linksrm_gif: 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 threshol

evalpts

a matrix or data.frame. It must include two columns named "time" and "region" that can be referred to as evalpts[,"time"]

params

vector of parameters of length $n^2+2n$, where $n$ is the number of regions, for the proposed LSRM in the following order:
$$\code{c}(a_1, \cdots, a_n, b_1, \cdots, b_n, c_{11}, c_{12}, c_{13}, \cdots, c_{nn}).$$

TT

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

tplus

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

eta

a scalar used in the stress calculations, see Details below.

`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.

`synopsis`

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

`Details`

The ground intensity for the $i$th region is assumed to have the form
$$\lambda_g(t,i | {\cal H}_t) = \exp\left{ a_i + b_i\left[t - \sum_{j=1}^n c_{ij} S_j(t)\right]\right}$$
with $\code{params} = \code{c}(a_1, \cdots, a_n, b_1, \cdots, b_n, c_{11}, c_{12}, c_{13}, \cdots, c_{nn})$; 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 $\eta = \code{eta} = 0.75$.
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)
Function Attributes{
[object Object],[object Object],[object Object]}
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.
}
General details about the structure of ground intensity functions are given in the topic gif.
models