Recently a likelihood-based methodology has been developed by the Collaboratory for the Study of Earthquake Predictability (CSEP) with a view to testing and ranking seismicity models. We analyze this approach from the standpoint of possible applications to hazard analysis. We arrive at the conclusion that model testing can be made more efficient by focusing on some integral characteristics of the seismicity distribution. This is achieved either in the likelihood framework but with economical and physically reasonable coarsening of the phase space or by choosing a suitable measure of closeness between empirical and model seismicity rate in this space.
The assessment of seismic hazard and risk are based on maps of long-term rate of damaging seismic events. There is a wide diversity of approaches to the making of such maps, which differ in the use of historical and low magnitude seismicity, seismotectonic regionalization, the Gutenberg-Richter law, smoothing techniques, and so on (see, e.g., Molchan et al., 1997;Giardini, 1999 and references therein). For this reason, the initiative of the U.S. branch of the Collaboratory for the Study of Earthquake Predictability (CSEP) is important; its purpose is to develop a statistical methodology for objective testing and ranking of seismicity models (Field, 2007). That program has been implemented as the Regional Earthquake Likelihood Models (RELM) project for California (Schorlemmer et al., 2010) and now the methodology is in a stage of active analysis and development (see e.g. Lombardi and Marzocchi, 2010;Werner et al., 2010;Rhoades et al., 2011;Zechar et al.,2010).
Below we examine the RELM methodology from the point of view of possible applications to hazard analysis, i.e., to the testing of long-term seismicity maps. We provide a brief description of basic elements of the methodology with a parallel discussion of its strong and weak points.
A seismicity map describes the mean rate of target events j This contradiction may cause appreciable difficulties in the testing of non-trivial timedependent forecasts (see more in Werner and Sornette (2008), Lombardi and Marzocchi (2010)).
In testing a long-term seismicity model, the hypothesis is reasonable for main shocks only, i.e., the catalog needs to be declustered. This operation is not unique.
Consequently, the statistical analysis should be weakly sensitive to the independence property of 0 H } { j as much as possible, focusing on important parameters of the ) ( measure.
Nearly all goodness-of-fit tests of model } { j with data } { j suggested by the RELM working group are based on the likelihood approach. The log-probability of
where is the rate of target events in G , ) ( g p and ) ( M q are normalized distributions of the events over space and over magnitude, respectively.
Taking the case of the trivial partition as represented by the single element , we arrive at the statistic
The -statistic in this case labeled as is then given by the relation (4); the distributions of and depend on the single parameter
is sufficient statistic for the analysis of .
If the partition deals with magnitude only then
and thestatistic , , is sufficient for the analysis of the parameters
. In practice we use this partition to analyse the frequency-magnitude law.
Finally, the space partition is based on
; the correspondingstatistic, , is sufficient for the analysis of the parameters
The respective -statistic for the conditional distribution of
The statistics and enable us to perform a separate analysis of the parameters and (see, e.g., Molchan and Podgaetskaya, 1973;Molchan et al., 1997;Werner et al., 2010). The necessity for the separate analysis is caused by many things: the small amount of data , catalog declustering, standardization of catalog magnitude, etc. (see, e.g., Kagan, 2010, andWerner et.al, 2011).
The significance of the L-test.
The Monte Carlo method can be used to find the distributions of all type statistics under . In the case the distribution ( 6) corresponds to the model of independent trials with n outcomes and probabilities .
The distribution of can be used to find the observed significance level for an observed -value, :
In the case of statistic , both small and large values are suspect, so a two-sided test is used:
This is a standard scheme for testing any hypothesis. The key point for applications in this scheme is the choice of the test statistic.
To answer this question, let us discuss some peculiarities of the RELM experiment:
-in general, the number of tested models ) (
for the same territory can be arbitrary. are not used. To be specific if the bins are small we could assume that the } { j j are equal within some space structures;
In other words, in the RELM experiment we have to deal with the statistical problem of a large number of degrees of freedom because usually . The advantage of the likelihood method in such conditions is not obvious. D.R. Cox and D.V.Hinkley (1974) in their book “Theoretical Statistics” tried to formulate some general principles underlying the theory of statistical inference. One of the obstacles that impede the use of likelihood theory is worded as follows: “in considering problems with many parameters one generally focuses on a small number of components, but to do this one needs principles that are outside the “pure” likelihood theory” (Section 2.4.VIII). In the framework of the likelihood approach we possess a good enough tool to focus on the essentials in the rate measure. The tool in question is the partition of the phase space. For purposes of seismic risk analysis the physica
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