fitPP.fun {NHPoisson}R Documentation

Fit a non homogeneous Poisson Process

Description

This function fits by maximum likelihood a NHPP where the intensity λ(t) is formulated as a function of covariates. It also calculates and plots approximate confidence intervals for λ(t).

Usage

fitPP.fun(covariates = NULL, beta, posE = NULL, inddat = NULL, POTob = NULL, 
namcovariates = NULL, n = NULL, tind = "TRUE", tim = NULL, modCI = "TRUE", 
CIty = "Delta", clevel = 0.95, tit = '', modSim = "FALSE", dplot = TRUE, 
xlegend = "topleft")

Arguments

covariates

Matrix of the covariates to be included in the linear predictor of the PP intensity (each column is a covariate).

beta

Numeric vector of the initial values for the estimation of the β parameters.

posE

Optional (see Details section). Numeric vector of the position of the PP occurrence points.

inddat

Optional (see Details section). Index vector equal to 1 for the observations used in the estimation process By default, all the observations are considered.

POTob

Optional (see Details section). List with elements T and thres that defines the PP resulting from a POT approach; see POTevents.fun for more details.

namcovariates

Optional. Vector of the names of the variables in covariates.

n

Optional. Number of observations in the observation period; it is only neccessary if POTob, inddat and covariates are NULL.

tind

Logical flag. If it is TRUE, an independent term is fitted in the linear predictor.

tim

Optional. Time vector of the observation period. By default, a vector 1,...n is considered.

modCI

Logical flag. If it is TRUE, confidence intervals for λ(t) values are calculated.

CIty

Label indicating the method to calculate the approximate confidence intervals for λ(t). It can be 'delta' for the delta method or 'transf' for transformed asymptotic intervals; see CIdelta.fun and CItran.fun for details.

clevel

Confidence level of the confidence intervals.

tit

Character string. A title for the plot.

modSim

Logical flag. If it is FALSE, information on the estimation process is shown on the screen. For simulation process, the option TRUE should be prefered.

dplot

Logical flag. If it is TRUE, the fitted intensity is plotted.

xlegend

Label indicating the position where the legend on the graph will be located.

Details

A Poisson process (PP) is usually specified by a vector containing the occurrence points of the process (t_i)_{i=1}^k, (argument posE). Since PP are often used in the framework of POT models, fitPP.fun also provides the possibility of using as input the series of the observed values in a POT model (x_i)_{i=1}^n and the threshold used to define the extreme events (argument POTob).

In the case of PP defined by a POT approach, the observations of the extreme events which are not defined as the occurrence point are not considered in the estimation. This is done through the argument inddat, see POTevents.fun. If the input is provided via argument POTob, index inddat is calculated automatically. See Coles (2001) for more details on the POT approach.

The estimation of the β covariance matrix is based on the asymptotic distribution of the MLE \hat β, and calculated as the inverse of the hessian. Confidence intervals for λ(t) can be calculated using two approaches, the delta method or a transformation of the confidence interval for the linear predictor ν(t)=\textbf{X(t)} β. The interval for ν(t) is also based on the asymptotic properties of the MLE \hat ν(t). See Casella (2002) for more details on ML theory and delta method.

Value

A list with elements

llik

Value of the loglikelihood function.

npar

Number of estimated parameters.

beta

Vector of the MLE \hat β.

inddat

Input argument.

VARbeta

Covariance matrix of the β parameters

lambdafit

Vector of the fitted intensity \hat λ(t).

LIlambda

Vector of lower extremes of the CI

LUlambda

Vector of upper extremes of the CI.

posE

Input argument.

namcovariates

Input argument.

tit

Input argument.

tind

Input argument.

Note

A homogeneous Poisson process (HPP) can be fitted as a particular case, using an intensity defined by only an independent term and no covariates.

References

Coles, S. (2001). An introduction to statistical modelling of extreme values. Springer.

Casella, G. and Berger, R.L., (2002). Statistical inference. Brooks/Cole.

See Also

POTevents.fun, globalval.fun, VARbeta.fun, CItran.fun, CIdelta.fun

Examples

#model fitted  using as input posE and inddat and  no confidence intervals 

data(BarTxTn)
covB<-cbind(cos(2*pi*BarTxTn$dia/365), sin(2*pi*BarTxTn$dia/365), 
	BarTxTn$TTx,BarTxTn$Txm31,BarTxTn$Txm31**2)
BarEv<-POTevents.fun(T=BarTxTn$Tx,thres=318, 
	date=cbind(BarTxTn$ano,BarTxTn$mes,BarTxTn$dia))


mod1B<-fitPP.fun(tind='TRUE',covariates=covB, 
	posE=BarEv$Px, inddat=BarEv$inddat,
	tit='BAR Tx; cos, sin, TTx, Txm31, Txm31**2', 
	beta=c(-100,1,10,0,0,0))

#model fitted  using as input  a list from POTevents.fun and with  confidence intervals 
tiempoB<-BarTxTn$ano+rep(c(0:152)/153,55)

mod2B<-fitPP.fun(tind='TRUE',covariates=covB, 
	POTob=list(T=BarTxTn$Tx, thres=318),
	tim=tiempoB, tit='BAR Tx; cos, sin, TTx, Txm31, Txm31**2', 
	beta=c(-100,1,10,0,0,0),CIty='Transf',modCI=TRUE,
	modSim=TRUE)

[Package NHPoisson version 1.0 Index]