FIT2086_MOC

📘 FIT2086: Modelling for Data Analysis

Map of Content

Index for FIT2086 Modelling for Data Analysis — the statistics spine of the DS degree; W0–1 revision is shared with FIT1058_MOC and FIT1043_MOC, not duplicated.

📊 Assessment Map

• Assignment 1 (10%, due W5) · Assignment 2 (20%, due W8) · Assignment 3 (20%, due W11) ➔ carry the whole in-semester half; all involve implementing models in R (LO5).
• Final exam (50%) ➔ (Studio 12 sample paper) nine sections — short answer · MLE · random variables · CI & $p$-values · regression · classification · logistic regression · bias & variance · machine learning — all hand-worked, with a formula sheet and $z$ table supplied ➔ FIT2086 Unit Cheatsheet.
• LO map ➔ LO1 EDA/descriptive (W1–2) · LO2 inferential models (W3–5) · LO3 predictive models (W6–9, W11) · LO4 sampling/simulation/testing (W3, W5, W10) · LO5 implement in R (W6–11) · LO6 interpret results (W4–11).
• LO thread so far ➔ W1–2 probability models ➔ W3–4 fit (MLE), judge (bias/var/MSE), bound (CI) [A1] ➔ W5 test ➔ W6–7 linear + logistic regression, predictor selection [A2] ➔ W8 complexity control (tests · IC · CV · ridge/lasso) ➔ W9 trees · forests · kNN ➔ W10 simulation (Monte Carlo · bootstrap · permutation) ➔ W11 unsupervised (k-means · mixtures/EM · imputation) [A3] ➔ W12 revision of L1–L9 + sample paper.

🧰 Toolkit Cheatsheets

• FIT2086 Unit Cheatsheet → the paper-exam crib: every hand formula with its precondition, 📄 marks what the formula sheet supplies, Studio 12 exam anatomy + model short answers
• R Toolkit (Cheatsheet) → dual-unit (FIT1043 + FIT2086); FIT2086 blocks: simulation/distributions · hypothesis tests · lm/step · glm/pROC · glmnet · trees/forests/kNN · resampling (hand loops + boot/boot.ci)

📅 Knowledge Index

Week 0 — Self-Study / Revision

• Measures of Centrality → Parent Framework: Statistical Modelling and Inference (dual-unit — statistic $s(\mathbf{y})$ framing)
• Measures of Spread and Boxplots → Parent Framework: Statistical Modelling and Inference (dual-unit — variance $v(\mathbf{y})$, percentile $Q(\mathbf{y},p)$)
• Association Between Variables → Parent Framework: Statistical Modelling and Inference (dual-unit — Pearson $R(\mathbf{x},\mathbf{y})$, correlation ≠ causation)
• Mathematics for Modelling (Log, Exp, Calculus) → Parent Framework: Statistical Modelling and Inference (log/exp/derivative/partial — the MLE toolkit)
• R Basics (Syntax, Types, Control Flow), R Vectors, R Data Frames and IO, R Visualisation (base graphics) + the cheatsheet.

Week 1 — Modelling, Probability & Random Variables

• Statistical Modelling and Inference → Parent Framework: FIT2086_MOC (hub: population/sample/model/inference)
• Random Variables and Probability Distributions (FIT2086) → Parent Framework: Statistical Modelling and Inference (exam-heavy: pmf/pdf/cdf/quantile/mode; joint→marginal→conditional; iid)
• R Simulation and Random Sampling → Parent Framework: R for Data Science ( set.seed, sample, d/p/q/r)
• (Cross-links: the FIT1058 probability cluster — Random Variable, Conditional Probability, Bayes’ Theorem, Expectation, Variance and Standard Deviation — holds the maths.)

Week 2 — Expectations & Probability Distributions

• Expectations and Covariance (FIT2086) → Parent Framework: Random Variables and Probability Distributions (FIT2086) (exam-heavy: $E[f(X)]$, linearity, $V=E[X^2]-E[X{]}^2$, cov/corr, WLLN, non-existence)
• Taylor Approximation of Expectations → Parent Framework: Expectations and Covariance (FIT2086) (derivation drill — 2nd order for $E$, 1st order for $V$)
• Parametric Probability Distributions → Parent Framework: Statistical Modelling and Inference (hub: $p(x\mid \theta )$, $\theta \in \Theta$ + the distribution zoo tables)
• Gaussian Distribution → Parent Framework: Parametric Probability Distributions (self-similarity, $\sigma$-rules, additivity)
• Binomial Distribution · Poisson Distribution · Uniform Distribution — the discrete pair plus continuous $U(a,b)$, with means and variances

Studio 1 (run in W2 — R foundations, heart.csv / Mushroom.csv / wine.csv)

• Categorical Summaries and Cross-Tabulation in R → Parent Framework: R for Data Science (factor/table/prop.table/pie — the cat–cat toolkit)
• R Basics (Syntax, Types, Control Flow) · R Data Frames and IO · Association Between Variables — merged: functions, logical referencing, the cor screening loop

Week 3 — Parameter Estimation, Maximum Likelihood & Estimator Quality

• Maximum Likelihood Estimation → Parent Framework: Statistical Modelling and Inference (exam-heavy derivation drill: likelihood → NLL → $\partial L/\partial \theta =0$)
• Sampling Distribution of an Estimator → Parent Framework: Statistical Modelling and Inference ($\hat{\theta }$ is an RV; $\bar{Y}\sim N(\mu ,{\sigma }^2/n)$; strength-of-assumptions ladder)
• Estimator Quality (Bias, Variance, MSE) → Parent Framework: Sampling Distribution of an Estimator (exam-heavy: $\mathrm{MSE}=b^2+\mathrm{Var}$, efficiency, consistency)
• (Reading: Ross Ch. 6 §6.1, 6.2, 6.4, 6.5 and Ch. 7 §7.1, 7.2, 7.7)

