Minimize the cost function of Regression Method

As much we minimize the cost function, regression line for prediction would be more accurate.

Lets start with simplified hypothesis where Θ₀ = 0.and h(X)= Θ₀ + Θ1 X = Θ1 X .

Minimize this cost function :

                           

h(x) : For fixed value of Θ1 this is function of x.

J( Θ) : Function of parameter Θ1.

We minimizing the cost function for reducing the gap between actual and predicted value. For visualizing it  explained it in below figure:

For different value of Θ1,  plot the hypothesis value h(x) and cost function J( Θ).

Plot of J(Θ) is shown as :

Calculate the minima of this cost function plot. That value consider as minimize cost function. In above figure minimum cost function value lie between 0 and 1.