Tuning for Classification with Parallel Factor Analysis

Fits Richard A. Harshman's Parallel Factor Analysis-1 (Parafac) model or Parallel Factor Analysis-2 (Parafac2) model to a three-way or four-way data array. Also fits a principal component analysis model (PCA) to a two-way matrix. Allows for different constraint options on multiple tensor modes. For PCA, allows for either an unrotated or varimax rotated solution. Uses component weights from a single mode of the model as predictors to tune parameters for one or more classification methods via a k-fold cross-validation procedure. Allows for additional features to be included. Supports binary and multiclass classification.

Usage

tunecpfa(x, y, z = NULL, model = c("parafac", "parafac2", "pca"), nfac = 1, 
         nfolds = 10, method = c("PLR", "SVM", "RF", "NN", "RDA", "GBM"), 
         family = c("binomial", "multinomial"), parameters = list(), 
         foldid = NULL, prior = NULL, cmode = NULL, parallel = FALSE, 
         cl = NULL, verbose = TRUE, compscale = TRUE, 
         pcarot = c("unrotated", "varimax"), ...)

Arguments

x

A three-way or four-way data array. For Parafac2, can be a list where each element is a matrix or three-way array. Array or list must contain only real numbers. See note below. For PCA, can be a two-way matrix, a three-way array, or a four-way array. If a three-way or four-way array for PCA, array is unfolded across all modes except the classification mode.

y

A vector containing at least two unique class labels. Should be a factor that contains two or more levels. For binary case, ensure the order of factor levels (left to right) is such that negative class is first and positive class is second.

z

A matrix containing additional features for classification. Must have number of rows equal to the number of class labels (i.e., equal to the length of y).

model

Character designating the component model to use, including: model = "parafac" to fit the Parafac model, model = "parafac2" to fit the Parafac2 model, or model = "pca" to fit the PCA model. One must be provided.

nfac

Vector containing integers that specify the number of components for each component model to fit. Default is nfac = 1.

nfolds

Numeric value specifying the number of folds for k-fold cross-validation. Must be 2 or greater. Default is nfolds = 10.

method

Character vector indicating classification methods to use. Possible methods include penalized logistic regression (PLR); support vector machine (SVM); random forest (RF); feed-forward neural network (NN); regularized discriminant analysis (RDA); and gradient boosting machine (GBM). If none selected, default is to use all methods.

family

Character value specifying binary classification (family = "binomial") or multiclass classification (family = "multinomial"). If not provided, number of levels of input y is used, where two levels is binary, and where three or more levels is multiclass.

parameters

List containing arguments related to classification methods. When specified, must contain one or more of the following:

alpha: Values for penalized logistic regression alpha parameter; default is alpha = seq(0, 1, length = 6). Must be numeric and contain only real numbers between 0 and 1, inclusive.
lambda: Optional user-supplied lambda sequence for cv.glmnet for penalized logistic regression. Default is NULL.
cost: Values for support vector machine cost parameter; default is cost = c(1, 2, 4, 8, 16, 32, 64). Must be numeric and contain only real numbers greater than 0.
gamma: Values for support vector machine gamma parameter; default is gamma = c(0, 0.01, 0.1, 1, 10, 100, 1000). Must be numeric and greater than or equal to 0.
ntree: Values for random forest number of trees parameter; default is ntree = c(100, 200, 400, 600, 800, 1600, 3200). Must be numeric and contain only integers greater than or equal to 1.
nodesize: Values for random forest node size parameter; default is nodesize = c(1, 2, 4, 8, 16, 32, 64). Must be numeric and contain only integers greater than or equal to 1.
size: Values for neural network size parameter; default is size = c(1, 2, 4, 8, 16, 32, 64). Must be numeric and contain only integers greater than or equal to 0.
decay: Values for neural network decay parameter; default is decay = c(0.001, 0.01, 0.1, 1, 2, 4, 8, 16). Must be numeric and contain only real numbers.
rda.alpha: Values for regularized discriminant analysis alpha parameter; default is rda.alpha = seq(0, 0.999, length = 6). Must be numeric and contain only real numbers between 0 (inclusive) and 1 (exclusive).
delta: Values for regularized discriminant analysis delta parameter; default is delta = c(0, 0.1, 1, 2, 3, 4). Must be numeric and contain only real numbers greater than or equal to 0.
eta: Values for gradient boosting machine eta parameter; default is eta = c(0.1, 0.3, 0.5, 0.7, 0.9). Must be numeric and contain only real numbers greater than 0 and less than 1.
max.depth: Values for gradient boosting machine max.depth parameter; default is max.depth = c(1, 2, 3, 4). Must be numeric and contain only integers greater than or equal to 1.
subsample: Values for gradient boosting machine subsample parameter; default is subsample = c(0.6, 0.7, 0.8, 0.9). Must be numeric and contain only real numbers greater than 0 and less than or equal to 1.
nrounds: Values for gradient boosting machine nrounds parameter; default is nrounds = c(100, 200, 300, 500). Must be numeric and contain only integers greater than or equal to 1.

