Factor regression model
Within statistical factor analysis, the factor regression model,[1] or hybrid factor model,[2] is a special multivariate model with the following form:
- [math]\displaystyle{ \mathbf{y}_n= \mathbf{A}\mathbf{x}_n+ \mathbf{B}\mathbf{z}_n +\mathbf{c}+\mathbf{e}_n }[/math]
where,
- [math]\displaystyle{ \mathbf{y}_n }[/math] is the [math]\displaystyle{ n }[/math]-th [math]\displaystyle{ G \times 1 }[/math] (known) observation.
- [math]\displaystyle{ \mathbf{x}_n }[/math] is the [math]\displaystyle{ n }[/math]-th sample [math]\displaystyle{ L_x }[/math] (unknown) hidden factors.
- [math]\displaystyle{ \mathbf{A} }[/math] is the (unknown) loading matrix of the hidden factors.
- [math]\displaystyle{ \mathbf{z}_n }[/math] is the [math]\displaystyle{ n }[/math]-th sample [math]\displaystyle{ L_z }[/math] (known) design factors.
- [math]\displaystyle{ \mathbf{B} }[/math] is the (unknown) regression coefficients of the design factors.
- [math]\displaystyle{ \mathbf{c} }[/math] is a vector of (unknown) constant term or intercept.
- [math]\displaystyle{ \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]\displaystyle{ \mathbf{y}_n= \mathbf{A}\mathbf{x}_n+ \mathbf{c}+\mathbf{e}_n }[/math]) and regression model ([math]\displaystyle{ \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 [2]
- [math]\displaystyle{ \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]\displaystyle{ \mathbf{D}=\begin{bmatrix} \mathbf{A} & \mathbf{B} \end{bmatrix} }[/math] is the loading matrix of the hybrid factor model and [math]\displaystyle{ \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.
References
- ↑ 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. PMID 21218139.
- ↑ 2.0 2.1 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. Archived from the original on 2011-11-23. https://web.archive.org/web/20111123144133/http://www.cmsworldwide.com/ICASSP2011/Papers/ViewPapers.asp?PaperNum=4439.
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