linksrm_gif {PtProcess}R Documentation

Ground Intensity for Linked Stress Release Model

Description

Calculates the value of the ground intensity of a Linked Stress Release Model (LSRM). This model allows for multiple linked regions, where the stress can be transferred between the regions.

Usage

linksrm_gif(data, evalpts, params, tplus=FALSE, eta=0.75)
linksrm_gif(data, evalpts=NULL, params, TT, eta=0.75)

linksrm1_gif(data, evalpts, params, tplus=FALSE, eta=0.75)
linksrm1_gif(data, evalpts=NULL, params, TT, eta=0.75)

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; "magnitude" which is the event magnitude less the magnitude threshold, i.e. Mk - M0; and "region" which are consecutively numbered starting at 1.
evalpts a matrix or data.frame. It must include two columns named "time" and "region" that can be referred to as evalpts[,"time"] and evalpts[,"region"], respectively. The function will be evaluated at these points.
params vector of parameters of length n^2+2n, where n is the number of regions, for the proposed LSRM in the following order:

c(a_1, ..., a_n, b_1, ..., b_n, c_{11}, c_{12}, c_{13}, ..., c_{nn}).

TT vector of length 2, being the time interval over which the integral of the ground intensity function is to be evaluated.
tplus logical, lambda_g(t,i|Ht) is evaluated as lambda_g(t^+,i|Ht) if TRUE, else lambda_g(t^-,i|Ht).
eta a scalar used in the stress calculations, see Details below.

Details

The ground intensity for the ith region is assumed to have the form

lambda_h(t,i) = exp{ a_i + b_i*[t - sum_j c_{ij} S_j(t)]}

with params = c(a_1, ..., a_n, b_1, ..., b_n, c_{11}, c_{12}, c_{13}, ..., c_{nn}); and

S_j(t) = sum_k 10^{eta(M_k-M_0)},

where the summation is taken over those events in region j with time t_k < t. This model has been discussed by Bebbington & Harte (2001, 2003). The default value of eta = 0.75.

The difference between linksrm_gif and linksrm1_gif is that the stress reduction matrices St1 and St2 (internal to both functions) are calculated every time that the function is called in the case of linksrm_gif. If the event history is not changing between successive calls (e.g. parameter estimation), then this is unnecessary. However, in a simulation, the history changes with the addition of each new event. The function linksrm1_gif checks to see whether the matrices St1 and St2 exist. If so, these existing matrices are used, and new ones are not calculated. Therefore when using linksrm1_gif for parameter estimation, one must check for the existence of such matrices, and delete upon starting to fit a new model, for example:

if (exists("St1")) rm(St1)

if (exists("St2")) rm(St2)

Value

The first usage returns a vector containing the values of lambda_g(t,i) evaluated at the specified “time-region” points. In the second usage, it returns a vector containing the value of the integral for each region.

Function Attributes

rate
is "increasing".
type
is "linksrm".
regions
is expression(sqrt(length(params) + 1) - 1).

Problems and Inconsistencies

It would be better if the objects St1 and St2 could be dealt with in a tidier manner. This is the only difference between linksrm_gif and linksrm1_gif.

See Also

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


[Package PtProcess version 3.1-3 Index]