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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Implementing logistic regression from scratch
The goal of this notebook is to implement your own logistic regression classifier. You will
Step1:  Load review dataset
For this assignment, we will use a subset of the Amazon product review dataset. The subset was chosen to contain similar numbers of positive and negative reviews, as the original dataset consisted primarily of positive reviews.
Step2:  One column of this dataset is 'sentiment', corresponding to the class label with +1 indicating a review with positive sentiment and -1 indicating one with negative sentiment.
Step3:  Let us quickly explore more of this dataset.  The 'name' column indicates the name of the product.  Here we list the first 10 products in the dataset.  We then count the number of positive and negative reviews.
Step4:  Note
Step5:  Now, we will perform 2 simple data transformations
Step6:  Now we proceed with Step 2. For each word in important_words, we compute a count for the number of times the word occurs in the review. We will store this count in a separate column (one for each word). The result of this feature processing is a single column for each word in important_words which keeps a count of the number of times the respective word occurs in the review text.
Note
Step7:  The SFrame products now contains one column for each of the 193 important_words. As an example, the column perfect contains a count of the number of times the word perfect occurs in each of the reviews.
Step8:  Now, write some code to compute the number of product reviews that contain the word perfect.
Hint
Step9:  Quiz Question. How many reviews contain the word perfect?
Convert SFrame to NumPy array
As you have seen previously, NumPy is a powerful library for doing matrix manipulation. Let us convert our data to matrices and then implement our algorithms with matrices.
First, make sure you can perform the following import.
Step10:  We now provide you with a function that extracts columns from an SFrame and converts them into a NumPy array. Two arrays are returned
Step11:  Let us convert the data into NumPy arrays.
Step12:  Are you running this notebook on an Amazon EC2 t2.micro instance? (If you are using your own machine, please skip this section)
It has been reported that t2.micro instances do not provide sufficient power to complete the conversion in acceptable amount of time. For interest of time, please refrain from running get_numpy_data function. Instead, download the binary file containing the four NumPy arrays you'll need for the assignment. To load the arrays, run the following commands
Step13:  Quiz Question
Step14:  Estimating conditional probability with link function
Recall from lecture that the link function is given by
Step15:  Aside. How the link function works with matrix algebra
Since the word counts are stored as columns in feature_matrix, each $i$-th row of the matrix corresponds to the feature vector $h(\mathbf{x}_i)$
Step16:  Compute derivative of log likelihood with respect to a single coefficient
Recall from lecture
Step17:  In the main lecture, our focus was on the likelihood.  In the advanced optional video, however, we introduced a transformation of this likelihood---called the log likelihood---that simplifies the derivation of the gradient and is more numerically stable.  Due to its numerical stability, we will use the log likelihood instead of the likelihood to assess the algorithm.
The log likelihood is computed using the following formula (see the advanced optional video if you are curious about the derivation of this equation)
Step18:  Checkpoint
Just to make sure we are on the same page, run the following code block and check that the outputs match.
Step19:  Taking gradient steps
Now we are ready to implement our own logistic regression. All we have to do is to write a gradient ascent function that takes gradient steps towards the optimum. 
Complete the following function to solve the logistic regression model using gradient ascent
Step20:  Now, let us run the logistic regression solver.
Step21:  Quiz Question
Step22:  Now, complete the following code block for Step 2 to compute the class predictions using the scores obtained above
Step23:  Quiz Question
Step24:  Measuring accuracy
We will now measure the classification accuracy of the model. Recall from the lecture that the classification accuracy can be computed as follows
Step25:  Quiz Question
Step26:  Now, word_coefficient_tuples contains a sorted list of (word, coefficient_value) tuples. The first 10 elements in this list correspond to the words that are most positive.
Ten "most positive" words
Now, we compute the 10 words that have the most positive coefficient values. These words are associated with positive sentiment.
Step27:  Quiz Question
Step28:  Quiz Question
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Python Code:
import graphlab
Explanation: Implementing logistic regression from scratch
The goal of this notebook is to implement your own logistic regression classifier. You will:
Extract features from Amazon product reviews.
Convert an SFrame into a NumPy array.
Implement the link function for logistic regression.
Write a function to compute the derivative of the log likelihood function with respect to a single coefficient.
Implement gradient ascent.
Given a set of coefficients, predict sentiments.
Compute classification accuracy for the logistic regression model.
Let's get started!
Fire up GraphLab Create
Make sure you have the latest version of GraphLab Create. Upgrade by
pip install graphlab-create --upgrade
See this page for detailed instructions on upgrading.
End of explanation
products = graphlab.SFrame('amazon_baby_subset.gl/')
Explanation: Load review dataset
For this assignment, we will use a subset of the Amazon product review dataset. The subset was chosen to contain similar numbers of positive and negative reviews, as the original dataset consisted primarily of positive reviews.
