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.
    """
                
    probs={}                 # empty dictionary
    for i in s:
        if i not in probs:     
            probs[i]=1
        else:
            probs[i] +=1       # if already in, add count
               
    return probs                             # this displays how many times each character appears
    count =0
    for i in range(len(s)):
        count+=1
        return count                # counts total number of characters
    for i in range(len(probs)):       
        probs[i]=((probs[i])/count)   # the new value for the key is now the probability. Number of times it appeared
                                    # divided by the total count.
    return probs



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#a='aaaadadad'
#count=0
#for i in range(len(a)):
   # count+=1
    
    
#a={' ': 1, 'a': 5, 'b': 3, 'h': 1, 't': 1, 'w': 1}
#char_probs(a)
#X=np.array(a)
#X



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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)









    



---------------------------------------------------------------------------
AssertionError                            Traceback (most recent call last)
<ipython-input-68-7884295ffbe7> 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.



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



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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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interact(entropy, d='Hi ')









    



---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-47-8fbf3945afbb> in <module>()
----> 1 interact(entropy, d='Hi there')

NameError: name 'entropy' is not defined



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assert True # use this for grading the pi digits histogram



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