# Linear Regression

> Linear regression draws the best straight line through your data. It is the "hello world" of machine learning.

**Type:** Build
**Languages:** Python
**Prerequisites:** Phase 1 (Linear Algebra, Calculus, Optimization), Phase 2 Lesson 1
**Time:** ~90 minutes

## Learning Objectives

- Derive the gradient descent update rules for mean squared error and implement linear regression from scratch
- Compare gradient descent and the normal equation in terms of computational complexity and when to use each
- Build a multiple linear regression model with feature standardization and interpret the learned weights
- Explain how Ridge regression (L2 regularization) prevents overfitting by penalizing large weights

## The Problem

You have data: house sizes and their sale prices. You want to predict the price of a new house given its size. You could eyeball it on a scatter plot, but you need a formula. You need a line that best fits the data so you can plug in any size and get a price prediction.

Linear regression gives you that line. More importantly, it introduces the entire ML training loop: define a model, define a cost function, optimize the parameters. Every ML algorithm follows this same pattern. Master it here with the simplest case, and you will recognize it everywhere.

This is not just for simple problems. Linear regression is used in production systems for demand forecasting, A/B test analysis, financial modeling, and as a baseline for every regression task.

## The Concept

### The Model

Linear regression assumes a linear relationship between input (x) and output (y):

```
y = wx + b
```

- `w` (weight/slope): how much y changes when x increases by 1
- `b` (bias/intercept): the value of y when x = 0

For multiple inputs (features), this extends to:

```
y = w1*x1 + w2*x2 + ... + wn*xn + b
```

Or in vector form: `y = w^T * x + b`

The goal: find the values of w and b that make the predicted y as close as possible to the actual y across all training examples.

### The Cost Function (Mean Squared Error)

How do you measure "as close as possible"? You need a single number that captures how wrong your predictions are. The most common choice is Mean Squared Error (MSE):

```
MSE = (1/n) * sum((y_predicted - y_actual)^2)
```

Why squared? Two reasons. First, it penalizes large errors more than small errors (an error of 10 is 100x worse than an error of 1, not 10x). Second, the squared function is smooth and differentiable everywhere, which makes optimization straightforward.

The cost function creates a surface. For a single weight w and bias b, the MSE surface looks like a bowl (a convex paraboloid). The bottom of the bowl is where MSE is minimized. Training means finding that bottom.

### Gradient Descent

Gradient descent finds the bottom of the bowl by taking steps downhill.

```mermaid
flowchart TD
    A[Initialize w and b randomly] --> B[Compute predictions: y_hat = wx + b]
    B --> C[Compute cost: MSE]
    C --> D[Compute gradients: dMSE/dw, dMSE/db]
    D --> E[Update parameters]
    E --> F{Cost low enough?}
    F -->|No| B
    F -->|Yes| G[Done: optimal w and b found]
```

The gradients tell you two things: which direction to move each parameter, and how much to move.

For MSE with y_hat = wx + b:

```
dMSE/dw = (2/n) * sum((y_hat - y) * x)
dMSE/db = (2/n) * sum(y_hat - y)
```

The update rule:

```
w = w - learning_rate * dMSE/dw
b = b - learning_rate * dMSE/db
```

The learning rate controls step size. Too large: you overshoot the minimum and diverge. Too small: training takes forever. Typical starting values: 0.01, 0.001, or 0.0001.

### The Normal Equation (Closed-Form Solution)

For linear regression specifically, there is a direct formula that gives the optimal weights without any iteration:

```
w = (X^T * X)^(-1) * X^T * y
```

This inverts a matrix to solve for w in one step. It works perfectly for small datasets. For large datasets (millions of rows or thousands of features), gradient descent is preferred because matrix inversion is O(n^3) in the number of features.

