I recently was the TA for a Monte Carlo methods course which involved sampling many statistical distributions. Often students would write standalone functions to draw samples which was prone to error. While knowing how to code up an exotic distribution can be useful, I encourage students not to re-invent the wheel every assignment. The scipy.stats package contains many probability distributions and useful functions when doing statistics.

%matplotlib inline
%config InlineBackend.figure_format = 'svg'
from scipy import stats
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

Normal Distribution

We will start by exploring the Normal distribution.

\[\mathcal{N}(x\;|\;\mu ,\sigma ^{2})={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\;e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}\]

All statistical distribution objects in scipy.stats package contain useful methods. The norm class has:

  • pdf(x) to calculate the Probability density function at x.
  • cdf(x) to calculate the Cumulative distribution function at x.
  • interval(alpha) to calculate the endpoints of the range that contains alpha percent of the distribution
  • rvs(size) to generate random variates of arbitrary size.
  • fit(data) to estimate the parameters for generic data.

and many more

The stats.norm class by default creates a standard normal distribution with mean 0 and variance of 1.

dist = stats.norm()

print('Distribution Mean: \t{0:.2f}'.format(dist.mean()))
print('Distribution Varience: \t{0:.2f}'.format(dist.var()))
Distribution Mean: 	0.00
Distribution Varience: 	1.00


Let’s plot the PDF and CDF to get a better understanding

fig, axs = plt.subplots(2,1, figsize=(5,6))

x = np.linspace(-5, 5, 200)

axs[0].plot(x, dist.pdf(x))
axs[0].set(title='Probability Density Function',

axs[1].plot(x, dist.cdf(x))
axs[1].set(title='Cumulative Distribution Function', 


Location and Shape Parameters

We can pass to the distribution object a parameter loc which controls the mean and scale which controls the standard deviation. Lets create a variety of distributions by varying these parameters

parameters =pd.DataFrame([
    [0,  0.2],
    [0,  1.0],
    [0,  5.0], 
    [-2, 0.5]], columns=('Mean', 'Varience'))
Mean Varience
0 0 0.2
1 0 1.0
2 0 5.0
3 -2 0.5
fig, axs = plt.subplots(2,1, figsize=(7,9))

for _, mean, var in parameters.itertuples():
    dist = stats.norm(loc=mean, scale=np.sqrt(var))
    axs[0].plot(x, dist.pdf(x), 
                label = r'$\mu = {},\quad \sigma^2 = {}$'.format(mean, var))
    axs[0].set(title='Probability Density Function',
    axs[1].plot(x, dist.cdf(x),
                label = r'$\mu = {},\quad \sigma^2 = {}$'.format(mean, var))
    axs[1].set(title='Cumulative Distribution Function', 


Random Variate Samples

To sample a distribution the rvs method is used. For reproducibility I will set the random_state.

dist = stats.norm()
samples = dist.rvs(1000, random_state=0)

plt.hist(samples, bins=50);


Numpy Random

If you are only interested in random variate samples sometimes it is easier to use the numpy.random module (See numpy docs) which also contains many distributions. By setting the seed we can create the same results as above

samples = np.random.normal(size=1000)
plt.hist(samples, bins=50);


Parameter Fitting

Often we are interested in fitting the parameters of a distribution to data we have collected. Lets generate some data first.

data = stats.norm.rvs(loc=5, scale=3, size=1000, random_state=42)

The fit method returns the shape parameters that best fit the given data

mu, std =
print('Best fit mean: \t{}'.format(mu))
print('Best fit std: \t{}'.format(std))
Best fit mean: 	5.0579961674669764
Best fit std: 	2.9361786232420632

We can use these values to overlay the fitted distribution onto a normalised histogram of the data

# Plotting the histogram of the data
plt.hist(data, bins=50, density=True, alpha=.8)

# Plotting the PDF of the fitted distribuiton
xmin, xmax = plt.xlim()
x = np.linspace(xmin, xmax, 100)
fitted_dist = stats.norm(mu, std)
plt.plot(x, fitted_dist.pdf(x), 'k--', linewidth=2)
plt.title("Fit: $\mu$ = {:.2f},  $\sigma$ = {:.2f}".format(mu, std));


Shaded Distributions

A common visualization is a shaded normal distribution to highlight the tails. Let’s take the fitted distribution from the previous example and fill in the 95% confidence regions. I made a simple fill_dist function to aid in the task.

def fill_dist(x1, x2, dist,  **kwargs):
    """Shade a distribution between x1, and x2"""
    x_range = np.linspace(x1, x2)
    plt.fill_between(x_range, dist.pdf(x_range), alpha=0.5, **kwargs)

