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."""
    a = np.array(a)
    s = np.sign(np.diff(a))
    d = np.diff(s)
    ind  = [i for i in range(len(d)) if d[i] == 2]
    if s[-1] == 1:
         ind.append(len(a)-1)
    return(np.array(ind))



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]))



In [10]:

    
find_peaks([2,2,2,1,2,2,2])









    Out[10]:





array([2])

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 [8]:

    
pi_int = np.array(list(pi_digits_str), dtype="int")
pks = find_peaks(pi_int)
pks_diff = np.diff(pks)
plt.hist(pks_diff, bins = range(0,max(pks_diff)+1))
min(pks_diff), max(pks_diff)









    Out[8]:





(2, 15)



In [6]:

    
pks









    Out[6]:





array([   1,    4,    7, ..., 9993, 9995, 9999])



In [7]:

    
assert True # use this for grading the pi digits histogram