brnn_ordinal: brnn_ordinal

In brnn: Bayesian Regularization for Feed-Forward Neural Networks

brnn_ordinal

Description

The brnn_ordinal function fits a Bayesian Regularized Neural Network for Ordinal data.

Usage

  brnn_ordinal(x, ...)
  
  ## S3 method for class 'formula'
brnn_ordinal(formula, data, contrasts=NULL,...)
  
  ## Default S3 method:
brnn_ordinal(x,
               y,
               neurons=2,
               normalize=TRUE,
               epochs=1000,
               mu=0.005,
               mu_dec=0.1,
               mu_inc=10,
               mu_max=1e10,
               min_grad=1e-10,
               change_F=0.01,
               change_par=0.01,
               iter_EM=1000,
               verbose=FALSE,
               ...)

Arguments

formula	A formula of the form y ~ x1 + x2 + ...
data	Data frame from which variables specified in formula are preferentially to be taken.
x	(numeric, n \times p) incidence matrix.
y	(numeric, n) the response data-vector (NAs not allowed).
neurons	positive integer that indicates the number of neurons.
normalize	logical, if TRUE will normalize inputs and output, the default value is TRUE.
epochs	positive integer, maximum number of epochs(iterations) to train, default 1000.
mu	positive number that controls the behaviour of the Gauss-Newton optimization algorithm, default value 0.005.
mu_dec	positive number, is the mu decrease ratio, default value 0.1.
mu_inc	positive number, is the mu increase ratio, default value 10.
mu_max	maximum mu before training is stopped, strict positive number, default value 1\times 10^{10}.
min_grad	minimum gradient.
change_F	the program will stop if the maximum (in absolute value) of the differences of the F function in 3 consecutive iterations is less than this quantity.
change_par	the program will stop iterations of the EM algorithm when the maximum of absolute values of differences between parameters in two consecutive iterations ins less than this quantity.
iter_EM	positive integer, maximum number of iteration for the EM algorithm.
verbose	logical, if TRUE will print iteration history.
contrasts	an optional list of contrasts to be used for some or all of the factors appearing as variables in the model formula.
...	arguments passed to or from other methods.

Details

The software fits a Bayesian Regularized Neural Network for Ordinal data. The model is an extension of the two layer network as described in MacKay (1992); Foresee and Hagan (1997), and Gianola et al. (2011). We use the latent variable approach described in Albert and Chib (1993) to model ordinal data, the Expectation maximization (EM) and Levenberg-Marquardt algorithm (Levenberg, 1944; Marquardt, 1963) to fit the model.

Following Albert and Chib (1993), suppose that Y_1,...,Y_n are observed and Y_i can take values on L ordered values. We are interested in modelling the probability p_{ij}=P(Y_i=j) using the covariates x_{i1},...,x_{ip}. Let

g(\boldsymbol{x}_i)= \sum_{k=1}^s w_k g_k (b_k + \sum_{j=1}^p x_{ij} \beta_j^{[k]}),

where:

• s is the number of neurons.

• w_k is the weight of the k-th neuron, k=1,...,s.

• b_k is a bias for the k-th neuron, k=1,...,s.

• \beta_j^{[k]} is the weight of the j-th input to the net, j=1,...,p.

• g_k(\cdot) is the activation function, in this implementation g_k(x)=\frac{\exp(2x)-1}{\exp(2x)+1}.

Let

Z_i=g(\boldsymbol{x}_i)+e_i,

where:

• e_i \sim N(0,1).

• Z_i is an unobserved (latent variable).

The output from the model for latent variable is related to observed data using the approach employed in the probit and logit ordered models, that is Y_i=j if \lambda_{j-1}<Z_i<\lambda_{j}, where \lambda_j are a set of unknown thresholds. We assign prior distributions to all unknown quantities (see Albert and Chib, 1993; Gianola et al., 2011) for further details. The Expectation maximization (EM) and Levenberg-Marquardt algorithm (Levenberg, 1944; Marquardt, 1963) to fit the model.

Value

object of class "brnn_ordinal". Mostly internal structure, but it is a list containing:

$theta	A list containing weights and biases. The first s components of the list contains vectors with the estimated parameters for the k-th neuron, i.e. (w_k, b_k, \beta_1^{[k]},...,\beta_p^{[k]})'.
$threshold	A vector with estimates of thresholds.
$alpha	\alpha parameter.
$gamma	effective number of parameters.

References

Albert J, and S. Chib. 1993. Bayesian Analysis of Binary and Polychotomus Response Data. JASA, 88, 669-679.

Foresee, F. D., and M. T. Hagan. 1997. "Gauss-Newton approximation to Bayesian regularization", Proceedings of the 1997 International Joint Conference on Neural Networks.

Gianola, D. Okut, H., Weigel, K. and Rosa, G. 2011. "Predicting complex quantitative traits with Bayesian neural networks: a case study with Jersey cows and wheat". BMC Genetics, 12,87.

Levenberg, K. 1944. "A method for the solution of certain problems in least squares", Quart. Applied Math., 2, 164-168.

MacKay, D. J. C. 1992. "Bayesian interpolation", Neural Computation, 4(3), 415-447.

Marquardt, D. W. 1963. "An algorithm for least-squares estimation of non-linear parameters". SIAM Journal on Applied Mathematics, 11(2), 431-441.

See Also

predict.brnn_ordinal

Examples

## Not run: 
#Load the library
library(brnn)

#Load the dataset
data(GLS)

#Subset of data for location Harare
HarareOrd=subset(phenoOrd,Loc=="Harare")

#Eigen value decomposition for GOrdm keep those 
#eigen vectors whose corresponding eigen-vectors are bigger than 1e-10
#and then compute principal components

evd=eigen(GOrd)
evd$vectors=evd$vectors[,evd$value>1e-10]
evd$values=evd$values[evd$values>1e-10]
PC=evd$vectors
rownames(PC)=rownames(GOrd)

#Response variable
y=phenoOrd$rating
gid=as.character(phenoOrd$Stock)

Z=model.matrix(~gid-1)
colnames(Z)=gsub("gid","",colnames(Z))

if(any(colnames(Z)!=rownames(PC))) stop("Ordering problem\n")

#Matrix of predictors for Neural net
X=Z%*%PC

#Cross-validation
set.seed(1)
testing=sample(1:length(y),size=as.integer(0.10*length(y)),replace=FALSE)
isNa=(1:length(y)%in%testing)
yTrain=y[!isNa]
XTrain=X[!isNa,]
nTest=sum(isNa)

neurons=2
	
fmOrd=brnn_ordinal(XTrain,yTrain,neurons=neurons,verbose=FALSE)

#Predictions for testing set
XTest=X[isNa,]
predictions=predict(fmOrd,XTest)
predictions



## End(Not run)

brnn documentation built on Nov. 10, 2023, 9:08 a.m.