# Algorithms Exercise 3

## Imports

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%matplotlib inline
from matplotlib import pyplot as plt
import numpy as np

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from IPython.html.widgets import interact

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:0: FutureWarning: IPython widgets are experimental and may change in the future.

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## Character counting and entropy

Write a function `char_probs` that takes a string and computes the probabilities of each character in the string:

• First do a character count and store the result in a dictionary.
• Then divide each character counts by the total number of character to compute the normalized probabilties.
• Return the dictionary of characters (keys) and probabilities (values).
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def char_probs(s):
"""Find the probabilities of the unique characters in the string s.

Parameters
----------
s : str
A string of characters.

Returns
-------
probs : dict
A dictionary whose keys are the unique characters in s and whose values
are the probabilities of those characters.
"""
chars = list(s)
chardic = {}
probs = {}
for ch in chars:
if ch in chardic:
chardic[ch]+=1
else:
chardic[ch]=1
for ch in chardic:
probs[ch]=chardic[ch]/len(chars)
return probs

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test1 = char_probs('aaaa')
assert np.allclose(test1['a'], 1.0)
test2 = char_probs('aabb')
assert np.allclose(test2['a'], 0.5)
assert np.allclose(test2['b'], 0.5)
test3 = char_probs('abcd')
assert np.allclose(test3['a'], 0.25)
assert np.allclose(test3['b'], 0.25)
assert np.allclose(test3['c'], 0.25)
assert np.allclose(test3['d'], 0.25)

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The entropy is a quantiative measure of the disorder of a probability distribution. It is used extensively in Physics, Statistics, Machine Learning, Computer Science and Information Science. Given a set of probabilities \$P_i\$, the entropy is defined as:

\$\$H = - \Sigma_i P_i \log_2(P_i)\$\$

In this expression \$\log_2\$ is the base 2 log (`np.log2`), which is commonly used in information science. In Physics the natural log is often used in the definition of entropy.

Write a funtion `entropy` that computes the entropy of a probability distribution. The probability distribution will be passed as a Python `dict`: the values in the `dict` will be the probabilities.

To compute the entropy, you should:

• First convert the values (probabilities) of the `dict` to a Numpy array of probabilities.
• Then use other Numpy functions (`np.log2`, etc.) to compute the entropy.
• Don't use any `for` or `while` loops in your code.
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def entropy(d):
"""Compute the entropy of a dict d whose values are probabilities."""
probs = np.array(list(d.values()))
probdist = probs*np.log2(probs)
H=-np.cumsum(probdist)[-1]
return H

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assert np.allclose(entropy({'a': 0.5, 'b': 0.5}), 1.0)
assert np.allclose(entropy({'a': 1.0}), 0.0)

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Use IPython's `interact` function to create a user interface that allows you to type a string into a text box and see the entropy of the character probabilities of the string.

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In [33]:

def show_entropy(s):
d=char_probs(s)
H=entropy(d)
print(H)
interact(show_entropy, s="Hello World");

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2.58496250072

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assert True # use this for grading the pi digits histogram

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