# Robust Estimation for the Weibull Process Applied to Eruption Records

## Abstract

The Weibull process is a parsimoniously parameterized nonhomogeneous Poisson process with monotonic trend, which has been widely used in reliability applications. It has also been used in volcanology to model the process of eruption onsets for a volcano with waning or waxing activity, and thus produce hazard forecasts. However, particularly in the latter application, problems with missing or spurious data can strongly influence the parameter estimates, which are usually obtained by maximizing the log likelihood function, and hence the future hazard. We show how theory developed for robust estimation of a nonhomogeneous Poisson process can be implemented for the Weibull process. The flank eruptions of Mt. Etna, in Sicily, is one of the most complete and best studied records of volcanism. Nevertheless, a number of different catalogs exist. We show how these can be at least partially reconciled by robust estimation, and how the more dubious regions of the catalogs can be identified.

### Keywords

Flank eruptions M-estimator Mt. Etna Nonhomogeneous Poisson process## 1 Introduction

Robustness is a desirable feature of any model, especially where certain data exert a great deal of leverage over the fitted model and its forecasts. In point estimation, for example, the median, or a trimmed estimate of the mean, are more robust than the mean. Methods for linear regression can be found in Maronna et al. (2006) and the references therein. These methods act to down-weight data of high leverage such as outliers, replacing the maximum likelihood estimate by one with more robust properties. With point processes, the theory and practice of robustness is less clear. Rather than sampling error or miss-specification problems, data can be contaminated by the inclusion of spurious points or, more commonly in applications, the omission of points through incomplete observation. Given a conditional intensity model, the likelihood theory of point processes is well defined (Daley and Vere-Jones 1988), but the parameter estimates in the model can be very sensitive to perturbations in the data. Thus, the estimated intensity may deviate significantly from the true intensity, with undesirable effects on forecast accuracy.

Robust inference for point processes is much less developed than for regression, due to the technical difficulties in dealing with dependent data and obtaining limit theorems or distributions of the functionals involved (Martin and Yohai 1986). While robust methods for autoregressive moving average (ARMA) time series models have been investigated (Künsch 1984; Martin and Yohai, 1986, 1991), the methods depend on the regularity of the time scale, and so do not generalize to the point process context. Hence, robustness in stochastic processes has largely been confined to simulation-based approaches. For example, Puente et al. (1993) compared four stochastic rainfall-runoff models, accepting the model which gave comparable statistics (in terms of variance and autocorrelation) to the historical data as robust. However, this is model validation rather than robustness, mimicking the potentially contaminated data rather than protecting against it. Simulation methods for point processes have been used by Ng and Cook (1999) to compare the robustness of parametric and semiparametric regression models for recurrent event data, and by Ng and Cook (2000) to show that multiplicative intensity models with random effects are robust to miss-specification of the mixing distribution and the baseline intensity function. Bebbington and Harte (2003) examined the robustness of a multivariate point process with regard to data accuracy through Monte Carlo simulation and refitting of the perturbed data, and Bebbington (2007) similarly investigated the robustness of parameter estimation and hidden state path estimation in a hidden Markov model for volcanic eruptions. Lawless and Nadeau (1995) discussed some simple robust nonparametric estimates of the cumulative mean function for identically distributed processes of recurrent events based on Poisson maximum likelihood estimates, while Grillenzoni (2008) developed a robust local polynomial regression estimator of the intensity functions of point processes, making use of the weighted-average M-estimates. To the best of our knowledge, only two papers have constructed robust estimators for parametric stochastic processes. Yoshida and Hayashi (1990) dealt with the one-dimensional case of a robust M-estimator for Poisson processes with periodic intensities, while Assunção and Guttorp (1999) considered more general measures of robustness, and robust M-estimators of vector parameters, for nonhomogeneous Poisson processes. Recall that *N*(*t*), the number of events in the time window (0,*t*], is a nonhomogeneous Poisson process if it has a Poisson distribution with mean \(\mu(t) = \int_{0}^{t} \lambda(s) \,\mathrm{d}s\). In this case, *λ*(*s*) is the intensity of the point process.

In volcanology, there is considerable interest in the probabilistic forecast of future eruption onsets (Marzocchi and Bebbington 2012). This requires a parameterized intensity model, rather than the kernel method of Grillenzoni (2008). While the problem of unobserved eruptions in a renewal process has been examined by Wang and Bebbington (2012), only a minority of volcanoes can be considered in a steady state at any given time (Wadge 1982). The activity of most active volcanoes is either waxing or waning (Bebbington 2010), which can be modeled by the Weibull process (Ho 1991) in the case of complete observation. However, it is well known (Guttorp and Thompson 1991; Deligne et al. 2010) that observation probability has generally increased over the last few centuries. Even in cases, such as Mt. Etna, where the record is considered complete since approximately AD 1600, recent research has shown that some of the earlier eruption dates may be spurious. As the earlier observations in a Weibull process analysis have greater influence on the maximum likelihood estimates (Bebbington and Lai 1996), a robust approach is desirable.

