Factor regression model

Within statistical factor analysis, the factor regression model,{{cite journal|last=Carvalho|first=Carlos M.|title=High-Dimensional Sparse Factor Modeling: Applications in Gene Expression Genomics|journal=Journal of the American Statistical Association|date=1 December 2008|volume=103|issue=484|pages=1438–1456|doi=10.1198/016214508000000869|pmc=3017385|pmid=21218139}} or hybrid factor model,{{cite journal|last=Meng |first=J. |title=Uncover cooperative gene regulations by microRNAs and transcription factors in glioblastoma using a nonnegative hybrid factor model |journal=International Conference on Acoustics, Speech and Signal Processing |year=2011 |url=http://www.cmsworldwide.com/ICASSP2011/Papers/ViewPapers.asp?PaperNum=4439 |url-status=dead |archiveurl=https://web.archive.org/web/20111123144133/http://www.cmsworldwide.com/ICASSP2011/Papers/ViewPapers.asp?PaperNum=4439 |archivedate=2011-11-23 }} is a special multivariate model with the following form:

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

where,

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

:$\backslash mathbf\{x\}\_n$ is the $n$-th sample $L\_x$ (unknown) hidden factors.

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

:$\backslash mathbf\{z\}\_n$ is the $n$-th sample $L\_z$ (known) design factors.

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

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

:$\backslash mathbf\{e\}\_n$ is a vector of (unknown) errors, often white Gaussian noise.