End of explanation
products['sentiment']
Explanation: One column of this dataset is 'sentiment', corresponding to the class label with +1 indicating a review with positive sentiment and -1 indicating one with negative sentiment.
End of explanation
products.head(10)['name']
print '# of positive reviews =', len(products[products['sentiment']==1])
print '# of negative reviews =', len(products[products['sentiment']==-1])
Explanation: Let us quickly explore more of this dataset.  The 'name' column indicates the name of the product.  Here we list the first 10 products in the dataset.  We then count the number of positive and negative reviews.
End of explanation
import json
with open('important_words.json', 'r') as f: # Reads the list of most frequent words
    important_words = json.load(f)
important_words = [str(s) for s in important_words]
print important_words
Explanation: Note: For this assignment, we eliminated class imbalance by choosing 
a subset of the data with a similar number of positive and negative reviews. 
Apply text cleaning on the review data
In this section, we will perform some simple feature cleaning using SFrames. The last assignment used all words in building bag-of-words features, but here we limit ourselves to 193 words (for simplicity). We compiled a list of 193 most frequent words into a JSON file. 
Now, we will load these words from this JSON file:
End of explanation
def remove_punctuation(text):
    import string
    return text.translate(None, string.punctuation) 
products['review_clean'] = products['review'].apply(remove_punctuation)
Explanation: Now, we will perform 2 simple data transformations:
Remove punctuation using Python's built-in string functionality.
Compute word counts (only for important_words)
We start with Step 1 which can be done as follows:
End of explanation
for word in important_words:
    products[word] = products['review_clean'].apply(lambda s : s.split().count(word))
Explanation: Now we proceed with Step 2. For each word in important_words, we compute a count for the number of times the word occurs in the review. We will store this count in a separate column (one for each word). The result of this feature processing is a single column for each word in important_words which keeps a count of the number of times the respective word occurs in the review text.
Note: There are several ways of doing this. In this assignment, we use the built-in count function for Python lists. Each review string is first split into individual words and the number of occurances of a given word is counted.
End of explanation
products['perfect']
Explanation: The SFrame products now contains one column for each of the 193 important_words. As an example, the column perfect contains a count of the number of times the word perfect occurs in each of the reviews.
End of explanation
def contains_perfect(count):
    return 1 if count >= 1 else 0
products['contains_perfect'] = products['perfect'].apply(contains_perfect)
products['contains_perfect'].sum()
Explanation: Now, write some code to compute the number of product reviews that contain the word perfect.
Hint: 
* First create a column called contains_perfect which is set to 1 if the count of the word perfect (stored in column perfect) is >= 1.
* Sum the number of 1s in the column contains_perfect.
End of explanation
import numpy as np
Explanation: Quiz Question. How many reviews contain the word perfect?
Convert SFrame to NumPy array
As you have seen previously, NumPy is a powerful library for doing matrix manipulation. Let us convert our data to matrices and then implement our algorithms with matrices.
First, make sure you can perform the following import.
End of explanation
def get_numpy_data(data_sframe, features, label):
    data_sframe['intercept'] = 1
    features = ['intercept'] + features
    features_sframe = data_sframe[features]
    feature_matrix = features_sframe.to_numpy()
    label_sarray = data_sframe[label]
    label_array = label_sarray.to_numpy()
    return(feature_matrix, label_array)
Explanation: We now provide you with a function that extracts columns from an SFrame and converts them into a NumPy array. Two arrays are returned: one representing features and another representing class labels. Note that the feature matrix includes an additional column 'intercept' to take account of the intercept term.
End of explanation
# Warning: This may take a few minutes...
feature_matrix, sentiment = get_numpy_data(products, important_words, 'sentiment') 
Explanation: Let us convert the data into NumPy arrays.
End of explanation
feature_matrix.shape
Explanation: Are you running this notebook on an Amazon EC2 t2.micro instance? (If you are using your own machine, please skip this section)
It has been reported that t2.micro instances do not provide sufficient power to complete the conversion in acceptable amount of time. For interest of time, please refrain from running get_numpy_data function. Instead, download the binary file containing the four NumPy arrays you'll need for the assignment. To load the arrays, run the following commands:
arrays = np.load('module-3-assignment-numpy-arrays.npz')
feature_matrix, sentiment = arrays['feature_matrix'], arrays['sentiment']
End of explanation
sentiment
Explanation: Quiz Question: How many features are there in the feature_matrix?
 Quiz Question: Assuming that the intercept is present, how does the number of features in feature_matrix relate to the number of features in the logistic regression model?
Now, let us see what the sentiment column looks like:
End of explanation
def prediction(score):
    return (1 / (1 + np.exp(-score)))
'''
produces probablistic estimate for P(y_i = +1 | x_i, w).
estimate ranges between 0 and 1.
'''
def predict_probability(feature_matrix, coefficients):
    # Take dot product of feature_matrix and coefficients  
    scores = np.dot(feature_matrix, coefficients)
    
