Many clustering models imply two kinds of earthquakes: spontaneous ones and those triggered by previous earthquakes. The pair-wise links from earlier to later earthquakes control the estimates of the clustering parameters. However, earthquake catalogs are limited in time, space, and magnitude, so that triggers of some cataloged earthquakes may be unknown. Thus some links are unrecognized and some triggered events appear spontaneous. Here we present a method for identifying such earthquakes and reducing the bias from missing links. We treat earthquakes probably affected by missing links as potential triggers, but we exclude them in evaluating modeled effects. We use an Epidemic-type Aftershock Sequence (ETAS) model to examine specifics. The most affected parameter is the proportion of spontaneous earthquakes. The most important missing links apparently follow earthquakes below the magnitude threshold, before the start, and outside the spatial boundaries of the catalog, in that order.
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 The obvious clustering of earthquakes in space and time implies that an earthquake increases the probability of another one nearby. This clustering is most often described by mathematical models of the pair-wise relationship of earlier and later events. By analogy with population studies, the earlier earthquake is often called a “parent”, and the later one a “child”. Some events, called “immigrants”, apparently arise spontaneously, without known parents, but immigrants may trigger children. Many models allow parents to have many children, but some allow only one parent per child. Most models specify a space- and time-dependent event rate based on the location, time, magnitude, and perhaps other properties of all previous events. Goodness-of-fit of a model is evaluated by the degree to which the child events occur when and where the calculated event rate is high.
 Many analyses evaluate goodness-of-fit by summing a log-likelihood score for some cataloged earthquakes in a specific space, time and magnitude window called the “target window”. We argue that some events outside this target window should be included in the evaluation because they may have spawned events in the target window. Thus, we introduce the “auxiliary window” which includes possible triggers of the events in the target window. The combination of the target window and the auxiliary window is called the “data window”. The triggering potential should be estimated for all earthquakes in the data window, but the model's success should be based only on quakes in the target window.
 One approach would be to assume the data window is fixed in advance, and ask what target window meets the criterion that events inside have their parents within the data window. Instead, we assume that the target window is fixed in advance, and we focus on the adequacy of the auxiliary window. How big is big enough? In general bigger is better, but really distant, early, or small events may have little influence within the target window, and practical considerations enter the decision. We provide here some tools for assessing the adequacy of auxiliary window and the value of expanding it.
 The optimum size of the auxiliary window depends on the temporal, spatial, and magnitude extent of triggering, the very properties that the model is designed to measure. Thus we have a “Catch-22”; we need to know the answer before we can choose the data needed to get the answer. For that reason we explore the behavior of model parameters for both real and simulated catalogs as a function of the auxiliary window size. We use the Epidemic-type Aftershock Sequence (ETAS) model introduced by Ogata [1988, 1998]; it allows both multiple children and multiple parents.
 Mathematically this model is expressed in the form of the conditional rate (or conditional intensity) [see, e.g., Daley and Vere-Jones, 2003]: λ(t,x,y∣Ht), defined by the expected seismicity rate at time t and location (x, y) given historical information Ht, the history of events prior to time t. If the background seismicity rate is stationary and the magnitude distribution is independent, as suggested by Zhuang et al. , the conditional rate function can be written
where η(x, y) is the background (spontaneous) seismicity rate, m0 is the minimum magnitude threshold, the parameter p indicates the temporal decay rate of primary (first generation) triggered events, the parameter shows the expected number of aftershocks per event, and the exponent a governs the effect of an earthquake's magnitude on its expected number of aftershocks. The sum covers all previous earthquakes in the catalog.
 We believe our results apply to clustering models in general [e.g., Kagan, 1991] but prefer the ETAS model for several reasons. It is widely used and its properties are widely explored. Wang et al. [2010a] showed that the ETAS model of Zhuang et al.  has some advantages over others and effectively distinguishes spontaneous from triggered events. Sornette and Werner [2005a, 2005b] studied the relationship between the lower magnitude threshold and the branching ratio, which shows the average number of directly triggered events per event [Helmstetter and Sornette, 2003], in the ETAS model. Wang et al. [2010b] studied the uncertainties in ETAS parameter estimates. The bias in parameter estimates introduced by neglecting earthquakes below a lower magnitude cutoff has been investigated by Schoenberg et al. .
