It’s important to plot distributions of variables when doing exploratory analysis. However, there may be times when you want to see the theoretical distribution on a plot, i.e. when you want to see how much your variable deviates from it, or when you want to decide on a distribution function visually.
Let’s take the normal (gaussian) distribution as an example. The probability density function (pdf) is:
$ f(x|\mu,\sigma^2)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}} $
Basically, if we have a range of $x$’s, a mean ($\mu$) and a standard deviation ($\sigma$), we can pass them onto this formula and get corresponding $y$ values, which we can then plot using the standard matplotlib plot() function:
Let’s setup the scene first:
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
plt.style.use('seaborn') # pretty matplotlib plots
plt.rcParams['figure.figsize'] = (12, 8)
Let’s get our x values, determine a mean and a standard deviation, and setup the formula for the normal pdf:
x = np.linspace(-5, 5, 5000)
mu = 0
sigma = 1
y = (1 / (np.sqrt(2 * np.pi * np.power(sigma, 2)))) * \
(np.power(np.e, -(np.power((x - mu), 2) / (2 * np.power(sigma, 2)))))
Now we can plot these using:
plt.plot(x, y);
Which is fine and dandy, but it gets quite cumbersome to write those formulas from scratch using numpy and scipy functions for every distribution we want. Some are even really hard to implement, take for example the cumulative distribution function (cdf) for the standard normal distribution:
$ \Phi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty }^{x}e^{-t^{2}/2}\,{\rm {d}}t $
Fortunately for us, the people at scipy provided nearly every kind of distribution function in the scipy.stats package. Using that, we can achieve the same result as above in a cleaner, less error-prone code. We can even plot the cdf on top of that:
import scipy.stats as ss
x = np.linspace(-5, 5, 5000)
mu = 0
sigma = 1
y_pdf = ss.norm.pdf(x, mu, sigma) # the normal pdf
y_cdf = ss.norm.cdf(x, mu, sigma) # the normal cdf
plt.plot(x, y_pdf, label='pdf')
plt.plot(x, y_cdf, label='cdf')
plt.legend();
For reuse, it may be a good idea to put these into a function:
import scipy.stats as ss
def plot_normal(x_range, mu=0, sigma=1, cdf=False, **kwargs):
'''
Plots the normal distribution function for a given x range
If mu and sigma are not provided, standard normal is plotted
If cdf=True cumulative distribution is plotted
Passes any keyword arguments to matplotlib plot function
'''
x = x_range
if cdf:
y = ss.norm.cdf(x, mu, sigma)
else:
y = ss.norm.pdf(x, mu, sigma)
plt.plot(x, y, **kwargs)
x = np.linspace(-5, 5, 5000)
plot_normal(x)
plot_normal(x, cdf=True)
plot_normal(x, -2, 1, color='red', lw=2, ls='-', alpha=0.5)
plot_normal(x, 2, 1.2, color='blue', lw=2, ls='-', alpha=0.5)
plot_normal(x, 0, 0.8, color='green', lw=2, ls='-', alpha=0.5);
Given this knowledge, we can now define a function for plotting any kind of distribution. The important bit is to be careful about the parameters of the corresponding scipy.stats function (Some distributions require more than a mean and a standard deviation). You can check those parameters on the official docs for scipy.stats.
The exponential distribution:
import scipy.stats as ss
def plot_exponential(x_range, mu=0, sigma=1, cdf=False, **kwargs):
'''
Plots the exponential distribution function for a given x range
If mu and sigma are not provided, standard exponential is plotted
If cdf=True cumulative distribution is plotted
Passes any keyword arguments to matplotlib plot function
'''
x = x_range
if cdf:
y = ss.expon.cdf(x, mu, sigma)
else:
y = ss.expon.pdf(x, mu, sigma)
plt.plot(x, y, **kwargs)
x = np.linspace(0, 5, 5000)
plot_exponential(x, 0, 1, color='red', lw=2, ls='-', alpha=0.5, label='pdf')
plot_exponential(x, 0, 1, cdf=True, color='blue', lw=2, ls='-', alpha=0.5, label='cdf')
plt.legend();
The F distribution:
import scipy.stats as ss
def plot_f(x_range, dfn, dfd, mu=0, sigma=1, cdf=False, **kwargs):
'''
Plots the f distribution function for a given x range, dfn and dfd
If mu and sigma are not provided, standard f is plotted
If cdf=True cumulative distribution is plotted
Passes any keyword arguments to matplotlib plot function
'''
x = x_range
if cdf:
y = ss.f.cdf(x, dfn, dfd, mu, sigma)
else:
y = ss.f.pdf(x, dfn, dfd, mu, sigma)
plt.plot(x, y, **kwargs)
x = np.linspace(0.001, 5, 5000)
plot_f(x, 10, 10, 0, 1, color='red', lw=2, ls='-', alpha=0.5, label='pdf')
plot_f(x, 10, 10, 0, 1, cdf=True, color='blue', lw=2, ls='-', alpha=0.5, label='cdf')
plt.legend();
The beta distribution:
import scipy.stats as ss
def plot_beta(x_range, a, b, mu=0, sigma=1, cdf=False, **kwargs):
'''
Plots the f distribution function for a given x range, a and b
If mu and sigma are not provided, standard beta is plotted
If cdf=True cumulative distribution is plotted
Passes any keyword arguments to matplotlib plot function
'''
x = x_range
if cdf:
y = ss.beta.cdf(x, a, b, mu, sigma)
else:
y = ss.beta.pdf(x, a, b, mu, sigma)
plt.plot(x, y, **kwargs)
x = np.linspace(0, 1, 5000)
plot_beta(x, 5, 2, 0, 1, color='red', lw=2, ls='-', alpha=0.5, label='pdf')
plot_beta(x, 5, 2, 0, 1, cdf=True, color='blue', lw=2, ls='-', alpha=0.5, label='cdf')
plt.legend();
As you see, we can extend these as far as we like. These theoretical distributions are important to assess visually and get yourself familiarized with. It’s a powerful tool in a data scientist’s belt to determine the distribution of any variable just by looking at its histogram or KDE.
Let me know what you think, hit me up at @emredjan.
