Description Usage Arguments Details Value Note References See Also Examples

This function fits by maximum likelihood a NHPP where the intensity *λ(t)*
is formulated as a function of covariates. It also calculates and plots
approximate confidence intervals for *λ(t)*.

1 2 3 4 5 |

`covariates` |
Matrix of the covariates to be included in the
linear predictor of the PP intensity (each column is a covariate). It is advisable to give
names to the columns of this matrix
(using |

`start` |
Named list of the initial values for the estimation of
the |

`fixed` |
Named list of the fixed |

`posE` |
Optional (see Details section). Numeric vector of the position of the PP occurrence points. |

`inddat` |
Optional (see Details section). Index vector equal to 1 for the observations used in the estimation process By default, all the observations are considered. |

`POTob` |
Optional (see Details section). List with elements T and thres
that defines the PP resulting from a POT approach;
see |

`nobs` |
Optional. Number of observations in the observation period; it is only neccessary if POTob, inddat and covariates are NULL. |

`tind` |
Logical flag. If it is TRUE, an independent term is fitted in the linear predictor. It cannot be a character string, so TRUE and not'TRUE' should be used. |

`tim` |
Optional. Time vector of the observation period. By default, a vector 1,...n is considered. |

`minfun` |
Label indicating the function to minimize the negative of the loglikelihood function. There are two possible values: "nlminb" (the default option) and "optim". In the last case, the method of optimization can be chosen with an additional method argument. |

`modCI` |
Logical flag. If it is TRUE, confidence intervals
for |

`CIty` |
Label indicating the method to calculate the approximate
confidence intervals for |

`clevel` |
Confidence level of the confidence intervals. |

`tit` |
Character string. A title for the plot. |

`modSim` |
Logical flag. If it is FALSE, information on the estimation process is shown on the screen. For simulation process, the option TRUE should be preferred. |

`dplot` |
Logical flag. If it is TRUE, the fitted intensity is plotted. |

`xlegend` |
Label indicating the position where the legend on the graph will be located. |

`lambdaxlim` |
Optional. Numeric vector of length 2, giving the lowest and highest values which determine the x range. |

`lambdaylim` |
Optional. Numeric vector of length 2, giving the lowest and highest values which determine the y range. |

`...` |
Further arguments to pass to |

A Poisson process (PP) is usually specified by a vector containing the occurrence
points of the process *(t_i)_{i=1}^k*, (argument posE).
Since PP are often used in the framework of POT models, `fitPP.fun`

also
provides the possibility of
using as input the series of the observed values in a POT model
*(x_i)_{i=1}^n* and the threshold used to define the extreme events
(argument POTob).

In the case of PP defined by a POT approach,
the observations of the extreme events which are
not defined as the occurrence point are not considered in the estimation. This is done
through the argument inddat, see `POTevents.fun`

. If the input is provided via argument POTob, index inddat
is calculated automatically. See Coles (2001) for more details on the POT approach.

The maximization of the loglikelihood function can be done using two different optimization routines,
`optim`

or `nlminb`

, selected in the argument `minfun`

. Depending on
the covariates included in the function, one routine can succeed to converge when the other fails.

This function allows us to keep fixed some *β* parameters (offset terms). This can be
used to specify an a priori known component to be included in the linear predictor during fitting. The fixed parameters
must be specified in the `fixed`

argument (and also in `start`

);
the fixed covariates must be included as columns of `covariates`

.

The estimation of the *\hat β* covariance matrix is based on the
asymptotic distribution of the MLE *\hat β*, and calculated as the inverse of the negative of the hessian matrix.
Confidence intervals for *λ(t)* can be calculated using two approaches
specified in the argument `CIty`

. See Casella (2002) for more details on ML theory and delta method.

An object of class `mlePP`

, which is a subclass of `mle`

.
Consequently, many of the generic functions with `mle`

methods, such as
`logLik`

or `summary`

, can be applied to the output of this function. Some other generic
functions related to fitted models, such as `AIC`

or `BIC`

, can also be applied to `mlePP`

objects.

A homogeneous Poisson process (HPP) can be fitted as a particular case, using an intensity defined by only an intercept and no covariate.

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.

Coles, S. (2001). *An introduction to statistical modelling of extreme
values.* Springer.

Casella, G. and Berger, R.L., (2002). *Statistical inference.* Brooks/Cole.

Kutoyants Y.A. (1998).*Statistical inference for spatial Poisson processes.*
Lecture notes in Statistics 134. Springer.

`POTevents.fun`

, `globalval.fun`

,
`VARbeta.fun`

, `CItran.fun`

, `CIdelta.fun`

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 | ```
#model fitted using as input posE and inddat and no confidence intervals
data(BarTxTn)
covB<-cbind(cos(2*pi*BarTxTn$dia/365), sin(2*pi*BarTxTn$dia/365),
BarTxTn$TTx,BarTxTn$Txm31,BarTxTn$Txm31**2)
BarEv<-POTevents.fun(T=BarTxTn$Tx,thres=318,
date=cbind(BarTxTn$ano,BarTxTn$mes,BarTxTn$dia))
mod1B<-fitPP.fun(covariates=covB,
posE=BarEv$Px, inddat=BarEv$inddat,
tit="BAR Tx; cos, sin, TTx, Txm31, Txm31**2",
start=list(b0=-100,b1=1,b2=-1,b3=0,b4=0,b5=0))
#model fitted using as input a list from POTevents.fun and with confidence intervals
tiempoB<-BarTxTn$ano+rep(c(0:152)/153,55)
mod2B<-fitPP.fun(covariates=covB,
POTob=list(T=BarTxTn$Tx, thres=318),
tim=tiempoB, tit="BAR Tx; cos, sin, TTx, Txm31, Txm31**2",
start=list(b0=-100,b1=1,b2=-1,b3=0,b4=0,b5=0),CIty="Delta",modCI=TRUE,
modSim=TRUE)
#model with a fixed parameter (b0)
mod1BF<-fitPP.fun(covariates=covB,
posE=BarEv$Px, inddat=BarEv$inddat,
tit="BAR Tx; cos, sin, TTx, Txm31, Txm31**2",
start=list(b0=-89,b1=1,b2=10,b3=0,b4=0,b5=0),
fixed=list(b0=-100))
``` |

```
Loading required package: stats4
Number of events: 137
Number of excesses over threshold 318 : 253
Number of observations not used in the estimation process: 116
Total number of time observations: 8415
Number of events: 137
Convergence code: 0
Convergence attained
Loglikelihood: -522.727
Estimated coefficients:
b0 b1 b2 b3 b4 b5
-89.289 2.534 1.425 -0.006 0.557 -0.001
Full coefficients:
b0 b1 b2 b3 b4 b5
-89.289 2.534 1.425 -0.006 0.557 -0.001
attr(,"TypeCoeff")
[1] "Fixed: No fixed parameters"
Number of observations not used in the estimation process: 116
Total number of time observations: 8415
Number of events: 137
Convergence code: 0
Convergence attained
Loglikelihood: -522.889
Estimated coefficients:
b1 b2 b3 b4 b5
2.674 1.488 -0.004 0.630 -0.001
Full coefficients:
b0 b1 b2 b3 b4 b5
-100.000 2.674 1.488 -0.004 0.630 -0.001
attr(,"TypeCoeff")
[1] "Fixed: b0"
```

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