The interval function returns the bounds which include a given percentage of the data

lower, upper = fitted_dist.interval(.95)
plt.plot(x, fitted_dist.pdf(x), 'k')
fill_dist(lower, upper, fitted_dist, color='C0', label='Included')
fill_dist(upper, xmax, fitted_dist, color='C3', label='Excluded')
fill_dist(xmin, lower, fitted_dist, color='C3')
plt.vlines(fitted_dist.mean(), 0, 
           fitted_dist.pdf(fitted_dist.mean()), linestyles='dashed')
plt.title('95% Confidence Region')
<matplotlib.legend.Legend at 0x11ab60eb8>


Other Distributions

There are many different distributions we can choose from such as the Chi Squared distribution.

\[P(x) = {\frac {1}{2^{k/2}\Gamma (k/2)}}\;x^{\frac {k}{2}-1}e^{-{\frac {x}{2}}}\]

Often it can be useful to vary a shape parameter and see how it affects the distribution. For example

x = np.linspace(0, 8, 100)
for i in range(1, 7):
    dist = stats.chi2(i)
    plt.plot(x, dist.pdf(x), label=r'$k={}$'.format(i))
    plt.ylim(-.01, 0.5)

plt.title('$\chi^2_k$ for k degrees of freedom')
Text(0.5,1,'$\\chi^2_k$ for k degrees of freedom')


Multivariate Normal

The final distribution I wanted to discuss was the multivariate_normal distribution. The probability density function is defined as \(f(x) = \frac{1}{\sqrt{(2 \pi)^k \det \Sigma}} \exp\left( -\frac{1}{2} (x - \mu)^T \Sigma^{-1} (x - \mu) \right),\) where \(\mu\) is the mean, \(\Sigma\) the covariance matrix, and \(k\) is the dimension of the space where \(x\) takes values.

mu = [2,1]
sigma = [[1.3, .3], 
          [0.3, .5]]
N = stats.multivariate_normal(mu, sigma)

I would like to create a contour plot of this distribution. First we will generate the grid of \(x\) and \(y\) coordinates using np.meshgrid.

x = np.linspace(-1, 5, 200)
y = np.linspace(-1, 3, 200)
X, Y = np.meshgrid(x, y)

From the documentation it reads

The inputs can be any shape of array, as long as the last axis labels the components.

In order to calculate our pdf at every \(xy\) point we need to stack the data into an \(NxMx2\) array. This can be done with np.dstack which concatenates 2D arrays along a third axis

samples = np.dstack((X, Y))

Now we can call plt.contourf with the pdf as the \(Z\) component

plt.contourf(X, Y, N.pdf(samples))
<matplotlib.contour.QuadContourSet at 0x11ab5d470>


Consider the following mixture of 2D distributions,

\[\pi(x,y) = 0.6\mathcal{N}(\mu_1, \Sigma_1) + 0.4\mathcal{N}(\mu_2, \Sigma_2)\]


\[\begin{align} \mu_1 &= \begin{bmatrix}3\\ 0\end{bmatrix}, \quad &\Sigma_1 &= \begin{bmatrix} 1.5 & 0 \\ 0 & 0.5 \end{bmatrix}\\ \mu_2 &= \begin{bmatrix}-1.25\\ 2.5\end{bmatrix}, \quad &\Sigma_2 &= \begin{bmatrix} 0.5 & -0.6 \\ -0.6 & 1.0 \end{bmatrix} \end{align}\]
mu1 = [3,0]
sigma1 = [[1.5, 0], 
          [0, 0.5]]
N1 = stats.multivariate_normal(mu1, sigma1)

mu2 = [-1.25,2.5]
sigma2 = [[0.5, -0.6], 
          [-0.6, 1.0]]
N2 = stats.multivariate_normal(mu2, sigma2)

Now we can define a function to sample the PDF

def pi(sample):
    """A mixed Gaussian model."""
    return 0.6*N1.pdf(sample) + 0.4*N2.pdf(sample)

Plotting in the same manner as before but now calling pi instead.

x = np.linspace(-4, 6, num=100)
y = np.linspace(-2, 6, num=100)
X, Y = np.meshgrid(x, y)
plt.contourf(x, y, pi(np.dstack((X, Y))), cmap='coolwarm')
<matplotlib.contour.QuadContourSet at 0x11b10c4a8>



Sampling and exploring a wide variety of statistical distributions is made easier with the scipy.stats module. Creating your own functions each time you want to draw a sample is error prone and inefficient. I think it’s important to know what tools are available and I hope this post shines some light on one useful example.

I hope to do more scipy related posts in the future.