cp_int2 {rrcov3way}

ATLD-ALS algorithm for Candecomp/Parafac (CP)

Description

Integrated algorithm combining ATLD and ALS for the minimization of the Candecomp/Parafac (CP) loss function.

Usage

cp_int2(
  X,
  n,
  m,
  p,
  ncomp,
  initconv = 0.01,
  conv = 1e-06,
  const = "none",
  start = "random",
  maxit = 5000,
  trace = FALSE
)

Arguments

X

A three-way array or a matrix. If X is a matrix (matricised threeway array), n, m and p must be given and are the number of A-, B- and C-mode entities respectively

n

Number of A-mode entities

m

Number of B-mode entities

p

Number of C-mode entities

ncomp

Number of components to extract

initconv

Convergence criterion for the initialization phase (ATLD), default is conv=1e-2.

conv

Convergence criterion, default is conv=1e-6.

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 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.

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.

trace

Logical, provide trace output.

Value

The result of the decomposition as a list with the following elements:

• fit Value of the loss function

• fp Fit value expressed as a percentage

• ss Sum of squares

• A Component matrix for the A-mode

• B Component matrix for the B-mode

• C Component matrix for the C-mode

• iter Number of iterations

• tripcos Minimal triple cosine between two components across the three component matrices, used to inspect degeneracy

• mintripcos Minimal triple cosine during the iterative algorithm observed at every 10 iterations, used to inspect degeneracy

• ftiter Matrix containing in each row the function value and the minimal triple cosine at every 10 iterations

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

Note

The argument const should be a three element character vector. Set const[j]="none" for unconstrained update in j-th mode weight matrix (the default), const[j]="orth" for orthogonal update in j-th mode weight matrix or const[j]="zerocor" for zero correlation between the extracted factors. The default is unconstrained update for all modes.

The loss function to be minimized is sum(k)|| X(k) - A D(k) B' ||^2, where D(k) is a diagonal matrix holding the k-th row of C.

Author(s)

Valentin Todorov, [e-mail]; Violetta Simonacci, [e-mail]

References

H.-L. Wu, M. Shibukawa, K. Oguma, An alternating trilinear decomposition algorithm with application to calibration of HPLC-DAD for simultaneous determination of overlapped chlorinated aromatic hydrocarbons, Journal of Chemometrics 12 (1998) 1–26.

Simonacci, V. and Gallo, M. (2020). An ATLD–ALS method for the trilinear decomposition of large third-order tensors, Soft Computing 24 13535–13546.

Todorov, V. and Simonacci, V. and Gallo, M. and Trendafilov, N. (2023). A novel estimation procedure for robust CANDECOMP/PARAFAC model fitting. Econometrics and Statistics. In press.

Examples

## Not run: 
## Example with the OECD data
 data(elind)
 dim(elind)

 res <- cp_int2(elind, ncomp=3)
 res$fp
 res$fp
 res$iter

 res <- cp_int2(elind, ncomp=3, const="orth")
 res$A

## End(Not run)