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 of evaluation points at which the ground intensity is to be calculated. The matrix must include two columns named "time" and "region" at which the intensity function will be evaluated.
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: newline if (exists("St1")) rm(St1) newline 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 3.1-0 Index]