### Multiple Linear Regression

With multiple features, the model becomes:

```
y = w1*x1 + w2*x2 + ... + wn*xn + b
```

Everything works the same: MSE is the cost function, gradient descent updates all weights simultaneously. The only difference is that you are fitting a hyperplane instead of a line.

Feature scaling matters here. If one feature ranges from 0 to 1 and another ranges from 0 to 1,000,000, gradient descent will struggle because the cost surface becomes elongated. Standardize features (subtract mean, divide by standard deviation) before training.

### Polynomial Regression

What if the relationship is not linear? You can still use linear regression by creating polynomial features:

```
y = w1*x + w2*x^2 + w3*x^3 + b
```

This is still "linear" regression because the model is linear in the weights (w1, w2, w3). You are just using nonlinear features of x.

Higher-degree polynomials can fit more complex curves but risk overfitting. A degree-10 polynomial will pass through every point in a 10-point dataset but predict poorly on new data.

### R-Squared Score

MSE tells you how wrong you are, but the number depends on the scale of y. R-squared (R^2) gives a scale-independent measure:

```
R^2 = 1 - (sum of squared residuals) / (sum of squared deviations from mean)
    = 1 - SS_res / SS_tot
```

- R^2 = 1.0: perfect predictions
- R^2 = 0.0: the model is no better than predicting the mean every time
- R^2 < 0.0: the model is worse than predicting the mean

### Regularization Preview (Ridge Regression)

When you have many features, the model can overfit by assigning large weights. Ridge regression (L2 regularization) adds a penalty:

```
Cost = MSE + lambda * sum(w_i^2)
```

The penalty term discourages large weights. The hyperparameter lambda controls the tradeoff: higher lambda means smaller weights and more regularization. This is covered in depth in a later lesson. For now, know that it exists and why it helps.

```figure
linear-regression-fit
```

## Build It

### Step 1: Generate sample data

```python
import random
import math

random.seed(42)

TRUE_W = 3.0
TRUE_B = 7.0
N_SAMPLES = 100

X = [random.uniform(0, 10) for _ in range(N_SAMPLES)]
y = [TRUE_W * x + TRUE_B + random.gauss(0, 2.0) for x in X]

print(f"Generated {N_SAMPLES} samples")
print(f"True relationship: y = {TRUE_W}x + {TRUE_B} (+ noise)")
print(f"First 5 points: {[(round(X[i], 2), round(y[i], 2)) for i in range(5)]}")
```

### Step 2: Linear regression from scratch with gradient descent

```python
class LinearRegression:
    def __init__(self, learning_rate=0.01):
        self.w = 0.0
        self.b = 0.0
        self.lr = learning_rate
        self.cost_history = []

    def predict(self, X):
        return [self.w * x + self.b for x in X]

    def compute_cost(self, X, y):
        predictions = self.predict(X)
        n = len(y)
        cost = sum((pred - actual) ** 2 for pred, actual in zip(predictions, y)) / n
        return cost

    def compute_gradients(self, X, y):
        predictions = self.predict(X)
        n = len(y)
        dw = (2 / n) * sum((pred - actual) * x for pred, actual, x in zip(predictions, y, X))
        db = (2 / n) * sum(pred - actual for pred, actual in zip(predictions, y))
        return dw, db

    def fit(self, X, y, epochs=1000, print_every=200):
        for epoch in range(epochs):
            dw, db = self.compute_gradients(X, y)
            self.w -= self.lr * dw
            self.b -= self.lr * db
            cost = self.compute_cost(X, y)
            self.cost_history.append(cost)
            if epoch % print_every == 0:
                print(f"  Epoch {epoch:4d} | Cost: {cost:.4f} | w: {self.w:.4f} | b: {self.b:.4f}")
        return self

    def r_squared(self, X, y):
        predictions = self.predict(X)
        y_mean = sum(y) / len(y)
        ss_res = sum((actual - pred) ** 2 for actual, pred in zip(y, predictions))
        ss_tot = sum((actual - y_mean) ** 2 for actual in y)
        return 1 - (ss_res / ss_tot)


print("=== Training Linear Regression (Gradient Descent) ===")
model = LinearRegression(learning_rate=0.005)
model.fit(X, y, epochs=1000, print_every=200)
print(f"\nLearned: y = {model.w:.4f}x + {model.b:.4f}")
print(f"True:    y = {TRUE_W}x + {TRUE_B}")
print(f"R-squared: {model.r_squared(X, y):.4f}")
```

