Before changing anything
Make a prediction
Live moduleFoundations
Linear Regression
Fit a line to editable data and see how residuals, mean squared error, and gradient descent describe the same learning process.
Central questionHow does a model learn the best line?
Experiment controls
Edit the data
Click empty space to add a point. Drag a point to move it. Shift-click a point to remove it.
Data space
Prediction line and residuals
TrainingTestPrediction
The vertical segments are residuals: the difference between an observed value and the model prediction.
Parameter space
Loss surface
Drag or use the arrow keys to rotate. Height represents training loss; the red marker is the current model and the yellow line records gradient-descent steps.
Connect the views
What the model is calculating
A line makes one prediction at every horizontal position. Learning means adjusting its slope and intercept so the residuals become small overall—not necessarily zero for every point.
Try this
Select One influential outlier, choose Show best fit, then drag the isolated point. Watch the line and test error respond.
ŷ = wx + b
Select or add a point to inspect its calculation.
MSE = (1/n) Σ (y − ŷ)²
For mean squared error, the gradient measures how a tiny change in each parameter would change the loss. One update is:
w ← w − α(∂MSE/∂w) b ← b − α(∂MSE/∂b)
- A very small learning rate moves safely but slowly.
- A very large learning rate can overshoot or diverge.
- The closed-form best fit and converged gradient descent solve the same least-squares problem here.
- A low training error does not guarantee a good model for unseen data.
Live moduleClassical ML
Logistic Regression
Turn a weighted score into a probability, then see how a decision threshold converts that probability into one of two classes.
Central questionHow does a model draw a probabilistic boundary?
Experiment controls
Add new points as
Edit the data
Choose a class, then click to add a point. Drag any point to move it. Shift-click to remove it.
Feature space
Probability map and decision boundary
Class 0Class 1Boundary
Colour shows the model's probability for Class 1. The solid line marks the selected decision threshold.
Score to probability
The sigmoid function
Select or add a point to inspect its score and probability.
Decisions after thresholding
Confusion matrix
Predicted class
Actual class
01
0
0True negative
0False positive
10False negative
0True positive
Move only the threshold to change decisions while leaving every learned probability unchanged.
Connect the views
From score to class
The weights tilt a smooth probability surface across the feature space. The threshold slices that surface to produce a decision boundary: one side becomes Class 0 and the other Class 1.
Try this
Select Overlapping classes, auto-step briefly, then move the threshold from 0.30 to 0.70. Compare precision, recall, and the two kinds of error.
z = w₁x₁ + w₂x₂ + b → p = 1 / (1 + e−z)
Select or add a point to inspect its calculation.
Predict Class 1 when p ≥ 0.50
Cross-entropy strongly penalises confident probabilities that are wrong. One gradient step changes the weights and bias in the direction that lowers this loss.
Loss = −(1/n) Σ [y log(p) + (1−y) log(1−p)]
- The decision boundary is linear even though the probability is produced by a curved sigmoid.
- Threshold choice changes decisions and metrics, but it does not retrain the model.
- Accuracy can hide poor performance on an imbalanced dataset.
- A nonlinear pattern reveals a limitation of a single straight boundary.
Live module
Central question
Experiment controls
Interactive view
Connect interaction and idea