Logistic regression is a method for classifying data into discrete outcomes. For example, the classification of an email as spam or not. It is a statistical model that in its basic form uses a logistic function to model a binary dependent variable, although many more complex extensions exist. In regression analysis, logistic regression (or logit regression) is estimating the parameters of a logistic model (a form of binary regression).

The classification problem is just like the regression problem, except that the values we now want to predict take only a small number of discrete values. For instance, if we are trying to build a spam classifier for email, then x(i) may be some features of a piece of email, and y may be 1 if it is a piece of spam mail, and 0 otherwise. Hence, yε{0,1}. 0 is also called the negative class, and 1 is a positive class, and they are sometimes also denoted by the symbols “-” and “+”.

Intuitively, it doesn’t make sense for hΘ(x) to take values larger than 1 or smaller than 0 when we know y ε {0,1}. To fix this, let’s change the form of our hypothesis hΘ(x) to satisfy 0 ≤ hΘ(x) ≤ 1. This is accomplished by plugging ΘTx into the “Sigmoid Function”, also called the “Logistic Function”.

Thus our logistics regression hypothesis function compared to simple linear regression looks like this:

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