Linear Regression

Linear Regression

Linear regression is the simplest supervised learning model. It learns a linear mapping from an input $x$ to an output $y$:

$\hat{y} = wx + b$

where $w$ is the weight (slope) and $b$ is the bias (intercept).

Mean Squared Error

To measure how well the model fits the data, we use the Mean Squared Error (MSE):

$\text{MSE} = \frac{1}{n} \sum_{i=1}^{n} (\hat{y}_i - y_i)^2$

A perfect model has MSE = 0. Higher MSE means worse predictions.

R-Squared

The coefficient of determination $R^2$ measures how much variance in $y$ is explained by the model:

$R^2 = 1 - \frac{SS_{\text{res}}}{SS_{\text{tot}}}$

where:

• $SS_{\text{res}} = \sum (\hat{y}_i - y_i)^2$ — residual sum of squares
• $SS_{\text{tot}} = \sum (y_i - \bar{y})^2$ — total sum of squares

$R^2 = 1$ means a perfect fit; $R^2 = 0$ means the model is no better than predicting the mean.

Your Task

Implement:

• predict(x, w, b) — returns $wx + b$
• mse_loss(y_pred, y_true) — mean of squared differences
• r_squared(y_pred, y_true) — coefficient of determination