Studio 2 (run in W3 — drills the W2 distributions; no new notes, merged into the existing four)

• Gaussian Distribution — $z$-table lookup with linear interpolation ($X\sim N(3,16)$ worked three ways); $Z_{\mu +k\sigma }=k$ is why the $\sigma$-rules are scale-free
• Binomial Distribution — term-by-term pmf interpretation; a named sequence has no $\left(\genfrac{}{}{0ex}{}{n}{m}\right)$; fair-coin tails by hand
• Poisson Distribution — additivity run backwards ($\lambda =6$/week ➔ ${\lambda }_i=6/7$/day); “at least one” $=1-e^{-\lambda }$
• Uniform Distribution — cdf derivation, $E[X]$ by integration, out-of-support probabilities are zero
• Parametric Probability Distributions — the family selection drill (15 described variables ➔ verdict + reason)
• R Simulation and Random Sampling — $O(n)$ running-mean accumulator + the WLLN convergence plot; why $\theta =0.9$ converges faster than $\theta =0.5$

Week 4 — Central Limit Theorem & Confidence Intervals

• Central Limit Theorem → Parent Framework: Sampling Distribution of an Estimator (exam-heavy: $\bar{Y}\stackrel{d}{\to }N(\mu ,{\sigma }^2/n)$; normal approximations)
• Confidence Intervals → Parent Framework: Statistical Modelling and Inference (exam-heavy: coverage vs the wrong ”$95\%$ probability” reading; the four cases $z$ / $t$ / difference / CLT-approximate)
• Student-t Distribution → Parent Framework: Parametric Probability Distributions ($\nu =n-1$, heavier tails, $t_{\alpha /2,\nu }>z_{\alpha /2}$ always)
• (Reading: Ross Ch. 6 §6.3 and Ch. 7 §7.3, 7.4, 7.5)

Studio 3 (run in W4 — drills the W3 estimation material; train.csv / test.csv)

• Plug-in Prediction and Held-Out Evaluation → Parent Framework: Maximum Likelihood Estimation (fit on train ➔ predict ➔ out-of-sample NLL)
• Monte Carlo Estimator Comparison → Parent Framework: Estimator Quality (Bias, Variance, MSE) (simulation loop for $b_{\theta }$/${\mathrm{Var}}_{\theta }$/MSE; RelMSE; mean vs median under contamination)
• Maximum Likelihood Estimation — the Bernoulli derivation $\hat{\theta }=m/n$; ML’s boundary overconfidence at $m\in \{0,n\}$
• Estimator Quality (Bias, Variance, MSE) — efficiency vs robustness: RelMSE is scale-free, moves with $n$ and the contamination fraction

Week 5 — Hypothesis Testing

• Hypothesis Testing → Parent Framework: Statistical Modelling and Inference (exam-heavy hub: Neyman–Pearson, $p$-value semantics + evidence grading, one vs two sided, $\alpha$, why the null can never be proved)
• Tests for Normal Means (z-test and t-test) → Parent Framework: Hypothesis Testing (exam-heavy hand skill: the 5-case selection matrix + 8 worked examples; pooled/Welch $t$ flagged optional)
• Tests for Bernoulli Populations → Parent Framework: Hypothesis Testing (CLT-based proportion $z$-test; null-supplied $\mathrm{se}$, pooled ${\hat{\theta }}_p$, exact binom.test/prop.test)
• Confidence Intervals — merged: proportion intervals, small-$n$ pooled/Welch difference intervals, and the interval–test pairing from the W5 summary tables
• (Reading: Lecture 5 Notes)

Studio 4 (run in W5 — drills the W4 CLT/CI material; train.csv / test.csv / SP500.csv)

• Confidence Intervals in R (calcCI) → Parent Framework: Confidence Intervals (A1 hand-and-R skill: calcCI anatomy ➔ heights train/test check ➔ SP500 pre/post-Lehman difference ➔ the reporting statement)
• Confidence Interval Coverage Simulation → Parent Framework: Confidence Intervals (generate ➔ build interval ➔ tally containment; exact $z$/$t$ vs the undercovering plug-in; Poisson $(\lambda ,n)$ coverage grid)
• Confidence Intervals — $z_{0.1}=1.281$ at $80\%$, the degenerate $100\%$ interval $(-\infty ,\infty )$, the plug-in proportion interval + the $n=12$ coin toss $(0.066,0.600)$
• Student-t Distribution — critical-value table extended to $n=5,10,50,100,1000$ ➔ $t\to z$ visibly by $n\approx 50$
• Estimator Quality (Bias, Variance, MSE) — the Bernoulli ${\hat{\theta }}_{ML}$ drill: $b=0$, $\mathrm{Var}=\mathrm{MSE}=\theta (1-\theta )/n$, consistency, and the CLT limit

