# 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 [15]:

def find_peaks(a):
"""Find the indices of the local maxima in a sequence."""
peaks = []
data = np.array(a)
deriv = np.diff(data)
if deriv[0] < 0:
peaks.append(0)
for i in range(1,len(deriv)):
if deriv[i]<0 and deriv[i-1]>0:
peaks.append(i)
if deriv[-1] >0:
peaks.append(len(data)-1)
return np.array(peaks)




In [16]:

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 [24]:

from sympy import pi, N
pi_digits_str = str(N(pi, 10001))[2:]
pi_dig_num=np.array(list(pi_digits_str))
pi_dig_num=pi_dig_num.astype(int)
peak=find_peaks(pi_dig_num)
peakdiff=np.diff(peak)




[1 4 1 ..., 6 7 8]




In [59]:

plt.hist(peakdiff, bins=100, width=1, color='k',edgecolor='b', align='right');
plt.xlabel("Distance between adjacent local maxima in pi")
plt.ylabel("Counts")
plt.title("Distance between adjacent local maxima in pi");







In [60]:

assert True # use this for grading the pi digits histogram




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