# tPCAladle: Ladle Estimate for tPCA In tensorBSS: Blind Source Separation Methods for Tensor-Valued Observations

## Description

For r-dimensional tensors, the Ladle estimate for tPCA assumes that for a given mode m, the last p_m - k_m modewise eigenvalues are equal. Combining information from the eigenvalues and eigenvectors of the m-mode covariance matrix the ladle estimator yields estimates for k_1,...,k_r.

## Usage

 `1` ```tPCAladle(x, n.boot = 200, ncomp = NULL) ```

## Arguments

 `x` array of an order at least two with the last dimension corresponding to the sampling units. `n.boot` number of bootstrapping samples to be used. `ncomp` vector giving the number of components among which the ladle estimator is to be searched for each mode. The default follows the recommendation of Luo and Li 2016.

## Details

The model here assumes that the eigenvalues of the m-mode covariance matrix are of the form lambda_{1,m} >= ... >= lambda_{k_m,m} > lambda_{k_m+1,m} = ... = λ_{p_m,m} and the goal is to estimate the value of k_m for all modes. The ladle estimate for this purpose combines the values of the scaled eigenvalues and the variation of the eigenvectors based on bootstrapping. The idea there is that for distinct eigenvales the variation of the eigenvectors is small and for equal eigenvalues the corresponding eigenvectors have large variation.

This measure is then computed assuming k_m=0,..., `ncomp[m]` and the ladle estimate for k_m is the value where the measure takes its minimum.

## Value

A list of class 'tladle' containing:

 `U` list containing the modewise rotation matrices. `D` list containing the modewise eigenvalues. `S` array of the same size as `x` containing the principal components. `ResMode` a list with the modewise results which are lists containing: modelabel for the mode. kthe estimated value of k. fnvector giving the measures of variation of the eigenvectors using the bootstrapped eigenvectors for the different number of components. phinnormalized eigenvalues. lambdathe unnormalized eigenvalues used to compute phin. gnthe main criterion for the ladle estimate - the sum of fn and phin. k is the value where gn takes its minimum. compvector from 0 to the number of dimensions to be evaluated. `xmu` the data location `data.name` string with the name of the input data `method` string `tPCA`.

Klaus Nordhausen

## References

Koesner, C, Nordhausen, K. and Virta, J. (2019), Estimating the signal tensor dimension using tensorial PCA. Manuscript.

Luo, W. and Li, B. (2016), Combining Eigenvalues and Variation of Eigenvectors for Order Determination, Biometrika, 103, 875–887. <doi:10.1093/biomet/asw051>

`tPCA`, `ggtladleplot`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25``` ```library(ICtest) n <- 200 sig <- 0.6 Z <- rbind(sqrt(0.7)*rt(n,df=5)*sqrt(3/5), sqrt(0.3)*runif(n,-sqrt(3),sqrt(3)), sqrt(0.3)*(rchisq(n,df=3)-3)/sqrt(6), sqrt(0.9)*(rexp(n)-1), sqrt(0.1)*rlogis(n,0,sqrt(3)/pi), sqrt(0.5)*(rbeta(n,2,2)-0.5)*sqrt(20) ) dim(Z) <- c(3, 2, n) U1 <- rorth(12)[,1:3] U2 <- rorth(8)[,1:2] U <- list(U1=U1, U2=U2) Y <- tensorTransform2(Z,U,1:2) EPS <- array(rnorm(12*8*n, mean=0, sd=sig), dim=c(12,8,n)) X <- Y + EPS TEST <- tPCAladle(X) TEST ggtladleplot(TEST) ```