Algorithms Exercise 3

Imports



In [1]:

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



In [2]:

    
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).



In [23]:

    
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.
    """
    a = len(s)
    b = list(s)
    dict1 = {'a':1, 'b':2, 'c':3, 'd':4, 'e':5, 'f':6, 'g':7, 'h':8, 'i':9, 'j':10, 'k':11, 'l':12, 'm':13, 'n':14, 'o':15, 'p':16, 'q':17, 'r':18, 's':19, 't':20, 'u':21, 'v':22, 'w':23, 'x':24, 'y':25, 'z':26}
    dict2 = {'a':0, 'b':0, 'c':0, 'd':0, 'e':0, 'f':0, 'g':0, 'h':0, 'i':0, 'j':0, 'k':0, 'l':0, 'm':0, 'n':0, 'o':0, 'p':0, 'q':0, 'r':0, 's':0, 't':0, 'u':0, 'v':0, 'w':0, 'x':0, 'y':0, 'z':0}
    for item in b:
        dict2[item] += 1
    for key in dict2:
        print 
    return dict2

print(char_probs('hibebe'))









    



{'q': 0, 't': 0, 'b': 2, 'c': 0, 'n': 0, 'z': 0, 'o': 0, 'u': 0, 'h': 1, 'r': 0, 'd': 0, 'j': 0, 'p': 0, 'a': 0, 'g': 0, 'x': 0, 'e': 2, 'f': 0, 'k': 0, 'v': 0, 'y': 0, 'm': 0, 'i': 1, 'l': 0, 's': 0, 'w': 0}



In [19]:

    
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)









    



---------------------------------------------------------------------------
AssertionError                            Traceback (most recent call last)
<ipython-input-19-7884295ffbe7> in <module>()
      1 test1 = char_probs('aaaa')
----> 2 assert np.allclose(test1['a'], 1.0)
      3 test2 = char_probs('aabb')
      4 assert np.allclose(test2['a'], 0.5)
      5 assert np.allclose(test2['b'], 0.5)

AssertionError:

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.



In [25]:

    
def entropy(d):
    """Compute the entropy of a dict d whose values are probabilities."""

d = {'1':2, '3':4}
for item in d:
    print(item)

3
1



In [ ]:

    
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.



In [ ]:

    
# YOUR CODE HERE
raise NotImplementedError()



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assert True # use this for grading the pi digits histogram