varimcoco: Varimax Rotation for Tucker3 and Tucker2
In ThreeWay: Three-Way Component Analysis

Description Usage Arguments Value Note Author(s) References See Also Examples

Performs varimax rotation of the core and component matrix rotations to simple structure.

1	varimcoco(A, B, C, H, wa_rel, wb_rel, wc_rel, rot1, rot2, rot3, nanal)

`A`	Columnwise orthomornal component matrix for the `A`-mode
`B`	Columnwise orthomornal component matrix for the `B`-mode
`C`	Columnwise orthomornal component matrix for the `C`-mode
`H`	Matricized core array (frontal slices)
`wa_rel`	relative weight (>=0) for the simplicity of `A`
`wb_rel`	relative weight (>=0) for the simplicity of `B`
`wc_rel`	relative weight (>=0) for the simplicity of `C`
`rot1`	binary indicator (1 if the `A`-mode is rotated, 0 otherwise, default 1)
`rot2`	binary indicator (1 if the `B`-mode is rotated, 0 otherwise, default 1)
`rot3`	binary indicator (1 if the `C`-mode is rotated, 0 otherwise, default 1)
`nanal`	Number of random starts, default 5

A list including the following components:

`AS`	Rotated component matrix for the `A`-mode
`BT`	Rotated component matrix for the `B`-mode
`CU`	Rotated component matrix for the `C`-mode
`K`	Rotated matricized core array (frontal slices)
`S`	Rotation matrix for the `A`-mode
`T`	Rotation matrix for the `B`-mode
`U`	Rotation matrix for the `C`-mode
`f`	Best solution for three-way orthomax function value
`f1`	Varimax value of `H`
`f2a`	Varimax value of `AS`
`f2b`	Varimax value of `BT`
`f2c`	Varimax value of `CU`
`func`	Function values upon convergence for all the runs of the orthomax algorithm

The simplicity values f1, f2a, f2b, f2c are based on ‘natural’ weigths and therefore comparable across matrices. When multiplied by the relative weights, they give the contribution to the overall simplicity value (they are I^2/p, J^2/q or K^2/r, respectively, times the sum of the variances of squared values).

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it
Henk A.L. Kiers h.a.l.kiers@rug.nl
Paolo Giordani paolo.giordani@uniroma1.it

H.A.L. Kiers (1998). Joint orthomax rotation of the core and component matrices resulting from three-mode principal components analysis. Journal of Classification 15:245–263.

orthmax2, varim

data(Bus)
# T3 solution
BusT3 <- T3funcrep(Bus, 7, 5, 37, 2, 2, 2, 0, 1e-6)
# Simplicity of A (with weight = 2.5), B (with weight = 2) and C (with weight = 1.5)
T3vmABC <- varimcoco(BusT3$A, BusT3$B, BusT3$C, BusT3$H, 2.5, 2, 1.5)
# Simplicity of only A (with weight = 2.5) and B (with weight = 2)
# rot3=0; the value of wc_rel (= 0) does not play an active role
T3vmAB <- varimcoco(BusT3$A, BusT3$B, BusT3$C, BusT3$H, 2.5, 2, 0, 1, 1, 0)
# simplicity repeatedly with different relative weights for A, B and C
T3vm <- list()
weight.a <- c(1, 3, 6)
weight.b <- c(0, 2, 5)
weight.c <- c(1, 4)
i <- 1
for (wa_rel in weight.a){
 for (wb_rel in weight.b){
  for (wc_rel in weight.c){
   T3vm[[i]] <- varimcoco(BusT3$A, BusT3$B, BusT3$C, 
    BusT3$H, wa_rel, wb_rel, wc_rel)
   i <- i+1
  }
 }
}