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  • Second-order residual analysis of spatio-temporal point processes and applications in model evaluation

    Jiancang Zhuang

    Institute of Statistical Mathematics, Tokyo, Japan.

    Summary.This paper gives first-order residual analysis for spatio-temporal point processes similar to the residual analy-

    sis developed by Baddeley et al. (2005) for spatial point process, and also proposes principles for second-order residual

    analysis based on the viewpoint of martingales. Examples are given for both first- and second-order residuals. In par-

    ticular, residual analysis can be used as a powerful tool in model improvement. Taking a spatio-temporal epidemic-type

    aftershock sequence (ETAS) model for earthquake occurrences as the baseline model, second-order residual analysis

    can be useful to identify many features of the data not implied in the baseline model, providing us with clues of how to

    formulate better models.

    1. Introduction and motivations

    Temporal, spatial and spatio-temporal point processes have been increasingly widely used in many fields, in-

    cluding epidemiology, biology, environmental sciences and geosciences. Among the associated statistical inference

    techniques, such as model specification, parameter estimation, model selection, testing goodness-of-fit and model

    evaluation, the tools used for testing goodness-of-fit and model evaluation are quite under-developed. This is one

    motivation for this article.

    Model selection procedures can be used in testing goodness-of-fit and model evaluation. Given several explicit

    models that are fitted to the same dataset, we can use some model selection criterion such as Akaike’s information

    criterion (AIC, see, e.g., Akaike, 1974) and cross validation (e.g., Stone, 1977) to find the best model among

    them. To find a model better than the current best model, one can always try several possible versions of new

  • 2 J. Zhuang

    models, fit them to the same dataset and use the model selection procedures again to see whether one of the new

    models becomes the best performing model. However, the above procedures are not always easy to implement.

    Formulating a model and fitting it to the dataset may involve heavy programming and computational tasks. Finally,

    model selection procedures only give us some quantities that indicate the overall fit of each model. It is hard to

    deduce from these quantities whether a model, even if it is not the best one, has some better properties than other

    models ranked higher by the model selection procedure. It is very helpful if the model improvement process can

    be simplified. Residual analysis developed in this article can be used for this purpose.

    To help motivate this work, we begin with a description of the dataset used in this study. The developments

    here are associated with seeking answers to a series of problems in modelling the phenomena associated with

    earthquake clusters. Although earthquake data come from the field of geophysics, similar problems also appear in

    the epidemiological modelling of contagious diseases, in biology, in ecology and in environmental sciences.

    The epicentres of earthquakes are not homogeneously distributed on the surface of the earth. In the globe,

    earthquakes mainly occur in the subduction zone between plate boundaries. Locally, earthquakes accumulate along

    active faults or in volcanic regions. Their depths range from several to 700 kilometers. Although an earthquake

    can be as big as M7, resulting in huge disasters, most earthquakes are so small that they can be detected only by

    sensitive seismometres.

    Seismicity is clustered in both space and time. The overlapping of earthquake clusters with one another and

    also with the background seismicity, complicates our analysis. For the purpose of long-term earthquake prediction,

    i.e., evaluating the risk of the occurrence of a powerful earthquake in about a 10-year time scale, a good estimate

    of the background seismicity rate is necessary. On the other hand, for short-term prediction (in a scale of an hour

    or a day), a good understanding of earthquake clusters is necessary.

    The earthquake catalogue consist of a list {(ti, xi, si)} and other associated information, where ti, xi and si

    record, respectively, the occurrence time, the epicentre location and magnitude of the ith event. Figure 1 shows

  • second-order residual analysis of point processes 3

    the shallow earthquakes (with depths less than 100 km) in the Japanese Meteorological Agency (JMA) catalogue

    used in this analysis. The time span of this catalogue is 01/01/1926 to 31/12/1999. In this article, we select

    the data in the polygon with vertices (134.0◦E, 31.9◦N), (137.9◦E, 33.0◦N), (143.1◦E, 33.2◦N), (144.9◦E, 35.2◦N),

    (147.8◦E, 41.3◦N), (137.8◦E, 44.2◦N), (137.4◦E, 40.2◦N), (135.1◦E, 38.0◦N) and (130.6◦E, 35.4◦N). The time

    period from the 10000th day after 01/01/1926 to 31/12/1991 is used as the target range in which to estimate the

    parameters through the method of maximum likelihood.

