svm() — Support Vector Machines

Finds the boundary with the widest gap between classes.

svm(X, y)

Key idea

Pick the dividing line with the WIDEST gap (the margin) between the two groups. (e.g. margin = 2 / ||w||)
Only the closest points — the support vectors — touch the gap and define the line.
A soft margin allows a few points to slip across the gap, set by a cost setting.
The kernel trick (a richer similarity measure) lets SVMs draw curved boundaries without heavy high-dimensional maths.

Definition

$w, b min \frac{1}{2} ∥ w ∥^{2} s.t. y_{i} (w^{⊤} x_{i} + b) \geq 1$

$w$: the boundary direction
$margin = 2/ ∥ w ∥$: the width of the gap
$y_{i}$: the class label (+1 or -1)

In plain words

Don't just separate the two groups — leave the biggest possible no-man's-land down the middle.

Where you'd use it

Text and image classification, especially with limited data.
Bioinformatics problems with many features and few samples.
Any task where a wide-margin, robust boundary matters.

Frequently asked questions

What does an SVM do?: Finds the widest-margin boundary between classes; the kernel lets that boundary bend.
Why tune C and gamma?: They trade margin width against fitting the training points — cross-validate to choose.

Statistics Playground

An interactive playground for probability and statistics. Pick a function, edit its parameters, and see the distribution curve, the computed value, and a plain-English interpretation — all computed in your browser (no server, no install). The functions mirror familiar R syntax (for example pnorm, dbinom, t.test) and the numbers come from numpy running under Pyodide.

The normal distribution

pnorm(q, mean = 0, sd = 1) — Cumulative probability for the normal distribution.
dnorm(x, mean = 0, sd = 1) — Height of the bell curve at a point (the density).
qnorm(p, mean = 0, sd = 1) — The inverse of pnorm; turns a probability back into a value.
rnorm(n, mean = 0, sd = 1) — Draws random values from a normal distribution.
scale(x) — Re-expresses values as z-scores (mean 0, sd 1).

Binomial & Poisson distributions

dbinom(x, size, prob) — Probability of exactly x successes in n trials.
pbinom(q, size, prob) — Probability of q or fewer successes.
dpois(x, lambda) — Probability of exactly x events at rate λ.
ppois(q, lambda) — Probability of q or fewer events at rate λ.

The chi-square distribution & hypothesis testing

pchisq(q, df) — Cumulative probability for the χ² distribution.
t.test(x, mu = 0) — One-sample t-test against a hypothesised mean.

Correlation & regression

cor(x, y) — Strength of a linear relationship, −1 to +1.
lm(y ~ x) — Fits a least-squares straight line through the points.

Descriptive statistics

mean(x) — The arithmetic average.
median(x) — The middle value of the data.
sd(x) — Standard deviation; typical distance of a value from the mean.
var(x) — Variance; the mean squared deviation from the mean.
IQR(x) — Span of the middle 50% of the data.
range(x) — The minimum and maximum values.
quantile(x) — Sample quantiles that split the data into equal slices.
summary(x) — The five-number summary plus the mean.
weighted.mean(x, w) — An average where each value carries a weight.

Combinatorics, sampling & visualisation

factorial(x) — x!, the ways to arrange x distinct items in order.
choose(n, k) — Ways to choose k items from n, order ignored.
sample(x, size, replace = FALSE) — Draw a random sample.
hist(x, breaks) — A frequency histogram of the data.
boxplot(x) — A box-and-whisker summary of the spread.