$\newcommand{\one}{\mathbf 1}\newcommand{\convd}{\stackrel{\text d}\to}\newcommand{\convp}{\stackrel{\textp}\to}\newcommand{\p}{\mathbf p}\newcommand{\0}{\mathbf 0}\newcommand{\X}{\mathbf X}\newcommand{\Mult}{\text{Mult}}\newcommand{\E}{\operatorname{E}}\newcommand{\e}{\varepsilon}\newcommand{\Var}{\operatorname{Var}}\newcommand{\R}{\mathbb R}\newcommand{\rank}{\operatorname{rank}}\newcommand{\H}{\mathcal H}$Let $X$ be a continuous random vector in $\mathbb R^n$ with distribution $P_X$…
$\newcommand{\vp}{\varphi}$$\newcommand{\E}{\operatorname{E}}$$\newcommand{\Exp}{\operatorname{Exp}}$In this post I’ll take a look at wrapped distributions. If $X$ has a continuous distribution on $\mathbb R$ with…
$\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{\F}{\mathscr F}$$\newcommand{\R}{\mathbb R}$$\newcommand{\A}{\mathscr A}$$\newcommand{\G}{\mathcal G}$$\newcommand{\E}{\operatorname E}$$\newcommand{\dp}{\,\text dP}$$\newcommand{\1}{\mathbf 1}$Let $(\Omega, \F, P)$ be a probability space and let $X : \Omega…
$\newcommand{\vp}{\varphi}$$\newcommand{\dmux}{\,\text d\mu(x)}$$\newcommand{\dmu}{\,\text d\mu}$$\newcommand{\E}{\operatorname{E}}$Let $\mu$ be a probability measure on $(\mathbb R, \mathbb B)$ where $\mathbb B$ is the Borel $\sigma$-algebra…
$\newcommand{\e}{\varepsilon}$In this post I’ll be exploring non-integer bases and the “natural numbers” they represent. I say “natural numbers” because the…