> ## Documentation Index
> Fetch the complete documentation index at: https://ai.teamjaaf.com/llms.txt
> Use this file to discover all available pages before exploring further.

# Linear Regression

> Predicting continuous values with linear models.

# 📈 Linear Regression

Linear regression models the relationship between a dependent variable $y$ and one or more features $x_1, \dots, x_n$ by fitting a linear function.

## Hypothesis

For an input vector $\mathbf{x} = [1, x_1, \dots, x_n]^T$, the model predicts

$$
\hat{y} = \mathbf{x}^T \boldsymbol{\theta} = \theta_0 + \theta_1 x_1 + \cdots + \theta_n x_n.
$$

## Loss Function

Parameters $\boldsymbol{\theta}$ are learned by minimizing the **mean squared error** (MSE):

$$
J(\boldsymbol{\theta}) = \frac{1}{m} \sum_{i=1}^{m} (\hat{y}^{(i)} - y^{(i)})^2.
$$

## Example (scikit-learn)

```python theme={null}
import numpy as np
from sklearn.linear_model import LinearRegression

# toy dataset
X = np.array([[1], [2], [3], [4]])
y = np.array([2, 3, 3, 5])

model = LinearRegression()
model.fit(X, y)

pred = model.predict([[5]])
print(pred)
```

## Interpretation

* Coefficients $\theta_j$ show how much the prediction changes per unit increase in feature $x_j$.
* The model is fast and interpretable but assumes a linear relationship between features and target.