Random walks in Python

 

Random Walks in Python using Matplotlib

In this blog, we’ll use Python to generate data for a random walk, and then use Matplotlib to create a visual representation of that data. A random walk is a path that has no clear direction but is determined by a series of random decisions, each of which is left entirely to chance. Take for example the path a confused ant would take if it took every step in a random direction. Random walk models are used in many real-world situations. Here is a real-world application of random walks.

Importing the modules

For this, we require two modules; Matplotlib and Random

import matplotlib.pyplot as plt

from random import choice

To make random decisions, we’ll store possible moves in a list and use the choice() function, from the random module, to decide which move to make each time a step is taken. Hence the random module.

 The RandomWalk() Function

def RandomWalk(num_points):

    x_values = [0]

    y_values = [0]

When calling the function, we will require to pass in the number of points that the random walk will make. Then, we make two lists to hold the x- and y-values, and we start each walk at the point (0, 0).

def RandomWalk(num_points):

    x_values = [0]

    y_values = [0]

 

    """Keep taking steps until the walk reaches 

the desired length."""

    while len(x_values) < num_points:

 

        """Decide which direction to go and how far 

to go in that direction."""

        x_direction = choice([1, -1])

        x_distance = choice([0, 1, 2, 3, 4])

        x_step = x_direction * x_distance

 

        y_direction = choice([1, -1])

        y_distance = choice([0,1,2,3,4])

        y_step = y_direction * y_distance

 

        """Reject a move that goes no-where"""

        if x_step == 0 and y_step == 0:

                continue

 

        x = x_values[-1] + x_step

        y = y_values[-1] + y_step

 

        x_values.append(x)

        y_values.append(y)

 

Next, we set up a loop that runs until the walk is filled with the correct number of points. The main part of the loop tells Python how to simulate four random decisions: will the walk go right or left? How far will it go in that direction? Will it go up or down? How far will it go in that direction? We use choice([1, -1]) to choose a value for x_direction, which returns either 1 for right movement or −1 for left. Next, choice([0, 1, 2, 3, 4]) tells Python how far to move in that direction (x_distance) by randomly selecting an integer between 0 and 4. (The inclusion of a 0 allows us to take steps along the y-axis as well as steps that have moved along both axes.). We need to determine the length of each step in the x and y directions by multiplying the direction of movement by the distance chosen. A positive result for x_step means move right, a negative result means move left, and 0 means move vertically. A positive result for y_step means move up, negative means move down, and 0 means move horizontally. If the value of both x_step and y_step is 0, the walk doesn’t go anywhere, so we continue the loop to ignore this move. To get the next x-value for the walk, we add the value in x_step to the last value stored in x_values and do the same for the y-values. When we have these values, we append them to x_values and y_values.

     x_values.append(x)

     y_values.append(y)

 

    """plot the points in the walk"""

    plt.style.use("classic")

    fig, ax = plt.subplots(figsize=(15,9))

    point_numbers = range(num_points)

    ax.scatter(x_values, y_values, c=point_numbers,

 cmap=plt.cm.Blues, edgecolors       ='none', s=15)

    

    """Emphasize the first and last points"""

    ax.scatter(0,0, c='green', edgecolors='none', s=100)

    ax.scatter(x_values[-1], y_values[-1], c='red', 

edgecolors='none', s=100)

 

    ax.get_xaxis().set_visible(False)

    ax.get_yaxis().set_visible(False)

 

    plt.show()

We proceed to create a scatter plot using Matplotlib. First, is to specify the style that we will use. The variable fig represents the entire figure or collection of plots that are generated. The variable ax represents a single plot in the figure and is the variable used most of the time. In the ax.scatter function we pass the x and y values to be plotted. We also pass the color maps. Pyplot module includes a set of built-in color maps. To use one of these cmaps, you need to specify how Pyplot should assign a color to each point in the data set. Here is how to assign each point a color based on its y-value.

ax.scatter(x_values, y_values, c=point_numbers, 

cmap=plt.cm.Blues, edgecolors='none', s=15)

Pass a list of y-values to c, and then tell pyplot which color map to use using the color map argument. This code colors the points with lower y-values light blue and colors the points with higher y-values dark blue. We also emphasize the starting points and ending points of our plot. Using green for beginning and red for the end of the walk.

RandomWalk(5000)

Finally, we call the function with the number of points we desire to be scattered on our random walk.