I have just downloaded a free ebook The Elements of Statistical Learning. It's been recommended by many so I decided to give it a go. But I found explanation on denoting variables confusing. In section 2.2, it says:
We will typically denote an input variable by the symbol $X$. If $X$ is a vector, its components can be accessed by subscripts $X_j$. Quantitative outputs will be denoted by $Y$, and qualitative outputs by $G$ (for group). We use uppercase letters such as $X$, $Y$ or $G$ when referring to the generic aspects of a variable. Observed values are written in lowercase; hence the $i$-th observed value of $X$ is written as $x_i$ (where $x_i$ is again a scalar or vector). Matrices are represented by bold uppercase letters; for example, a set of $N$ input $p$-vectors $x_i$, $i = 1,\dotsb,N$ would be represented by the $N×p$ matrix $\textbf{X}$. In general, vectors will not be bold, except when they have $N$ components; this convention distinguishes a $p$-vector of inputs $x_i$ for the $i$th observation from the $N$-vector $\textbf{x}_j$ consisting of all the observations on variable $X_j$. Since all vectors are assumed to be column vectors, the ith row of $X$ is $x^T_i$, the vector transpose of $x_i$.
What do $N$, $p$, $i$ and $j$ represent here? The most confusing is $i$ and $j$. I try to show how I understood it, although not 100% sure if I am correct.
I feel like $\textbf{X}$ (the matrix not generic variable) is defined as
$\textbf{X}=[x_1, x_2, \dotsb,x_i]^T$
with observed values $x_i$ where $i=1,\dotsb,N$ and $x_i=[v_{i1},v_{i2},\dotsb,v_{ip}]$, and $v_{ip}$ is some $p$ith scalar value from the observed vector $x_i$ I still don't understand what $j$ correspond to?
