varimcoco {ThreeWay}

Varimax Rotation for Tucker3 and Tucker2

Description

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

Usage

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

Arguments

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

Value

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

Note

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).

Author(s)

Maria Antonietta Del Ferraro [e-mail]
Henk A.L. Kiers [e-mail]
Paolo Giordani [e-mail]

References

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.

See Also

orthmax2, varim

Examples

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
  }
 }
}