pp.fit {ismev}

Maximum-likelihood Fitting for the Point Process Model

Description

Maximum-likelihood fitting for the point process model, including generalized linear modelling of each parameter.

Usage

pp.fit(xdat, threshold, npy = 365, ydat = NULL, mul = NULL, sigl =
    NULL, shl = NULL, mulink = identity, siglink = identity, shlink =
    identity, muinit = NULL, siginit = NULL, shinit = NULL, show = TRUE,
    method = "Nelder-Mead", maxit = 10000, ...)

Arguments

xdat

A numeric vector of data to be fitted.

threshold

The threshold; a single number or a numeric vector of the same length as xdat.

npy

The number of observations per year/block.

ydat

A matrix of covariates for generalized linear modelling of the parameters (or NULL (the default) for stationary fitting). The number of rows should be the same as the length of xdat.

mul, sigl, shl

Numeric vectors of integers, giving the columns of ydat that contain covariates for generalized linear modelling of the location, scale and shape parameters repectively (or NULL (the default) if the corresponding parameter is stationary).

mulink, siglink, shlink

Inverse link functions for generalized linear modelling of the location, scale and shape parameters repectively.

muinit, siginit, shinit

numeric giving initial parameter estimates. See Details section for information on default (NULL) initial values.

show

Logical; if TRUE (the default), print details of the fit.

method

The optimization method (see optim for details).

maxit

The maximum number of iterations.

...

Other control parameters for the optimization. These are passed to components of the control argument of optim.

Details

For non-stationary fitting it is recommended that the covariates within the generalized linear models are (at least approximately) centered and scaled (i.e.\ the columns of ydat should be approximately centered and scaled). Otherwise, the numerics may become unstable.

As of version 1.32, a more accurate estimate of the exceedance rate, in the face of covariates, is used (at the expense of computational efficiency). In particular, when including covariates, parameter estimates may differ from those in Coles (2001).

Let m=mean(xdat) and s=sqrt(6*var(xdat))/pi. Then, initial values assigend when 'muinit' is NULL are m - 0.57722 * s (stationary case). When 'siginit' is NULL, the initial value is taken to be s, and when 'shinit' is NULL, the initial value is taken to be 0.1. When covariates are introduced (non-stationary case), these same initial values are used by default for the constant term, and zeros for all other terms. For example, if a GEV( mu(t)=mu0+mu1*t, sigma, xi) is being fitted, then the initial value for mu0 is m - 0.57722 * s, and 0 for mu1.

Value

A list containing the following components. A subset of these components are printed after the fit. If show is TRUE, then assuming that successful convergence is indicated, the components nexc, nllh, mle and se are always printed.

trans

An logical indicator for a non-stationary fit.

model

A list with components mul, sigl and shl.

link

A character vector giving inverse link functions.

threshold

The threshold, or vector of thresholds.

npy

The number of observations per year/block.

nexc

The number of data points above the threshold.

data

The data that lie above the threshold. For non-stationary models, the data is standardized.

conv

The convergence code, taken from the list returned by optim. A zero indicates successful convergence.

nllh

The negative logarithm of the likelihood evaluated at the maximum likelihood estimates.

vals

A matrix with four columns containing the maximum likelihood estimates of the location, scale and shape parameters, and the threshold, at each data point.

gpd

A matrix with three rows containing the maximum likelihood estimates of corresponding GPD location, scale and shape parameters at each data point.

mle

A vector containing the maximum likelihood estimates.

cov

The covariance matrix.

se

A vector containing the standard errors.

Warning

Different optimization methods may result in wildly different parameter estimates.

Note

This code is adapted by Eric Gilleland from code originally written for S-Plus by Stuart Coles, and ported to R by Alec Stephenson. See details section above.

References

Beirlant J, Goegebeur Y, Segers J and Teugels J. (2004). Statistics of Extremes, Wiley, Chichester, England.

Coles, Stuart (2001). An Introduction to Statistical Modeling of Extreme Values. Springer-Verlag, London.

See Also

pp.diag, optim, pp.fitrange, mrl.plot, gpd.fit

Examples

data(rain)
pp.fit(rain, 10)