Week 6 — Linear Regression & Model Selection

• Linear Regression (FIT2086) → Parent Framework: Statistical Modelling and Inference (exam-heavy hub: supervised setup, simple→multiple, least squares, $\text{RSS}$/$\text{TSS}$/$R^2$, prediction)
• Least Squares as Maximum Likelihood → Parent Framework: Linear Regression (FIT2086) (derivation drill: Gaussian errors ⟹ LS $=$ ML)
• Predictor Transformations (Indicators, Polynomials, Interactions) → Parent Framework: Linear Regression (FIT2086) ($K-1$ indicators, $\log$/polynomial, $x_j{x}_k$ — linear in $\mathbf{\beta }$, not in $x$)
• Model Selection and Information Criteria (AIC, BIC) → Parent Framework: Linear Regression (FIT2086) (exam-heavy: $L+\alpha (n,k_M)$, AIC vs BIC, stepwise)
• Multiple Regression and Stepwise Selection in R → Parent Framework: R for Data Science (LO5 hand skill: lm ➔ summary ➔ step(…, k = log(n)))
• Bias-Variance Tradeoff (Underfitting vs Overfitting) — merged: under/overfitting restated as omitting important vs including spurious predictors; generalisation; polynomial degree $=$ predictor-set choice
• (Reading: Ross Ch. 9)

Studio 5 (run in W6 — drills the W5 hypothesis-testing material; bpdata.csv / SP500.csv)

• Hypothesis Testing in R (t.test, binom.test, prop.test) → Parent Framework: Hypothesis Testing (exam + A2 hand skill: mu / alternative / conf.level / var.equal; rv$p.value; binom.test vs prop.test)
• Tests for Normal Means (z-test and t-test) — merged: one- vs two-sided on bpdata; three routes to one difference on S&P
• Tests for Bernoulli Populations — merged: CLT $z$ vs exact binom.test/prop.test; the sensitivity sweep
• Hypothesis Testing — merged: ${\sigma }^2\uparrow \Rightarrow p\uparrow$; the threshold scales with the consequences of a wrong call
• Confidence Intervals — merged: a one-sided alternative returns a bound $(-\infty ,u)$, not a range; the $90/95/99\%$ width sweep on bpdata

Week 7 — Classification & Logistic Regression

• Classification and Conditional Class Probabilities → Parent Framework: Statistical Modelling and Inference (hub: $\mathbb{P}(Y=y\mid \mathbf{x})$, curse of dimensionality)
• Logistic Regression → Parent Framework: Classification and Conditional Class Probabilities (exam-heavy: log-odds, Bernoulli NLL, ${\beta }_j$ reading)
• Classification Evaluation (Confusion Matrix and Metrics) — merged (now dual-unit): $\arg \max$ rule, CA/TPR/TNR, class-frequency baseline
• ROC and AUC → Parent Framework: Classification Evaluation (Confusion Matrix and Metrics) (hand skill: threshold $T$ sweep → sensitivity/specificity trade-off → ROC → AUC by counting ordered pairs)
• Logarithmic Loss → Parent Framework: Classification Evaluation (Confusion Matrix and Metrics) (scores the probabilities, not the labels; $=$ NLL of future data; $0.501$ vs $0.99$ is the whole point)
• Model Selection and Information Criteria (AIC, BIC) — merged: the penalty transfers unchanged to logistic regression as $L+k{\alpha }_n$, with ${\alpha }_n=1$ (AIC), $\frac{3}{2}$ (KIC, new), $\frac{1}{2}\log n$ (BIC)
• (Terms to revise, from the lecture: odds/log-odds · logistic regression · classification accuracy · specificity/sensitivity · AUC · logarithmic loss)

Week 8 — Model Selection & Penalized Regression

• Bias-Variance Tradeoff (Underfitting vs Overfitting) — merged: ${\text{MSE}}_f={\text{bias}}^2+\text{variance}$, irreducible ${\sigma }^2$, the simulation
• Multiple Testing and the Bonferroni Correction → Parent Framework: Hypothesis Testing (exam hand skill: $\alpha p$ false positives, threshold $\alpha /p$, FWER vs FDR)
• Model Selection and Information Criteria (AIC, BIC) — merged: NLL-scale AIC/KIC/BIC/RIC, when each over/underfits, the polynomial-order plot, problems with conventional selection
• Cross-Validation → Parent Framework: Model Selection and Information Criteria (AIC, BIC) (MSPE estimate; $K$-fold / repeated / LOO; LOO ≈ AIC)
• Penalized Regression (Ridge and Lasso) → Parent Framework: Linear Regression (FIT2086) (exam-heavy: ridge ${\ell }_2$ vs lasso ${\ell }_1$, $\lambda$ by CV)

Studio 7 (run in W8 — drills the W7 logistic-regression material; gene.*.csv / pima.*.csv)

• Logistic Regression in R (glm, pROC, step) → Parent Framework: Logistic Regression (LO5 hand skill: glm ➔ type = "response" ➔ my.pred.stats ➔ step)
• Logistic Regression — merged: null/residual deviance $=2L$, interaction reading on the log-odds, the pruned SNP12/SNP56 equation drill
• Predictor Transformations (Indicators, Polynomials, Interactions) — merged: judge a new term by $p$-value and deviance drop (log(BMI) keep, PLAS^2 drop, SKIN*AGE keep)
• (Terms to revise: under/overfitting · irreducible error · Bonferroni · AIC/BIC · CV · instability · ridge/lasso)

Week 9 — Trees & Nearest Neighbour Methods

• Decision Trees and Regression Trees — merged (now dual-unit with FIT1043): $L$ disjoint regions, Bernoulli/Normal leaf models, complexity $L$, strengths/weaknesses, three near-equal trees ⟹ instability
• Decision Tree Learning (Likelihood Splits, Pruning, CV) → Parent Framework: Decision Trees and Regression Trees (exam hand skill: leaf-NLL splits, pruning, CV over $L$)
• Random Forest — merged (dual-unit): bootstrap sample + random candidate variables per split, average means/probabilities, low bias with reduced variance, variable importance
• k-Nearest Neighbours → Parent Framework: Classification and Conditional Class Probabilities (exam hand skill: Euclidean distance, standardise, vote vs average vs kernel-weighted average, LOO CV for $k$)
• Cross-Validation — merged: the general complexity parameter $\gamma$ (predictor count, $\lambda$, $L$, $k$) and ${\gamma }^{\ast }=\arg {\min }_{\gamma }\text{CV}(\gamma )$
• Machine Learning — merged: the statistician’s framing — algorithmic, flexible, few assumptions, prediction over interpretation
• (Terms to revise, from the lecture: cross-validation · decision tree · split and leaf · random forest · $k$-nearest neighbours method)

