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Logistic Regression
Log-Odds

In Linear Regression we multiply the coefficients of our features by their respective feature values and add the intercept, resulting in our prediction, which can range from -∞ to +∞. In Logistic Regression, we make the same multiplication of feature coefficients and feature values and add the intercept, but instead of the prediction, we get what is called the log-odds.

The log-odds are another way of expressing the probability of a sample belonging to the positive class, or a student passing the exam. In probability, we calculate the odds of an event occurring as follows:

Odds=P(event occurring)P(event not occurring)Odds = \frac{P(event\ occurring)}{P(event\ not\ occurring)}

The odds tell us how many more times likely an event is to occur than not occur. If a student will pass the exam with probability 0.7, they will fail with probability 1 - 0.7 = 0.3. We can then calculate the odds of passing as:

Odds of passing=0.70.3=2.33Odds\ of\ passing = \frac{0.7}{0.3} = 2.\overline{33}

The log-odds are then understood as the logarithm of the odds!

Log odds of passing=log(2.33)=0.847Log\ odds\ of\ passing = log(2.\overline{33}) = 0.847

For our Logistic Regression model, however, we calculate the log-odds, represented by z below, by summing the product of each feature value by its respective coefficient and adding the intercept. This allows us to map our feature values to a measure of how likely it is that a data sample belongs to the positive class.

z=b0+b1x1++bnxnz = b_{0}+b_{1}x_{1} + \cdots + b_{n}x_{n}
  • b_0 is the intercept
  • b_1, b_2, … b_n are the coefficients of the features x_1, x_2, … x_n

This kind of multiplication and summing is known as a dot product.

We can perform a dot product using numpy‘s np.dot() method! Given feature matrix features, coefficient vector coefficients, and an intercept, we can calculate the log-odds in numpy as follows:

log_odds = np.dot(features, coefficients) + intercept

np.dot() will take each row, or student, in features and multiply each individual feature value by its respective coefficient in coefficients, summing the result, as shown below.

We then add in the intercept to get the log-odds!

Instructions

1.

Let’s create a function log_odds that takes features, coefficients and intercept as parameters. For now return features.

2.

Update log_odds to return the dot product of features and coefficients.

3.

Update the return statement of log-odds by adding the intercept after the dot product.

4.

With the log_odds function you created, let’s calculate the log-odds of passing for the Introductory Machine Learning students. Use hours_studied as the features, calculated_coefficients as the coefficients and intercept as the intercept. Store the result in calculated_log_odds, and print it out.

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