foldid

Vector containing fold IDs for k-fold cross-validation. Must be of class integer. Should contain integers from 1 through the number of folds. If not provided, fold IDs are generated randomly for observations using integers of 1 through the number of folds nfolds.

prior

Prior probabilities of class membership. If specified, the probabilities should be in the order of the factor levels of input y. If unspecified, the observed class proportions for input y are used. Based on prior, inverse probability weights are calculated to account for class imbalance. Note that RF and RDA ignore prior and use uniform priors to handle imbalance.

cmode

Integer value of 1, 2, or 3 (or 4 if x is a four-way array) specifying the mode whose component weights will be predictors for classification. Defaults to the last mode of the input array (i.e., defaults to 3 for a three-way array, and to 4 for a four-way array). If model = "parafac2", last mode will be used. If model = "pca", cmode is set to the first mode.

parallel

Logical indicating if parallel computing should be implemented. Defaults to FALSE, which implements sequential computing.

cl

Cluster for parallel computing, which is used when parallel = TRUE. Note that if parallel = TRUE and cl = NULL, then the cluster is defined as makeCluster(max(1L, detectCores() - 1L)).

verbose

If TRUE, progress is printed.

compscale

Logical indicating whether to scale each column of the estimated classification component weights matrix (i.e., the features/predictors). If TRUE, each column is scaled to have mean zero and unit variance. If FALSE, no scaling is performed.

pcarot

Character indicating whether to apply a varimax rotation or leave PCA loadings unrotated when model is set to pca. Ignored when model is not pca. Defaults to "unrotated" if not specified.

...

Additional arguments to be passed to function parafac for fitting a Parafac model or function parafac2 for fitting a Parafac2 model. Example: can impose different constraints on different modes of the input array using the argument const. See help file for function parafac or for function parafac2 for additional details.

Details

After fitting a Parafac or Parafac2 model with the training set using package multiway (see parafac or parafac2 in multiway for details), or after fitting a PCA model using the singular value decomposition, the estimated classification mode weight matrix is passed to one or more classification methods. The methods include: penalized logistic regression (PLR); support vector machine (SVM); random forest (RF); feed-forward neural network (NN); regularized discriminant analysis (RDA); and gradient boosting machine (GBM).

Package glmnet fits models for PLR. PLR tunes penalty parameter lambda while the elastic net parameter alpha is set by the user (see the help file for function cv.glmnet in package glmnet). For SVM, package e1071 is used with a radial basis kernel. Penalty parameter cost and radial basis parameter gamma are used (see svm in package e1071). For RF, package randomForest is used and implements Breiman's random forest algorithm. The number of predictors sampled at each node split is set at the default of sqrt(R), where R is the number of Parafac or Parafac2 components. Two tuning parameters allowed are ntree, the number of trees to be grown, and nodesize, the minimum size of terminal nodes (see randomForest in package randomForest). For NN, package nnet fits a single-hidden-layer, feed-forward neural network model. Penalty parameters size (i.e., number of hidden layer units) and decay (i.e., weight decay) are used (see nnet). For RDA, package rda fits a shrunken centroids regularized discriminant analysis model. Tuning parameters include rda.alpha, the shrinkage penalty for the within-class covariance matrix, and delta, the shrinkage penalty of class centroids towards the overall dataset centroid. For GBM, package xgboost fits a gradient boosting machine model. Four tuning parameters are allowed: (1) eta, the learning rate; (2) max.depth, the maximum tree depth; (3) subsample, the fraction of samples per tree; and (4) nrounds, the number of boosting trees to build.

For all six methods, k-fold cross-validation is implemented to tune classification parameters where the number of folds is set by argument nfolds.

Value

Returns an object of class tunecpfa with the following elements:

opt.model: List containing optimal model for tuned classification methods for each component model that was fit.
opt.param: Data frame containing optimal parameters for tuned classification methods.
kcv.error: Data frame containing KCV misclassification error for optimal parameters for tuned classification methods.
est.time: Data frame containing times for fitting the component model and for tuning classification methods.
model: Character designating the component model that was used, including: model = "parafac" for the Parafac model, model = "parafac2" for the Parafac2 model, or model = "pca" for the PCA model.
method: Numeric indicating classification methods used. Value of '1' indicates 'PLR'; value of '2' indicates 'SVM'; value of '3' indicates 'RF'; value of '4' indicates 'NN'; value of '5' indicates 'RDA'; and value of '6' indicates 'GBM'.
x: Two-way matrix, or three-way or four-way data array or list used in argument x. If x was a three-way or four-way array, and if model was pca, returns the flattened two-way matrix.
y: Factor containing class labels used. Note that output y is recoded such that the input labels of y are converted to numeric integers from 0 through the number of levels, which are then applied as labels for output y.
z: Matrix containing additional features for classification.
Aweights: List containing estimated A weights for each component model that was fit. For PCA, contains the loadings.
Bweights: List containing estimated B weights for each component model that was fit. Null if model was pca.
Cweights: List containing estimated C weights for each Parafac or Parafac2 model that was fit. Null if input argument x was a three-way array or if model was pca.
Phi: If model = "parafac2", a list containing estimated Phi from the Parafac2 model. Phi is the common cross product matrix shared by all levels of the last mode (see help file for function parafac2 in package multiway for additional details). NULL if model = "parafac" or if model = "pca".
const: Constraints used in fitting the Parafac, Parafac2, or PCA model. When model was pca, contains the value used in input pcarot to indicate whether a rotation was applied to the solution.
cmode: Integer value of 1, 2, or 3 (or 4 if x is a four-way array) specifying mode whose component weights were predictors for classification.
family: Character value specifying whether classification was binary (family = "binomial") or multiclass (family = "multinomial").
xdim: Numeric value specifying number of levels for each mode of input x. If model = "parafac2", the number of levels for the first mode is designated as NA because the number of levels can differ across levels of the last mode. If model = "pca" and if input x was a 3-way or 4-way array, xdim contains the number of levels of the matrix that is the unfolded version of the input array.
lxdim: Integer specifying the number of modes of the output x. Identical to the number of modes of input x, except when input x was a 3-way or 4-way array and when model was pca. In this case, lxdim is 2.
train.weights: List containing classification component weights for each fit Parafac or Parafac2 model, for possibly different numbers of components. The weights used to train classifiers.
priorweights: List containing three elements based on input prior: (1) weight, inverse probability weights per observation, used to balance classes for PLR, NN, and GBM; (2) frac, inverse probability weights per class, used to balance classes for SVM; and (3) pricorrect, uniform priors used to balance classes for RF and RDA.
scenters: List containing center of the scale for each Parafac or Parafac2 model that was fit. Note that, for each component, estimated classification weights are mean-centered before being passed to classification methods. Returns FALSE if argument compscale = FALSE.
sscales: List containing standard devition of the scale for each Parafac or Parafac2 model that was fit. Note that, for each component, estimated classification weights are scaled to have a variance of 1 before being passed to classification methods. Returns FALSE if argument compscale = FALSE.
pcacenter: Numeric vector containing centers for each feature used in PCA. NULL if model is not pca.

Note

For fitting the Parafac model, if argument cmode is not NULL, input array x is reshaped with function aperm such that the cmode dimension of x is ordered last. Estimated mode A and B (and mode C for a four-way array) weights that are outputted as Aweights and Bweights (and Cweights) reflect this permutation. For example, if x is a four-way array and cmode = 2, the original input modes 1, 2, 3, and 4 will correspond to output modes 1, 3, 4, 2. Here, output A = input 1; B = 3, and C = 4 (i.e., the second mode specified by cmode has been moved to the D mode/last mode). For model = "parafac2", classification mode is assumed to be the last mode (i.e., mode C for three-way array and mode D for four-way array). For PCA, if argument cmode is not NULL, and if x is a 3-way or 4-way array, the array is reshaped with aperm such that the cmode dimension of x is ordered first. Then, the array is unfolded into a two-way matrix. If x is already input as a two-way matrix, the matrix is reshaped if cmode is 2, such that the matrix is transposed. Note that for PCA, Aweights contains the PCA model loadings.

In addition, note that the following combination of arguments will give an error: nfac = 1, family = "multinomial", method = "PLR". The issue arises from providing glmnet::cv.glmnet input x with a matrix that has a single column. The issue is resolved for family = "binomial" because a column of 0s is appended to the single column, but this solution does not appear to work for the multiclass case. As such, this combination of arguments is not currently allowed. This issue will be resolved in a future update.

Applications of this function to real datasets can be explored at the following repository: https://github.com/matthewasisgress/multiway-classification/.

Author

Matthew A. Asisgress <mattgress@protonmail.ch>

References

Breiman, L. (2001). Random forests. Machine Learning, 45(1), 5-32.

Chen, T., He, T., Benesty, M., Khotilovich, V., Tang, Y., Cho, H., Chen, K., Mitchell, R., Cano, I., Zhou, T., Li, M., Xie, J., Lin, M., Geng, Y., Li, Y., Yuan, J. (2025). xgboost: Extreme gradient boosting. R Package Version 1.7.9.1.

Corporation, M. and Weston S (2022). doParallel: foreach parallel adaptor for the 'parallel' package. R Package Version 1.0.17.