Studio 8 (run in W9 — drills the W8 selection/shrinkage material; gene.*.csv / pima.*.csv / wrappers.R)

• Shrinkage Estimator of the Mean → Parent Framework: Estimator Quality (Bias, Variance, MSE) (derivation drill: $\hat{\mu }(c)=\frac{n}{n+c}\bar{Y}$ — bias, variance, MSE)
• Penalized Regression in R (glmnet) → Parent Framework: Penalized Regression (Ridge and Lasso) (LO5 hand skill: cv.glmnet.f, lambda.min, lasso vs BIC)
• Multiple Testing and the Bonferroni Correction — merged: the gene drill — 12 SNPs pass $0.05$ vs $5$ expected by chance, $0$ pass Bonferroni, RIC k = 2*log(p) keeps SNP56 alone ($p=0.0018$)
• Penalized Regression (Ridge and Lasso) — merged: the one-parameter shrinkage case and the Studio 8 small-$n$ evidence

Week 10 — Simulation-Based Statistical Methods

• Monte Carlo Simulation (Empirical Probabilities) → Parent Framework: Statistical Modelling and Inference (exam hand skill: average of indicators, resolution $1/m$)
• Pseudo-Random Number Generators → Parent Framework: Monte Carlo Simulation (Empirical Probabilities) (seed ➔ deterministic sequence ➔ reproducible; uniforms first; the LCG $X_{n+1}=(a{X}_n+c)\mathrm{mod}m$ hand trace)
• Bootstrap → Parent Framework: Sampling Distribution of an Estimator (exam-heavy: exact bootstrap by hand, resampling algorithm, percentile CI, bagging)
• Permutation Tests → Parent Framework: Hypothesis Testing (exam-heavy: shuffle $y$ under $H_0$, $p=\frac{1+\text{count}}{m+1}$)
• (Terms to revise, from the lecture: simulation · pseudo-random numbers · bootstrap distribution · bootstrap algorithm · permutation test)

Studio 9 (run in W10 — drills the W9 trees/forests/kNN material; diabetes.*.csv / wrappers.R)

• Trees, Forests and kNN in R (rpart, randomForest, kknn) → Parent Framework: Decision Trees and Regression Trees (LO5 hand skill: rpart, randomForest, train.kknn, test RMSE)
• Random Forest — merged: bagging lineage, out-of-bag error, %IncMSE as permutation importance, ntree
• k-Nearest Neighbours — merged: the six kknn kernels; CV picked gaussian, $k=24$
• Decision Trees and Regression Trees — merged: the pruned diabetes tree as instability evidence (different top leaf, near-equal prediction)

Week 11 — Introduction to Unsupervised Learning

• Machine Learning Styles (Supervised vs Unsupervised) — merged (now dual-unit): $p(Y\mid X)$ vs joint $p(X_1,\dots ,X_q)$, intrinsic classification
• k-means Clustering — merged (now tri-unit): objective $D={\sum }_i{\min }_k{d}_k(i{)}^2$ (WCSS), local minima + k-means++, cityblock/Hamming, the limitations list, the 5-person height/weight $K=2$ trace
• Mixture Models (EM Algorithm) → Parent Framework: Machine Learning Styles (Supervised vs Unsupervised) (exam hand skill: posterior $r_{i,k}$ by Bayes, EM, $K$ by IC)
• Multivariate Normal Distribution → Parent Framework: Gaussian Distribution (density, ${\mu }_j$/${\Sigma }_{i,j}$, spherical/diagonal/elliptical, read $\mathbf{\Sigma }$ from a scatter)
• Missing Data Imputation and Recommender Systems → Parent Framework: Machine Learning Styles (Supervised vs Unsupervised) (hand skill: univariate vs MVN vs kNN imputation; the Netflix $k=1$ trace)
• k-Nearest Neighbours · Model Selection and Information Criteria (AIC, BIC) — merged: kNN reused for imputation; IC/MML choose $K$
• (Terms to revise, from the lecture: clustering · k-means algorithm · mixture modelling · missing data · imputation · recommender systems · collaborative filtering)

Studio 10 (run in W11 — drills the W10 resampling material; bpdata.csv / pima.*.csv / studio10.prediction.stats.R / perm.log.reg.R)

• Resampling Inference in R (boot, boot.ci, perm.log.reg) → Parent Framework: Bootstrap (LO5 hand skill: boot ➔ boot.ci basic/BCa; permutation vs Wald $p$-values)
• Bootstrap — merged: $t$ vs basic vs BCa on BSA, median se and its spiky bootstrap distribution, the AUC interval inherits training optimism
• Permutation Tests — merged: global null vs conditional Wald test, Monte Carlo noise at $R=1000$, reading INS/BP null histograms, the shipped code’s $b/R$ zeros

Week 12 — Revision

• FIT2086 Unit Cheatsheet → Parent Framework: FIT2086_MOC (new: the L1–L9 revision lecture as one crib, syllabus-ordered, W10–11 appended)
• (No new concepts — the lecture’s worked examples (Poisson $\hat{\lambda }=5.4$, Pima BMI CI and $z$-test) already live in Maximum Likelihood Estimation, Confidence Intervals, Tests for Normal Means (z-test and t-test).)

