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 [7]:

    
a=[1,2,3,4,5,3]
for x in a:
    print(x)









    



  File "<ipython-input-7-1a3235487279>", line 4
    .index()?
    ^
SyntaxError: invalid syntax



In [27]:

    
def find_peaks(a):
    """Find the indices of the local maxima in a sequence."""
    a=list(a)
    index=[]
    first=a[0]
    
    if first>a[1]:
        index.append(0)
        
    start=1
    for x in a[1:len(a)-1]:
        prev=a[a.index(x)-1]
        foll=a[a.index(x)+1]
        if x> prev and x>foll:
            index.append(a.index(x,start))
        start+=1
    last=a[len(a)-1]
    if last>a[len(a)-2]:
        index.append(len(a)-1)
    return(np.array(index))
        
find_peaks([3,2,1,0])









    Out[27]:





array([0])



In [28]:

    
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 [29]:

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



In [46]:

    
pi_digits_list=np.array([int(x) for x in pi_digits_str])
pi_digits_list
x=np.diff(find_peaks(pi_digits_list))
plt.hist(x,20)
plt.title('Distance between local maxes in the digits of $\pi$')
plt.xlabel('Distance between Maxima')
plt.ylabel('Number of Maxima')
plt.xticks(range(0,30,2))
plt.xlim(right=22)









    Out[46]:





(0.0, 22)



In [ ]:

    
assert True # use this for grading the pi digits histogram