Logistic Regression

Logistic regression is named for the function used at the core of the method, the logistic function.

The logistic function, also called the sigmoid function was developed by statisticians to describe properties of population growth in ecology, rising quickly and maxing out at the carrying capacity of the environment. It’s an S-shaped curve that can take any real-valued number and map it into a value between 0 and 1, but never exactly at those limits.

$$\frac{1} {(1 + e^{-value})}$$

Logistic regression models the probability of the default class (e.g. the first class).

Logistic regression is a linear method, but the predictions are transformed using the logistic function. The impact of this is that we can no longer understand the predictions as a linear combination of the inputs as we can with linear regression

Binary Logistic Regression models how binary response variable Y depends on a set of k explanatory variables, X=(X1, X2, ... Xk).

$$logit(\pi) = log({\frac{\pi}{1-\pi}})=\beta_0+\beta_{x_i}+....+\beta_0+\beta_{x_{k^,}}$$

Which models the log odds of probability of "success" as a function of explanatory variables.

• Random component: The distribution of Y is assumed to be Binomial(n,π), where π is a probability of "success".