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









    



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

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)

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)

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









    



2.58496250072



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assert True # use this for grading the pi digits histogram



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