### Step 3: Normal equation (closed-form solution)

```python
class LinearRegressionNormal:
    def __init__(self):
        self.w = 0.0
        self.b = 0.0

    def fit(self, X, y):
        n = len(X)
        x_mean = sum(X) / n
        y_mean = sum(y) / n
        numerator = sum((X[i] - x_mean) * (y[i] - y_mean) for i in range(n))
        denominator = sum((X[i] - x_mean) ** 2 for i in range(n))
        self.w = numerator / denominator
        self.b = y_mean - self.w * x_mean
        return self

    def predict(self, X):
        return [self.w * x + self.b for x in X]

    def r_squared(self, X, y):
        predictions = self.predict(X)
        y_mean = sum(y) / len(y)
        ss_res = sum((actual - pred) ** 2 for actual, pred in zip(y, predictions))
        ss_tot = sum((actual - y_mean) ** 2 for actual in y)
        return 1 - (ss_res / ss_tot)


print("\n=== Normal Equation (Closed-Form) ===")
model_normal = LinearRegressionNormal()
model_normal.fit(X, y)
print(f"Learned: y = {model_normal.w:.4f}x + {model_normal.b:.4f}")
print(f"R-squared: {model_normal.r_squared(X, y):.4f}")
```

### Step 4: Multiple linear regression

```python
class MultipleLinearRegression:
    def __init__(self, n_features, learning_rate=0.01):
        self.weights = [0.0] * n_features
        self.bias = 0.0
        self.lr = learning_rate
        self.cost_history = []

    def predict_single(self, x):
        return sum(w * xi for w, xi in zip(self.weights, x)) + self.bias

    def predict(self, X):
        return [self.predict_single(x) for x in X]

    def compute_cost(self, X, y):
        predictions = self.predict(X)
        n = len(y)
        return sum((pred - actual) ** 2 for pred, actual in zip(predictions, y)) / n

    def fit(self, X, y, epochs=1000, print_every=200):
        n = len(y)
        n_features = len(X[0])
        for epoch in range(epochs):
            predictions = self.predict(X)
            errors = [pred - actual for pred, actual in zip(predictions, y)]
            for j in range(n_features):
                grad = (2 / n) * sum(errors[i] * X[i][j] for i in range(n))
                self.weights[j] -= self.lr * grad
            grad_b = (2 / n) * sum(errors)
            self.bias -= self.lr * grad_b
            cost = self.compute_cost(X, y)
            self.cost_history.append(cost)
            if epoch % print_every == 0:
                print(f"  Epoch {epoch:4d} | Cost: {cost:.4f}")
        return self

    def r_squared(self, X, y):
        predictions = self.predict(X)
        y_mean = sum(y) / len(y)
        ss_res = sum((actual - pred) ** 2 for actual, pred in zip(y, predictions))
        ss_tot = sum((actual - y_mean) ** 2 for actual in y)
        return 1 - (ss_res / ss_tot)


random.seed(42)
N = 100
X_multi = []
y_multi = []
for _ in range(N):
    size = random.uniform(500, 3000)
    bedrooms = random.randint(1, 5)
    age = random.uniform(0, 50)
    price = 50 * size + 10000 * bedrooms - 1000 * age + 50000 + random.gauss(0, 20000)
    X_multi.append([size, bedrooms, age])
    y_multi.append(price)


def standardize(X):
    n_features = len(X[0])
    means = [sum(X[i][j] for i in range(len(X))) / len(X) for j in range(n_features)]
    stds = []
    for j in range(n_features):
        variance = sum((X[i][j] - means[j]) ** 2 for i in range(len(X))) / len(X)
        stds.append(variance ** 0.5)
    X_scaled = []
    for i in range(len(X)):
        row = [(X[i][j] - means[j]) / stds[j] if stds[j] > 0 else 0 for j in range(n_features)]
        X_scaled.append(row)
    return X_scaled, means, stds


y_mean_val = sum(y_multi) / len(y_multi)
y_std_val = (sum((yi - y_mean_val) ** 2 for yi in y_multi) / len(y_multi)) ** 0.5
y_scaled = [(yi - y_mean_val) / y_std_val for yi in y_multi]

X_scaled, x_means, x_stds = standardize(X_multi)

print("\n=== Multiple Linear Regression (3 features) ===")
print("Features: house size, bedrooms, age")
multi_model = MultipleLinearRegression(n_features=3, learning_rate=0.01)
multi_model.fit(X_scaled, y_scaled, epochs=1000, print_every=200)

print(f"\nWeights (standardized): {[round(w, 4) for w in multi_model.weights]}")
print(f"Bias (standardized): {multi_model.bias:.4f}")
print(f"R-squared: {multi_model.r_squared(X_scaled, y_scaled):.4f}")
```

