$\newcommand{\X}{\mathcal X}$$\newcommand{\dl}{\,\text d\lambda}$$\newcommand{\dx}{\,\text dx}$$\newcommand{\one}{\mathbf 1}$I’m going to take a look at functional linear regression, where I’ll find the best linear…
$\newcommand{\bern}{\text{Bern}}$$\newcommand{\vp}{\varphi}$$\newcommand{\e}{\varepsilon}$$\newcommand{\P}{\mathcal P}$$\newcommand{\Cov}{\text{Cov}}$$\newcommand{\E}{\text{E}}$$\newcommand{\one}{\mathbf 1}$$\newcommand{\s}{\mathbf s}$$\newcommand{\Disc}{\text{Disc}}$In this post I’m going to explore correlations between finitely-supported discrete variables. Bernoulli case I’ll begin with…
$\newcommand{\hb}{\hat\beta}$$\newcommand{\hbl}{\hat\beta_\lambda}$$\newcommand{\tb}{\tilde \beta}$$\newcommand{\L}{\mathcal L}$$\newcommand{\l}{\lambda}$$\newcommand{\e}{\varepsilon}$$\newcommand{\0}{\mathbf 0}$$\newcommand{\Lam}{\Lambda}$$\newcommand{\g}{\gamma}$$\newcommand{\D}{\mathcal D}$$\newcommand{\ht}{\hat\theta}$$\newcommand{\a}{\alpha}$In this post I’m going to explore ridge regression as a constrained optimization, and I’ll do…
$\newcommand{\e}{\varepsilon}$$\newcommand{\1}{\mathbf 1}$$\newcommand{\one}{\mathbf 1}$$\newcommand{\0}{\mathbf 0}$In this post I’m going to work with simple linear regression in matrix form. In most places…
In this post I want to take a look at interpolation with a single polynomial. I’ll prove the existence and…
$\newcommand{\var}{\text{Var}}$$\newcommand{\cov}{\text{Cov}}$$\newcommand{\E}{\text{E}}$$\newcommand{\one}{\mathbf 1}$$\newcommand{\0}{\mathbf 0}$$\newcommand{\a}{\alpha}$$\newcommand{\l}{\lambda}$$\newcommand{\s}{\sigma}$$\newcommand{\e}{\varepsilon}$$\newcommand{\t}{\theta}$In this post I’m going to consider a linear model of the form $y = \mu\one + Z\a…
$\newcommand{\a}{\alpha}$$\newcommand{\L}{\mathcal L}$$\newcommand{\one}{\mathbf 1}$$\newcommand{\x}{\mathbf x}$$\newcommand{\0}{\mathbf 0}$Let $G = (V,E)$ be an undirected and unweighted graph on $n$ vertices. I’ll have $A\in\{0,1\}^{n\times…
$\newcommand{\one}{\mathbf 1}$In this post I’m going to consider the following situation: there is a latent unobserved continuous-state Markov chain $Z…
$\newcommand{\e}{\varepsilon}$$\newcommand{\E}{\text{E}}$$\newcommand{\C}{\mathcal C}$$\newcommand{\0}{\mathbf 0}$Consider the linear model $y = X\beta+\e$ with $\E\e = \0$, $\E \e\e^T = \sigma^2 I$, and $X\in\mathbb…
In the previous post I looked at binary relations on finite sets; in this post I’m going to prove a…