Start with data
Past house sales pair size, location and price. The target is a number, not a category.
MACHINE LEARNING / REGRESSION
LINEAR REGRESSION
Finding the straight-line relationship hidden inside data — the line that best explains a continuous target.
SupervisedRegressionLinear
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01 Overview
01 / UNDERSTAND
What problem does a straight line solve?
Linear Regression finds the line that best connects an input to a continuous outcome. It turns a cloud of observations into a relationship we can explain, evaluate and use.
Look for a trend
Points scatter, but they can still tilt. A slope describes the average change in the target.
Make it explainable
Each coefficient has a readable meaning: how much the prediction changes per unit of input.
REAL-WORLD QUESTIONCan previous house sales teach us the mathematical relationship between size and price?
02 / VISUALIZE
See Linear Regression in Action
Move the parameters and watch the line, residuals and error respond immediately.
Drag a point to change the data
03 / INTUITION
From scattered data to a useful line
Observe paired values.
Choose β₀ and β₁.
Compare y to ŷ.
Minimize total error.
Linear Regression searches for the line that minimizes prediction error across the observed data.
04 / MATHEMATICS
The Mathematics Behind Linear Regression
ŷ = β₀ + β₁x
One line. Four ideas. Select a symbol to focus its meaning.
Explore the Mathematics +
A residual is \(e_i = y_i - \hat{y}_i\). Least squares minimizes \(J = \frac{1}{n}\sum_i (y_i-\hat{y}_i)^2\), producing a smooth convex loss surface. For a full matrix, the normal equation is \(\beta = (X^T X)^{-1}X^T y\). Gradient descent follows \(\beta \leftarrow \beta - \eta\nabla J\) until the loss stops improving.
Show Full Derivation +
Centre x and y, compute \(s_{xx}=\sum(x_i-\bar{x})^2\) and \(s_{xy}=\sum(x_i-\bar{x})(y_i-\bar{y})\). Then \(\beta_1=s_{xy}/s_{xx}\) and \(\beta_0=\bar{y}-\beta_1\bar{x}\). Squaring the residuals makes the optimum unique for a non-degenerate dataset.
05 / TRAINING
How the model learns
LOSS CURVEError falls as the line learns
See the training algorithm +
Start with β₀ = β₁ = 0, compute the gradient of the mean squared error, step downhill with learning rate η, and repeat until the loss stabilizes. The normal equation reaches the same optimum directly when the matrix is well-conditioned.
06 / PRACTICE
Work it through
WORKED EXAMPLE
Five observations, one equation
| x | y |
|---|---|
| 1 | 1 |
| 2 | 3 |
| 3 | 2 |
| 4 | 5 |
| 5 | 5 |
- Average: x̄ = 3, ȳ = 3
- Slope: β₁ = sxy / sxx = 12 / 10 = 1.2
- Intercept: β₀ = ȳ − β₁x̄ = −0.6
ŷ = −0.6 + 1.2x For x = 6, predict ŷ = 6.6
07 / APPLY
Know when it earns its place
Great choice when
✓ The relationship is roughly linear
✓ A continuous prediction is needed
✓ Interpretability matters
✓ A fast baseline is useful
Avoid or be careful when
× The relationship is highly nonlinear
× Extreme outliers dominate
× Complex interactions matter
× Assumptions are seriously violated
60-SECOND RECAP
Data↓Assume relationship↓Fit line↓Minimize error↓Evaluate↓Predict
Remember: Linear Regression learns the straight-line function that best represents the relationship between input and output variables.