bgeva: 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

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. s terms are used to specify smooth functions of predictors. See the examples below and the documentation of mgcv for further details on GAM formula specifications.

data

An optional data frame, list or environment containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which bgeva is called.

tau

Shape parameter of the GEV distribution. It must be provided.

Hes

If FALSE, then the Fisher (rather than the observed) information matrix is employed.

gIM

Different versions of GEV distribution. Options are a and b.

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 gamma=1.4 achieves this.

aut.sp

If TRUE, then automatic multiple smoothing parameter selection is carried out. If FALSE, then smoothing parameters are set to the values obtained from the univariate fits.

If TRUE, then a fully parametric model with regression splines is fitted. See the example below.

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) (start.vo=2) or from a combination of options 1 and 2 (start.vo=3).

rinit

Starting trust region radius. The trust region radius is adjusted as the algorithm proceeds. See the documentation of trust for further details.

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: maxit: maximum number of iterations of the magic algorithm; tol: tolerance to use in judging convergence; step.half: if a trial step fails then the method tries halving it up to a maximum of step.half times; rank.tol: constant used to test for numerical rank deficiency of the problem. See the documentation of magic in mgcv for further details.

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.

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).

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.

Examples

Run this code


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)


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