Logistic regression algorithm

Let's dig into the internals and implement a logistic regression algorithm. Since R's glm() function is very complex, you'll stick to a implementing simple logistic regression for a single dataset.

Rather than using sum of squares as the metric, we want to use likelihood. However, log-likelihood is more computationally stable, so we'll use that instead. Actually, there is one more change: since we want to maximize log-likelihood, but optim() defaults to finding minimum values, it is easier to calculate the negative log-likelihood.

The log-likelihood value for each observation is $$ log(y_{pred}) * y_{actual} + log(1 - y_{pred}) * (1 - y_{actual}) $$

The metric to calculate is minus the sum of these log-likelihood contributions.

The explanatory values (the time_since_last_purchase column of churn) are available as x_actual. The response values (the has_churned column of churn) are available as y_actual.

Set the intercept to one.
Set the slope to 0.5.
Calculate the predicted y-values as the intercept plus the slope times the actual x-values, all transformed with the logistic distribution CDF.
Calculate the log-likelihood for each term as the log of the predicted y-values times the actual y-values, plus the log of one minus the predicted y-values times one minus the actual y-values.
Calculate minus the sum of the log-likelihoods for each term.

Exercise

Logistic regression algorithm

Instructions 1/3