    # Compute P(y_i = +1 | x_i, w) using the link function
    predictions = np.apply_along_axis(prediction, 0, scores)
    
    # return predictions
    return predictions
Explanation: Estimating conditional probability with link function
Recall from lecture that the link function is given by:
$$
P(y_i = +1 | \mathbf{x}_i,\mathbf{w}) = \frac{1}{1 + \exp(-\mathbf{w}^T h(\mathbf{x}_i))},
$$
where the feature vector $h(\mathbf{x}_i)$ represents the word counts of important_words in the review  $\mathbf{x}_i$. Complete the following function that implements the link function:
End of explanation
dummy_feature_matrix = np.array([[1.,2.,3.], [1.,-1.,-1]])
dummy_coefficients = np.array([1., 3., -1.])
correct_scores      = np.array( [ 1.*1. + 2.*3. + 3.*(-1.),          1.*1. + (-1.)*3. + (-1.)*(-1.) ] )
correct_predictions = np.array( [ 1./(1+np.exp(-correct_scores[0])), 1./(1+np.exp(-correct_scores[1])) ] )
print 'The following outputs must match '
print '------------------------------------------------'
print 'correct_predictions           =', correct_predictions
print 'output of predict_probability =', predict_probability(dummy_feature_matrix, dummy_coefficients)
Explanation: Aside. How the link function works with matrix algebra
Since the word counts are stored as columns in feature_matrix, each $i$-th row of the matrix corresponds to the feature vector $h(\mathbf{x}_i)$:
$$
[\text{feature_matrix}] =
\left[
\begin{array}{c}
h(\mathbf{x}_1)^T \
h(\mathbf{x}_2)^T \
\vdots \
h(\mathbf{x}_N)^T
\end{array}
\right] =
\left[
\begin{array}{cccc}
h_0(\mathbf{x}_1) & h_1(\mathbf{x}_1) & \cdots & h_D(\mathbf{x}_1) \
h_0(\mathbf{x}_2) & h_1(\mathbf{x}_2) & \cdots & h_D(\mathbf{x}_2) \
\vdots & \vdots & \ddots & \vdots \
h_0(\mathbf{x}_N) & h_1(\mathbf{x}_N) & \cdots & h_D(\mathbf{x}_N)
\end{array}
\right]
$$
By the rules of matrix multiplication, the score vector containing elements $\mathbf{w}^T h(\mathbf{x}_i)$ is obtained by multiplying feature_matrix and the coefficient vector $\mathbf{w}$.
$$
[\text{score}] =
[\text{feature_matrix}]\mathbf{w} =
\left[
\begin{array}{c}
h(\mathbf{x}_1)^T \
h(\mathbf{x}_2)^T \
\vdots \
h(\mathbf{x}_N)^T
\end{array}
\right]
\mathbf{w}
= \left[
\begin{array}{c}
h(\mathbf{x}_1)^T\mathbf{w} \
h(\mathbf{x}_2)^T\mathbf{w} \
\vdots \
h(\mathbf{x}_N)^T\mathbf{w}
\end{array}
\right]
= \left[
\begin{array}{c}
\mathbf{w}^T h(\mathbf{x}_1) \
\mathbf{w}^T h(\mathbf{x}_2) \
\vdots \
\mathbf{w}^T h(\mathbf{x}_N)
\end{array}
\right]
$$
Checkpoint
Just to make sure you are on the right track, we have provided a few examples. If your predict_probability function is implemented correctly, then the outputs will match:
End of explanation
def feature_derivative(errors, feature):     
    # Compute the dot product of errors and feature
    derivative = np.dot(feature, errors)
    