Studio 12 (run in W12 — sample exam questions)

• FIT2086 Unit Cheatsheet — exam anatomy (section ➔ LO ➔ drilling note), formula-sheet coverage, the short-answer recipe + six model answers
• Tests for Normal Means (z-test and t-test) — merged: the cholesterol $z$-test by table lookup, $p\approx 0.077$, CI–test agreement
• Expectations and Covariance (FIT2086) — merged: two independent Poissons — label each line linearity vs independence
• Classification and Conditional Class Probabilities — merged: odds from a conditional; judge a predictor by comparing both conditionals
• Logistic Regression — merged: non-linearity fix (transform or switch to tree/forest); “is logistic regression appropriate?” off a scatter
• (Already drilled, no change: Exponential MLE ➔ Maximum Likelihood Estimation Practice 1 · $\bar{Y}$ unbiased/consistent ➔ Estimator Quality (Bias, Variance, MSE) · tree print-out ➔ Trees, Forests and kNN in R (rpart, randomForest, kknn) Practice 1 · kNN vote ➔ k-Nearest Neighbours.)

Revision sheets (L1–6 formula summary + W7–9 worked examples)

• Model Selection and Information Criteria (AIC, BIC) — merged: hand-score a four-model AIC/BIC table ($n=100$); AIC ties, BIC picks the smaller model
• Bias-Variance Tradeoff (Underfitting vs Overfitting) — merged: diagnose under/overfit from training vs held-out error (polynomial CV MSE · tree vs forest accuracy gap)
• Random Variables and Probability Distributions (FIT2086) · Expectations and Covariance (FIT2086) · Linear Regression (FIT2086) · Penalized Regression (Ridge and Lasso) · Classification and Conditional Class Probabilities — merged one line each: discrete median · $V[cX-dY]=c^{2}V[X]+d^{2}V[Y]$ · ”$R^2$ = % variation explained” · why the intercept is unpenalised · odds straight from the joint row
• FIT2086 Unit Cheatsheet — all of the above + $U(k)$, the $n<30$ / $n>30$ case split, the $p\le \alpha$ rule
• (Everything else in both decks is already drilled — no change.)

🔭 Outline topics not seen in the material

• Dirichlet · exploratory vs confirmatory analysis · Bayesian classification & inverse probability (closest: the posterior $r_{i,k}$ in Mixture Models (EM Algorithm)).

🧭 Suggested Reading Order

(read left→right · bold = assessment-critical)

• W0 — revision: Measures of Centrality → Measures of Spread and Boxplots → Association Between Variables → Mathematics for Modelling (Log, Exp, Calculus) (needed for every MLE derivation)
• W1 — modelling & probability: Statistical Modelling and Inference (the hub) → Random Variables and Probability Distributions (FIT2086) (sum/product rules, iid, pdf/CDF/quantile) → R Simulation and Random Sampling (simulate it in R)
• W2 — expectations & distributions: Expectations and Covariance (FIT2086) (every summary is an $E$) → Taylor Approximation of Expectations (derivation drill) → Parametric Probability Distributions (the zoo tables) → Gaussian Distribution → Binomial Distribution → Poisson Distribution → Uniform Distribution
• W3 — estimation: Maximum Likelihood Estimation (the derivation drill) → Sampling Distribution of an Estimator ($\hat{\theta }$ as an RV) → Estimator Quality (Bias, Variance, MSE) (compare estimators)
• W4 — CLT & intervals: Central Limit Theorem (shape for free) → Student-t Distribution (unknown ${\sigma }^2$) → Confidence Intervals (A1 hand skill) → Plug-in Prediction and Held-Out Evaluation (Studio 3, in R) → Monte Carlo Estimator Comparison (Studio 3, in R)
• W5 — testing: Hypothesis Testing (the logic + $p$-value) → Tests for Normal Means (z-test and t-test) (exam hand skill) → Tests for Bernoulli Populations (proportions) → Confidence Intervals in R (calcCI) (Studio 4, A1 skill) → Confidence Interval Coverage Simulation (Studio 4, in R)
• W6 — regression & selection: Linear Regression (FIT2086) (the hub) → Least Squares as Maximum Likelihood (derivation drill) → Predictor Transformations (Indicators, Polynomials, Interactions) (build the columns) → Bias-Variance Tradeoff (Underfitting vs Overfitting) (why prune) → Model Selection and Information Criteria (AIC, BIC) (exam hand skill) → Multiple Regression and Stepwise Selection in R (LO5, in R) → Hypothesis Testing in R (t.test, binom.test, prop.test) (Studio 5, in R)
• W7 — classification: Classification and Conditional Class Probabilities (why the joint dies) → Logistic Regression (the model + the derivation) → Classification Evaluation (Confusion Matrix and Metrics) (CA, TPR, TNR) → ROC and AUC (exam hand skill) → Logarithmic Loss (confidence, not labels)
• W8 — selection & shrinkage: Bias-Variance Tradeoff (Underfitting vs Overfitting) (the decomposition) → Multiple Testing and the Bonferroni Correction (why tests mislead) → Model Selection and Information Criteria (AIC, BIC) (four penalties) → Cross-Validation (estimate MSPE) → Penalized Regression (Ridge and Lasso) (stable selection) → Logistic Regression in R (glm, pROC, step) (Studio 7, in R)
• W9 — trees & neighbours: Machine Learning (the framing) → Decision Trees and Regression Trees (leaves + leaf models) → Decision Tree Learning (Likelihood Splits, Pruning, CV) (NLL split drill) → Random Forest (average away instability) → k-Nearest Neighbours (vote/average by hand) → Cross-Validation (one $\gamma$ recipe) → Shrinkage Estimator of the Mean (Studio 8 derivation) → Penalized Regression in R (glmnet) (Studio 8, in R)
• W10 — simulation & resampling: Monte Carlo Simulation (Empirical Probabilities) (average indicators) → Pseudo-Random Number Generators (why a seed reproduces) → Bootstrap (variability without assumptions) → Permutation Tests ($p$-values without assumptions) → Trees, Forests and kNN in R (rpart, randomForest, kknn) (Studio 9, in R)
• W11 — unsupervised learning: Machine Learning Styles (Supervised vs Unsupervised) (joint, not conditional) → k-means Clustering (WCSS trace by hand) → Multivariate Normal Distribution (read $\mathbf{\Sigma }$) → Mixture Models (EM Algorithm) (posterior by hand) → Missing Data Imputation and Recommender Systems (kNN fill by hand) → Resampling Inference in R (boot, boot.ci, perm.log.reg) (Studio 10, in R)
• W12 — revision: FIT2086 Unit Cheatsheet (one re-read of the whole paper half; then attempt Studio 12 cold)

