ML | Why Logistic Regression in Classification ?

Using Linear Regression, all predictions >= 0.5 can be considered as 1 and rest all < 0.5 can be considered as 0. But then the question arises why classification can’t be performed using it?

Problem –

Suppose we are classifying a mail as spam or not spam and our output is y, it can be 0(spam) or 1(not spam). In case of Linear Regression, h_θ(x) can be > 1 or < 0. Although our prediction should be in between 0 and 1, the model will predict value out of the range i.e. maybe > 1 or < 0.

So, that’s why for a Classification task, Logistic/Sigmoid Regression plays its role.

Here, we plug θ^Tx into logistic function where θ are the weights/parameters and x is the input and h_θ(x) is the hypothesis function. g() is the sigmoid function.

It means that y = 1 probability when x is parameterized to θ

To get the discrete values 0 or 1 for classification, discrete boundaries are defined. The hypothesis function cab be translated as

Decision Boundary is the line that distinguishes the area where y=0 and where y=1. These decision boundaries result from the hypothesis function under consideration.

Understanding Decision Boundary with an example –
Let our hypothesis function be

Then the decision boundary looks like

Let out weights or parameters be –

So, it predicts y = 1 if

And that is the equation of a circle with radius = 1 and origin as the center. This is the Decision Boundary for our defined hypothesis.

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