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Multivariate Lab

MAST90138 · The University of Melbourne · 2023 Semester 2

Multivariate Lab:
Seeds & Rainfall

Two assignments from Multivariate Statistics for Data Science, rebuilt as interactive labs. Explore covariance and eigen-decomposition, run a principal component analysis of wheat kernels in your browser, and follow how 365 days of rainfall tell northern and southern Australian weather stations apart.

Start with the rainfall story Open the covariance playground

Every result is reproduced from the original R code. Where the 2023 submission had a bug, an As submitted / Corrected switch shows both.

JanFebMarAprMayJunJulAugSepOctNovDec365 daysof rainfallNorthSouth

The coursework

What the assignments asked

Both were individual assignments in R Markdown, submitted in the second semester of 2023.

Assignment 1 · submitted 11 September 2023

Covariance, eigenvectors and PCA

  • Decide which entries make a 2 × 2 matrix a valid covariance matrix.
  • Find eigenvalues and orthonormal eigenvectors of a 2 × 2 matrix by hand.
  • Write the wheat data's sample covariance as S=ΓΛΓ⊤S=\Gamma\Lambda\Gamma^\top.
  • Run a PCA: variance explained, ψ, scree plot, PC scatter by variety, correlation circle.

Assignment 3 · submitted 22 October 2023

High-dimensional classification

  • Explain why QDA and logistic regression fail with 365 predictors and 150 stations.
  • Logistic regression on q principal components, q chosen by hand-written LOOCV.
  • A random forest with m tuned by out-of-bag error; Gini importance by day.
  • A single tree on 50 PLS components, interpreted through correlation curves.

What I built

Three labs

The statistics are ported to TypeScript and checked against R in the test suite: a Jacobi eigen-solver, R's glm.fit with LINPACK's pivoting QR, kernel PLS and the rpart tree-growing algorithm. Your browser recomputes the wheat PCA, every logistic fit, the LOOCV curve (on request) and the PLS tree; the rainfall PCA and PLS scores and the random forests come from R as small JSON files.

Assignment 1 · Problem 1

Covariance & eigen playground

Drag the entries of a 2 × 2 matrix, see when it is a valid covariance, and watch Σ = ΓΛΓᵀ and its ellipse update live.

Corrects the 2023 answer: a = 3 and b ≥ 9, not b ≥ 3a.

Open lab
Assignment 1 · Problem 2

Wheat seeds PCA explorer

Scree plot, cumulative variance, PC scores by variety and the correlation circle, recomputed in your browser for three versions of the analysis.

PC1 of S keeps 82.9% of the variance (as asked); after standardising, three PCs keep 98.7%.

Open lab
Assignment 3

Rainfall classifier story

PCA + logistic regression with hand-written LOOCV, a 5,000-tree random forest and a tree on PLS components, each with an as-submitted / corrected switch.

Fixing the LOOCV and test-scaling bugs takes the test errors from 4 and 16 down to 1 and 1.

Open lab

Key results

What the analysis found

82.9%
wheat variance on PC1 of S
covariance PCA, as asked; 98.7% with 3 PCs after standardising
q = 3
PCs chosen by corrected LOOCV
7/150 LOOCV errors (2023 loop: q = 27)
3/41
random-forest test errors
late-January rainfall is the most informative
16 → 1
PLS-tree test errors after the fix
test set standardised like the training set

About this project

Subject, stack and credits

Subject
MAST90138 Multivariate Statistics for Data Science
Institution
The University of Melbourne, Master of Data Science, 2023 Semester 2
Author
Sunchuangyu (Rin) Huang (individual assignments)
Submitted
A1: 11 September 2023 · A3: 22 October 2023
Original stack
R, R Markdown (knitr → PDF), MASS, randomForest, pls, rpart, ggplot2
Revived stack
Next.js 16, TypeScript, Tailwind CSS 4, KaTeX, Web Workers, Vitest parity tests against R

The original submissions are kept unchanged in the repository for reference. Assignment specifications and the course rainfall files are not published here; tasks are paraphrased and only class-level aggregates and projections of the rainfall data are shown.

Source: github.com/rNLKJA/Unimelb-Master-2023-MAST90138-Assignments (private repository for now)

Methods, errata and reproducibility