5.3 - NumPy Application: Random Walks

Ha Khanh Nguyen (hknguyen)

• 1. What are Random Walks?
• 2. Using NumPy for Simulating Random Walks
• 3. Simulating Many Random Walks at Once

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1. What are Random Walks?

• Here is the Wikipedia page on Random Walk!
• In short, a random walk is a stochastic process.
• In this example, we will consider a simple random walk starting at 0 with steps of 1 and -1 occuring with equal probability.

import random

position = 0
walk = [position]
steps = 1000

for i in range(steps):
    step = 1 if random.randint(0, 1) else -1
    position += step
    walk.append(position)

• Now let's plot our walk!

• Before we can do plotting, first, you might need to install the matplotlib package by running the following command in the command line prompt (Terminal on Mac, Anaconda Prompt on Windows).

conda install matplotlib

import matplotlib.pyplot as plt

plt.plot(walk[:100])

[<matplotlib.lines.Line2D at 0x7fe2c2f8d5b0>]

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2. Using NumPy for Simulating Random Walks

• walk is simply the cumulative sum of the random steps and could be evaluated as an array expression.

import numpy as np
np.random.seed(430)

nsteps = 1000
draws = np.random.randint(0, 2, size=nsteps)
steps = np.where(draws > 0, 1, -1)
walk = steps.cumsum()

• Note: we have to use np.random.randint() to generate an array of random integers. random.randint() only returns 1 number at the time.
• Since we use 2 different random functions, even if we set the same seed, we won't get the same result (because the mechanisms inside the functions work differently).

• Here is the plot of new generated walk:

plt.plot(walk)

[<matplotlib.lines.Line2D at 0x7fe2c30934c0>]

• From this we can begin to extract statistics like the minimum and maximum value along the walk’s trajectory:

walk.min()

-23

walk.max()

15

• First crossing time = the step at which the random walk reaches a particular value. This is a more advanced statistic.
• Let's say we want to know how long it takes the random walk to get at least 10 steps away from the starting point in either direction.

(np.abs(walk) >= 10).argmax()

11

• np.abs(walk) >= 10 gives us a boolean array indicating where the walk has reached or exceeded 10.
• argmax() returns the first index of the maximum value of the array. In this case, it returns the index of the first True value.

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3. Simulating Many Random Walks at Once

• Say our goal is to generate 5000 random walks at once! How do we do that?

nwalks = 5000
nsteps = 1000
draws = np.random.randint(0, 2, size=(nwalks, nsteps))
steps = np.where(draws > 0, 1, -1)
walks = steps.cumsum(1) # sum across the columns
walks

array([[ -1,   0,   1, ..., -28, -27, -26],
       [ -1,   0,  -1, ...,  54,  55,  56],
       [ -1,  -2,  -3, ...,   2,   3,   2],
       ...,
       [ -1,   0,  -1, ..., -22, -21, -22],
       [  1,   2,   1, ...,  48,  49,  48],
       [  1,   0,   1, ..., -38, -39, -38]])

• Out of these walks, let's compute the minimum crossing time to 30 or -30!
• Now, note that not all of 5000 walks reach 30.

hits30 = (np.abs(walks) >= 30).any(1)
hits30

array([ True,  True, False, ...,  True,  True,  True])

• any(1) returns True if at least 1 of the values of the row is True.

# number of walks that hit 30 or -30
hits30.sum()

3336

# estimate for probability a walk hitting 30 in either direction
hits30.sum()/nwalks

0.6672

• Use the Boolean array hist30 to select only the walks that actually hit 30 or -30! Then use argmax() across axis 1 (the column) to get the crossing times:

crossing_times = (np.abs(walks[hits30]) >= 30).argmax(1)
crossing_times.mean()

511.1636690647482

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This lecture note is modified from Chapter 4 of Wes McKinney's Python for Data Analysis 2nd Ed.