# Chapter 13: Statistics¶

Robert Johansson

Source code listings for Numerical Python - Scientific Computing and Data Science Applications with Numpy, SciPy and Matplotlib (ISBN 978-1-484242-45-2).

## Imports¶

In [1]:
from scipy import stats

In [2]:
from scipy import optimize

In [3]:
import numpy as np
import random

In [4]:
%matplotlib inline
import matplotlib.pyplot as plt

In [5]:
import seaborn as sns

In [6]:
sns.set(style="whitegrid")


## Descriptive statistics¶

In [7]:
x = np.array([3.5, 1.1, 3.2, 2.8, 6.7, 4.4, 0.9, 2.2])

In [8]:
np.mean(x)

Out[8]:
3.1
In [9]:
np.median(x)

Out[9]:
3.0
In [10]:
x.min(), x.max()

Out[10]:
(0.9, 6.7)
In [11]:
x.var()

Out[11]:
3.0700000000000007
In [12]:
x.std()

Out[12]:
1.7521415467935233
In [13]:
x.var(ddof=1)

Out[13]:
3.5085714285714293
In [14]:
x.std(ddof=1)

Out[14]:
1.8731181032095732

## Random numbers¶

In [15]:
random.seed(123456789)

In [16]:
random.random()

Out[16]:
0.6414006161858726
In [17]:
random.randint(0, 10)  # 0 and 10 inclusive

Out[17]:
8
In [18]:
np.random.seed(123456789)

In [19]:
np.random.rand()

Out[19]:
0.532833024789759
In [20]:
np.random.randn()

Out[20]:
0.8768342101492541
In [21]:
np.random.rand(5)

Out[21]:
array([0.71356403, 0.25699895, 0.75269361, 0.88387918, 0.15489908])
In [22]:
np.random.randn(2, 4)

Out[22]:
array([[ 3.13325952,  1.15727052,  1.37591514,  0.94302846],
[ 0.8478706 ,  0.52969142, -0.56940469,  0.83180456]])
In [23]:
np.random.randint(10, size=10)

Out[23]:
array([0, 3, 8, 3, 9, 0, 6, 9, 2, 7])
In [24]:
np.random.randint(low=10, high=20, size=(2, 10))

Out[24]:
array([[12, 18, 18, 17, 14, 12, 14, 10, 16, 19],
[15, 13, 15, 18, 11, 17, 17, 10, 13, 17]])
In [25]:
fig, axes = plt.subplots(1, 3, figsize=(12, 3))

axes[0].hist(np.random.rand(10000))
axes[0].set_title("rand")
axes[1].hist(np.random.randn(10000))
axes[1].set_title("randn")
axes[2].hist(np.random.randint(low=1, high=10, size=10000), bins=9, align='left')
axes[2].set_title("randint(low=1, high=10)")

fig.tight_layout()
fig.savefig("ch13-random-hist.pdf")

In [26]:
#random.sample(range(10), 5)

In [27]:
np.random.choice(10, 5, replace=False)

Out[27]:
array([9, 0, 5, 8, 1])
In [28]:
np.random.seed(123456789)

In [29]:
np.random.rand()

Out[29]:
0.532833024789759
In [30]:
np.random.seed(123456789); np.random.rand()

Out[30]:
0.532833024789759
In [31]:
np.random.seed(123456789); np.random.rand()

Out[31]:
0.532833024789759
In [32]:
prng = np.random.RandomState(123456789)

In [33]:
prng.randn(2, 4)

Out[33]:
array([[ 2.212902  ,  2.1283978 ,  1.8417114 ,  0.08238248],
[ 0.85896368, -0.82601643,  1.15727052,  1.37591514]])
In [34]:
prng.chisquare(1, size=(2, 2))

Out[34]:
array([[1.26859720e+00, 2.02731988e+00],
[2.52605129e-05, 3.00376585e-04]])
In [35]:
prng.standard_t(1, size=(2, 3))

Out[35]:
array([[ 0.59734384, -1.27669959,  0.09724793],
[ 0.22451466,  0.39697518, -0.19469463]])
In [36]:
prng.f(5, 2, size=(2, 4))

