PtProcess (version 3.1-1)

mpp: Marked Point Process Object

Description

Creates a marked point process model object with class "mpp".

Usage

mpp(data, gif, marks, params, gmap, mmap, TT)

Arguments

data

a data.frame containing the history of the process, denoted below as ${\cal H}_t$. It should contain all variables that are required to evaluate the gif function and the mark distribution, t

gif

ground intensity function. See topic gif for further details.

marks

mark distribution. See topic marks for further details.

params

numeric vector of all model parameters.

gmap

expression, maps the model parameters (params) into the parameter sub-space of the ground intensity function; see Details below.

mmap

expression, maps the model parameters (params) into the parameter sub-space of the mark distribution; see Details below.

TT

vector of length 2, being the time interval over which the integral of the ground intensity function is to be evaluated.

Details

Let $\lambda_g(t|{\cal H}_t)$ denote the ground intensity function and $f(y|{\cal H}_t)$ denote the joint mark densities, where $y \in {\cal Y}$. The log-likelihood of a marked point process is given by $$\log L = \sum_i \log \lambda_g(t_i|{\cal H}_{t_i}) + \sum_i \log f(y_i|{\cal H}_{t_i}) - \int \lambda_g(t|{\cal H}_t) dt,$$ where the summation is taken over those events contained in the interval (TT[1], TT[2]), and the integral is also taken over that interval. However, all events in the data frame data before $t$, even those before TT[1], form the history of the process ${\cal H}_t$. This allows an initial period for the process to reach a steady state or equilibrium.

The parameter spaces of the ground intensity function and mark distribution are not necessarily disjoint, and can have common parameters. Hence, when the model parameters are estimated, these relationships must be known, and are specified by the arguments gmap and mmap. The mapping expressions can also contain arithmetic expressions. The $i$th element in the params argument is addressed in the expressions as params[i]. Here is an example of a five parameter model, where the gif has 4 parameters, and the mark distribution has 2, with mappings specified as: gmap = expression(c(params[1:3], exp(params[4]+params[5])))

mmap = expression(c(log(params[2]/3), params[5])) Note the inclusion of the combine (c) function, because the expression must create a vector of parameters. Care must be taken specifying these expressions are they are embedded directly into the code of various functions.

Examples

data(Tangshan)

#   increment magnitudes a fraction so none are zero
Tangshan[,"magnitude"] <- Tangshan[,"magnitude"] + 0.01

dmagn_mark <- function(x, data, params){
    #  Gamma distribution
    #  exponential density when params[7]=0
    #   See topic "marks" for further discussion
    lambda <- etas_gif(data, x[,"time"], params=params[1:5])
    y <- dgamma(x[,"magnitude"], shape=1+sqrt(lambda)*params[7],
                rate=params[6], log=TRUE)
    return(y)
}

TT <- c(0, 4018)
params <- c(0.007, 2.3, 0.98, 0.008, 0.94, 1.25, 0.2)

x <- mpp(data=Tangshan,
         gif=etas_gif,
         mark=list(dmagn_mark, NULL),
         params=params,
         gmap=expression(params[1:5]),
         mmap=expression(params[1:7]),
         TT=TT)

allmap <- function(y, p){
    #    one to one mapping, all p positive
    y$params <- exp(p)
    return(y)
}

#    Parameters must be positive. Transformed so that nlm
#    can use entire real line (no boundary problems, see
#    topic "neglogLik" for further explanation).
z <- nlm(neglogLik, log(params), object=x, pmap=allmap,
         print.level=2, iterlim=500, typsize=abs(params))

x1 <- allmap(x, z$estimate)

#    print parameter estimates
print(x1$params)

print(logLik(x))
print(logLik(x1))
plot(x1, log=TRUE)