CPfunc function

Algorithm for the Candecomp/Parafac (CP) model

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

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

Arguments

• 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

Returns

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

Note

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.

CPfunc is the same as CPfuncrep except that all printings are available.

References

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.

Author(s)

Maria Antonietta Del Ferraro mariaantonietta.delferraro@yahoo.it

Henk A.L. Kiers h.a.l.kiers@rug.nl

Paolo Giordani paolo.giordani@uniroma1.it

See Also

CP, CPfuncrep

Examples

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 <- CPfunc(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 <- CPfunc(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 <- CPfunc(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 <- CPfunc(TVdata, 30, 16, 15, 2, 1, 1, 1, 1, 1e-6, 10000)
# unconstrained CP solution using two components (user starting point [start=2])
TVcp <- CPfunc(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))