Factor regression model: Difference between revisions

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Within [[statistics|statistical]] [[factor analysis]], the '''factor regression model''',<ref>{{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}}</ref> or hybrid factor model,<ref name="meng2011">{{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 }}</ref> is a special [[Multivariate normal distribution|multivariate]] [[Mathematical model|model]] with the following form:

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 (unknown) 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 (known) design factors.

:<math> \mathbf{c} </math> is a [[vector (mathematics and physics)|vector]] of (unknown) [[constant term]] or intercept.

:<math> \mathbf{e}_n </math> is a vector of (unknown) errorerrors, oroften [[white Gaussian noise]].

== Relationship between factor regression model, factor model and regression model ==

== Factor Regression Model, Factor Analysis 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>), thus called factor regression model.

Alternatively, the model can be viewed as a singlespecial kind of [[factor analysis]] model, exceptthe that, part of thehybrid factor matrix aremodel already<ref known.name="meng2011"/>

::<math>

\begin{bmatrix}

1 & 9 & 13 \\

20 & 55 & 6

\end{bmatrix}.

</math>

Alternatively, the model can be viewed as a single [[factor analysis]] model, except that, part of the factor matrix are already known.

:<math>

\begin{align}

Line 42 ⟶ 35:

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 ==

[https://www2.stat.duke.edu/~mw/mwsoftware/BFRM/index.html Open source software to perform factor regression is available].

==References==

{{Reflist}}

[[Category:Factor analysis]]

[[Category:Latent variable models]]

[[Category:Regression models]]