Cortes, C. and Vapnik, V. (1995). Support-vector networks. Machine Learning, 20(3), 273-297.

Friedman, J. H. (2001). Greedy function approximation: a gradient boosting machine. Annals of Statistics, 29(5), 1189-1232.

Friedman, J. H. (1989). Regularized discriminant analysis. Journal of the American Statistical Association, 84(405), 165-175.

Friedman, J., Hastie, T., and Tibshirani, R. (2010). Regularization paths for generalized linear models via coordinate descent. Journal of Statistical Software, 33(1), 1-22.

Gaujoux, R. (2025). doRNG: Generic reproducible parallel backend for 'foreach' loops. R Package Version 1.8.6.2.

Guo, Y., Hastie, T., and Tibshirani, R. (2007). Regularized linear discriminant analysis and its application in microarrays. Biostatistics, 8(1), 86-100.

Guo, Y., Hastie, T., and Tibshirani, R. (2023). rda: Shrunken centroids regularized discriminant analysis. R Package Version 1.2-1.

Harshman, R. (1970). Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multimodal factor analysis. UCLA Working Papers in Phonetics, 16, 1-84.

Harshman, R. (1972). PARAFAC2: Mathematical and technical notes. UCLA Working Papers in Phonetics, 22, 30-44.

Harshman, R. and Lundy, M. (1994). PARAFAC: Parallel factor analysis. Computational Statistics and Data Analysis, 18, 39-72.

Helwig, N. (2017). Estimating latent trends in multivariate longitudinal data via Parafac2 with functional and structural constraints. Biometrical Journal, 59(4), 783-803.

Helwig, N. (2025). multiway: Component models for multi-way data. R Package Version 1.0-7.

Liaw, A. and Wiener, M. (2002). Classification and regression by randomForest. R News 2(3), 18–22.

Meyer, D., Dimitriadou, E., Hornik, K., Weingessel, A., and Leisch, F. (2024). e1071: Misc functions of the Department of Statistics, Probability Theory Group (Formerly: E1071), TU Wien. R Package Version 1.7-16.

Ripley, B. (1994). Neural networks and related methods for classification. Journal of the Royal Statistical Society: Series B (Methodological), 56(3), 409-437.

Venables, W. and Ripley, B. (2002). Modern applied statistics with S. Fourth Edition. Springer, New York. ISBN 0-387-95457-0.

Zou, H. and Hastie, T. (2005). Regularization and variable selection via the elastic net. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 67(2), 301-320.

Examples

########## Parafac example with 3-way array and binary response ##########
if (FALSE) { # \dontrun{
# set seed and simulate a three-way array connected to a binary response
set.seed(5)

# define list of arguments specifying distributions for A and G weights
techlist <- list(distA = list(dname = "poisson", 
                              lambda = 3),                 # for A weights
                 distG = list(dname = "gamma", shape = 2, 
                              scale = 4))                  # for G weights

# define target correlation matrix for columns of C mode weights matrix
cormat <- matrix(c(1, .6, .6, .6, 1, .6, .6, .6, 1), nrow = 3, ncol = 3)

# simulate a three-way array connected to a response
data <- simcpfa(arraydim = c(11, 12, 100), model = "parafac", nfac = 3, 
                nclass = 2, nreps = 1e2, onreps = 10, corresp = rep(.6, 3), 
                meanpred = rep(2, 3), modes = 3, corrpred = cormat,
                technical = techlist, smethod = "eigende")
                
# initialize
alpha <- seq(0, 1, length = 2)
gamma <- c(0, 0.01)
cost <- c(1, 2)
ntree <- c(100, 200)
nodesize <- c(1, 2)
size <- c(1, 2)
decay <- c(0, 1)
rda.alpha <- c(0.1, 0.6)
delta <- c(0.1, 2)
eta <- c(0.3, 0.7)
max.depth <- c(1, 2)
subsample <- c(0.75)
nrounds <- c(100)
method <- c("PLR", "SVM", "RF", "NN", "RDA", "GBM")
family <- "binomial"
parameters <- list(alpha = alpha, gamma = gamma, cost = cost, ntree = ntree,
                   nodesize = nodesize, size = size, decay = decay, 
                   rda.alpha = rda.alpha, delta = delta, eta = eta,
                   max.depth = max.depth, subsample = subsample,
                   nrounds = nrounds)
model <- "parafac"
nfolds <- 3
nstart <- 3

# constrain first mode weights to be orthogonal
const <- c("orthog", "uncons", "uncons")

# fit Parafac models and use third mode to tune classification methods
tune.object <- tunecpfa(x = data$X, y = data$y, z = data$Cmat, model = model, 
                        nfac = 3, nfolds = nfolds, method = method, 
                        family = family, parameters = parameters, 
                        parallel = FALSE, const = const, nstart = nstart)

# print tuning object
tune.object
} # }