The object of this paper is to apply the theory of Assunção and Guttorp (1999) to the Weibull process, and hence develop a robust estimation method for the intensity. The result is used to look at the record of flank eruptions at Mount Etna volcano (Sicily), where a number of different published catalogs exist, to see if they can be reconciled using the robust estimates. In so doing, we will also examine whether the early record is consistent with the more recent data, or if it is likely that several eruption onsets are missing. The remainder of the paper is organized as follows: The Weibull process is described next. In Sect. 3, we briefly review the robust estimation theory of Assunção and Guttorp (1999). The implementation of this theory to the Weibull process, in the form of an algorithm, is presented in Sect. 4, although the more mathematical details are deferred to the Appendix. A simulation exercise, focusing on determining a key parameter in the algorithm, follows in Sect. 5. Section 6 presents an application to a number of catalogs of flank eruptions from Mt. Etna (Sicily).

## 2 The Weibull Process

*ξ*=(

*β*,

*θ*), and includes the homogeneous Poisson process as a special (

*β*=1) case. The nomenclature follows from the fact that Eq. (1) is the hazard rate for the Weibull distribution, but apart from the time to the first point, the intervals between points are not Weibull-distributed. The process is sometimes called the power-law process. If

*β*>1, the intensity of Eq. (1) is increasing; conversely, if

*β*<1, the intensity is decreasing. The maximum likelihood estimates (MLEs) are

*t*

_{1}<

*t*

_{2}<⋯<

*t*

_{n}in an observation window (0,

*T*). Statistical tests are outlined by Bain et al. (1985). The problem of missing data has been examined using the EM algorithm (Yu et al. 2008) under the restrictive assumption that only the first

*r*−1 events are missing and the

*r*th is known.

## 3 Robust Estimator

*n*observed occurrence times 0<

*t*

_{1}<

*t*

_{2}<⋯<

*t*

_{n}≤

*T*in a Weibull process,

*N*, with intensity function given by Eq. (1), the MLEs of Eq. (2) of the parameters are obtained by solving the log likelihood equation where

*l*(

*t*;

*ξ*)=

*∂*log

*λ*(

*t*;

*ξ*)/

*∂ξ*,

*M*(

*t*;

*ξ*)=

*N*(d

*t*)−

*λ*(

*t*;

*ξ*)d

*t*, and

*A*is the sample space. When the observed data are contaminated, the MLEs can be altered significantly. The most common form of contamination in volcanology is missing data, which will result in under-estimation of the intensity when using the MLE. Another possible source of contamination may be the addition of points of another process to the underlying process. For instance, eruptions from other volcanoes may be mistaken as being eruptions of the volcano of interest, resulting in the addition of spurious points to the observed data. Measurement errors, in particular a few days delay in observing the onset of an eruption, may also be common. However, for the time scale, we are considering (in years) this last issue is insignificant, although it is discussed by Assunção and Guttorp (1999). Hence, in the presence of contamination, one would like a robust inferential procedure analogous to that in the wider statistical theory, for example, one based on M-estimates. An M-estimate for the Weibull process can be obtained by substituting for the

*l*(⋅;⋅) function some robustifying function

*ψ*(⋅;⋅), which does not have to be related to the intensity function of the process. Thus instead of solving Eq. (3), an M-estimator is defined as the solution of the equation (Andersen et al. 1993; Assunção and Guttorp 1999). A robustifying

*ψ*(⋅;⋅) function should result in a bounded influence functional, which is a heuristic description of the effect of an infinitesimal contamination on the estimate at a point, standardized by the mass of the contamination (Assunção and Guttorp 1999). In particular, to protect against both missing and spurious data, one should choose a bounded

*ψ*function. Assunção and Guttorp (1999) proposed a

*ψ*function defined as for some constant

*b*, where and The effect is to bound the likelihood term

*l*(

*t*;

*ξ*) outside the region {

*t*:

*l*

^{∗ ′}

*B*

^{−1}

*l*

^{∗}>

*b*

^{2}}, and thus down-weight the information extracted there. The function

*ψ*also remains close enough to

*l*(

*t*;

*ξ*) to validate the asymptotic properties of the estimator (Assunção and Guttorp 1999). In particular, for the Weibull process, we consider for the volcanic eruption record, the