### Step 5: Polynomial regression

```python
class PolynomialRegression:
    def __init__(self, degree, learning_rate=0.01):
        self.degree = degree
        self.weights = [0.0] * degree
        self.bias = 0.0
        self.lr = learning_rate

    def make_features(self, X):
        return [[x ** (d + 1) for d in range(self.degree)] for x in X]

    def predict(self, X):
        features = self.make_features(X)
        return [sum(w * f for w, f in zip(self.weights, row)) + self.bias for row in features]

    def fit(self, X, y, epochs=1000, print_every=200):
        features = self.make_features(X)
        n = len(y)
        for epoch in range(epochs):
            predictions = [sum(w * f for w, f in zip(self.weights, row)) + self.bias for row in features]
            errors = [pred - actual for pred, actual in zip(predictions, y)]
            for j in range(self.degree):
                grad = (2 / n) * sum(errors[i] * features[i][j] for i in range(n))
                self.weights[j] -= self.lr * grad
            grad_b = (2 / n) * sum(errors)
            self.bias -= self.lr * grad_b
            if epoch % print_every == 0:
                cost = sum(e ** 2 for e in errors) / n
                print(f"  Epoch {epoch:4d} | Cost: {cost:.6f}")
        return self

    def r_squared(self, X, y):
        predictions = self.predict(X)
        y_mean = sum(y) / len(y)
        ss_res = sum((actual - pred) ** 2 for actual, pred in zip(y, predictions))
        ss_tot = sum((actual - y_mean) ** 2 for actual in y)
        return 1 - (ss_res / ss_tot)


random.seed(42)
X_poly = [x / 10.0 for x in range(0, 50)]
y_poly = [0.5 * x ** 2 - 2 * x + 3 + random.gauss(0, 1.0) for x in X_poly]

x_max = max(abs(x) for x in X_poly)
X_poly_norm = [x / x_max for x in X_poly]
y_poly_mean = sum(y_poly) / len(y_poly)
y_poly_std = (sum((yi - y_poly_mean) ** 2 for yi in y_poly) / len(y_poly)) ** 0.5
y_poly_norm = [(yi - y_poly_mean) / y_poly_std for yi in y_poly]

print("\n=== Polynomial Regression (degree 2 vs degree 5) ===")
print("True relationship: y = 0.5x^2 - 2x + 3")

print("\nDegree 2:")
poly2 = PolynomialRegression(degree=2, learning_rate=0.1)
poly2.fit(X_poly_norm, y_poly_norm, epochs=2000, print_every=500)
print(f"  R-squared: {poly2.r_squared(X_poly_norm, y_poly_norm):.4f}")

print("\nDegree 5:")
poly5 = PolynomialRegression(degree=5, learning_rate=0.1)
poly5.fit(X_poly_norm, y_poly_norm, epochs=2000, print_every=500)
print(f"  R-squared: {poly5.r_squared(X_poly_norm, y_poly_norm):.4f}")

print("\nDegree 2 fits the true curve well. Degree 5 fits training data slightly better")
print("but risks overfitting on new data.")
```

