condmixt.foldtrain {condmixt} R Documentation

## Training of conditional mixtures and evaluation of the negative log-likelihood on validation data

### Description

This function can be used to parallelize n-fold cross-validation. Training is done on a training set and the negative log-likelihood is evaluated on validation data. This is repeated for a set of hyper-parameter values. Model selection can be performed based on the evaluation of each set of hyper-parameters on validation data.

### Usage

condhparetomixt.foldtrain(xtrain, ytrain, xtest, ytest, hp, nstart = 1, ...)
condhparetomixt.foldtrain.tailpen(xtrain, ytrain, xtest, ytest, hp, nstart = 1, ...)
condhparetomixt.dirac.foldtrain.tailpen(xtrain, ytrain, xtest, ytest, hp, nstart = 1, ...)
condgaussmixt.foldtrain(xtrain, ytrain, xtest, ytest, hp, nstart = 1, ...)
condgaussmixt.dirac.foldtrain(xtrain, ytrain, xtest, ytest, hp, nstart = 1, ...)
condlognormixt.foldtrain(xtrain, ytrain, xtest, ytest, hp, nstart = 1, ...)
condlognormixt.dirac.foldtrain(xtrain, ytrain, xtest, ytest, hp, nstart = 1, ...)
condbergamixt.foldtrain(xtrain, ytrain, xtest, ytest, hp,nstart = 1, ...)


### Arguments

 xtrain Matrix of explanatory (independent) variables of dimension d x ntrain, d is the number of variables and ntrain is the number of training examples (patterns) ytrain Vector of ntrain dependent variables xtest Matrix of explanatory (independent) variables of dimension d x ntest, d is the number of variables and ntest is the number of test/validation examples (patterns) ytest Vector of ntest dependent variables hp Matrix of hyper-parameters where the columns represent the different hyper-parameters and the rows the sets of values for each hyper-parameter. For condhparetomixt, condgaussmixt and lognormixt together with the version with a dirac component, the hyper-parameters are the number of hidden units and the number of components. When a tail penalty is included for the hybrid Pareto conditional mixture, there are possibly five other hyper-parameters: lambda, w, beta, mu and sigma which controls the tail penalty, see in the Details section below. nstart Number of minimization re-starts, default is one. ... optional arguments for the optimizer nlm

### Details

condhparetomixt indicates a mixture with hybrid Pareto components, condgaussmixt for Gaussian components, condlognormixt for Log-Normal components, condbergam for a Bernoulli-Gamma two component mixture, tailpen indicates that a penalty is added to the log-likelihood to guide the tail index parameter estimation, dirac indicates that a discrete dirac component at zero is included in the mixture

In order to drive the tail index estimation, a penalty is introduced in the log-likelihood. The goal of the penalty is to include a priori information which in our case is that only a few mixture components have a heavy tail index which should approximate the tail of the underlying distribution while most other mixture components have a light tail and aim at modelling the central part of the underlying distribution.

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

### Value

Returns a vector of negative log-likelihood values evaluated on the test set corresponding to each set of hyper-parameters.

Julie Carreau

### 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

condmixt.train,condmixt.nll, condmixt.init

### Examples

# generate train data
ntrain <- 200
xtrain <- runif(ntrain)
ytrain <- rfrechet(ntrain,loc = 3*xtrain+1,scale =
0.5*xtrain+0.001,shape=xtrain+1)
plot(xtrain,ytrain,pch=22) # plot train data
qgen <- qfrechet(0.99,loc = 3*xtrain+1,scale = 0.5*xtrain+0.001,shape=xtrain+1)
points(xtrain,qgen,pch="*",col="orange")

# generate test data
ntest <- 200
xtest <- runif(ntest)
ytest <- rfrechet(ntest,loc = 3*xtest+1,scale =
0.5*xtest+0.001,shape=xtest+1)

# 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)

# train and test a mixture with hybrid Pareto components
condhparetomixt.foldtrain(t(xtrain),ytrain,t(xtest),ytest,hp,nstart=2,iterlim=100)


[Package condmixt version 1.1 Index]