Factor regression model

The factor regression model,^[1] or hybrid factor model,^[2] is a special multivariate model with the following form.

{\mathbf {y}}_{n}={\mathbf {A}}{\mathbf {x}}_{n}+{\mathbf {B}}{\mathbf {z}}_{n}+{\mathbf {c}}+{\mathbf {e}}_{n}

where,

{\mathbf {y}}_{n}

is the

n

-th

G\times 1

(known) observation.

{\mathbf {x}}_{n}

is the

n

-th sample

L_{x}

(unknown) hidden factors.

\mathbf {A}

is the (unknown) loading matrix of the hidden factors.

{\mathbf {z}}_{n}

is the

n

-th sample

L_{z}

(known) design factors.

\mathbf {B}

is the (unknown) regression coefficients of the design factors.

{\mathbf {c}}

is a vector of (unknown) constant term or intercept.

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

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

{\begin{aligned}&{\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{aligned}}

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

Software

Factor regression software is available from here.^[3]

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.
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.
↑ Wang, Quanli. "BFRM". BFRM.

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