NHPoisson (version 3.4)

CalcRes.fun: Calculate NHPP residuals on overlapping intervals

Description

This function calculates raw and scaled residuals of a NHPP based on overlapping intervals. The scaled residuals can be Pearson or any other type of scaled residuals defined by the function \(h(t)\).

Usage

CalcRes.fun(mlePP, lint, h = NULL, typeRes = NULL)

Value

A list with elements

RawRes

Numeric vector of the raw residuals.

ScaRes

A list with elements ScaRes (vector of the scaled residuals) and typeRes (name of the type of scaled residuals).

emplambda

Numeric vector of the empirical estimator of the PP intensity on the considered intervals.

fittedlambda

Numeric vector of the sum of the intensities \(\hat \lambda(t)\) on the considered intervals, divided by the length of the interval.

lintV

Numeric vector of the exact length of each interval. The exact length is defined as the number of observations in each interval used in the estimation (observations with inddat=1).

lint

Input argument.

typeI

Label indicating the type of intervals used to calculate the residuals, 'Overlapping'.

h

Input argument.

mlePP

Input argument.

Arguments

mlePP

An object of class mlePP-class; usually, the output from fitPP.fun.

lint

Length of the intervals to calculate the residuals.

h

Optional. Weight function to calculate the scaled residuals. By default, Pearson residuals with \(h(t)=1/\sqrt{\hat \lambda(t)}\) are calculated.

typeRes

Optional. Label indicating the type of scaled residuals. By default, Pearson residuals are calculated and label is 'Pearson'.

Details

The raw residuals are based on the increments of the raw process \(R(t)=N_t-\int_0^t\hat\lambda(u)du\) in overlapping intervals \((l_1, l_2)\) centered on t:

$$r'(l_1, l_2)=R(l_2)-R(l_1)=\sum_{t_i \in (l_1,l_2)}I_{t_i}-\int_{l_1}^{l_2} \hat \lambda(u)du.$$

Residuals \(r'(l_1, l_2)\) are made 'instantaneous' dividing by the length of the intervals (specified by the argument lint), \(r(l_1, l_2)=r'(l_1,l_2)/(l_2-l_1)\). A residual is calculated for each time in the observation period.

The function also calculates the residuals scaled with the function \(h(t)\)

$$r_{sca}(l_1, l_2)=\sum_{t_i \in (l_1,l_2)}h(t_i)-\int_{l_1}^{l_2} h(u)\hat \lambda(u)du.$$

By default, Pearson residuals with \(h(t)=1/\sqrt{\hat \lambda(t)}\) are calculated.

References

Abaurrea, J., Asin, J., Cebrian, A.C. and Centelles, A. (2007). Modeling and forecasting extreme heat events in the central Ebro valley, a continental-Mediterranean area. Global and Planetary Change, 57(1-2), 43-58.

Baddeley, A., Turner, R., Moller, J. and Hazelton, M. (2005). Residual analysis for spatial point processes. Journal of the Royal Statistical Society, Series B 67,617-666.

Brillinger, D. (1994). Time series, point processes and hybrids. Can. J. Statist., 22, 177-206.

Cebrian, A.C., Abaurrea, J. and Asin, J. (2015). NHPoisson: An R Package for Fitting and Validating Nonhomogeneous Poisson Processes. Journal of Statistical Software, 64(6), 1-24.

Lewis, P. (1972). Recent results in the statistical analysis of univariate point processes. In Stochastic point processes (Ed. P. Lewis), 1-54. Wiley.

See Also

unifres.fun, graphres.fun

Examples

X1<-rnorm(1000)
X2<-rnorm(1000)

modE<-fitPP.fun(tind=TRUE,covariates=cbind(X1,X2), 
	posE=round(runif(40,1,1000)), inddat=rep(1,1000),
	tim=c(1:1000), tit="Simulated example",start=list(b0=1,b1=0,b2=0),
	dplot=FALSE,modCI=FALSE,modSim=TRUE)


#Residuals, based on overlapping intervals of length 50, from the fitted NHPP modE  

ResE<-CalcRes.fun(mlePP=modE, lint=50)