condmixt-package {condmixt} R Documentation

## Conditional Density Estimation with Neural Network Conditional Mixtures

### Description

Neural network conditional mixtures are mixtures whose parameters depends on explanatory variables through a neural network. In other words, for a given set of values of explanatory variables, a neural network will compute the corresponding mixture parameters. Thus, the parameters of the mixture and hence the shape of its density is modified by the values of the explanatory variables. There are two hyper-parameters, the number of hidden units of the neural network and the number of components of the mixture, which control the complexity of the conditional mixtures. In this package, a number of types of conditional mixtures are implemented, which differ in their type of components and which allow the possibility to include a discrete component in the mixture.

### Details

 Package: condmixt Type: Package Version: 1.0 Date: 2012-04-06 License: GPL-2 LazyLoad: yes

Functions for conditional mixtures with - hybrid Pareto components start with condhparetomixt - Gaussian components start with condgaussmixt - Log-Normal components start with condlognormixt

If a discrete dirac component at zero is added, for instance to model the presence of zeros for "no rain" events in a rainfall time-series, then functions related to such mixtures start with something like condhparetomixt.dirac depending on the type of components for the continuous part.

One special mixture, which has only two-components (a discrete, dirac or Bernoulli, component and a Gamma component) is implemented under the name condbergamixt. It has been introduced in Williams(1998).

Finally, hybrid Pareto components have a tail index parameter which might be difficult to estimate, specially in the conditional, non-stationnary case. To alleviate this issue, a penalty term might be added to the log-likelihood in order to guide maximum likelihood estimation of the tail indexes. The functions which employ a tail penalty term have a name ending with tailpen.

The goal of the tail penalty for hybrid Pareto mixtures 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.

### Author(s)

Julie Carreau

Maintainer: Julie Carreau <julie.carreau@univ-montp2.fr>

### References

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

Carreau J. and Vrac, M. (2011) Stochastic Downscaling of Precipitation with Neural Network Conditional Mixture Models, 47, Water Resources Research

Carreau, J. and Bengio, Y. (2009), A Hybrid Pareto Model for Asymmetric Fat-tailed Data: the Univariate Case, 12, Extremes

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

Williams, M.P. (1998) Modelling Seasonality and Trends in Daily Rainfall Data, 10, Advances in Neural Information and Processing Systems

condmixt.init, condmixt.nll, condmixt.fit