Interactive concept guides
Machine learning, one concept at a time
Short, hands-on pages that let you move the numbers yourself, a threshold, a penalty strength, and watch the consequence happen in front of you instead of reading about it secondhand.
Neural networks
Activation Functions
Drag a point along Sigmoid, ReLU, GELU and friends and watch the gradient it hands back to backprop, then see why over half a batch of neurons can go permanently dead under plain ReLU.
➕Trigonometry
Angle Sum Formulas
Slide two angles apart on a number line and watch sin(a+b) and sin a + sin b land in completely different places, then watch the real formula get built out of two stacked right triangles, live.
🧩Deep Learning
Backpropagation
Drag the weights of a tiny network, watch a forward pass flow left to right into a prediction, then watch the error flow backward through the exact same chain rule, node by node, and click through real gradient-descent steps.
🌳Ensemble Learning
Bagging, Boosting & Stacking
Grow bootstrapped trees and watch their average smooth out, add weak learners one at a time, then blend two different models better than either one alone.
🏹Model fitting
Bias-Variance Tradeoff
Fit the same noisy points with a straight line, then a wiggly degree-9 curve, and watch training error keep falling while testing error draws a U, with a dartboard showing exactly what bias and variance look like.
🔔Statistics
Central Limit Theorem
Pick a lopsided population, draw samples from it over and over, and watch the distribution of their means turn into a bell curve no matter how strange the population looked.
🔗Calculus
Chain Rule
Slide x and watch a nudge get resized by the inner function, then resized again by the outer one, and shrink the nudge itself to watch the chain rule's product prediction stop being an approximation at all.
🎚️Data preparation
Class Imbalance & Resampling
Watch a classifier hit 89% accuracy while catching only 32% of loan defaults, then fix it three ways: undersampling, oversampling, and SMOTE, and see why SMOTE wins on every metric.
🎲Probability
Conditional Probability & Bayes' Theorem
Drag the same medical-test picture apart both ways: condition on disease status, then flip it, and see why a 99%-accurate test on a rare disease is still a coin flip.
📏Statistics
Confidence Intervals
Draw sample after sample and build a fresh interval from each one, then watch roughly 95% of them actually contain the true mean, and a few honestly miss, exactly what "95% confident" is supposed to mean and almost never does in practice.
🔀Model evaluation
Cross-Validation
Flag credit card fraud with a held-out test set and 5-fold CV, then watch plain shuffling leave whole folds with zero fraud cases, and stratified folds fix it every time.
🌌Clustering
DBSCAN
Tune eps and minPts and watch points flip between core, border, and noise, then see it separate two interleaving moons that K-Means never can.
🌲Classification
Decision Tree
Drag a split threshold and watch information gain rise and fall in real time, then grow a full tree two ways: entropy and Gini pick a genuinely different root split on the same data.
∫Calculus
Derivatives & Integration
Watch a secant line rotate into a tangent as two points get infinitely close, then watch a pile of rectangles under the same curve converge on its exact area, the same limit, twice.
🔶Linear Algebra
Determinant as Area Scaling
Drag the same grid-warping matrix and watch the shaded unit square's signed area track det(M) exactly, then watch a circle scale by that identical factor, proving it's not a unit-square coincidence.
📏Foundations
Distance Metrics
Drag two points and watch Euclidean and Manhattan disagree, morph a unit circle into a diamond and a square as Minkowski's p sweeps from 1 to infinity, then stretch a vector and watch cosine similarity refuse to care.
🔻Linear Algebra
Dot Product & Projection
Drag two vectors and watch the angle between them, the dot product, and the projection shadow update together, then watch the shadow flip behind the origin the instant the angle passes 90°.
🧭Linear Algebra
Eigenvalues & Eigenvectors
Drag a vector around the unit circle and watch it fight its own transformation, until it snaps onto a direction that only stretches, never turns.
🗻Statistics
Empirical Rule (68-95-99.7)
Drag a value under a normal curve and watch which of the 68/95/99.7 bands it lands in, then generate real random samples and check the exact same percentages empirically.
🎰Probability
Expected Value & Variance
Drag a die's own probability bars and watch its expected value slide like a balance point while the spread bracket around it grows and shrinks, live, right along with it.
⚖️Data preparation
Feature Scaling & Normalization
Watch a distance calculation get hijacked by whichever feature happens to use bigger numbers, and see the nearest neighbor itself change once every feature is put on equal footing.
⛰️Optimization
Gradient Descent
Watch a line get fit one step at a time: a ball rolling down its own error surface, and what happens when the steps are too big.
🎛️Model selection
Hyperparameter Tuning
Sweep k, max depth, alpha, and n_components across a regressor, a classifier, a decision tree, K-Means, and PCA, and watch the same shape happen every time: training performance keeps improving while validation performance turns around.
🧪Statistics
Hypothesis Testing
Drag a critical value between two overlapping bell curves and watch Type I and Type II error trade off in real time, then watch a bigger sample buy power without touching the false-alarm rate.
📍Clustering
K-Means Clustering
Place a few centroids yourself and watch points sort themselves into groups: no labels, just repeating one simple rule.
📌Classification · Regression
K-Nearest Neighbors
No training at all: just look up the K closest points and vote or average. Watch small K chase noise and big K smooth it away.
