index.Rmd

--- 
title: "Partial least squares (PLS) and friends"
author: "JDM and friends"
date: "`r Sys.Date()`"
site: bookdown::bookdown_site
documentclass: book
#bibliography: [book.bib]
description: |
  This is a bookdown to explain main concepts of PLS and its related methods.
#link-citations: yes
#github-repo: rstudio/bookdown-demo
---

# About


## Notation

- $a, b$: scalars, i.e., elements of $\mathbb{R}$, the set of real numbers.

- $X, Y$: matrices; $X \in \mathbb{R}^{n\times p}$ is an $n$ by $p$ matrix.

- $\boldsymbol{a}, \boldsymbol{b}$: column vectors (i.e., vectors are by default column vectors); $\boldsymbol{a} \in \mathbb{R}^{n \times 1}$ has $n$ elements, i.e., $\boldsymbol{a} = (a_1,\dots,a_n)^\top$.

- $\lvert a \rvert$: length of vector $\boldsymbol{a}$, i.e., $\lvert \boldsymbol{a} \rvert = \left( \sum_{i=1}^n a_i^2 \right)^{1/2}$.

- $\angle (\boldsymbol{a}, \boldsymbol{b})$: angle (between $-90^\circ$ and $90^\circ$) between two vectors.

- $\bar{\boldsymbol{a}}$: average of elements in $\boldsymbol{a}$, i.e., $\bar{\boldsymbol{a}} = n^{-1} \sum_{i=1}^n a_i$.

- $var(\boldsymbol{a})$: variance of elements in $\boldsymbol{a}$, i.e., $var(\boldsymbol{a}) = n^{-1} \sum_{i=1}^{n} (a_i - \bar{\boldsymbol{a}})^2$.

- $X^\top, \boldsymbol{a}^\top$: transpose of matrix and vectors.

- $\equiv$: always equivalent to, or equivalent by definition; e.g., $c \equiv \boldsymbol{a}^\top \boldsymbol{b}$ says that $c$ is defined as the inner product between $\boldsymbol{a}$ and $\boldsymbol{b}$.