].set_ylabel('Counts / N')
280+
for xd in [xdata, xdata2]:
281+
ax['no_norm'].hist(xd, bins=xbins, histtype='step')
282+
ax['density'].hist(xd, bins=xbins, histtype='step', density=True)
283+
ax['weight'].hist(xd, bins=xbins, histtype='step',
284+
weights=1 / len(xd) * np.ones(len(xd)),
285+
label=f'N={len(xd)}')
286+
287+
288+
ax['no_norm'].set(ylabel='Counts', xlabel='x bins [$V$]',
289+
title='No normalization')
290+
ax['density'].set(ylabel='Probability density [$V^{-1}$]', xlabel='x bins [$V$]',
291+
title='Density=True')
292+
ax['weight'].set(ylabel='Counts / N', xlabel='x bins [$V$]',
293+
title='Weight = 1/N')
239294
ax['weight'].legend(fontsize='small')
240-
ax['weight'].set_title('Weight = 1/N')
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242296
plt.show()
243297

@@ -253,3 +307,4 @@
253307
# - `matplotlib.axes.Axes.set_xlabel`
254308
# - `matplotlib.axes.Axes.set_ylabel`
255309
# - `matplotlib.axes.Axes.legend`
310+
#

‎galleries/tutorials/index.rst

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Original file line numberDiff line numberDiff line change
@@ -95,6 +95,7 @@ a :ref:`FAQ <faq-index>` in our :ref:`user guide <users-guide-index>`.
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/tutorials/images
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/tutorials/lifecycle
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/tutorials/artists
98+
/tutorials/histogram_normalization
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.. only:: html
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@@ -134,6 +135,7 @@ Intermediate
134135
- :ref:`arranging_axes`
135136
- :ref:`autoscale`
136137
- :ref:`imshow_extent`
138+
- :ref:`histogram_normalization`
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138140
Advanced
139141
^^^^^^^^

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