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Linear Models and Statistical Learning

Year 3 · 4 sections · 40 concepts · 64 practice questions

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What's covered

free1 · The general linear model10 concepts

Overview 1 · The plan of this section
Proposition 1.1 · The hat matrix
Lemma 1.2 · Cochran's theorem (projection form)
Theorem 1.3 · Sampling distribution in the normal linear model
Theorem 1.4 · The F test for nested models
Example 1.5 · One-way ANOVA as a linear model
Proposition 1.6 · Confidence and prediction intervals
Example 1.7 · Collinearity and variance inflation
Definition 1.8 · Model selection criteria
Check 1.9 · Check: inference in a linear model

2 · Diagnostics9 concepts

Overview 2 · The plan of this section
Proposition 2.1 · Properties of residuals
Proposition 2.2 · Leverage
Proposition 2.3 · Deleting a case
Definition 2.4 · Cook's distance
Example 2.5 · Reading residual plots
Definition 2.6 · Transformations and Box–Cox
Proposition 2.7 · Weighted and generalised least squares
Check 2.8 · Check: diagnosing a regression

3 · Generalised linear models11 concepts

Overview 3 · The plan of this section
Definition 3.1 · Exponential dispersion families
Proposition 3.2 · Mean and variance from the cumulant function
Example 3.3 · Standard families
Definition 3.4 · Generalised linear models and links
Proposition 3.5 · The score equations
Theorem 3.6 · Iteratively reweighted least squares
Example 3.7 · Logistic regression
Example 3.8 · Poisson regression and offsets
Definition 3.9 · Deviance
Check 3.10 · Check: fitting and interpreting a GLM

4 · Regularisation and model assessment10 concepts

Overview 4 · The plan of this section
Proposition 4.1 · The bias–variance decomposition
Definition 4.2 · Ridge regression
Proposition 4.3 · Ridge as shrinkage along principal directions
Theorem 4.4 · Ridge can beat least squares
Proposition 4.5 · The lasso and soft thresholding
Remark 4.6 · Why the lasso gives sparse solutions
Proposition 4.7 · Cross-validation and its shortcut
Definition 4.8 · Effective degrees of freedom
Check 4.9 · Check: choosing a regularised model

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Proposition 1.1 · The hat matrix

$H$ projects onto the columns of $X$ ; $\operatorname{tr}H = p$ ; residuals live in the $(n - p)$ -dimensional complement.

Proof

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