    It is easy to see from Figure 1 that earthquakes are clustered. Typically, the spatio-temporal ETAS (epidemic

    type aftershock sequence) model is used to describe the behavior of earthquake clustering (Kagan, 1991; Rathbun,

    1994; Musmeci and Vere-Jones, 1992; Ogata, 1988, 1998, 2004; Ogata et al 2003; Zhuang et al., 2002, 2004;

    Console and Murru, 2002; Console et al., 2003; Helmstetter and Sornette, 2003a, 2003b; Helmstetter et al., 2003).

    In this model, seismicity is classified into two components, the background and the cluster. Background seismicity

    is modelled as a Poisson process that is temporally stationary but not spatially homogeneous. Once an event

    occurs, no matter if it is a background event or if it is generated (triggered) by another previous event, it produces

    (triggers) its own children according to certain rules. Such a model is a continuous-type branching processes with

    immigration (background). This model can by defined completely by the conditional intensity function (hazard

    rate conditional on a given history Ft up to current time t, see Appendix A or Daley and Vere-Jones, 2003, Chapter

    7, for more details) as

    λ(t, x, s) = lim ∆t↓0, ∆x↓0,∆s↓0

    Pr{N((t, t + ∆t] × (x, x + ∆x] × (s, s + ∆s]) ≥ 1|Ft}

    ∆t ∆x∆s .

    The conditional intensity function of the ETAS model used in this paper takes the form given by Ogata (1998),

    i.e.,

    λ(t, x, s) = γ(s)

    [

    u(x) + ∑

    i: ti

  • 4 J. Zhuang

    where

    κ(s) = A exp[αs]; (2)

    γ(s) = β exp[−βs] H(s);

    f(x | s) = q − 1

    πCeαs

    (

    1 + ‖x‖2

    Ceαs

    )−q

    , (3)

    and

    g(t) = p − 1

    c

    (

    1 + t

    c

    )−p

    H(t), p > 1,

    H being the Heaviside function. In the above, the magnitude distribution γ(s) is based on the Gutenberg-Richter

    law (Gutenberg and Richter, 1956), the expected number of children κ(s) is based on Yamanaka and Shimazaki

    (1990), and the time density g(t) is based on the modified Omori formula (Omori, 1898; Utsu, 1969), all being

    empirical laws in seismicity studies. The background rate and the parameters A, α, β, c, p and C in the model

    can be estimated by an iterative algorithm (Zhuang et al., 2002, 2004, see Appendix C).

    However, there are many questions about the above model formulation. For example:

    1◦ Is the background process stationary?

    2◦ Are background events and triggered events different, for example, in magnitude distribution or in triggering

    offspring?

    3◦ Does the magnitude distribution of triggered events depend on the magnitudes of their parent events?

    4◦ Is it reasonable to apply the same exponential function eαs in both κ(s) and f(x|s)?

    In previous studies, residual analysis has been carried out by transforming the point process into a standard

    Poisson process (Ogata, 1988). Schoenberg (2004) uses the thinned residuals to analyse the goodness-of-fit of the

    ETAS model to Californian earthquake data. Baddeley et al. (2004) have made more remarkable and general

  • second-order residual analysis of point processes 5

    developments. But their residual analysis methods are all of the first order and far from being sufficient for solving

    problems where second-order properties such as clustering and inhibition are concerned. To answer these, it is

    necessary for us to generalise the concepts of residual analysis to higher orders.

    Zhuang et al. (2004) developed a stochastic reconstruction method to test the above hypotheses associated with

    earthquake clusters, using the ETAS model as the reference model. Their method is based purely on intuition

    rather than on a strict theoretical basis. As we show in this article, their method can be validated by using the

    tools of residual analysis. Providing a theoretical basis for the stochastic reconstruction method that can also be

    applied to a wider range of point-process models is another motivation of this article.

    In the ensuing sections, we first review the first-order residuals developed by Baddeley et al. (2005) and then

    propose principles for second-order residuals. The uses and powers of these resi