🎯 Learning Outcomes (key skills per week)

• W0 ➔
  • classify data (nominal/ordinal/discrete/continuous)
  • compute + interpret centrality ($\bar{y}$, median, mode) and spread (range, $s(\mathbf{y})$, $v(\mathbf{y})$, percentiles/IQR, boxplots)
  • read Pearson correlation $R(\mathbf{x},\mathbf{y})$ and know $R\approx 0\ne$ independence
  • apply the log/exp identities and differentiate (power/log/exp, product, chain, partial) toward log-likelihoods
• W1 ➔
  • separate population/sample/model · sampling/inference/model-checking · three sources of randomness
  • state a pmf ($p(x)\ge 0$, $\sum p=1$); apply inclusion–exclusion
  • joint $\to$ marginal (sum rule) $\to$ conditional (product rule); test independence; write the iid product
  • continuous: $f$ is not a probability, $P(X=x)=0$, split piecewise integrals
  • derive cdf $\leftrightarrow$ pdf, survival $1-F$, quantile $Q(p)=F^{-1}(p)$, mode $=\arg \max p(x)$
  • in R: set.seed, sample, the d/p/q/r family
• W2 ➔
  • compute $E[X]$, $E[f(X)]$, $V[X]=E[X^2]-E[X{]}^2$ from a pmf
  • apply linearity $E[cf(X)+d]$ and $V[cX+d]=c^{2}V[X]$; know products need independence
  • read population $\mathrm{cov}$/$\mathrm{corr}$ and why $\mathrm{corr}=0\ne$ independence
  • state the WLLN; test whether $E[X]$ exists at all
  • derive $E[f(X)]\approx f(\mu )+\frac{{\sigma }^2}{2}{f}^{\prime \prime }(\mu )$ and $V[f(X)]\approx {\sigma }^2(f^{\prime }(\mu ){)}^2$
  • select a family by support: $N(\mu ,{\sigma }^2)$ · $Be(\theta )$ · $Bin(\theta ,n)$ · $U(a,b)$ · $Pois(\lambda )$, with each mean/variance
  • (Studio 1) subset a data frame by condition; write an R function returning a list; factor + table + prop.table; screen predictors with cor
• W3 ➔
  • separate the three inference tasks: point / interval / hypothesis testing
  • derive an MLE cold: $\prod p(y_i\mid \theta )$ → $L=-\log p$ → $\partial L/\partial \theta =0$ → $\hat{\theta }$
  • quote $\hat{\mu }=\bar{y}$, ${\hat{\sigma }}_{ML}^2=\frac{1}{n}\sum (y_i-\bar{y}{)}^2$, ${\hat{\lambda }}_{Poi}=\bar{y}$, ${\hat{\beta }}_{Exp}=1/\bar{y}$
  • use the plug-in distribution $p(y\mid \hat{\theta })$ for probability statements
  • derive $\bar{Y}\sim N(\mu ,{\sigma }^2/n)$ and say what naming the family buys
  • compute $b_{\theta }$, ${\mathrm{Var}}_{\theta }$, $\mathrm{MSE}=b^2+\mathrm{Var}$; rank by efficiency; test consistency
  • (Studio 2) read a $z$-table by interpolation; justify a family choice from a data description; code an $O(n)$ running mean and read a convergence plot
• W4 ➔
  • state the CLT for sums and derive $\bar{Y}\stackrel{d}{\to }N(\mu ,{\sigma }^2/n)$ from it
  • approximate $Bin(\theta ,n)$ by $N(n\theta ,n\theta (1-\theta ))$ and $Poi(\lambda )$ by $N(\lambda ,\lambda )$
  • recognise an estimator as an average ⟹ asymptotically normal
  • derive the $z$ interval by inverting $\frac{\hat{\mu }-\mu }{\sigma /\sqrt{n}}\sim N(0,1)$
  • select among the four CI cases: $z$ known ${\sigma }^2$ · $t_{\alpha /2,n-1}$ unknown · difference of means · CLT-approximate $\hat{\theta }\pm z_{\alpha /2}\sqrt{v(\hat{\theta })/n}$
  • state coverage correctly (procedure, not the one interval) and read a difference CI containing zero
  • (Studio 3) derive the Bernoulli MLE $\hat{\theta }=m/n$ and explain ML’s boundary overconfidence
  • (Studio 3) fit on train, predict with pnorm, check against empirical proportions, rank models by held-out NLL
  • (Studio 3) code a simulation study returning bias/variance/MSE, and read RelMSE for efficiency vs robustness
• W5 ➔
  • set up $H_0$ vs $H_A$ and identify one-sided vs two-sided from the wording
  • define a $p$-value as $P(\text{data this extreme}\mid H_0)$ and grade it ($0.01$ / $0.05$)
  • select the test: $\sigma$ known $z$ · $\sigma$ unknown small-$n$ $t_{n-1}$ · large-$n$ approximate $z$ · two-sample difference
  • compute $z$/$t$ by hand and bracket $p$ with tabulated criticals
  • test a proportion via the CLT, pooling ${\hat{\theta }}_p$ for two Bernoulli samples
  • explain why a large $p$ never proves $H_0$, and why $\alpha$-thresholding is deprecated
  • (Studio 4) state how CI width scales with $\sigma$ and $n$ ($\times 4$ data to halve the width); read $z$ at any $\alpha$
  • (Studio 4) write calcCI cold and report an interval as a sentence about the population
  • (Studio 4) build a difference-of-means CI for two groups and read it against zero
