KPCA {HDMFA}

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)

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.

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).

Value

\eqn{k_1}

The estimated row factor number.

\eqn{k_2}

The estimated column factor number.

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
   
   KPCA(X, 8, alpha = 0)