CalcRes.fun: Calculate NHPP residuals on overlapping intervals
In NHPoisson: Modelling and Validation of Non Homogeneous Poisson Processes

Description Usage Arguments Details Value References See Also Examples

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).

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

`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/√{\hat λ(t)} are calculated.
`typeRes`	Optional. Label indicating the type of scaled residuals. By default, Pearson residuals are calculated and label is 'Pearson'.

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

r'(l_1, l_2)=R(l_2)-R(l_1)=∑_{t_i \in (l_1,l_2)}I_{t_i}-\int_{l_1}^{l_2} \hat λ(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)=∑_{t_i \in (l_1,l_2)}h(t_i)-\int_{l_1}^{l_2} h(u)\hat λ(u)du.

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

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 λ(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.

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.

unifres.fun, graphres.fun

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)