Numpy Assignment Solution

import numpy as np
import matplotlib.pyplot as plt
import random

np.random.seed(12345)

Random Walks

The simulation of random walks provides an illustrative application of utilizing array operations. Let’s first consider a simple random walk starting at 0 with steps of 1 and –1 occurring with equal probability.

Here is a pure Python way to implement a single random walk with 1,000 steps using the built-in random module:

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)

plt.figure()
plt.plot(walk[:100])

[<matplotlib.lines.Line2D at 0x11eb25e20>]

Question 1

Now let's use numpy instead of using for loop to generate the array walk. You are expected to use three lines of code to generate a single random walk with 2,000 steps and save the position after each step in a numpy array walk.

• On line 1, generate 2,000 random binary numbers of 0 or 1 using np.random.randint and save it in an array steps.
• On line 2, use mask to change all 0 in steps to -1, which means walking to the -1 direction.
• On line 3, use cumsum method to calculate the current positions of the walk and save it to an array walk.

nsteps = 2000
# line 1: 
steps = np.random.randint(0, 2, size=nsteps)
# line 2
steps[steps==0] = -1
# line 3
walk = steps.cumsum()

Question 2

Replace the question mark and print out the statistics of the array walk on each line.

print("The most left (minimal) position of the walk is ", walk.min())
print("The most right (maximal) position of the walk is ", walk.max())
print("The average position of the walk is ", walk.mean())
print("The median position of the walk is ", np.median(walk))
print("The variance of the position of the walk is ", walk.var())

The most left (minimal) position of the walk is  -3
The most right (maximal) position of the walk is  40
The average position of the walk is  16.114
The median position of the walk is  16.0
The variance of the position of the walk is  79.79500399999999

Question 3

A more complicated statistic is the first crossing time, the step at which the random walk reaches a particular value. Here we might want to know how long it took the random walk to get at least 10 steps away from the origin 0 in either direction.

You are expected to use two lines of code to return the first crossing time that this walk needs to cross either 10 or -10.

• On line 1, use a boolean condition to generate a boolean array walk10 from walk to indicate if or not the walk crosses 10 or -10.

• On line 2, use np.argmax to return the index of the first True from walk10.

Note the method above may not be efficient since we only want to know the first crossing time in this question.

# line 1
walk10 = np.abs(walk) >= 10
# line 2
walk10.argmax()

37

Question 4

Simulating Many Random Walks at Once

If your goal was to simulate many random walks, say 5,000 of them, you can generate all of the random walks with minor modifications to the preceding code.

Please modify your previous code a little bit to compute all 5,000 random walks with 1000 steps in each and save the result in a two dimensional array walks, in which each row is one walk and each column represnts current position of the walk.

nwalks = 5000
nsteps = 1000
# line 1: modify the size to generate a two dimensional random array. 
steps = np.random.randint(0, 2, size=(nwalks, nsteps))
# line 2
steps[steps==0] = -1
# line 3: collapes steps along the column to calculate the cumulative sum of each row.
walks = steps.cumsum(axis = 1)

Question 5

Replace the question mark and print out the statistics of the array walks on each line.

print("The maximum value obtained over all of the walks is: ", walks.max())
print("The minimum value obtained over all of the walks is: ", walks.min())

The maximum value obtained over all of the walks is:  138
The minimum value obtained over all of the walks is:  -133

Out of these walks, let’s compute the first crossing time to 30 or –30 for each walk. This is slightly tricky because not all 5,000 of them reach 30. Please answer following two questions:

Question 6

• How many of these 5,000 walks cross 30 or -30 eventually? Please use numpy methods only and do not use for loop. (hint: you may use np.abs, np.any, and np.sum, and note that you can use np.any along axis.)

hits30 = (np.abs(walks) >= 30).any(axis = 1)
hits30
hits30.sum() # Number that hit 30 or -30

3411

Question 7

• What is the avarage of first crossing time of the walks who eventually cross 30 or -30? Please use numpy methods only and do not use for loop. （hint: you may use boolean index and np.argmax, and note that you can use argmax along axis.）

# method 1
crossing_times = (np.abs(walks) >= 30).argmax(axis = 1)
crossing_times[crossing_times != 0].mean() # do not count 0

499.00996775139254

# method 2
hits30 = (np.abs(walks) >= 30).any(axis = 1)
crossing_times = (np.abs(walks[hits30]) >= 30).argmax(axis = 1) # slice out walks who eventually cross 30/-30
crossing_times.mean()

499.00996775139254