White noise

Random walk

Imagine that you repeatedly toss a coin. If it lands heads, you move one step forward. If it lands tails you move one step back. This is a random walk. The function below simulates this process.

import random
import numpy as np
import matplotlib.pyplot as plt
import matplotlib
from pylab import rcParams
matplotlib.font_manager.FontProperties(family='Helvetica',size=11)
rcParams['figure.figsize'] = 14/2.54, 14/2.54


def random_walk(w0,n):
    # n step random walk
    w = w0
    positions = []
    for k in range(n):
        positions.append(w)
        e = random.choice([-1, 1])
        w += e

    return np.array(positions)

Let’s simulate this nine times and plot the output.

def plotOverTime(ax, w):
    n=len(w)
    t=np.arange(n)
    ax.plot(t,w, '-',color='k')
    ax.set_xlabel('Time: t')

    ax.spines['top'].set_visible(False)
    ax.spines['right'].set_visible(False)
    ax.set_xticks(np.arange(0,n,step=25))
    ax.set_yticks(np.arange(-n,n+1,step=10))
    ax.set_xlim(0,n)
    ax.set_ylim(-2*np.sqrt(n),2*np.sqrt(n))

fig,axs=plt.subplots(3,3)
n=100
w=random_walk(0,n)
for axsr in axs:
    for j,ax in enumerate(axsr):
        w=random_walk(0,n)
        plotOverTime(ax, w)
        if j==0:
            ax.set_ylabel('Position: w')

Notice that the random walk doesn’t move very far. Over 100 time steps, it typically remains within 20 steps of its starting position.

Normal noise

In the above example the noise is generated by a coin toss. These outcomes are distributed accordining to, what is known as, the Bernoulli distribution. Other distributions can be used to generate noise. A common choice is the Normal (bell-shaped) distribution.

def normal_walk(w0,n,std):
    # n step random walk with Normal distribution
    w = w0
    positions = []
    for k in range(n):
        positions.append(w)
        e = np.random.normal(0, std)
        w += e

    return np.array(positions)

fig,axs=plt.subplots(3,3)
n=100
w=random_walk(0,n)
for axsr in axs:
    for j,ax in enumerate(axsr):
        w=normal_walk(0,n,1)
        plotOverTime(ax, w)
        if j==0:
            ax.set_ylabel('Position: w')

Notice that the step size changes each time step now, producing a less jerky movement.