 Given an earthquake catalog, parameters in the ETAS models (1) can be estimated by maximizing the log-likelihood function,
where θ = (K0, p, c, d, q, γ) is the parameter vector for (1) [Daley and Vere-Jones, 2003] and S = [x0, x1] × [y0, y1]. Here, [T0, T1] × [m1, mmax] × S represents the target space-time-magnitude window. The distinction between the target and auxiliary windows is best seen in equations (1) and (6): all events are used in (1), but only those in the target window appear in (6). Qualitatively, events in both windows contribute to the forecast earthquake rate λ(t, x, y∣Ht), but only those in the target window contribute to the log-likelihood L(θ).
 The target and auxiliary windows are subject to choice. In this study we fixed the target window and varied the size of the auxiliary window. Based on California catalog comparisons by Kagan et al.  and Wang et al. , we chose the Advanced National Seismic System (ANSS) earthquake catalog. The target space-time-magnitude window including 1122 events is:
We varied the start of the auxiliary time window in 5 year steps from 1940 to 1980; it always ended in 1980. We varied the minimum auxiliary magnitude threshold from 3.8 to 3.0 in 0.1 unit steps and the size of the space window by 2 degree steps in both longitude and latitude.
 To measure the bias introduced by this estimation procedure, including data limitations, we generated synthetic catalogs from an ETAS model with known parameters. For every operation we performed on the actual catalog, we did the same on twelve synthetic catalogs; each covered a larger area, time, and magnitude range than the actual data. Simulating over an expanded data window assured that the simulated quakes were consistent with the assumed ETAS parameters over the actual data window. The simulation parameters are shown in Table 2 We estimated ETAS parameters for the simulated catalogs as we did for the actual data, using the same auxiliary and target windows. For simplicity in the simulations, we assumed that the spontaneous seismicity rate η(x, y) was spatially uniform.
2. Analysis and Results
 To simulate the earthquake catalogs, we used an algorithm by Zhuang et al.  slightly modified by Wang et al. [2010b]. Tables 1 and 2 show the parameter estimates of the ETAS model (2.4) when different data windows were considered but with the same target window. We define the spontaneous proportion μ as the ratio of the total number of spontaneous earthquakes to the total event number. Here μ is about 0.3079 when the data window in Table 1 is used while it is about 0.1410 for the data window in Table 2.
Table 1. Estimated Parameters of the ETAS Model (3.4) for a Specific Southern California Earthquake Catalog When Data Window Is the Same as the Target Windowa
Target space-time-magnitude window [N32°, N37°] × [W114°, W122°] × [1980, 2009] × [3.8, 8.0]. Data window is the same as the target window. The spontaneous proportion μ is 0.3080.
Table 2. Estimated Parameters of the ETAS Model (3.4) for a Specific Southern California Earthquake Catalog When Data Window Is Not the Same as the Target Windowa
 First, the data window was set equal to the target window: there was no auxiliary window. Earthquakes outside the target range were ignored. Then auxiliary windows of different sizes were used to explore the influence of events that had been ignored. The changes in the ETAS parameter estimates following the increases in the auxiliary window are shown in Figure 1. The vertical axis in each plot of Figure 1 shows the ratio of the estimated parameters to those assumed in the simulations rather than the parameter values themselves. We estimated 8 parameters in total, but for brevity we show only the 4 that most influence the calculated conditional rate: the spontaneous proportione μ, the aftershock productivity exponent e a, the direct aftershock decay exponent e p and ee d6, defined as D* exp(γ* (6.0 − m0)), a measure of the triggering range of a magnitude 6.0 earthquake.
 In Figure 1, each plot shows the effect of varying one sub-space of the auxiliary window (space, time, or magnitude) on one of the important parameters. The curves composed of linear segments show the parameter values for the twelve simulated catalogs, while the dots are for the actual earthquake data. The following effects may be inferred from these diagrams. The minimum range of parameter values from different simulations, which usually occurs for the largest data window, is a measure of the random variation (uncertainty) in parameter estimates. Wang et al. [2010b] discussed the random estimation errors and showed that the repeatability of simulations is a much more reliable error estimate than the asymptotic uncertainty from the Hessian. The degree to which the several curves on a subplot cross each other is a measure of randomness in the parameter estimation procedure. If the curves do not cross, the variation between them is probably inherent in the simulations rather than in the parameter estimation procedure. When all the curves have a consistent slope, the relevant parameter value is clearly biased by lack of information about earthquakes outside the data window. When the curves converge to a constant value near 1.0 as the data window is expanded, earthquakes outside it have little effect so there is little bias from missing data. When the simulation curves differ strongly from the actual data curve, the simulations are missing an essential feature of the real data. Note that when the data window for one subspace is varied, the data window for each of the others is held fixed at its smallest value: the size of the target window. Thus, all the parameter estimates shown for the spatial window, for example, will be biased because data gets ignored from before the beginning of the target time window, and for earthquakes below the target magnitude threshold.
