condmixt (version 1.1)

condhparetomixt.cvtrain.tailpen: Cross-validation of the conditinal mixture with hybrid Pareto components with a tail penalty added to the negative log-likelihood for training.

Description

K-fold cross-validation is performed in order to select hyper-parameters of the conditional mixture with hybrid Pareto components. A tail penalty is added to the negative log-likelihood in order to guide the tail index parameters estimation. Performance is evaluated by computing the negative log-likelihood, without penalty.

Usage

condhparetomixt.cvtrain.tailpen(x, y, hp, nfold = 5, nstart = 1, ...)

Arguments

x

Matrix of explanatory (independent) variables of dimension d x n, d is the number of variables and n is the number of examples (patterns)

y

Vector of n dependent variables

hp

Matrix nhp x 7 whose rows represent a set of values for the following hyper-parameters : number of hidden unit, number of component, lambda, w, beta, mu and sigma. The last five hyper-parameters control the tail penalty, see in the Details section below.

nfold

Number of folds for the cross-validation, default is 5.

nstart

Number of optimization re-starts for each training, default is one. Optimization is re-started from different initial values in order to avoir local minima.

…

Extra arguments passed to nlm.

Value

Returns a vector of negative log-likelihood values evaluated out-of-sample by cross-validation. Elements in the vector correspond to each set of hyper-parameters evaluated.

Details

The penalty term is given by the logarithm of the following two-component mixture, as a function of a tail index parameter xi :

w beta exp(-beta xi) + (1-w) exp(-(xi-mu)^2/(2 sigma^2))/(sqrt(2 pi) sigma)

where the first term is the prior on the light tail component and the second term is the prior on the heavy tail component.

The extra hyper-parameters for the penalty terms are as follows : - lambda : penalty parameter which controls the trade-off between the penalty and the negative log-likelihood, takes on positive values. If zero, no penalty - w : penalty parameter in [0,1] which is the proportion of components with light tails, 1-w being the proportion of components with heavy tails - beta : positive penalty parameter which indicates the importance of the light tail components (it is the parameter of an exponential which represents the prior over the light tail components) - mu : penalty parameter in (0,1) which represents the a priori value for the heavy tail index of the underlying distribution - sigma : positive penalty parameter which controls the spread around the a priori value for the heavy tail index of the underlying distribution

References

Bishop, C. (1995), Neural Networks for Pattern Recognition, Oxford

Carreau, J. and Bengio, Y. (2009), A Hybrid Pareto Mixture for Conditional Asymmetric Fat-Tailed Distributions, 20, IEEE Transactions on Neural Networks

See Also

condmixt.foldtrain,condmixt.train,condmixt.nll, condmixt.init

Examples

# NOT RUN {
n <- 200
x <- runif(n) # x is a random uniform variate
 # y depends on x through the parameters of the Frechet distribution
y <- rfrechet(n,loc = 3*x+1,scale = 0.5*x+0.001,shape=x+1)

plot(x,y,pch=22)
# 0.99 quantile of the generative distribution
qgen <- qfrechet(0.99,loc = 3*x+1,scale = 0.5*x+0.001,shape=x+1)
points(x,qgen,pch="*",col="orange")

# create a matrix with sets of values for the number of hidden units and
# the number of components
hp <- matrix(c(2,4,2,2),nrow=2,ncol=2)

# keep tail penalty parameters constant
hp <- cbind(hp, rep(10,2),rep(0.5,2),rep(20,2),rep(0.2,2),rep(0.5,2))

condhparetomixt.cvtrain.tailpen(t(x), y, hp, nfold = 2, nstart = 2, iterlim=100)
# }