    # Return the derivative
    return derivative
Explanation: Compute derivative of log likelihood with respect to a single coefficient
Recall from lecture:
$$
\frac{\partial\ell}{\partial w_j} = \sum_{i=1}^N h_j(\mathbf{x}_i)\left(\mathbf{1}[y_i = +1] - P(y_i = +1 | \mathbf{x}_i, \mathbf{w})\right)
$$
We will now write a function that computes the derivative of log likelihood with respect to a single coefficient $w_j$. The function accepts two arguments:
* errors vector containing $\mathbf{1}[y_i = +1] - P(y_i = +1 | \mathbf{x}_i, \mathbf{w})$ for all $i$.
* feature vector containing $h_j(\mathbf{x}_i)$  for all $i$. 
Complete the following code block:
End of explanation
def compute_log_likelihood(feature_matrix, sentiment, coefficients):
    indicator = (sentiment==+1)
    scores = np.dot(feature_matrix, coefficients)
    logexp = np.log(1. + np.exp(-scores))
    
    # Simple check to prevent overflow
    mask = np.isinf(logexp)
    logexp[mask] = -scores[mask]
    
    lp = np.sum((indicator-1)*scores - logexp)
    return lp
Explanation: In the main lecture, our focus was on the likelihood.  In the advanced optional video, however, we introduced a transformation of this likelihood---called the log likelihood---that simplifies the derivation of the gradient and is more numerically stable.  Due to its numerical stability, we will use the log likelihood instead of the likelihood to assess the algorithm.
The log likelihood is computed using the following formula (see the advanced optional video if you are curious about the derivation of this equation):
$$\ell\ell(\mathbf{w}) = \sum_{i=1}^N \Big( (\mathbf{1}[y_i = +1] - 1)\mathbf{w}^T h(\mathbf{x}_i) - \ln\left(1 + \exp(-\mathbf{w}^T h(\mathbf{x}_i))\right) \Big) $$
We provide a function to compute the log likelihood for the entire dataset.
End of explanation
dummy_feature_matrix = np.array([[1.,2.,3.], [1.,-1.,-1]])
dummy_coefficients = np.array([1., 3., -1.])
dummy_sentiment = np.array([-1, 1])
correct_indicators  = np.array( [ -1==+1,                                       1==+1 ] )
correct_scores      = np.array( [ 1.*1. + 2.*3. + 3.*(-1.),                     1.*1. + (-1.)*3. + (-1.)*(-1.) ] )
correct_first_term  = np.array( [ (correct_indicators[0]-1)*correct_scores[0],  (correct_indicators[1]-1)*correct_scores[1] ] )
correct_second_term = np.array( [ np.log(1. + np.exp(-correct_scores[0])),      np.log(1. + np.exp(-correct_scores[1])) ] )
correct_ll          =      sum( [ correct_first_term[0]-correct_second_term[0], correct_first_term[1]-correct_second_term[1] ] ) 
print correct_scores
print correct_indicators
print correct_first_term
print correct_second_term
print 'The following outputs must match '
print '------------------------------------------------'
print 'correct_log_likelihood           =', correct_ll
print 'output of compute_log_likelihood =', compute_log_likelihood(dummy_feature_matrix, dummy_sentiment, dummy_coefficients)
Explanation: Checkpoint
Just to make sure we are on the same page, run the following code block and check that the outputs match.
End of explanation
from math import sqrt
def logistic_regression(feature_matrix, sentiment, initial_coefficients, step_size, max_iter):
    coefficients = np.array(initial_coefficients) # make sure it's a numpy array
    for itr in xrange(max_iter):
        # Predict P(y_i = +1|x_i,w) using your predict_probability() function
        # YOUR CODE HERE
        predictions = predict_probability(feature_matrix, coefficients)
        
        # Compute indicator value for (y_i = +1)
        indicator = (sentiment==+1)
        
        # Compute the errors as indicator - predictions
        errors = indicator - predictions
        for j in xrange(len(coefficients)): # loop over each coefficient
            
            # Recall that feature_matrix[:,j] is the feature column associated with coefficients[j].
            # Compute the derivative for coefficients[j]. Save it in a variable called derivative
            # YOUR CODE HERE
            derivative = feature_derivative(errors, feature_matrix[:, j])
            
            # add the step size times the derivative to the current coefficient
            coefficients[j] += (step_size * derivative)
        