💠Model fitting
L1 vs. L2: The Geometry of Sparsity
Drag the unconstrained optimum and watch why the L1 penalty's diamond-shaped constraint region snaps a coefficient exactly to zero, while the L2 penalty's circular region only ever shrinks it.
📉Statistics
Law of Large Numbers
Run the same three populations from the Central Limit Theorem page out to a thousand draws each and watch six independent running averages, all started from wildly different places, collapse onto the same true mean.
📐Trigonometry
Law of Sines & Cosines
Drag any vertex of a triangle and watch the Law of Sines' three ratios stay perfectly equal live, then slide a single side in the classic SSA setup and watch one triangle split into two, or vanish entirely.
🔎Calculus
Limits
Drag a point toward a discontinuity from both sides at once and watch the two one-sided approaches agree on a hole, disagree on a jump, or both fly off to infinity.
🪄Classification · Dimensionality reduction
Linear Discriminant Analysis
Rotate a single projection axis and watch two classes blur together or snap apart, then see why PCA's favorite direction and LDA's are almost never the same one.
📏Regression
Linear Regression
Drag a line through the noise and compare it to the exact least-squares answer, then watch one dragged point bend the whole fit.
📈Classification
Logistic Regression
Drag the line or run gradient descent yourself and watch a straight score get squashed into a probability by the sigmoid.
🌦️Probability
Markov Chains
Drag a weather chain's transition probabilities and watch tomorrow's forecast get computed from today's, over and over, until it settles into a steady-state no matter which day you started counting from.
🔲Linear Algebra
Matrix Transformations
Drag a matrix's two column vectors and watch a grid and the unit square shear, rotate, and scale in real time, then meet the same matrices again as presets on the Eigenvectors page.
🎯Statistics
Measures of Central Tendency
Drag points around and watch the mean chase every one of them while the median barely flinches, then see all three come apart once a distribution skews.
📐Statistics
Measures of Dispersion
Drag one point far out and watch the range and standard deviation balloon while the IQR barely moves, then read a box plot across four different spreads.
🕸️Deep Learning
Multi-Layer Perceptron
Train a tiny 2-3-1 network live on the exact same three datasets that broke the single Perceptron, and watch a hidden layer bend a straight line into a curve that finally solves XOR.
📧Classification
Naive Bayes Classifier
Toggle words in a message on and off and watch a spam filter multiply per-word evidence together live, the same Bayes' theorem as before, just with several clues instead of one.
🏁Optimization
Optimizers Race
Drop six optimizers onto the same loss surface from the same starting point and watch plain gradient descent zigzag down a narrow ravine while momentum and Adam glide past it, then watch them all crawl or escape a flat saddle.
🗻Calculus
Partial Derivatives & Gradient
Drag a point across a contour plot and watch the gradient arrow update live, always pointing straight uphill, while two side slices show the two plain slopes it's built from.
🧠Classification · Neural networks
Perceptron
Watch a decision line snap toward every point it gets wrong, converge in two passes on separable data, oscillate forever on messy data, and fail outright on XOR, no straight line ever will.
🔭Classification
Precision-Recall Curve & F1
The same classifier, the same threshold slider as ROC & AUC, but scored on a balanced population and a heavily imbalanced one: watch ROC AUC barely blink while the PR curve and F1 both collapse.
🗜️Dimensionality Reduction
Principal Component Analysis
Drag a scatter of points and watch its two principal components recompute live, try to beat PCA by hand-rotating a candidate direction, then compress the whole cloud down to one dimension and see exactly what gets lost.
🎛️Model fitting
Regularization
Watch a model overfit a small dataset, then watch Ridge, Lasso, and Elastic Net pull it back under control as the penalty grows.
📈Classification
ROC & AUC
Why the default 0.5 decision threshold is only correct when a false positive and a false negative cost the same.
📋Study design
Sampling Techniques
Survey the same 32 students eight different ways and watch convenience, purposive, and snowball sampling badly distort the true class-year mix, while stratified sampling nails it.
🖼️Matrix Factorization
Singular Value Decomposition
Watch any matrix split live into a rotation, a stretch, and another rotation, then keep only the biggest few singular values of a small image and watch it sharpen from a blur into the real picture.
⚔️Classification · Regression
SVM & the Margin
Drag the decision line yourself and watch the margin shrink and grow, then see the widest possible street an SVM finds instead.
🧮Linear Algebra
Systems of Linear Equations
Drag two lines by the points that define them and watch their solution point slide, then pull them parallel and watch it vanish, or coincident and watch every point become a solution at once.
🌊Calculus
Taylor Series
Slide the number of terms up and watch a polynomial hug sine, cosine, or eˣ tighter with every one added, then watch that same trick fail outside its radius of convergence no matter how many terms you add.
🌀Trigonometry
Unit Circle
Drag an angle around the unit circle and watch sine, cosine, and tangent trace out live as synced curves, then flip between degrees and radians without either curve so much as flickering.
💥Deep Learning
Vanishing/Exploding Gradients
Stack a chain of neurons deep and watch the backward gradient shrink toward zero or blow up exponentially depending only on one shared weight and one activation function's own derivative ceiling.
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