Linear Regression

Linear regression is used to predict real-valued output $y \in \mathbb{R}$ for a given input data point $x \in \mathbb{R}^D$ . Linear regression establishes a relationship of dependent variable $y$ with the features of the input data with an assumption that the expected value of the output(dependent variable) is a linear function of the input ( $\mathbb{E}[y|x]=w^T x$ ).

Let's assume our training dataset is $X \in \mathbb{R}^{N \times D}$ where $N$ is the number of data points and $D$ is the number of dimension or number of features in our dataset. From now on we will write our dataset as $[x_1, x_2,..., x_D]$ where each $x_i$ for $i=1,...,D$ is a column vector.

We can write the output $y$ as:

$y = w_0 + w_{11} x_1 + w_{12} x_1^2 + ... + w_{21} x_2 + w_{22} x_2^2 + ...$

or we can write it as:

$y = w_0 + \phi_1(x_1) + \phi_2(x_2) + ... = w_0 + \sum_{i=1}^{D} \phi_i(x_i)$

Before computing the final weights for this equation, we need to figure out what degree we should choose. We usually select the degree for which we get less mean squared error(MSE).

The most common form of linear regression is degree 1 form:

$y = w_0 + w_1 x_1 + w_2 x_2 + ... = w_0 + \sum_{i=1}^{D} w_i x_i$

There are two ways by which we can estimate the parameters:

Normal equation: Weight vector is estimated by matrix multiplication of psuedo-inverse of feature matrix and the label vector
Gradient descent method: Loss minimization

Loss function is given by: $J(w) = (Xw-y)^T(Xw-y)$

gradient of loss w.r.t weight vector: $\nabla_w J(w) = 2(X^TXw - X^Ty)$

from the above equation we get the ML-estimate for $w$ :

$\hat{w} = (X^TX)^{-1}X^Ty$

In case of Ridge Regression, the loss function becomes:

$J(w) = (Xw-y)^T(Xw-y) + \lambda ||w||_{2}^{2}$

now, $\nabla_w J(w) = 2(X^TXw - X^Ty + \lambda w)$

and $\hat{w} = (X^TX+\lambda I)^{-1}X^Ty$

$y$

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