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

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.
    """
    # YOUR CODE HERE
    #raise NotImplementedError()
    s=s.replace(' ','')
    l = [i for i in s]
    dic={i:l.count(i) for i in l}
    prob = [(dic[i]/len(l)) for i in dic]
    result = {i:prob[j] for i in l for j in range(len(prob))}
    return result



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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."""
    # YOUR CODE HERE
    #raise NotImplementedError()
    s = char_probs(d)
    z = [(i,s[i]) for i in s]
    w=np.array(z)
    P = np.array(w[::,1])
    np.log2(P[1])
entropy('haldjfhasdf')









    



---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-113-4190199d8962> in <module>()
      8     P = np.array(w[::,1])
      9     np.log2(P[1])
---> 10 entropy('haldjfhasdf')

<ipython-input-113-4190199d8962> in entropy(d)
      7     w=np.array(z)
      8     P = np.array(w[::,1])
----> 9     np.log2(P[1])
     10 entropy('haldjfhasdf')

TypeError: Not implemented for this type



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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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# YOUR CODE HERE
raise NotImplementedError()



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assert True # use this for grading the pi digits histogram