srm_gif {PtProcess}

Conditional Intensity for Stress Release Model

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

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. M_i - M_0.

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|{\cal H}_t) is evaluated as \lambda_g(t^+|{\cal H}_t) if TRUE, else \lambda_g(t^-|{\cal H}_t).

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_i 10^{0.75(M_i-M_0)}

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

Value

Two usages are as follows.

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.

See Also

General details about the structure of conditional intensity functions are given in the topic gif.

Examples

#  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)