# Binary Generalized Extreme Value Additive Modelling

### Description

`bgeva`

can be used to fit regression models for binary rare events where the link function is the quantile function of
the Generalized Extreme Value random variable. The linear predictor can be flexibly specified using parametric and
regression spline components. Regression
spline bases are extracted from the package `mgcv`

. Multi-dimensional smooths are available
via the use of penalized thin plate regression splines (isotropic). The current implementation does not support scale invariant tensor
product smooths.

### Usage

1 2 3 4 5 6 7 | ```
bgeva(formula.eq, data=list(), tau=-0.25, Hes=TRUE, gIM="a", iterlimSP=50,
pr.tol=1e-6,
gamma=1, aut.sp=TRUE, fp=FALSE, start.v=NULL, start.vo=1,
rinit=1, rmax=100, fterm=sqrt(.Machine$double.eps),
mterm=sqrt(.Machine$double.eps),
control=list(maxit=50,tol=1e-6,step.half=25,
rank.tol=sqrt(.Machine$double.eps)))
``` |

### Arguments

`formula.eq` |
A GAM formula. |

`data` |
An optional data frame, list or environment containing the variables in the model. If not found in |

`tau` |
Shape parameter of the GEV distribution. It must be provided. |

`Hes` |
If |

`gIM` |
Different versions of GEV distribution. Options are |

`iterlimSP` |
A positive integer specifying the maximum number of loops to be performed before the smoothing parameter estimation step is terminated. |

`pr.tol` |
Tolerance to use in judging convergence of the algorithm when automatic smoothing parameter selection is used. |

`gamma` |
It is an inflation factor for the model degrees of freedom in the UBRE score. Smoother models can be obtained setting
this parameter to a value greater than 1. Typically |

`aut.sp` |
If |

`fp` |
If |

`start.v` |
Starting values for the parameters can be provided here. |

`start.vo` |
Default is 1 meaning that starting values are obtained from fitting a logistic model. Otherwise, these can be set as described in Calabrese and Osmetti (2013) ( |

`rinit` |
Starting trust region radius. The trust region radius is adjusted as the algorithm proceeds. See the documentation
of |

`rmax` |
Maximum allowed trust region radius. This may be set very large. If set small, the algorithm traces a steepest descent path. |

`fterm` |
Positive scalar giving the tolerance at which the difference in objective function values in a step is considered close enough to zero to terminate the algorithm. |

`mterm` |
Positive scalar giving the tolerance at which the two-term Taylor-series approximation to the difference in objective function values in a step is considered close enough to zero to terminate the algorithm. |

`control` |
It is a list containing iteration control constants with the following elements: |

### Details

The Binary Generalized Extreme Value Additive model has the quantile function of the Generalized Extreme Value (GEV) random variable as link function. The linear predictor is flexibly specified using parametric components and smooth functions of covariates. Replacing the smooth components with their regression spline expressions yields a fully parametric univariate GEV model. In principle, classic maximum likelihood estimation can be employed. However, to avoid overfitting, penalized likelihood maximization has to be employed instead. Here the use of penalty matrices allows for the suppression of that part of smooth term complexity which has no support from the data. The trade-off between smoothness and fitness is controlled by smoothing parameters associated with the penalty matrices. Smoothing parameters are chosen to minimize the approximate Un-Biased Risk Estimator (UBRE).

Automatic smoothing parameter selection is integrated using a performance-oriented iteration approach (Gu, 1992; Wood, 2004). At each iteration, (i) the penalized weighted least squares problem is solved, then (ii) the smoothing parameters of that problem estimated by approximate UBRE. Steps (i) and (ii) are iterated until convergence.

Full details can be found in Calabrese, Marra and Osmetti (2013).

### Value

The function returns an object of class `bgeva`

as described in `bgevaObject`

.

### WARNINGS

Any automatic smoothing parameter selection procedure is not likely to work well when the data have low information content. In binary models, this
issue is especially relevant the number of observations low. Here, convergence failure is
typically associated with an infinite cycling between the two steps detailed above. If this occurs, as some practical solutions, one might
either (i) lower the total number of parameters to estimate by reducing the dimension of the regression spline
bases, (ii) set the smoothing parameters to the values obtained from the univariate fits (`aut.sp=FALSE`

), or (iii) set the smoothing
parameters to the values obtained from the non-converged algorithm. The default option is (iii).

The GEV distribution may not be defined for certain combinations of parameter and covariate values. In such cases, a sub-design matrix is formed. This consists of the rows (of the original design matrix) for which the distributrion is defined.

### Author(s)

Maintainer: Giampiero Marra giampiero.marra@ucl.ac.uk

### References

Calabrese R., Marra G., Osmetti S.A. (2013), Bankruptcy Prediction of Small and Medium Enterprises Using a Flexible Binary Generalized Extreme Value Model. *Submitted*.

Calabrese R., Osmetti S.A. (2013), Modelling SME Loan Defaults as Rare Events: The Generalized Extreme Value Regression Model. *Journal of Applied Statistics*.

Gu C. (1992), Cross validating non-Gaussian data. *Journal of Computational and Graphical Statistics*, 1(2), 169-179.

Wood S.N. (2004), Stable and efficient multiple smoothing parameter estimation for generalized additive models. *Journal of the American Statistical Association*, 99(467), 673-686.

### See Also

`plot.bgeva`

, `bgeva-package`

, `bgevaObject`

, `summary.bgeva`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 | ```
library(bgeva)
##########
## EXAMPLE
##########
set.seed(0)
n <- 1500
x1 <- round(runif(n))
x2 <- runif(n)
x3 <- runif(n)
f1 <- function(x) (cos(pi*2*x)) + sin(pi*x)
f2 <- function(x) (x+exp(-30*(x-0.5)^2))
y <- as.integer(rlogis(n, location = -6 + 2*x1 + f1(x2) + f2(x3), scale = 1) > 0)
dataSim <- data.frame(y,x1,x2,x3)
out <- bgeva(y ~ x1 + s(x2) + s(x3))
bg.checks(out)
summary(out)
plot(out,scale=0,pages=1,shade=TRUE)
#
#
``` |