# CPSPPOTevents.fun: Calculates the occurrence times of the three indicator... In IndTestPP: Tests of Independence Between Point Processes in Time

## Description

This function calculates the occurrence times and other characteristics (length, maximum and mean intensity) of the extreme events of the three indicator processes of a bivariate Common Poisson shock process (CPSP) obtained from a POT approach.

## Usage

 `1` ```CPSPPOTevents.fun(X, Y, thresX, thresY, date = NULL) ```

## Arguments

 `X` Numeric vector. Series (x_i) whose threshold exceedances define the first CPSP marginal process. `Y` Numeric vector. Series (y_i) whose threshold exceedances define the second CPSP marginal process. `thresX` Numeric value. Threshold used to define the extreme events in (x_i). `thresY` Numeric value. Threshold used to define the extreme events in (y_i). `date` Optional. A vector or matrix indicating the date of each observation.

## Details

A CPSP N can be decomposed into three independent indicator processes: N_{(1)}, N_{(2)} and N_{(12)}, the processes of the points occurring only in the first marginal process, only in the second and in both of them (simultaneous points). In the CPSP resulting from applying a POT approach, the events in N_{(1)} are a run of consecutive days where (x_i) exceeds its extreme threshold but (y_i) does not exceed its extreme threshold. An extreme event in N_{(2)} is defined analogously. A simultaneous event, or event in N_{(12)}, is a run where both series exceed their thresholds.

For the events defined in each indicator process, three magnitudes (length, maximum intensity and mean intensity) are calculated together with the initial point and the point of maximum excess of each event. In the N_{(12)}, the maximum and the mean intensity of both (x_i) and (y_i) are also calculated. The occurrence point of each event is located at the time of the maximum sum of the excesses over the threholds (where an excess is 0 if the observation does not exceed its corresponding threshold). According to this definition, the occurrence point in N_{(1)} is the point with maximum intensity in (x_i) within the run.

The vectors `inddatX`, `inddatY` and `inddatXY`, elements of the output list, mark the observations that should be used in the estimation of each indicator process. The observations in an extreme event which are not the occurrence point are marked with 0 and treated as non observed in the estimation process. The rest are marked with 1 and must be included in the likelihood function.

## Value

A list with components

 `ImX ` Vector of mean excesses (over the threshold) of the extreme events in N_{(1)}. `IxX` Vector of maximum excesses (over the threshold) of the extreme events in N_{(1)}. `LX ` Vector of lengths of the extreme events in N_{(1)}. `PxX ` Vector of points of maximum excess of the extreme events in N_{(1)}. `PiX ` Vector of initial points of the extreme events in N_{(1)}. `inddatX ` Index equal to 1 in the observations which should be used in the estimation process of N_{(1)}, and to 0 in the others. `ImY ` Vector of mean excesses (over the threshold) of the extreme events in N_{(2)}. `IxY` Vector of maximum excesses (over the threshold) of the extreme events in N_{(2)}. `LY ` Vector of lengths of the extreme events in N_{(2)}. `PxY ` Vector of points of maximum excess of the extreme events in N_{(2)}. `PiY ` Vector of initial points of the extreme events in N_{(2)}. `inddatY ` Index equal to 1 in the observations which should be used in the estimation process of N_{(2)} and to 0 in the others. `ImXYx ` Vector of mean excesses of the series (x_i) in N_{(12)}. `IxXYx ` Vector of maximum excesses the series (x_i) in N_{(12)}. `ImXYy ` Vector of mean excesses of the series (y_i) in N_{(12)}. `IxXYy ` Vector of maximum excesses the series (y_i) in N_{(12)}. `LXY ` Vector of lengths of the extreme events in N_{(12)}. `PxXY ` Vector of points of maximum excess of the extreme events in N_{(12)}. `PiXY ` Vector of initial points of the extreme events in N_{(12)}. `inddatXY ` Index equal to 1 in the observations which should be used in the estimation process of N_{(12)} and to 0 in the others. `X` Input argument. `Y` Input argument. `thresX` Input argument. `thresY` Input argument. `date` Input argument.

## References

Abaurrea, J. Asin, J. and Cebrian, A.C. (2015). A Bootstrap Test of Independence Between Three Temporal Nonhomogeneous Poisson Processes and its Application to Heat Wave Modeling. Environmental and Ecological Statistics, 22(1), 127-144.

`CPSPpoints.fun`
 ```1 2 3 4``` ```data(BarTxTn) dateB<-cbind(BarTxTn\$ano,BarTxTn\$mes,BarTxTn\$diames) BarBivEv<-CPSPPOTevents.fun(X=BarTxTn\$Tx,Y=BarTxTn\$Tn,thresX=318, thresY=220, date=dateB) ```