Factor regression model

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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


 \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[edit]

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]

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

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.


Factor regression software is available from here.[3]


  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. ^ a b 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.