### Step 6: Ridge regression (L2 regularization)

```python
class RidgeRegression:
    def __init__(self, n_features, learning_rate=0.01, alpha=1.0):
        self.weights = [0.0] * n_features
        self.bias = 0.0
        self.lr = learning_rate
        self.alpha = alpha

    def predict_single(self, x):
        return sum(w * xi for w, xi in zip(self.weights, x)) + self.bias

    def predict(self, X):
        return [self.predict_single(x) for x in X]

    def fit(self, X, y, epochs=1000, print_every=200):
        n = len(y)
        n_features = len(X[0])
        for epoch in range(epochs):
            predictions = self.predict(X)
            errors = [pred - actual for pred, actual in zip(predictions, y)]
            mse = sum(e ** 2 for e in errors) / n
            reg_term = self.alpha * sum(w ** 2 for w in self.weights)
            cost = mse + reg_term
            for j in range(n_features):
                grad = (2 / n) * sum(errors[i] * X[i][j] for i in range(n))
                grad += 2 * self.alpha * self.weights[j]
                self.weights[j] -= self.lr * grad
            grad_b = (2 / n) * sum(errors)
            self.bias -= self.lr * grad_b
            if epoch % print_every == 0:
                print(f"  Epoch {epoch:4d} | Cost: {cost:.4f} | L2 penalty: {reg_term:.4f}")
        return self


print("\n=== Ridge Regression (L2 Regularization) ===")
print("Same data as multiple regression, with alpha=0.1")
ridge = RidgeRegression(n_features=3, learning_rate=0.01, alpha=0.1)
ridge.fit(X_scaled, y_scaled, epochs=1000, print_every=200)
print(f"\nRidge weights: {[round(w, 4) for w in ridge.weights]}")
print(f"Plain weights: {[round(w, 4) for w in multi_model.weights]}")
print("Ridge weights are smaller (shrunk toward zero) due to the L2 penalty.")
```

## Use It

Now the same thing with scikit-learn, which is what you will actually use in production.

```python
from sklearn.linear_model import LinearRegression as SklearnLR
from sklearn.linear_model import Ridge
from sklearn.preprocessing import PolynomialFeatures, StandardScaler
from sklearn.model_selection import train_test_split
from sklearn.metrics import mean_squared_error, r2_score
import numpy as np

np.random.seed(42)
X_sk = np.random.uniform(0, 10, (100, 1))
y_sk = 3.0 * X_sk.squeeze() + 7.0 + np.random.normal(0, 2.0, 100)

X_train, X_test, y_train, y_test = train_test_split(X_sk, y_sk, test_size=0.2, random_state=42)

lr = SklearnLR()
lr.fit(X_train, y_train)
y_pred = lr.predict(X_test)

print("=== Scikit-learn Linear Regression ===")
print(f"Coefficient (w): {lr.coef_[0]:.4f}")
print(f"Intercept (b): {lr.intercept_:.4f}")
print(f"R-squared (test): {r2_score(y_test, y_pred):.4f}")
print(f"MSE (test): {mean_squared_error(y_test, y_pred):.4f}")

poly = PolynomialFeatures(degree=2, include_bias=False)
X_poly_sk = poly.fit_transform(X_train)
X_poly_test = poly.transform(X_test)

lr_poly = SklearnLR()
lr_poly.fit(X_poly_sk, y_train)
print(f"\nPolynomial degree 2 R-squared: {r2_score(y_test, lr_poly.predict(X_poly_test)):.4f}")

scaler = StandardScaler()
X_train_scaled = scaler.fit_transform(X_train)
X_test_scaled = scaler.transform(X_test)

ridge = Ridge(alpha=1.0)
ridge.fit(X_train_scaled, y_train)
print(f"Ridge R-squared: {r2_score(y_test, ridge.predict(X_test_scaled)):.4f}")
print(f"Ridge coefficient: {ridge.coef_[0]:.4f}")
```

