To see how well the parameter estimation method works, an MMHPSD process can be simulated, and then the model can be fitted to the simulated data set. The method proposed above can be used to estimate the parameters. By comparing the estimated parameters with the true parameters, one
can see how the parameter estimation method performs.
Algorithm 3.6.1 Given the initial statey1 =j, the parametersQ,λl,νlandηl,l= 1,· · · , r, and
the history data setH ⊂ {t≤0}, the following steps can be carried out to generatenevents from an MMHPSD process.
1. Seti=s= 1andti=ts= 0.
2. Useri =qj+λj+νjηjPtl<tie−η(ti−tl)as the rate and generate an inter-event timeτifrom
the exponential distribution.
3. Setts+1=ts+τi. Generate a uniform random variableU ∈(0,1).
– IfU > qj/ri, then addti+1 =ts+1into the history. Seti=i+ 1,s=s+ 1and go to
Step 2.
– IfU ≤qj/ri, then this point is a state transition point. Use(qjk/qj)1≤k≤r to generate
the next statey2. Setj=y2,s=s+ 1and go to Step 2.
Ifi=n, then stop.
47
Chapter 4
Simulation Study and an Application of
the MMHPSD
4.1
Introduction
As many geophysical processes occur in a self-exciting way, in which the events already occurred often trigger new ones, and as the underlying dynamics for these processes might be represented as being governed by a Markov chain, Chapter 3 introduced the Markov-modulated Hawkes process with stepwise decay. In this model the hidden process switches among some finite states of a contin- uous Markov chain and in each state the observed events follow a self-exciting Hawkes process with a stepwise decay rate. A parameter estimation method is also developed by using the EM algorithm for this model. Before putting this into application, the parameter estimation algorithm needs to be validated. Once we have established that the parameter estimation from the EM algorithm performs reasonably well, an exploratory data analysis of the model on earthquake sequences can be carried out to study how this model captures seismicity rate changes.
There have been many investigations on seismicity rate changes before and after the magni- tude 7.3 Landers earthquake on June 28, 1992, the Big Bear earthquake of magnitude 6.4 which occurred three hours after the Landers main shock, and the 1999 magnitude 7.1 Hector Mine earth- quake thought to have been triggered by aftershocks of the Landers earthquake (Felzer et al., 2002). A significant seismicity rate increase following the Landers earthquake has been observed as far as 600km away from the Landers source region (Hill et al., 1993, 1995). Wyss and Wiemer (2000) investigated the seismicity rate changes for Landers using declustered data, comparing the data for the 12 years before Landers earthquake to the 7 years following. They concluded that the 1992 Lan-
ders earthquake shut off the production of small earthquakes in some regions (the volumes south of the future Hector Mine rupture and north of Big Bear) while increasing the seismicity in the neighboring regions (the volume surrounding the future Hector Mine hypocenter and north of Lan- ders). They also detected that on average more small earthquakes were produced after this shock. Gomberg et al. (2001) detected an increase of seismicity rate following the Hector Mine earthquake within 250km from the main shock. Marsan (2003) observed seismicity shadows east of the Joshua Tree rupture, which occurred on April 22, 1992 with a magnitude of 6.1, following the Landers earthquake. This correlates well with the stress shadows modeled by King et al. (1994) and Mc- Closkey et al. (2003). Ogata et al. (2003) used residual analysis of the Epidemic Type Aftershock Sequence (ETAS) model on the Landers aftershock sequence. This analysis revealed relative qui- escence about 6 months after the main shock, which lasted nearly 7 years leading up to the Hector Mine earthquake. They also detected relative quiescence in the aftershock sequence of the Joshua Tree earthquake for a period leading up to the Landers rupture. Marsan and Nalbant (2005) ob- served seismicity shadows developing after a few days of the Landers earthquake in the region of the Joshua Tree earthquake, which are sometimes preceded by instances of early triggering. The MMHPSD will be applied to the sequence of data collected from Joshua Tree, Landers, Big Bear and Hector Mine to examine how this model captures the seismicity rate changes in the selected area.
In this chapter, first, the performance of the EM algorithm for the parameter estimation of the MMHPSD is evaluated. The simulation algorithm of this model was provided in Chapter 3. An arbitrary set of parameters is used to simulate 100 sequences of MMHPSD events, and the MMH- PSD is then refitted to each of the simulated sequences. The parameters are estimated using the EM algorithm for the 100 sequences and the histogram of the parameter estimates is plotted to examine how the EM algorithm works for the parameter estimation of this model. Another simulation study is conducted via a simulated ETAS sequence. The estimated intensity function of the MMHPSD is compared with the true ETAS intensity to check how well the model captures the simulated data. The estimated parameters for this simulated earthquake catalogue are used to conduct a consistency test for the parameter estimation of the MMHPSD. After the simulation studies, a case study of the model is carried out using the earthquake data around Landers. A discussion section concludes the chapter.
4.2. EVALUATION OFPARAMETERESTIMATION ALGORITHM 49