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Theory

Christoph Molnar

A solution for classification is logistic regression. Instead of fitting a straight line or hyperplane, the logistic regression model uses the logistic function to squeeze the output of a linear equation between 0 and 1. The logistic function is defined as:

\text{logistic}(\eta)=\frac{1}{1+exp(-\eta)}

And it looks like this:

A logistic function. It outputs numbers between 0 and 1. At input 0, it outputs 0.5.
The logistic function. It outputs numbers between 0 and 1. At input 0, it outputs 0.5.

The step from linear regression to logistic regression is kind of straightforward. In the linear regression model, we have modelled the relationship between outcome and features with a linear equation:

\hat{y}^{(i)}=\beta_{0}+\beta_{1}x^{(i)}_{1}+\ldots+\beta_{p}x^{(i)}_{p}

For classification, we prefer probabilities between 0 and 1, so we wrap the right side of the equation into the logistic function. This forces the output to assume only values between 0 and 1.

P(y^{(i)}=1)=\frac{1}{1+exp(-(\beta_{0}+\beta_{1}x^{(i)}_{1}+\ldots+\beta_{p}x^{(i)}_{p}))}

Let us revisit the tumor size example again. But instead of the linear regression model, we use the logistic regression model:

A logistic regression model finding the correct decision boundary between malignant and benign depending on tumor size. The line is the logistic function shifted and squeezed to fit the data.
The logistic regression model finds the correct decision boundary between malignant and benign depending on tumor size.

Classification works better with logistic regression and we can use 0.5 as a threshold in both cases. The inclusion of additional points does not really affect the estimated curve.

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