Factor regression model explained

Within statistical factor analysis, the factor regression model,^[1] or hybrid factor model,^[2] is a special multivariate model with the following form:

y _{n=} A x _{n+} B z _{n} + c + e _{n} where,y _{n} is the n -th G x 1 (known) observation.x _{n} is the n -th sample L _{x} (unknown) hidden factors.A is the (unknown) loading matrix of the hidden factors.z _{n} is the n -th sample L _{z} (known) design factors.B is the (unknown) regression coefficients of the design factors.c is a vector of (unknown) constant term or intercept.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 (

y _{n=} A x _{n+} c + e _{n} ) and regression model (y _{n=} B z _{n} + c + e _{n} ).

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

\begin{align} & y _{n} = A x _{n+} B z _{n} + c + e _{n} \\ = & \begin{bmatrix} A & B \end{bmatrix} \begin{bmatrix} x _{n} \\ z _{n\end{bmatrix}} + c + e _{n} \\ = & D f _{n} + c + e _{n \end{align}} where, D = \begin{bmatrix} A & B \end{bmatrix} is the loading matrix of the hybrid factor model and f _{n=\begin{bmatrix} x} _{n} \\ z _{n\end{bmatrix}} are the factors, including the known factors and unknown factors.

Software

Open source software to perform factor regression is available.

Notes and References

1. Carvalho. Carlos M.. High-Dimensional Sparse Factor Modeling: Applications in Gene Expression Genomics. Journal of the American Statistical Association. 1 December 2008. 103. 484. 1438–1456. 10.1198/016214508000000869. 3017385. 21218139.
2. Meng . J. . 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 . 2011 . dead . https://web.archive.org/web/20111123144133/http://www.cmsworldwide.com/ICASSP2011/Papers/ViewPapers.asp?PaperNum=4439 . 2011-11-23 .