Algorithms Exercise 2

Imports



In [59]:

    
%matplotlib inline
from matplotlib import pyplot as plt
import seaborn as sns
import numpy as np

Peak finding

Write a function find_peaks that finds and returns the indices of the local maxima in a sequence. Your function should:

Properly handle local maxima at the endpoints of the input array.
Return a Numpy array of integer indices.
Handle any Python iterable as input.



In [60]:

    
def find_peaks(a):
    """Find the indices of the local maxima in a sequence."""
    leest = []
    if a[0] > a[1]:
        leest.append(0)
    for x in range(1,len(a)-1):
        if (a[x-1]<a[x]) and (a[x]>a[x+1]):
            leest.append(x)
    if a[len(a)-1] > a[len(a)-2]:
        leest.append(len(a)-1)
    return leest



In [61]:

    
p1 = find_peaks([2,0,1,0,2,0,1])
assert np.allclose(p1, np.array([0,2,4,6]))
p2 = find_peaks(np.array([0,1,2,3]))
assert np.allclose(p2, np.array([3]))
p3 = find_peaks([3,2,1,0])
assert np.allclose(p3, np.array([0]))

Here is a string with the first 10000 digits of $\pi$ (after the decimal). Write code to perform the following:

Convert that string to a Numpy array of integers.
Find the indices of the local maxima in the digits of $\pi$.
Use np.diff to find the distances between consequtive local maxima.
Visualize that distribution using an appropriately customized histogram.



In [62]:

    
from sympy import pi, N
pi_digits_str = str(N(pi, 10001))[2:]



In [63]:

    
x = np.zeros(10000, dtype=int )
for a in range(len(pi_digits_str)):
    x[a] = int(pi_digits_str[a])
    
ind = find_peaks(x)
dis = np.diff(ind)

plt.hist(dis, bins=20, range=(0,15))
plt.title('Distribution of local Maxima in Pi')
plt.xlabel('Distantce between local maxima')
plt.ylabel('Frequency')
plt.show()



In [64]:

    
assert True # use this for grading the pi digits histogram