Factor regression model

The factor regression model,^[1] or hybrid factor model,^[2] is a special multivariate model with the following form.

$\mathbf{y}_n= \mathbf{A}\mathbf{x}_n+ \mathbf{B}\mathbf{z}_n +\mathbf{c}+\mathbf{e}_n$

where,

$\mathbf{y}_n$ is the $n$-th $G \times 1$ (known) observation.

$\mathbf{x}_n$ is the $n$-th sample $L_x$ (unknown) hidden factors.

$\mathbf{A}$ is the (unknown) loading matrix of the hidden factors.

$\mathbf{z}_n$ is the $n$-th sample $L_z$ (known) design factors.

$\mathbf{B}$ is the (unknown) regression coefficients of the design factors.

$\mathbf{c}$ is a vector of (unknown) constant term or intercept.

$\mathbf{e}_n$ 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 ($\mathbf{y}_n= \mathbf{A}\mathbf{x}_n+ \mathbf{c}+\mathbf{e}_n$) and regression model ($\mathbf{y}_n= \mathbf{B}\mathbf{z}_n +\mathbf{c}+\mathbf{e}_n$).

Alternatively, the model can be viewed as a special kind of factor model, the hybrid factor model ^[2]

$\begin{align} & \mathbf{y}_n = \mathbf{A}\mathbf{x}_n+ \mathbf{B}\mathbf{z}_n +\mathbf{c}+\mathbf{e}_n \\ = & \begin{bmatrix} \mathbf{A} & \mathbf{B} \end{bmatrix} \begin{bmatrix} \mathbf{x}_n \\ \mathbf{z}_n\end{bmatrix} +\mathbf{c}+\mathbf{e}_n \\ = & \mathbf{D}\mathbf{f}_n +\mathbf{c}+\mathbf{e}_n \end{align}$

where, $\mathbf{D}=\begin{bmatrix} \mathbf{A} & \mathbf{B} \end{bmatrix}$ is the loading matrix of the hybrid factor model and $\mathbf{f}_n=\begin{bmatrix} \mathbf{x}_n \\ \mathbf{z}_n\end{bmatrix}$ are the factors, including the known factors and unknown factors.

Software

Factor regression software is available from here.^[3]

References

1. Carvalho, Carlos M. (1 December 2008). "High-Dimensional Sparse Factor Modeling: Applications in Gene Expression Genomics". Journal of the American Statistical Association 103 (484): 1438–1456. doi:10.1198/016214508000000869.
2. Meng, J. (2011). "Uncover cooperative gene regulations by microRNAs and transcription factors in glioblastoma using a nonnegative hybrid factor model". International Conference on Acoustics, Speech and Signal Processing.
3. Wang, Quanli. "BFRM". BFRM.