ENNreg {evreg}

Training the ENNreg model

Description

ENNreg trains the ENNreg model using batch or minibatch learning procedures.

Usage

ENNreg(
  X,
  y,
  init = NULL,
  K = NULL,
  batch = TRUE,
  nstart = 100,
  c = 1,
  lambda = 0.9,
  xi = 0,
  rho = 0,
  eps = NULL,
  nu = 1e-16,
  optimProto = TRUE,
  verbose = TRUE,
  options = list(maxiter = 1000, rel.error = 1e-04, print = 10),
  opt.rmsprop = list(batch_size = 100, epsi = 0.001, rho = 0.9, delta = 1e-08, Dtmax =
    100)
)

Arguments

X

Input matrix of size n x p, where n is the number of objects and p the number of attributes.

y

Vector of length n containing observations of the response variable.

init

Initial model generated by ENNreg_init (default=NULL).

K

Number of prototypes (default=NULL; must be supplied if initial model is not supplied).

batch

If TRUE (default), batch learning is used; otherwise, online learning is used.

nstart

Number of random starts of the k-means algorithm (default: 100, used only if initial model is not supplied).

c

Multiplicative coefficient applied to scale parameter gamma (defaut: 1, used only if initial model is not supplied)

lambda

Parameter of the loss function (default=0.9)

xi

Regularization coefficient penalizing precision (default=0).

rho

Regularization coefficient shrinking the solution towards a linear model (default=0).

eps

Parameter of the loss function (if NULL, set to 0.01 times the standard deviation of y).

nu

Parameter of the loss function to avoid a division par zero (default=1e-16).

optimProto

If TRUE (default), the initial prototypes are optimized.

verbose

If TRUE (default) intermediate results are displayed.

options

Parameters of the optimization procedure (see details).

opt.rmsprop

Parameters of the RMSprop algorithm (see details).

Details

If batch=TRUE, function harris from package evclust is used for optimization. Otherwise, the RMSprop minibatch learning algorithm is used. The three parameters in list options are:

maxiter

Maximum number of iterations (default: 100).

rel.error

Relative error for stopping criterion (default: 1e-4).

print

Number of iterations between two displays (default: 10).

Additional parameters for the RMSprop, used only if batch=FALSE, are contained in list opt.rmsprop. They are: '

batch_size

Minibatch size.

epsi

Global learning rate.

rho

Decay rate.

delta

Small constant to stabilize division by small numbers.

Dtmax

The algorithm stops when the loss has not decreased in the last Dtmax iterations.

Value

An object of class "ENNreg" with the following components:

loss

Value of the loss function.

param

Parameter values.

K

Number of prototypes.

pred

Predictions on the training set (a list containing the prototype unit activations, the output means, variances and precisions, as well as the lower and upper expectations).

References

Thierry Denoeux. An evidential neural network model for regression based on random fuzzy numbers. In "Belief functions: Theory and applications (proc. of BELIEF 2022)", pages 57-66, Springer, 2022.

Thierry Denoeux. Quantifying prediction uncertainty in regression using random fuzzy sets: the ENNreg model. IEEE Transactions on Fuzzy Systems, Vol. 31, Issue 10, pages 3690-3699, 2023.

See Also

predict.ENNreg, ENNreg_init, ENNreg_cv, ENNreg_holdout

Examples

# Boston dataset

library(MASS)
X<-as.matrix(scale(Boston[,1:13]))
y<-Boston[,14]
set.seed(220322)
n<-nrow(Boston)
ntrain<-round(0.7*n)
train <-sample(n,ntrain)
fit <- ENNreg(X[train,],y[train],K=30)
plot(y[train],fit$pred$mux,xlab="observed response",ylab="predicted response")