Your from-scratch implementation and scikit-learn produce the same results. The difference: scikit-learn handles edge cases, numerical stability, and performance optimizations. Use the library for production. Use the from-scratch version to understand what is happening.

## Ship It

This lesson produces:
- `outputs/skill-regression.md` - a skill for choosing the right regression approach based on the problem

## Exercises

1. Implement batch gradient descent, stochastic gradient descent (SGD), and mini-batch gradient descent. Compare convergence speed on the same dataset. Which converges fastest? Which has the smoothest cost curve?
2. Generate data from a cubic function (y = ax^3 + bx^2 + cx + d + noise). Fit polynomials of degree 1, 3, and 10. Compare training R^2 and test R^2. At what degree does overfitting become obvious?
3. Implement Lasso regression (L1 regularization: penalty = alpha * sum(|w_i|)). Train on the multi-feature housing data. Compare which weights go to zero vs Ridge. Why does L1 produce sparse solutions while L2 does not?

## Key Terms

| Term | What people say | What it actually means |
|------|----------------|----------------------|
| Linear regression | "Draw a line through data" | Find weight w and bias b that minimize the sum of squared differences between wx+b and actual y values |
| Cost function | "How bad the model is" | A function that maps model parameters to a single number measuring prediction error, which optimization minimizes |
| Mean squared error | "Average of squared errors" | (1/n) * sum of (predicted - actual)^2, penalizing large errors disproportionately |
| Gradient descent | "Walk downhill" | Iteratively adjust parameters in the direction that reduces the cost function, using partial derivatives |
| Learning rate | "Step size" | A scalar that controls how much parameters change per gradient descent step |
| Normal equation | "Solve it directly" | The closed-form solution w = (X^T X)^-1 X^T y that gives optimal weights without iteration |
| R-squared | "How good the fit is" | The fraction of variance in y explained by the model, ranging from negative infinity to 1.0 |
| Feature scaling | "Make features comparable" | Transforming features to similar ranges (e.g., zero mean, unit variance) so gradient descent converges faster |
| Regularization | "Penalize complexity" | Adding a term to the cost function that shrinks weights, preventing overfitting |
| Ridge regression | "L2 regularization" | Linear regression with a penalty of lambda * sum(w_i^2) added to MSE |
| Polynomial regression | "Fitting curves with linear math" | Linear regression on polynomial features (x, x^2, x^3, ...), still linear in the weights |
| Overfitting | "Memorizing training data" | Using a model so complex that it fits noise in training data and fails on new data |

## Further Reading

- [An Introduction to Statistical Learning (ISLR)](https://www.statlearning.com/) -- free PDF, chapters 3 and 6 cover linear regression and regularization with practical R examples
- [The Elements of Statistical Learning (ESL)](https://hastie.su.domains/ElemStatLearn/) -- free PDF, the more mathematical companion to ISLR with deeper treatment of ridge and lasso
- [Stanford CS229 Lecture Notes on Linear Regression](https://cs229.stanford.edu/main_notes.pdf) -- Andrew Ng's notes deriving the normal equation and gradient descent from first principles
- [scikit-learn LinearRegression documentation](https://scikit-learn.org/stable/modules/linear_model.html) -- practical reference for LinearRegression, Ridge, Lasso, and ElasticNet with code examples
