Algorithms Exercise 2

Imports



In [1]:

    
%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 [2]:

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



In [3]:

    
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 [4]:

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



In [5]:

    
ints = [int(a) for a in pi_digits_str]



In [6]:

    
diff = np.diff(find_peaks(ints))



In [7]:

    
plt.hist(diff,np.arange(0,15));
plt.xlim(2,15);
plt.xlabel('Number of digits between maxima');
plt.ylabel('Occurence');
plt.title('Occurences of Maxima spacing for 10,000 digits of Pi');



In [8]:

    
assert True # use this for grading the pi digits histogram



In [ ]: