srm_gif: Conditional Intensity for Stress Release Model In PtProcess: Time Dependent Point Process Modelling

Description

This function calculates the value of the conditional intensity of a Stress Release Model (SRM). Spatial coordinates of the events are not taken into account.

Usage

 `1` ```srm_gif(data, evalpts, params, TT=NA, tplus=FALSE) ```

Arguments

 `data` a data frame containing the event history, where each row represents one event. There must be columns named “time”, usually the number of days from some origin; and “magnitude” which is the event magnitude less the magnitude threshold, i.e. Mi - M0. `evalpts` a `vector`, `matrix` or `data.frame`. If a vector, the elements will be assumed to represent the required evaluation times. Other objects must include a column named `"time"` that can be referred to as `evalpts[,"time"]`, at which the intensity function will be evaluated. `params` vector of parameters for the proposed SRM model in the order (a, b, c). `TT` vector of length 2, being the time interval over which the integral of the conditional intensity function is to be evaluated. `tplus` logical, lambda_g(t|Ht) is evaluated as lambda_g(t^+|Ht) if `TRUE`, else lambda_g(t^-|Ht).

Details

Vere-Jones (1978) proposed the stress release model, being a stochastic version of elastic rebound theory (Reid, 1910). The SRM assumes a deterministic increase in stress over time, and a stochastic release through earthquake events. The conditional intensity function is

lambda_g(t) = exp{a + b[t - cS(t)]},

where

S(t) = sum{10^[0.75(M_i-M_0)]}

and the summation is taken over those i such that ti < t, where ti denotes the event times.

Value

Two usages are as follows.

 ```1 2``` ```srm_gif(data, evalpts, params, tplus=FALSE) srm_gif(data, evalpts=NULL, params, TT) ```

The first usage returns a vector containing the values of lambda_g(t) evaluated at the specified points. In the second usage, it returns the value of the integral.

Function Attributes

`rate`

is `"increasing"`.

Problems and Inconsistencies

Runs much slower than `linksrm_gif`. Should set up matrices `St1` and `St2` as in `linksrm_gif`.

References

Cited references are listed on the PtProcess manual page.

General details about the structure of conditional intensity functions are given in the topic `gif`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15``` ```# Treating North China as one region data(NthChina) p <- c(-2.46, 0.0113, 0.851) times <- seq(0, 517, 0.5) par.default <- par(mfrow=c(2,1), mar=c(4.1, 4.1, 0.5, 1)) plot(times+1480, srm_gif(NthChina, times, params=p), type="l", ylab=expression(lambda[g](t)), xlab="", xlim=c(1480, 2000)) plot(NthChina\$time+1480, NthChina\$magnitude+6, type="h", xlim=c(1480, 2000), ylim=c(5.8, 8.6), xlab="Year", ylab="Magnitude") par(par.default) ```