  • (Studio 4) derive $b$, $\mathrm{Var}$, MSE, consistency and the CLT limit for the Bernoulli ${\hat{\theta }}_{ML}$
  • (Studio 4) simulate coverage and explain why plug-in intervals undercover at small $n$
• W6 ➔
  • separate classification (categorical target) from regression (numerical target)
  • write $\mathbb{E}[Y_i\mid {\mathbf{x}}_i]={\beta }_0+{\sum }_j{\beta }_j{x}_{i,j}$; read each ${\beta }_j$ as a per-unit change
  • fit by least squares; state $\sum e_i=0$, $\mathrm{corr}(\mathbf{x},\mathbf{e})=0$, and the $p<n$ requirement
  • compute $R^2=1-\text{RSS}/\text{TSS}$ and say why it can never select a model
  • derive $L=\frac{n}{2}\log (2\pi {\sigma }^2)+\frac{\text{RSS}}{2{\sigma }^2}$ ⟹ LS $=$ ML; quote ${\hat{\sigma }}_{ML}^2$ vs ${\hat{\sigma }}_u^2$
  • build $K-1$ indicators, polynomial terms, and interaction columns
  • rank predictors by $t_j$, test $H_0:{\beta }_j=0$, score models by $L+\alpha (n,k_M)$
  • contrast AIC ($k_M$, overfits) with BIC ($\frac{k_M}{2}\log n$, underfits); $\ge 3$ is significant
  • search all-subsets $2^p$ vs forward/backward stepwise; run step(…, k = log(n)) in R
  • (Studio 5) run t.test with the right mu, alternative, conf.level and var.equal
  • (Studio 5) explain why ${\sigma }^2\uparrow$ weakens the evidence, and read a $p$ of $0.9$ correctly
  • (Studio 5) judge a large effect with a borderline $p$ — demand a larger trial, not a verdict
  • (Studio 5) compare approximate $z$, Welch $t$ and pooled $t$ on one difference, and say which interval is overconfident
  • (Studio 5) test one and two proportions by hand, then check against binom.test / prop.test
• W7 ➔
  • separate classification (categorical $Y$) from regression; predict by $\arg {\max }_y\mathbb{P}(Y=y\mid \mathbf{x})$
  • count $2^{p+1}$ joint cells — the curse of dimensionality that kills direct estimation
  • convert $\eta \to$ odds $e^{\eta }\to$ probability $\frac{1}{1+e^{-\eta }}$, and read ${\beta }_j$ as log-odds per unit
  • derive the Bernoulli NLL ${\sum }_i[-y_i{\eta }_i+\log (1+e^{{\eta }_i})]$; state convex, no closed form
  • compute $\text{CA}$, $\text{TPR}$, $\text{TNR}$ from a confusion matrix, against the class-frequency baseline
  • compute AUC by hand from ranked scores, and say what log-loss adds that AUC cannot
• W8 ➔
  • define $\text{bias}(x_0)$ and $\text{variance}(x_0)$ over repeated training samples ${\hat{f}}_{\mathcal{D}}$
  • state ${\text{MSE}}_f={\text{bias}}^2+\text{variance}$ and add ${\sigma }^2$ for predicting a new $Y^{\prime }$ — irreducible
  • map underfitting ➔ high bias/low variance and overfitting ➔ low bias/high variance; say what more $n$ does and doesn’t fix
  • compute expected false positives $\alpha p$ and the Bonferroni threshold $\alpha /p$; distinguish FWER from FDR
  • write AIC $L+k$, KIC $L+\frac{3}{2}k$, BIC $L+\frac{k}{2}\log n$, RIC $L+k\log p$ and say which over/underfits
  • describe $K$-fold, repeated $K$-fold and LOO CV, and choose a model by smallest CV error
  • explain why all-or-nothing subset selection is statistically unstable
  • write the penalised LS objective, justify standardising, and read a $\lambda$ path
  • contrast ridge (${\ell }_2$, stable, no zeros) with lasso (${\ell }_1$, sparse, biased large coefficients); pick $\lambda$ by CV then refit on all data
  • explain penalisation as adding bias to cut variance, and why it helps with multicollinearity and $n<p$
  • (Studio 7) fit glm(…, family = binomial) and read deviance, AIC $=$ deviance $+2k$, and each $z$-test
  • (Studio 7) predict probabilities with type = "response", threshold at $\frac{1}{2}$, and score CA/sens/spec/AUC/log-loss on test data
  • (Studio 7) prune with step at k = 2 / 3 / log(n) and argue BIC (interpretable) vs KIC (predictive)
  • (Studio 7) add log(), I(x^2) and a*b terms and justify each from its $p$-value and deviance drop
  • (Studio 7) write a pruned model’s log-odds equation and read each coefficient’s effect
• W9 ➔
  • explain how a tree partitions predictor space into $L$ leaves and predicts with each leaf’s Bernoulli/Normal model
  • compute a leaf’s minimised NLL $-n_1\log \frac{n_1}{n}-n_0\log \frac{n_0}{n}$ and pick the purer split
  • explain why splitting never raises the NLL ⟹ size trees by IC, pruning, or $K$-fold CV over $L$
  • explain how bootstrap samples and random candidate variables let a forest cut variance, and what it costs in interpretability