Out[36]:
array([[ 0.77372119,  0.1213796 ,  1.64779052,  1.21399831],
[ 0.45471421, 17.64891848,  1.48620557,  2.55433261]])
In [37]:
prng.binomial(10, 0.5, size=10)

Out[37]:
array([8, 3, 4, 2, 4, 5, 4, 4, 7, 5])
In [38]:
prng.poisson(5, size=10)

Out[38]:
array([7, 1, 3, 4, 6, 4, 9, 7, 3, 6])

# Probability distributions and random variables¶

In [39]:
np.random.seed(123456789)

In [40]:
X = stats.norm(1, 0.5)

In [41]:
X.mean()

Out[41]:
1.0
In [42]:
X.median()

Out[42]:
1.0
In [43]:
X.std()

Out[43]:
0.5
In [44]:
X.var()

Out[44]:
0.25
In [45]:
[X.moment(n) for n in range(5)]

Out[45]:
[1.0, 1.0, 1.25, 1.75, 2.6875]
In [46]:
X.stats()

Out[46]:
(array(1.), array(0.25))
In [47]:
X.pdf([0, 1, 2])

Out[47]:
array([0.10798193, 0.79788456, 0.10798193])
In [48]:
X.cdf([0, 1, 2])

Out[48]:
array([0.02275013, 0.5       , 0.97724987])
In [49]:
X.rvs(10)

Out[49]:
array([2.106451  , 2.0641989 , 1.9208557 , 1.04119124, 1.42948184,
0.58699179, 1.57863526, 1.68795757, 1.47151423, 1.4239353 ])
In [50]:
stats.norm(1, 0.5).stats()

Out[50]:
(array(1.), array(0.25))
In [51]:
stats.norm.stats(loc=2, scale=0.5)

Out[51]:
(array(2.), array(0.25))
In [52]:
X.interval(0.95)

Out[52]:
(0.020018007729972975, 1.979981992270027)
In [53]:
X.interval(0.99)

Out[53]:
(-0.2879146517744502, 2.28791465177445)
In [54]:
def plot_rv_distribution(X, axes=None):
"""Plot the PDF, CDF, SF and PPF of a given random variable"""
if axes is None:
fig, axes = plt.subplots(1, 3, figsize=(12, 3))

x_min_999, x_max_999 = X.interval(0.999)
x999 = np.linspace(x_min_999, x_max_999, 1000)

x_min_95, x_max_95 = X.interval(0.95)
x95 = np.linspace(x_min_95, x_max_95, 1000)

if hasattr(X.dist, 'pdf'):
axes[0].plot(x999, X.pdf(x999), label="PDF")
axes[0].fill_between(x95, X.pdf(x95), alpha=0.25)
else:
x999_int = np.unique(x999.astype(int))
axes[0].bar(x999_int, X.pmf(x999_int), label="PMF")
axes[1].plot(x999, X.cdf(x999), label="CDF")
axes[1].plot(x999, X.sf(x999), label="SF")
axes[2].plot(x999, X.ppf(x999), label="PPF")

for ax in axes:
ax.legend()

return axes

In [55]:
fig, axes = plt.subplots(3, 3, figsize=(12, 9))

X = stats.norm()
plot_rv_distribution(X, axes=axes[0, :])
axes[0, 0].set_ylabel("Normal dist.")
X = stats.f(2, 50)
plot_rv_distribution(X, axes=axes[1, :])
axes[1, 0].set_ylabel("F dist.")
X = stats.poisson(5)
plot_rv_distribution(X, axes=axes[2, :])
axes[2, 0].set_ylabel("Poisson dist.")

fig.tight_layout()
fig.savefig("ch13-distributions.pdf")

