CalcResD.fun: Calculate NHPP residuals on disjoint 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 disjoint intervals. The scaled residuals can be Pearson or any other type of scaled residuals defined by the function h(t).

1 2	CalcResD.fun(mlePP, h = NULL, nint = NULL, lint = NULL, typeRes = NULL, modSim = "FALSE")

`mlePP`	An object of class `mlePP-class`; usually, the output from `fitPP.fun`.
`lint`	Optional. 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'.
`modSim`	Logical flag. If it is FALSE, some information on the intervals is shown on the screen.
`nint`	Number of intervals used to calculate the residuals. Intervals with the same length are considered. Only one of lint or nint must be specified.

The intervals used to calculate the residuals can be specified either by nint or lint; only one of the arguments must be provided. If nint is specified, intervals of equal length are calculated.

The raw residuals are based on the increments of the raw process R(t)=N_t-\int_0^t\hatλ(u)du in disjoint 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).

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.
`nint`	Input argument.
`pm`	Numeric vector of the mean point of the intervals.
`typeI`	Label indicating the type of intervals used to calculate the residuals, 'Disjoint' .
`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.

CalcRes.fun, 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 20 disjoint intervals of length 50, from the fitted NHPP modE  

ResDE<-CalcResD.fun(mlePP=modE,lint=50)