Tucker3 {rrcov3way}

Robust Tucker3 estimator for compositional data

Description

Compute a robust Tucker3 model for compositional data

Usage

Tucker3(X, P = 2, Q = 2, R = 2, 
    center = FALSE, center.mode = c("A", "B", "C", "AB", "AC", "BC", "ABC"), 
    scale = FALSE, scale.mode = c("B", "A", "C"), 
    conv = 1e-06, start="svd",
    robust = FALSE, coda.transform=c("none", "ilr", "clr"), 
    ncomp.rpca = 0, alpha = 0.75, robiter=100, crit=0.975, trace = FALSE)

Arguments

X

3-way array of data

P

Number of A-mode components

Q

Number of B-mode components

R

Number of C-mode components

center

Whether to center the data

center.mode

If scaling the data, on which mode to do this

scale

Whether to scale the data

scale.mode

If centering the data, on which mode to do this

conv

Convergence criterion, defaults to 1e-6

start

Initial values for the A, B and C components. Can be "svd" for starting point of the algorithm from SVD's, "random" for random starting point (orthonormalized component matrices), or a list containing user specified components.

robust

Whether to apply a robust estimation

coda.transform

If the data are a composition, use an ilr or clr transformation. Default is non-compositional data, i.e. coda.transform="none"

ncomp.rpca

Number of components for robust PCA

alpha

Measures the fraction of outliers the algorithm should resist. Allowed values are between 0.5 and 1 and the default is 0.75

robiter

Maximal number of iterations for robust estimation

crit

Cut-off for identifying outliers, default crit=0.975

trace

Logical, provide trace output

Details

The function can compute four versions of the Tucker3 model:

1. Classical Tucker3,

2. Tucker3 for compositional data,

3. Robust Tucker3 and

4. Robust Tucker3 for compositional data.

This is controlled through the parameters robust=TRUE and coda.transform="ilr".

Value

An object of class "tucker3" which is basically a list with components:

fit

Fit value

fp

Fit percentage

A

Orthogonal loading matrix for the A-mode

B

Orthogonal loading matrix for the B-mode

Bclr

Orthogonal loading matrix for the B-mode, clr transformed. Available only if coda.transform="ilr", otherwise NULL

C

Orthogonal loading matrix for the C-mode

GA

Core matrix, which describes the relation between A, B and C, unfolded in A-form. The largest squared elements of the core matrix indicate the most important factors in the model of X.

iter

Number of iterations

rd

Residual distances

sd

Score distances

flag

The observations whose residual distance RD is larger than cutoff.RD can be considered as outliers and receive a flag equal to zero. The regular observations receive a flag 1

robust

The paramater robust, whether robust method is used or not

coda.transform

The input paramater coda.transform, what trnasofrmation for compositional data was used

La

Diagonal matrix containing the intrinsic eigenvalues for A-mode

Lb

Diagonal matrix containing the intrinsic eigenvalues for B-mode

Lc

Diagonal matrix containing the intrinsic eigenvalues for C-mode

Author(s)

Valentin Todorov [e-mail] and Maria Anna Di Palma [e-mail] and Michele Gallo [e-mail]

References

Tucker, L.R. (1966). Some mathematical notes on three-mode factor analysis. Psychometrika, 31: 279–311.

Egozcue J.J., Pawlowsky-Glahn, V., Mateu-Figueras G. and Barcel'o-Vidal, C. (2003). Isometric logratio transformations for compositional data analysis. Mathematical Geology, 35(3): 279–300.

Examples

#############
##
## Example with the UNIDO Manufacturing value added data

data(va3way)
dim(va3way)

## Treat quickly and dirty the zeros in the data set (if any)
va3way[va3way==0] <- 0.001

##
res <- Tucker3(va3way)
res
print(res$fit)
print(res$A)

## Print the core matrix
print(res$GA)

## Distance-distance plot
plot(res, which="dd", main="Distance-distance plot")

## Paired component plot, mode A
plot(res, which="comp", main="Paired component plot (mode A)")

## Paired component plot, mode B
plot(res, which="comp", mode="B", main="Paired component plot (mode B)")

## Joint biplot
plot(res, which="jbplot", main="Joint biplot")

## Trajectory
plot(res, which="tjplot", choices=c(1:4), arrows=FALSE, main="Trajectory biplot")