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:

𝐲_{n} = 𝐀 𝐱_{n} + 𝐁 𝐳_{n} + 𝐜 + 𝐞_{n}

where,

𝐲_{n}

is the

n

-th

G \times 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{matrix} 𝐲_{n} = 𝐀 𝐱_{n} + 𝐁 𝐳_{n} + 𝐜 + 𝐞_{n} \\ = & [\begin{matrix} 𝐀 & 𝐁 \end{matrix}] [\begin{matrix} 𝐱_{n} \\ 𝐳_{n} \end{matrix}] + 𝐜 + 𝐞_{n} \\ = & 𝐃 𝐟_{n} + 𝐜 + 𝐞_{n} \end{matrix}

where, $𝐃 = [\begin{matrix} 𝐀 & 𝐁 \end{matrix}]$ is the loading matrix of the hybrid factor model and $𝐟_{n} = [\begin{matrix} 𝐱_{n} \\ 𝐳_{n} \end{matrix}]$ 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. PMC 3017385. PMID 21218139.
1 2 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.

[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. PMC 3017385. PMID 21218139.

[meng2011-2] 1 2 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.

[1]

[2]