# 12.1 - Notations and Terminology

12.1 - Notations and Terminology

## Notation

Collect all of the variables X's into a vector $$\mathbf{X}$$ for each individual subject. Let $$\mathbf{X_i}$$ denote observable trait i. These are the data from each subject and are collected into a vector of traits.

$$\textbf{X} = \left(\begin{array}{c}X_1\\X_2\\\vdots\\X_p\end{array}\right) = \text{vector of traits}$$

This is a random vector, with a population mean. Assume that vector of traits $$\mathbf{X}$$ is sampled from a population with population mean vector:

$$\boldsymbol{\mu} = \left(\begin{array}{c}\mu_1\\\mu_2\\\vdots\\\mu_p\end{array}\right) = \text{population mean vector}$$

Here, $$\mathrm { E } \left( X _ { i } \right) = \mu _ { i }$$ denotes the population mean of variable i.

Consider m unobservable common factors $$f _ { 1 } , f _ { 2 } , \dots , f _ { m }$$. The $$i^{th}$$ common factor is $$f _ { i }$$. Generally, m is going to be substantially less than p.

The common factors are also collected into a vector,

$$\mathbf{f} = \left(\begin{array}{c}f_1\\f_2\\\vdots\\f_m\end{array}\right) = \text{vector of common factors}$$

## Model

Our factor model can be thought of as a series of multiple regressions, predicting each of the observable variables $$X_{i}$$ from the values of the unobservable common factors $$f_{i}$$ :

\begin{align} X_1 & =  \mu_1 + l_{11}f_1 + l_{12}f_2 + \dots + l_{1m}f_m + \epsilon_1\\ X_2 & =  \mu_2 + l_{21}f_1 + l_{22}f_2 + \dots + l_{2m}f_m + \epsilon_2 \\ &  \vdots \\ X_p & =  \mu_p + l_{p1}f_1 + l_{p2}f_2 + \dots + l_{pm}f_m + \epsilon_p \end{align}

Here, the variable means $$\mu_{1}$$ through $$\mu_{p}$$ can be regarded as the intercept terms for the multiple regression models.

The regression coefficients $$l_{ij}$$ (the partial slopes) for all of these multiple regressions are called factor loadings. Here, $$l_{ij}$$ = loading of the $$i^{th}$$ variable on the $$j^{th}$$ factor. These are collected into a matrix as shown here:

$$\mathbf{L} = \left(\begin{array}{cccc}l_{11}& l_{12}& \dots & l_{1m}\\l_{21} & l_{22} & \dots & l_{2m}\\ \vdots & \vdots & & \vdots \\l_{p1} & l_{p2} & \dots & l_{pm}\end{array}\right) = \text{matrix of factor loadings}$$

And finally, the errors $$\varepsilon _{i}$$ are called the specific factors. Here, $$\varepsilon _{i}$$ = specific factor for variable i. The specific factors are also collected into a vector:

$$\boldsymbol{\epsilon} = \left(\begin{array}{c}\epsilon_1\\\epsilon_2\\\vdots\\\epsilon_p\end{array}\right) = \text{vector of specific factors}$$

In summary, the basic model is like a regression model. Each of our response variables X is predicted as a linear function of the unobserved common factors $$f_{1}$$, $$f_{2}$$ through $$f_{m}$$. Thus, our explanatory variables are $$f_{1}$$ , $$f_{2}$$ through $$f_{m}$$. We have m unobserved factors that control the variation in our data.

We will generally reduce this into matrix notation as shown in this form here:

$$\textbf{X} = \boldsymbol{\mu} + \textbf{Lf}+\boldsymbol{\epsilon}$$

Note! In general, we want m << p.

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