# Sampling from common distributions

From the ground up!
ML
Author

Nipun Batra

Published

April 16, 2020

import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

### PRNG

#### Uniform distribution

I am pasting the code from vega-vis

export default function(seed) {
// Random numbers using a Linear Congruential Generator with seed value
// Uses glibc values from https://en.wikipedia.org/wiki/Linear_congruential_generator
return function() {
seed = (1103515245 * seed + 12345) % 2147483647;
return seed / 2147483647;
};
}
def random_gen(seed, num):
out = np.zeros(num)
out[0] =  (1103515245 * seed + 12345) % 2147483647
for i in range(1, num):
out[i] = (1103515245 * out[i-1] + 12345) % 2147483647
return out/2147483647


plt.hist(random_gen(0, 5000), density=True, alpha=0.5, label='Our implementation');
plt.hist(random_gen(0, 5000), density=True, alpha=0.5, label='Our implementation');
plt.hist(np.random.random(5000), density=True, alpha=0.4, label='Numpy implementation');
plt.legend()

### Inverse transform sampling

#### Exponential distribution

Borrowing from Wikipedia.

The probability density function (pdf) of an exponential distribution is $f(x ; \lambda)=\left\{\begin{array}{ll} \lambda e^{-\lambda x} & x \geq 0 \\ 0 & x \leq 0 \end{array}\right.$

The exponential distribution is sometimes parametrized in terms of the scale parameter $$\beta=1 / \lambda:$$ $f(x ; \beta)=\left\{\begin{array}{ll} \frac{1}{\beta} e^{-x / \beta} & x \geq 0 \\ 0 & x<0 \end{array}\right.$

The cumulative distribution function is given by $F(x ; \lambda)=\left\{\begin{array}{ll} 1-e^{-\lambda x} & x \geq 0 \\ 0 & x<0 \end{array}\right.$

from scipy.stats import expon
rvs = [expon(scale=s) for s in [1/1., 1/2., 1/3.]]
x = np.arange(0, 10, 0.1)
for i, lambda_val in enumerate([1, 2, 3]):
plt.plot(x, rvs[i].pdf(x), lw=2, label=r'$\lambda=%s$' %lambda_val)
plt.legend()

For the purposes of this notebook, I will be looking only at the standard exponential or set the scale to 1.

Let us now view the CDF of the standard exponential.

fig, ax = plt.subplots(nrows=2, sharex=True)
ax[0].plot(x, expon().pdf(x), lw=2)
ax[0].set_title("PDF")
ax[1].set_title("CDF")
ax[1].plot(x, expon().cdf(x), lw=2,)
r = expon.rvs(size=1000)
plt.hist(r, normed=True, bins=100)
plt.plot(x, expon().pdf(x), lw=2)
/home/nipunbatra-pc/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:1: MatplotlibDeprecationWarning:
The 'normed' kwarg was deprecated in Matplotlib 2.1 and will be removed in 3.1. Use 'density' instead.
"""Entry point for launching an IPython kernel.

Inverse of the CDF of exponential

The cumulative distribution function is given by $F(x ; \lambda)=\left\{\begin{array}{ll} 1-e^{-\lambda x} & x \geq 0 \\ 0 & x<0 \end{array}\right.$

Let us consider only $$x \geq 0$$.

Let $$u = F^{-1}$$ be the inverse of the CDF of $$F$$.

$u = 1-e^{-\lambda x} \\ 1- u = e^{-\lambda x} \\ \log(1-u) = -\lambda x\\ x = -\frac{\log(1-u)}{\lambda}$

def inverse_transform(lambda_val, num_samples):
u = np.random.random(num_samples)
x = -np.log(1-u)/lambda_val
return x
plt.hist(inverse_transform(1, 5000), bins=100, normed=True,label='Generated using our function');
plt.plot(x, expon().pdf(x), lw=4, label='Generated using scipy')
plt.legend()
/home/nipunbatra-pc/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:1: MatplotlibDeprecationWarning:
The 'normed' kwarg was deprecated in Matplotlib 2.1 and will be removed in 3.1. Use 'density' instead.
"""Entry point for launching an IPython kernel.

#### Drawing samples from Laplace distribution

A random variable has a Laplace $$(\mu, b)$$ distribution if its probability density function is $f(x | \mu, b)=\frac{1}{2 b} \exp \left(-\frac{|x-\mu|}{b}\right)$

$F^{-1}(u)=\mu-b \operatorname{sgn}(u-0.5) \ln (1-2|u-0.5|)$

def inverse_transform_laplace(b, mu, num_samples):
u = np.random.random(num_samples)
x = mu-b*np.sign(u-0.5)*np.log(1-2*np.abs(u-0.5))
return x
from scipy.stats import laplace
plt.hist(inverse_transform_laplace(1, 0, 5000),bins=100, density=True, label='Generated using \nour function');
x_n = np.linspace(-10, 10, 100)
plt.plot(x_n, laplace().pdf(x_n), lw=4, label='Generated from\n scipy')
plt.legend()