# CalcResD.fun: Calculate NHPP residuals on disjoint intervals In NHPoisson: Modelling and Validation of Non Homogeneous Poisson Processes

## Description

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).

## Usage

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

## Arguments

 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.

## Details

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.

## 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 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.

## 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.

CalcRes.fun, unifres.fun, graphres.fun
  1 2 3 4 5 6 7 8 9 10 11 12 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)