Parafac function

Robust Parafac estimator for compositional data

Compute a robust Parafac model for compositional data

Parafac(X, ncomp = 2, center = FALSE, 
    center.mode = c("A", "B", "C", "AB", "AC", "BC", "ABC"),
    scale=FALSE, scale.mode=c("B", "A", "C"), 
    const="none", conv = 1e-06, start="svd", maxit=10000, 
    optim=c("als", "atld", "int2"),
    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

• ncomp: Number of components

• center: Whether to center the data

• center.mode: If centering the data, on which mode to do this

• scale: Whether to scale the data

• scale.mode: If scaling the data, on which mode to do this

• const: Optional constraints for each mode. Can be a three element character vector or a single character, one of "none" for no constraints (default), "orth" for orthogonality constraints, "nonneg" for nonnegativity constraints or "zerocor" for zero correlation between the extracted factors. For example, const="orth" means orthogonality constraints for all modes, while const=c("orth", "none", "none") sets the orthogonality constraint only for mode A.

• 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 nonnegative matrices in case of nonnegativity constraint), or a list containing user specified components.

• maxit: Maximum number of iterations, default is maxit=10000.

• optim: How to optimize the CP loss function, default is to use ALS, i.e. optim="als". Other optins are ATLD (optim="atld") and INT2 (optim="INT2"). Please note that ATLD cannot be used with the robust option.

• 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 Parafac model:

1. Classical Parafac,
2. Parafac for compositional data,
3. Robust Parafac and
4. Robust Parafac for compositional data.

This is controlled though the paramters robust=TRUE and coda.transform=c("none", "ilr").

Returns

An object of class "parafac" which is basically a list with components: - fit: Fit value

• fp: Fit percentage

• ss: Sum of squares

• A: Orthogonal loading matrix for the A-mode

• B: Orthogonal loading matrix for the A-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

• Xhat: (Robustly) reconstructed array

• const: Optional constraints (same as the input parameter)

• iter: Number of iterations

• rd: Residual distances

• sd: Score distances

• flag: The observations whose residual distance rd is larger than cutoff.rd or score distance sd is larger than cutoff.sd, can be considered 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: Which coda transformation is used, can be coda.transform=c("none", "ilr", "clr").

References

Harshman, R.A. (1970). Foundations of Parafac procedure: models and conditions for an "explanatory" multi-mode factor analysis. UCLA Working Papers in Phonetics, 16: 1--84.

Engelen, S., Frosch, S. and Jorgensen, B.M. (2009). A fully robust PARAFAC method analyzing fluorescence data. Journal of Chemometrics, 23(3): 124--131.

Kroonenberg, P.M. (1983).Three-mode principal component analysis: Theory and applications (Vol. 2), DSWO press.

Rousseeuw, P.J. and Driessen, K.V. (1999). A fast algorithm for the minimum covariance determinant estimator. Technometrics, 41(3): 212--223.

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

Author(s)

Valentin Todorov valentin.todorov@chello.at and Maria Anna Di Palma madipalma@unior.it and Michele Gallo mgallo@unior.it

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 <- Parafac(va3way)
res
print(res$fit)
print(res$A)

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

data(ulabor)
res <- Parafac(ulabor, robust=TRUE, coda.transform="ilr")
res

## Plot Orthonormalized A-mode component plot
plot(res, which="comp", mode="A", main="Component plot, A-mode")

## Plot Orthonormalized B-mode component plot
plot(res, which="comp", mode="B", main="Component plot, B-mode")

## Plot Orthonormalized C-mode component plot
plot(res, which="comp", mode="C", main="Component plot, C-mode")