  • apply kNN: Euclidean distance on standardised predictors, then vote / average / kernel-weighted average
  • choose $k$, distance and kernel by LOO CV
  • (Studio 8) derive bias, variance and MSE of $\hat{\mu }(c)=\frac{n}{n+c}\bar{Y}$, find where it beats $\bar{Y}$, and prove it is the ridge solution
  • (Studio 8) count false discoveries against $\alpha p$, apply Bonferroni, and run RIC stepwise with k = 2*log(p)
  • (Studio 8) fit lasso/ridge with cv.glmnet.f, read the lambda.min coefficients, and compare with BIC stepwise on test data
  • (Studio 8) explain why penalisation beats stepwise when $n$ is small relative to $p$
• W10 ➔
  • write $\mathbb{P}((X_1,\dots ,X_p)\in Z)\approx \frac{1}{m}{\sum }_{i=1}^{m}I({\mathbf{X}}^{(i)}\in Z)$ and prove convergence via indicator ➔ Bernoulli ➔ WLLN
  • state the resolution $1/m$ and size $m\gg 1/p$ for a rare event
  • approximate $\mathbb{E}[X]$, $\mathbb{V}[X]$, quantiles and $p(x)$ from simulated draws
  • explain a PRNG as a seeded deterministic recurrence; trace an LCG $X_{n+1}=(a{X}_n+c)\mathrm{mod}m$
  • enumerate an exact bootstrap ($M=n^n$) and compute its bias and variance by hand
  • run the bootstrap algorithm: resample rows with replacement, refit, take percentiles for a CI
  • say why bootstrapping training MSE does not remove its optimism, and how bagging cuts a tree’s variance
  • justify a permutation test from $p(Y,X)=p(Y)p(X)$ and compute $p=\frac{1+\text{count}}{m+1}$
  • choose CV vs bootstrap vs permutation test from the question asked
  • (Studio 9) read an rpart print-out and follow a new individual to its leaf
  • (Studio 9) size a tree with learn.tree.cv and compare test RMSE against a lasso benchmark
  • (Studio 9) read a forest’s OOB error, % Var explained and %IncMSE; tune ntree
  • (Studio 9) choose $k$ and kernel with train.kknn and predict with fitted(kknn(…))
  • (Studio 9) score tree/forest classifiers by probability ([, 2], type = "prob") and kNN by accuracy only
• W11 ➔
  • contrast supervised $p(Y\mid X_1,\dots ,X_p)$ with unsupervised structure in $p(X_1,\dots ,X_q)$
  • run k-means by hand: assign to the nearest centroid, re-centre, stop when nothing moves
  • state the WCSS objective ${\sum }_i{\min }_k{d}_k(i{)}^2$, why seeds give local minima, and what k-means++ changes
  • list k-means’ limitations: hard assignment, mixed attributes, no statistical reading, choosing $K$
  • write the mixture density ${\sum }_k{\alpha }_{k}p({\mathbf{y}}_i\mid {\mathbf{\theta }}_k)$ and the conditional-independence product ${\prod }_{j}p(y_{i,j}\mid {\mathbf{\theta }}_{k,j})$
  • compute posterior memberships $r_{i,k}={\alpha }_k{p}_k/{\sum }_l{\alpha }_l{p}_l$ and assign by the largest
  • describe EM (E-step soft assign, M-step weighted ML) as a generalisation of k-means
  • choose $K$ by an information criterion / MML and say why the search needs split/merge/restart moves
  • read $\mathbf{\mu }$ and $\mathbf{\Sigma }$ of a multivariate normal off a scatter; name the spherical/diagonal/elliptical structures
  • compare univariate, multivariate-normal and kNN imputation; impute a ratings matrix with kNN on co-observed columns
  • (Studio 10) write a boot statistic function(data, indices) and get basic and BCa intervals with boot.ci
  • (Studio 10) explain the spiky bootstrap distribution of a median
  • (Studio 10) bootstrap a refitted model’s AUC and say why the interval does not remove training optimism
  • (Studio 10) run a coefficient-wise permutation test on a glm and explain why its $p$-values differ from the Wald $p$-values
• W12 ➔
  • answer a definition question as what it is + one or two properties, in 2–3 sentences
  • separate what the formula sheet supplies (known-$\sigma$ CI and $z$-test, differentiation rules, $z$ table) from what must be recalled (pmfs, means/variances, MLEs, $V[\bar{Y}]$, $R^2$, IC penalties)
  • read a $p$-value off the $P(Z<-|z|)$ table at the nearest row, double it for two-sided, and grade the evidence
  • label every $E$/$V$ step as linearity (always) or independence (needs it)
  • judge a predictor by comparing $P(Y=1\mid x)$ across its values, and judge logistic regression’s fit by whether a straight boundary separates the classes
  • score a model table by AIC and BIC by hand, and diagnose under/overfitting from the training vs held-out error gap