In [56]:
def plot_dist_samples(X, X_samples, title=None, ax=None):
""" Plot the PDF and histogram of samples of a continuous random variable"""
if ax is None:
fig, ax = plt.subplots(1, 1, figsize=(8, 4))

x_lim = X.interval(.99)
x = np.linspace(*x_lim, num=100)

ax.plot(x, X.pdf(x), label="PDF", lw=3)
ax.hist(X_samples, label="samples", normed=1, bins=75)
ax.set_xlim(*x_lim)
ax.legend()

if title:
ax.set_title(title)
return ax

In [57]:
fig, axes = plt.subplots(1, 3, figsize=(12, 3))
X = stats.t(7.0)
plot_dist_samples(X, X.rvs(2000), "Student's t dist.", ax=axes[0])
X = stats.chi2(5.0)
plot_dist_samples(X, X.rvs(2000), r"$\chi^2$ dist.", ax=axes[1])
X = stats.expon(0.5)
plot_dist_samples(X, X.rvs(2000), "exponential dist.", ax=axes[2])
fig.tight_layout()
fig.savefig("ch13-dist-sample.pdf")

/Users/rob/miniconda3/envs/py3.6/lib/python3.6/site-packages/matplotlib/axes/_axes.py:6521: MatplotlibDeprecationWarning:
The 'normed' kwarg was deprecated in Matplotlib 2.1 and will be removed in 3.1. Use 'density' instead.
alternative="'density'", removal="3.1")
/Users/rob/miniconda3/envs/py3.6/lib/python3.6/site-packages/matplotlib/axes/_axes.py:6521: MatplotlibDeprecationWarning:
The 'normed' kwarg was deprecated in Matplotlib 2.1 and will be removed in 3.1. Use 'density' instead.
alternative="'density'", removal="3.1")
/Users/rob/miniconda3/envs/py3.6/lib/python3.6/site-packages/matplotlib/axes/_axes.py:6521: MatplotlibDeprecationWarning:
The 'normed' kwarg was deprecated in Matplotlib 2.1 and will be removed in 3.1. Use 'density' instead.
alternative="'density'", removal="3.1")

In [58]:
X = stats.chi2(df=5)

In [59]:
X_samples = X.rvs(500)

In [60]:
df, loc, scale = stats.chi2.fit(X_samples)

In [61]:
df, loc, scale

Out[61]:
(4.728645123391404, 0.03257330219133387, 1.0734482977974253)
In [62]:
Y = stats.chi2(df=df, loc=loc, scale=scale)

In [63]:
fig, ax = plt.subplots(1, 1, figsize=(8, 3))

x_lim = X.interval(.99)
x = np.linspace(*x_lim, num=100)

ax.plot(x, X.pdf(x), label="original")
ax.plot(x, Y.pdf(x), label="recreated")
ax.legend()

fig.tight_layout()
fig.savefig("ch13-max-likelihood-fit.pdf")

In [64]:
fig, axes = plt.subplots(1, 2, figsize=(12, 4))

x_lim = X.interval(.99)
x = np.linspace(*x_lim, num=100)

axes[0].plot(x, X.pdf(x), label="original")
axes[0].plot(x, Y.pdf(x), label="recreated")
axes[0].legend()

axes[1].plot(x, X.pdf(x) - Y.pdf(x), label="error")
axes[1].legend()

fig.tight_layout()
fig.savefig("ch13-max-likelihood-fit.pdf")


# Hypothesis testing¶

In [65]:
np.random.seed(123456789)

In [66]:
mu, sigma = 1.0, 0.5

In [67]:
X = stats.norm(mu-0.2, sigma)

In [68]:
n = 100

In [69]:
X_samples = X.rvs(n)

In [70]:
z = (X_samples.mean() - mu)/(sigma/np.sqrt(n))

In [71]:
z

Out[71]:
-2.8338979550098298
In [72]:
t = (X_samples.mean() - mu)/(X_samples.std(ddof=1)/np.sqrt(n))

In [73]:
t

Out[73]:
-2.9680338545657845
In [74]:
stats.norm().ppf(0.025)

Out[74]:
-1.9599639845400545
In [75]:
2 * stats.norm().cdf(-abs(z))

Out[75]:
0.004598401329075357
In [76]:
2 * stats.t(df=(n-1)).cdf(-abs(t))

Out[76]:
0.003758647967422721
In [77]:
t, p = stats.ttest_1samp(X_samples, mu)

In [78]:
t

Out[78]:
-2.968033854565784
In [79]:
p

Out[79]:
0.003758647967422721
In [80]:
fig, ax = plt.subplots(figsize=(8, 3))

sns.distplot(X_samples, ax=ax)
x = np.linspace(*X.interval(0.999), num=100)
ax.plot(x, stats.norm(loc=mu, scale=sigma).pdf(x))

fig.tight_layout()
fig.savefig("ch13-hypothesis-test-dist-sample-mean.pdf")

