$\newcommand{\R}{\mathbb R}\newcommand{\e}{\varepsilon}\newcommand{\0}{\mathbf 0}\newcommand{\E}{\operatorname{E}}\newcommand{\Var}{\operatorname{Var}}$Consider a linear regression model $y = X\beta+\e$ with $X\in\R^{n\times p}$ known and full column rank. Under common…
$\newcommand{\R}{\mathcal R}\newcommand{\C}{\mathcal C}\newcommand{\Rbb}{\mathbb R}\newcommand{\0}{\mathbf 0}\newcommand{\one}{\mathbb 1}\newcommand{\span}{\operatorname{span}}\newcommand{\rank}{\operatorname{rank}}$In this post I’m going to prove the Steinitz exchange lemma, that the row rank…
$\newcommand{\one}{\mathbf 1}\newcommand{\convd}{\stackrel{\text d}\to}\newcommand{\convp}{\stackrel{\textp}\to}\newcommand{\p}{\mathbf p}\newcommand{\0}{\mathbf 0}\newcommand{\X}{\mathbf X}\newcommand{\Mult}{\text{Mult}}\newcommand{\E}{\operatorname{E}}\newcommand{\e}{\varepsilon}\newcommand{\Var}{\operatorname{Var}}\newcommand{\R}{\mathbb R}\newcommand{\rank}{\operatorname{rank}}\newcommand{\H}{\mathcal H}$Let $X$ be a continuous random vector in $\mathbb R^n$ with distribution $P_X$…
$\newcommand{\e}{\varepsilon}$$\newcommand{\Var}{\operatorname{Var}}$$\newcommand{\x}{\mathbf x}$$\newcommand{\one}{\mathbf 1}$$\newcommand{\0}{\mathbf 0}$In my previous post I stopped right before working out the REML objective function so I’ll start…
$\newcommand{\Var}{\operatorname{Var}}$$\newcommand{\Cov}{\operatorname{Cov}}$$\newcommand{\one}{\mathbf 1}$$\newcommand{\0}{\mathbf 0}$$\newcommand{\e}{\varepsilon}$$\newcommand{\E}{\operatorname{E}}$In this post I’m going to discuss leverage scores in linear regression. In particular, I’ll show how the…
$\newcommand{\e}{\varepsilon}$$\newcommand{\Var}{\operatorname{Var}}$$\newcommand{\x}{\mathbf x}$$\newcommand{\one}{\mathbf 1}$$\newcommand{\0}{\mathbf 0}$In this post I will derive the point estimates for mixed models. I’ll focus on interpretation so…
$\newcommand{\R}{\mathbb R}$$\newcommand{\y}{\mathbf y}$$\newcommand{\thb}{\tilde{\hat\beta}}$$\newcommand{\L}{\Lambda}$$\newcommand{\tr}{\operatorname{tr}}$In this post I’ll look at updating a linear regression with new data. One place where this can…
$\newcommand{\R}{\mathbb R}$$\newcommand{\e}{\varepsilon}$$\newcommand{\H}{\mathcal H}$$\newcommand{\0}{\mathbf 0}$In this post I’m going to prove that the squared exponential kernel is positive definite. I’ll finish…
$\newcommand{\one}{\mathbf 1}$$\newcommand{\0}{\mathbf 0}$$\newcommand{\tr}{\operatorname{tr}}$$\newcommand{\e}{\varepsilon}$In this post I’m going to look at some relationships between the diagonal of a positive semidefinite (PSD)…
$\newcommand{\0}{\mathbf 0}$$\newcommand{\one}{\mathbf 1}$In this post I’m going to work out the distribution of the area of a random parallelogram formed…