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?


1 Answer 1


Maybe an example would help? Say we have 10 peoples' heights and weights. Now, we want to predict shoe size for each person.

N = 10
i = a particular person
Y = shoe size
p = 2 (height and weight)
j = 1 is height, j = 2 is weight. (or vice versa)
$x_i$ is the $i^{th}$ person
$X_j$ is the $j^{th}$ attribute across everyone. e.g. everyones' heights.
$X_{8,2}$ is the eighth person's weight


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