        # Checking whether log likelihood is increasing
        if itr <= 15 or (itr <= 100 and itr % 10 == 0) or (itr <= 1000 and itr % 100 == 0) \
        or (itr <= 10000 and itr % 1000 == 0) or itr % 10000 == 0:
            lp = compute_log_likelihood(feature_matrix, sentiment, coefficients)
            print 'iteration %*d: log likelihood of observed labels = %.8f' % \
                (int(np.ceil(np.log10(max_iter))), itr, lp)
    return coefficients
Explanation: Taking gradient steps
Now we are ready to implement our own logistic regression. All we have to do is to write a gradient ascent function that takes gradient steps towards the optimum. 
Complete the following function to solve the logistic regression model using gradient ascent:
End of explanation
coefficients = logistic_regression(feature_matrix, sentiment, initial_coefficients=np.zeros(194),
                                   step_size=1e-7, max_iter=301)
Explanation: Now, let us run the logistic regression solver.
End of explanation
# Compute the scores as a dot product between feature_matrix and coefficients.
scores = np.dot(feature_matrix, coefficients)
Explanation: Quiz Question: As each iteration of gradient ascent passes, does the log likelihood increase or decrease?
Predicting sentiments
Recall from lecture that class predictions for a data point $\mathbf{x}$ can be computed from the coefficients $\mathbf{w}$ using the following formula:
$$
\hat{y}_i = 
\left{
\begin{array}{ll}
      +1 & \mathbf{x}_i^T\mathbf{w} > 0 \
      -1 & \mathbf{x}_i^T\mathbf{w} \leq 0 \
\end{array} 
\right.
$$
Now, we will write some code to compute class predictions. We will do this in two steps:
* Step 1: First compute the scores using feature_matrix and coefficients using a dot product.
* Step 2: Using the formula above, compute the class predictions from the scores.
Step 1 can be implemented as follows:
End of explanation
def get_prediction(score):
    if score > 0:
        return 1
    else:
        return -1
predictions = np.zeros(shape=(scores.shape[0],))
idx = 0
for score in scores:
    predictions[idx] = int(get_prediction(score))
    idx += 1
Explanation: Now, complete the following code block for Step 2 to compute the class predictions using the scores obtained above:
End of explanation
pos = (predictions == 1).sum()
neg = (predictions == -1).sum()
print ("number of positive predicted reviews: {}".format(pos))
print ("number of negative predicted reviews: {}".format(neg))
Explanation: Quiz Question:  How many reviews were predicted to have positive sentiment?
End of explanation
sentiment = products['sentiment'].to_numpy()
num_mistakes = sentiment - predictions
correct = len(sentiment) - np.count_nonzero(num_mistakes)
accuracy = float(correct) / len(sentiment)
print "-----------------------------------------------------"
print '# Reviews   correctly classified =', len(products) - num_mistakes
print '# Reviews incorrectly classified =', num_mistakes
print '# Reviews total                  =', len(products)
print "-----------------------------------------------------"
print 'Accuracy = %.2f' % accuracy
Explanation: Measuring accuracy
We will now measure the classification accuracy of the model. Recall from the lecture that the classification accuracy can be computed as follows:
$$
\mbox{accuracy} = \frac{\mbox{# correctly classified data points}}{\mbox{# total data points}}
$$
Complete the following code block to compute the accuracy of the model.
End of explanation
coefficients = list(coefficients[1:]) # exclude intercept
word_coefficient_tuples = [(word, coefficient) for word, coefficient in zip(important_words, coefficients)]
word_coefficient_tuples = sorted(word_coefficient_tuples, key=lambda x:x[1], reverse=True)
Explanation: Quiz Question: What is the accuracy of the model on predictions made above? (round to 2 digits of accuracy)
Which words contribute most to positive & negative sentiments?
Recall that in Module 2 assignment, we were able to compute the "most positive words". These are words that correspond most strongly with positive reviews. In order to do this, we will first do the following:
* Treat each coefficient as a tuple, i.e. (word, coefficient_value).
* Sort all the (word, coefficient_value) tuples by coefficient_value in descending order.
End of explanation
word_coefficient_tuples[:10]
Explanation: Now, word_coefficient_tuples contains a sorted list of (word, coefficient_value) tuples. The first 10 elements in this list correspond to the words that are most positive.
Ten "most positive" words
Now, we compute the 10 words that have the most positive coefficient values. These words are associated with positive sentiment.
End of explanation
word_coefficient_tuples_descending = sorted(word_coefficient_tuples, key = lambda x: x[1], reverse=False)
Explanation: Quiz Question: Which word is not present in the top 10 "most positive" words?
Ten "most negative" words
Next, we repeat this exercise on the 10 most negative words.  That is, we compute the 10 words that have the most negative coefficient values. These words are associated with negative sentiment.
End of explanation
word_coefficient_tuples_descending[0:10]
Explanation: Quiz Question: Which word is not present in the top 10 "most negative" words?
End of explanation
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