ML | Normal Equation in Linear Regression

Normal Equation is an analytical approach to Linear Regression with a Least Square Cost Function. We can directly find out the value of θ without using Gradient Descent. Following this approach is an effective and a time-saving option when are working with a dataset with small features.

Normal Equation is a follows :

In the above equation,
θ : hypothesis parameters that define it the best.
X : Input feature value of each instance.
Y : Output value of each instance.

Maths Behind the equation –

Given the hypothesis function
where,
n : the no. of features in the data set.
x0 : 1 (for vector multiplication)

Notice that this is dot product between θ and x values. So for the convenience to solve we can write it as :

The motive in Linear Regression is to minimize the cost function :

  J(\Theta) = \frac{1}{2m} \sum_{i = 1}^{m} \frac{1}{2} [h_{\Theta}(x^{(i)}) - y^{(i)}]^{2}

where,
xi : the input value of iih training example.
m : no. of training instances
n : no. of data-set features
yi : the expected result of ith instance



Let us representing cost function in a vector form.

we have ignored 1/2m here as it will not make any difference in the working. It was used for the mathematical convenience while calculation gradient descent. But it is no more needed here.


xij : value of jih feature in iih training example.

This can further be reduced to  X\theta - y
But each residual value is squared. We cannot simply square the above expression. As the square of a vector/matrix is not equal to the square of each of its values. So to get the squared value, multiply the vector/matrix with its transpose. So, the final equation derived is

Therefore, the cost function is

So, now getting the value of θ using derivative







So, this is the finally derived Normal Equation with θ giving the minimum cost value.

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