 There are four important points in Figure 1. (1) Except for the spontaneous proportion μ, the random variation in parameter estimates from simulation exceeds the estimation bias caused by the limited auxiliary window. Generally, the lack of spatial data outside a target window causes no significant bias in ETAS parameter estimation. The result would undoubtedly be different if the rupture of the largest earthquake exceeded the target spatial window. (2) The bias in the spontaneous proportion μ caused by time limitation is significant for real data only. The curvature of μ for real data as a function of the auxiliary time window indicates that the further the event is from the target time window, the weaker its triggering power for target events. The spontaneous proportion is biased when we neglect early earthquakes, and it converges to a stable level when the auxiliary time window is large enough. (3) In contrast with (2), bias in the spontaneous proportion μ caused by magnitude limitation is significant for both real and simulated data. The non-convergent curve of μ indicates that the weaker triggering power of smaller individual events is overcome by their increased number. The dramatic decrease in the spontaneous proportion here also shows the importance of small earthquake triggering. As we expected, more supposedly “spontaneous” earthquakes are found to be triggered when more small earthquakes are included in the auxiliary window. This finding also indicates that current earthquake locations strongly depend on past events. (4) For all parameters, estimates from real and simulation data are inconsistent. Possible reasons, in order of importance, are as follows: (i) real earthquake data are incomplete; (ii) real data have random errors but simulated data don't; (iii) magnitude distributions of real and simulation data differ in specific data windows; (iv) real spontaneous earthquakes are spatially inhomogeneous, whereas we assumed a uniform spatial distribution in our simulations; and (v) simulated earthquakes follow the ETAS model by construction, but real ones may not. In the future, effects (i) and (ii) could be tested by simulating the kinds of errors that occur in real data. Effect (iii) could be addressed by examining the magnitude distributions carefully, and (iv) could be addressed by using a spatially variable background in the simulations. Examining (v) requires a better clustering model, obviously a big job.
 There are very few events offshore to the west or in Arizona to the east of our target window, so the event number in the auxiliary window does not increase much when the longitude of the window significantly increases. Thus, the relatively small dependence of our results on the size of the auxiliary spatial window may not apply in other regions.
3. Discussion and Conclusion
 Our recommended method to reduce bias from missing links is simple; use as much auxiliary data as feasible. However, the cost of computation, the time required to assemble all data, and the work required to interpret them all increase with the size of the auxiliary catalog. Then the relevant question is whether the auxiliary window is large enough, or on the other side of the coin, whether shrinking it would involve significant bias. The exploratory techniques employed in this paper help us answer that question. What if the auxiliary window is limited by available data rather than convenience? Then, of course, the same exploratory techniques can be used to examine its sufficiency. If the auxiliary window is not sufficient, it may be appropriate to reduce the size of the target window.
 Clustering models can only recognize an earthquake as triggered when its parents are in the catalog, so it is no surprise that the estimated rate of “spontaneous” earthquakes decreases as the auxiliary window increases. Unfortunately, we saw no limit to this effect in the magnitude dimension. As we decreased the lower threshold, the spontaneous rate continued to decrease. Note that the magnitude threshold for the auxiliary and target windows needn't be the same, and in fact we used different thresholds. Our target window had a threshold of 3.8, while that of the auxiliary varied from 2.0 to 3.8. There may be a limit below which small earthquakes do not participate in triggering, but if so we have not found it. Thus, the estimated rate of spontaneous events generally depends on both the auxiliary and target magnitude thresholds, and we recommend that both be reported in the results of any clustering study.
 The authors appreciate support from the National Science Foundation through grant EAR-7032928556, as well as from the Southern California Earthquake Center (SCEC). SCEC is funded by NSF Cooperative Agreement EAR-0529922 and USGS Cooperative Agreement 07HQAG0008. The SCEC contribution for this paper is 1454. Finally, we are grateful to Kathleen J. Jackson for editing this manuscript.