HDMFA (version 0.1.1)

KPCA: Estimating the Pair of Factor Numbers via Eigenvalue Ratios Corresponding to \(\alpha\)-PCA

Description

The function is to estimate the pair of factor numbers via eigenvalue ratios corresponding to \(\alpha\)-PCA.

Usage

KPCA(X, kmax, alpha = 0)

Value

\(k_1\)

The estimated row factor number.

\(k_2\)

The estimated column factor number.

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.

kmax

The user-supplied maximum factor numbers. Here it means the upper bound of the number of row factors and column factors.

alpha

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

Author

Yong He, Changwei Zhao, Ran Zhao.

Details

The function KPCA uses the eigenvalue-ratio idea to estimate the number of factors. In details, the number of factors \(k_1\) is estimated by $$\hat{k}_1 = \arg \max_{j \leq k_{max}} \frac{\lambda _j (\hat{\bold{M}}_R)}{\lambda _{j+1} (\hat{\bold{M}}_R)},$$ where \(k_{max}\) is a given upper bound. \(k_2\) is defined similarly with respect to \(\hat{\bold{M}}_C\). See the function alpha_PCA for the definition of \(\hat{\bold{M}}_R\) and \(\hat{\bold{M}}_C\). For more details, see Chen & Fan (2021).

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
   
   KPCA(X, 8, alpha = 0)