In [81]:
n = 50

In [82]:
mu1, mu2 = np.random.rand(2)

In [83]:
mu1, mu2

Out[83]:
(0.24764580637159606, 0.42145435527527897)
In [84]:
X1 = stats.norm(mu1, sigma)

In [85]:
X1_sample = X1.rvs(n)

In [86]:
X2 = stats.norm(mu2, sigma)

In [87]:
X2_sample = X2.rvs(n)

In [88]:
t, p = stats.ttest_ind(X1_sample, X2_sample)

In [89]:
t

Out[89]:
-1.4283175246005888
In [90]:
p

Out[90]:
0.15637981059673237
In [91]:
mu1, mu2

Out[91]:
(0.24764580637159606, 0.42145435527527897)
In [92]:
sns.distplot(X1_sample)
sns.distplot(X2_sample)

Out[92]:
<matplotlib.axes._subplots.AxesSubplot at 0x1a2522c128>

# Nonparameteric methods¶

In [93]:
np.random.seed(0)

In [94]:
X = stats.chi2(df=5)

In [95]:
X_samples = X.rvs(100)

In [96]:
kde = stats.kde.gaussian_kde(X_samples)

In [97]:
kde_low_bw = stats.kde.gaussian_kde(X_samples, bw_method=0.25)

In [98]:
x = np.linspace(0, 20, 100)

In [99]:
fig, axes = plt.subplots(1, 3, figsize=(12, 3))

axes[0].hist(X_samples, normed=True, alpha=0.5, bins=25)
axes[1].plot(x, kde(x), label="KDE")
axes[1].plot(x, kde_low_bw(x), label="KDE (low bw)")
axes[1].plot(x, X.pdf(x), label="True PDF")
axes[1].legend()
sns.distplot(X_samples, bins=25, ax=axes[2])

fig.tight_layout()
fig.savefig("ch13-hist-kde.pdf")

/Users/rob/miniconda3/envs/py3.6/lib/python3.6/site-packages/matplotlib/axes/_axes.py:6521: MatplotlibDeprecationWarning:
The 'normed' kwarg was deprecated in Matplotlib 2.1 and will be removed in 3.1. Use 'density' instead.
alternative="'density'", removal="3.1")

In [100]:
kde.resample(10)

Out[100]:
array([[ 1.10979087,  0.4379679 , 14.20879078,  5.94683846,  1.78490438,
5.58416739,  4.18349885,  2.78527976,  0.68112826,  7.7643985 ]])
In [101]:
def _kde_cdf(x):
return kde.integrate_box_1d(-np.inf, x)

In [102]:
kde_cdf = np.vectorize(_kde_cdf)

In [103]:
fig, ax = plt.subplots(1, 1, figsize=(8, 3))

sns.distplot(X_samples, bins=25, ax=ax)
x = np.linspace(0, 20, 100)
ax.plot(x, kde_cdf(x))

fig.tight_layout()

In [104]:
def _kde_ppf(q):
return optimize.fsolve(lambda x, q: kde_cdf(x) - q, kde.dataset.mean(), args=(q,))[0]

In [105]:
kde_ppf = np.vectorize(_kde_ppf)

In [106]:
kde_ppf([0.05, 0.95])

Out[106]:
array([ 0.39074674, 11.94993578])
In [107]:
X.ppf([0.05, 0.95])

Out[107]:
array([ 1.14547623, 11.07049769])

# Versions¶

In [108]:
%reload_ext version_information

In [109]:
%version_information numpy, scipy, matplotlib, seaborn

Out[109]:
SoftwareVersion
Python3.6.8 64bit [GCC 4.2.1 Compatible Clang 4.0.1 (tags/RELEASE_401/final)]
IPython7.5.0
OSDarwin 18.2.0 x86_64 i386 64bit
numpy1.16.3
scipy1.2.1
matplotlib3.0.3
seaborn0.9.0
Mon May 06 15:41:15 2019 JST