CPfuncrep: Algorithm for the Candecomp/Parafac (CP) model
In ThreeWay: Three-Way Component Analysis

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

Alternating Least Squares algorithm for the minimization of the Candecomp/Parafac loss function.

1	CPfuncrep(X, n, m, p, r, ort1, ort2, ort3, start, conv, maxit, A, B, C)

`X`	Matrix (or data.frame coerced to a matrix) of order (`n` `x` `mp`) containing the matricized array (frontal slices)
`n`	Number of `A`-mode entities
`m`	Number of `B`-mode entities
`p`	Number of `C`-mode entities
`r`	Number of extracted components
`ort1`	Type of constraints on `A` (1 for no constraints, 2 for orthogonality constraints, 3 for zero correlations constraints)
`ort2`	Type of constraints on `B` (1 for no constraints, 2 for orthogonality constraints, 3 for zero correlations constraints)
`ort3`	Type of constraints on `C` (1 for no constraints, 2 for orthogonality constraints, 3 for zero correlations constraints)
`start`	Starting point (0 for starting point of the algorithm from SVD's, 1 for random starting point (orthonormalized component matrices), 2 for user specified components
`conv`	Convergence criterion
`maxit`	Maximal number of iterations
`A`	Optional (necessary if start=2) starting value for `A`
`B`	Optional (necessary if start=2) starting value for `B`
`C`	Optional (necessary if start=2) starting value for `C`

A list including the following components:

`A`	Component matrix for the `A`-mode
`B`	Component matrix for the `B`-mode
`C`	Component matrix for the `C`-mode
`f`	Loss function value
`fp`	Fit value expressed as a percentage
`iter`	Number of iterations
`tripcos`	Minimal triple cosine between two components across three component matrices (to inspect degeneracy)
`mintripcos`	Minimal triple cosine during the iterative algorithm observed at every 10 iterations (to inspect degeneracy)
`ftiter`	Matrix containing in each row the function value and the minimal triple cosine at every 10 iterations
`cputime`	Computation time

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.
CPfuncrep is the same as CPfunc except that all printings are suppressed. Thus, CPfuncrep can be helpful for simulation experiments.

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

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

CP, CPfunc

data(TV)
TVdata=TV[[1]]
# permutation of the modes so that the A-mode refers to students
TVdata <- permnew(TVdata, 16, 15, 30)
TVdata <- permnew(TVdata, 15, 30, 16)
# unconstrained CP solution using two components 
# (rational starting point by SVD [start=0])
TVcp <- CPfuncrep(TVdata, 30, 16, 15, 2, 1, 1, 1, 0, 1e-6, 10000)
# constrained CP solution using two components with orthogonal A-mode  
# component matrix (rational starting point by SVD [start=0])
TVcp <- CPfuncrep(TVdata, 30, 16, 15, 2, 2, 1, 1, 0, 1e-6, 10000)
# constrained CP solution using two components with orthogonal A-mode 
# component matrix and zero correlated C-mode component matrix 
# (rational starting point by SVD [start=0])
TVcp <- CPfuncrep(TVdata, 30, 16, 15, 2, 2, 1, 3, 0, 1e-6, 10000)
# unconstrained CP solution using two components 
# (random orthonormalized starting point [start=1])
TVcp <- CPfuncrep(TVdata, 30, 16, 15, 2, 1, 1, 1, 1, 1e-6, 10000)
# unconstrained CP solution using two components (user starting point [start=2])
TVcp <- CPfuncrep(TVdata, 30, 16, 15, 2, 1, 1, 1, 2, 1e-6, 10000, 
 matrix(rnorm(30*2),nrow=30), matrix(rnorm(16*2),nrow=16), 
 matrix(rnorm(15*2),nrow=15))