Description Usage Arguments Details Value Author(s) References See Also Examples

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.

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

`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 |

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

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.

Julie Carreau

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.foldtrain`

,`condmixt.train`

,`condmixt.nll`

, `condmixt.init`

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 | ```
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)
``` |

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