Английская Википедия:Factor regression model

Содержание

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Within statistical factor analysis, the factor regression model, or hybrid factor model, is a special multivariate model with the following form:

<math> \mathbf{y}_n= \mathbf{A}\mathbf{x}_n+ \mathbf{B}\mathbf{z}_n +\mathbf{c}+\mathbf{e}_n </math>

where,

<math> \mathbf{y}_n </math> is the <math>n</math>-th <math> G \times 1 </math> (known) observation.

<math> \mathbf{x}_n </math> is the <math>n</math>-th sample <math> L_x </math> (unknown) hidden factors.

<math> \mathbf{A} </math> is the (unknown) loading matrix of the hidden factors.

<math> \mathbf{z}_n </math> is the <math>n</math>-th sample <math> L_z </math> (known) design factors.

<math> \mathbf{B} </math> is the (unknown) regression coefficients of the design factors.

<math> \mathbf{c} </math> is a vector of (unknown) constant term or intercept.

<math> \mathbf{e}_n </math> is a vector of (unknown) errors, often white Gaussian noise.

Relationship between factor regression model, factor model and regression model

The factor regression model can be viewed as a combination of factor analysis model (<math> \mathbf{y}_n= \mathbf{A}\mathbf{x}_n+ \mathbf{c}+\mathbf{e}_n </math>) and regression model (<math> \mathbf{y}_n= \mathbf{B}\mathbf{z}_n +\mathbf{c}+\mathbf{e}_n </math>).

Alternatively, the model can be viewed as a special kind of factor model, the hybrid factor model

<math>

\begin{align} & \mathbf{y}_n = \mathbf{A}\mathbf{x}_n+ \mathbf{B}\mathbf{z}_n +\mathbf{c}+\mathbf{e}_n \\ = & \begin{bmatrix} \mathbf{A} & \mathbf{B} \end{bmatrix} \begin{bmatrix} \mathbf{x}_n \\ \mathbf{z}_n\end{bmatrix} +\mathbf{c}+\mathbf{e}_n \\ = & \mathbf{D}\mathbf{f}_n +\mathbf{c}+\mathbf{e}_n \end{align} </math> where, <math> \mathbf{D}=\begin{bmatrix} \mathbf{A} & \mathbf{B} \end{bmatrix} </math> is the loading matrix of the hybrid factor model and <math> \mathbf{f}_n=\begin{bmatrix} \mathbf{x}_n \\ \mathbf{z}_n\end{bmatrix} </math> are the factors, including the known factors and unknown factors.

Software

Open source software to perform factor regression is available.