Algorithms Exercise 2

Imports



In [2]:

    
%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 [20]:

    
np.array(range(5)).max()

list(range(1,5))
find_peaks([2,0,1,0,2,0,1])









    Out[20]:





[0, 2, 4, 6]



In [21]:

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



In [22]:

    
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 [23]:

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



In [42]:

    
first_10000=np.array(list(pi_digits_str), dtype=int)
peaks=find_peaks(first_10000)
differences=np.diff(peaks)
plt.figure(figsize=(10,10))
plt.hist(differences, 20, (1,20))
plt.title('Hoe Far Apart the Local Maxima of the First 10,0000 Digits of $\pi$ Are')
plt.ylabel('Number of Occurences')
plt.xlabel('Distance Apart')
plt.tight_layout()



In [ ]:

    
assert True # use this for grading the pi digits histogram