alpha_PCA {HDMFA}

Statistical Inference for High-Dimensional Matrix-Variate Factor Model

Description

This function is to fit the matrix factor model via the \alpha-PCA method by conducting eigen-analysis of a weighted average of the sample mean and the column (row) sample covariance matrix through a hyper-parameter \alpha.

Usage

alpha_PCA(X, m1, m2, alpha = 0)

Arguments

X

Input an array with T \times p_1 \times p_2, where T is the sample size, p_1 is the the row dimension of each matrix observation and p_2 is the the column dimension of each matrix observation.

m1

A positive integer indicating the row factor numbers.

m2

A positive integer indicating the column factor numbers.

alpha

A hyper-parameter balancing the information of the first and second moments (\alpha \geq -1 ). The default is 0.

Details

For the matrix factor models, Chen & Fan (2021) propose an estimation procedure, i.e. \alpha-PCA. The method aggregates the information in both first and second moments and extract it via a spectral method. In detail, for observations \bold{X}_t, t=1,2,\cdots,T, define

\hat{\bold{M}}_R = \frac{1}{p_1 p_2} \left( (1+\alpha) \bar{\bold{X}} \bar{\bold{X}}^\top + \frac{1}{T} \sum_{t=1}^T (\bold{X}_t - \bar{\bold{X}}) (\bold{X}_t - \bar{\bold{X}})^\top \right),

\hat{\bold{M}}_C = \frac{1}{p_1 p_2} \left( (1+\alpha) \bar{\bold{X}}^\top \bar{\bold{X}} + \frac{1}{T} \sum_{t=1}^T (\bold{X}_t - \bar{\bold{X}})^\top (\bold{X}_t - \bar{\bold{X}}) \right),

where \alpha \in [-1,+\infty], \bar{\bold{X}} = \frac{1}{T} \sum_{t=1}^T \bold{X}_t, \frac{1}{T} \sum_{t=1}^T (\bold{X}_t - \bar{\bold{X}}) (\bold{X}_t - \bar{\bold{X}})^\top and \frac{1}{T} \sum_{t=1}^T (\bold{X}_t - \bar{\bold{X}})^\top (\bold{X}_t - \bar{\bold{X}}) are the sample row and column covariance matrix, respectively. The loading matrices \bold{R} and \bold{C} are estimated as \sqrt{p_1} times the top k_1 eigenvectors of \hat{\bold{M}}_R and \sqrt{p_2} times the top k_2 eigenvectors of \hat{\bold{M}}_C, respectively. For details, see Chen & Fan (2021).

Value

The return value is a list. In this list, it contains the following:

F

The estimated factor matrix of dimension T \times m_1\times m_2.

R

The estimated row loading matrix of dimension p_1\times m_1, satisfying \bold{R}^\top\bold{R}=p_1\bold{I}_{m_1}.

C

The estimated column loading matrix of dimension p_2\times m_2, satisfying \bold{C}^\top\bold{C}=p_2\bold{I}_{m_2}.

Author(s)

Yong He, Changwei Zhao, Ran Zhao.

References

Chen, E. Y., & Fan, J. (2021). Statistical inference for high-dimensional matrix-variate factor models. Journal of the American Statistical Association, 1-18.

Examples

   set.seed(11111)
   T=20;p1=20;p2=20;k1=3;k2=3
   R=matrix(runif(p1*k1,min=-1,max=1),p1,k1)
   C=matrix(runif(p2*k2,min=-1,max=1),p2,k2)
   X=array(0,c(T,p1,p2))
   Y=X;E=Y
   F=array(0,c(T,k1,k2))
   for(t in 1:T){
     F[t,,]=matrix(rnorm(k1*k2),k1,k2)
     E[t,,]=matrix(rnorm(p1*p2),p1,p2)
     Y[t,,]=R%*%F[t,,]%*%t(C)
   }
   X=Y+E
   
   #Estimate the factor matrices and loadings
   fit=alpha_PCA(X, k1, k2, alpha = 0)
   Rhat=fit$R 
   Chat=fit$C
   Fhat=fit$F
   
   #Estimate the common component
   CC=array(0,c(T,p1,p2))
   for (t in 1:T){
   CC[t,,]=Rhat%*%Fhat[t,,]%*%t(Chat)
   }
   CC