*ψ*function acts to reduce the influence of relatively dense or sparse portions of the record. The constant

*b*controls how the unusual observations (relative to the remainder of the data) are weighted in the estimation procedure. Although the weighting in Eq. (5) is in an implicit form, we can see it is by a factor of \(b/\sqrt{l^{*\,\prime}B^{-1}l^{*}} \le1\). Hence, the smaller

*b*is, the less influence the unusual observations have on the estimates. As can be seen from Eqs. (5) and (6), as

*b*increases, the definition of unusual becomes narrower, and hence the estimates converge to a limit. This limit will be the same as the maximum likelihood estimates. The Weibull process used here considers observations unusual depending on where they fall relative to the expected progression. A long repose could cause the process

*N*(

*t*) to be below expectations, while a number of short reposes might elevate

*N*(

*t*) above the value to be expected from the remainder of the data. Hence, long reposes, especially in the later part of the record (assuming an increasing rate of onsets) would be considered unusual. Similarly, short reposes, especially in the earlier part of the record, or successive short reposes, would be natural candidates to have their influence reduced.

The choice of *b* determines both the efficiency, as in the size of the region where *ψ* differs from *l*(*t*;*ξ*), and the robustness of the estimates. A smaller *b* will decrease the efficiency, but provide more robust estimates through down weighting larger regions, increasing the protection against deviations from the model. Krasker and Welsch (1982) and Stefanski et al. (1986) discussed the choice of the constant *b* for regression and generalized linear models, but the theory that there is a minimum bound for *b* cannot be generalized to the point process content. Assunção and Guttorp (1999) suggested that *b* should not be too small—in their case, *b*≥0.1 for examples with approximately 200 points, one parameter, and a baseline function—but adopted the approach of selecting *b* empirically via a simulation exercise. We will also use simulation for this purpose, but the choice of the *b* value in the point process context, which is similar to the choice of a bandwidth in kernel estimation, deserves further study.

## 4 Estimation Algorithm

*ψ*in Eq. (5) is given implicitly, and concerns a minimization of a constant and a function which is again implicit. It is thus impossible to get a direct solution from Eq. (4), and an iterative estimation method needs to be adopted to obtain the M-estimates. The method proposed in Stefanski et al. (1986) can be borrowed to calculate the M-estimates. The basic idea is to first guess an initial value for the parameter

*ξ*, say

*ξ*

_{0}. Set

*B*

_{0}(

*ξ*

_{0})=∫

*l*

^{∗}

*l*

^{∗ ′}

*λ*d

*t*as the initial value for the

*B*matrix. Then we recursively obtain

*B*

_{i+1}(

*ξ*

_{0}) from substituting

*B*

_{i}(

*ξ*

_{0}) into Eq. (6) until {

*B*

_{i}:

*i*=0,1,…} converges to a limiting value \(\widehat{B}\). Then we set

*ξ*

_{0}and \(\widehat{B}\) as the initial values, and substitute them into Eq. (4). The Newton–Raphson method can then produce the M-estimates. The remainder of this section contains a more detailed algorithm. First, let us define and let tol

_{m},

*m*=1,2,3 be a series of tolerances.

- (1)Given the observed process 0<
*t*_{1}≤*t*_{2}≤⋯≤*t*_{n}≤*T*, and an initial value*ξ*_{0}=(*β*_{0},*θ*_{0}) for the M-estimator, let and set*i*=1. - (2)Set where
*ξ*_{−1}=*ξ*_{0}and, with*ξ*_{i−1}fixed, recursively solve*B*_{i,k+1}(*ξ*_{i−1})=*J*(*ξ*_{i−1},*B*_{ik}(*ξ*_{i−1})) until |*B*_{ik}−*B*_{i,k−1}|<tol_{1}, and set*B*_{i}(*ξ*_{i−1})=*B*_{ik}(*ξ*_{i−1}), and*j*=1. - (3)Calculate with
*B*=*B*_{i}(*ξ*_{i−1}), where*D*(⋅) stands for total differential. Then set \(\widehat{\xi}_{i,j}= \xi_{i-1}-K_{2j}^{-1}K_{1j}\).- Open image in new window
- Set
*j*=*j*+1 and with*B*=*B*_{i}(*ξ*_{i−1}). Then set \(\widehat{\xi}_{i,j}=\widehat{\xi }_{i,j-1}-K_{2j}^{-1}K_{1j}\). - Open image in new window
Repeat Step Open image in new window until \(|\widehat{\xi }_{i,m}-\widehat{\xi}_{i,m-1}|< \mbox{tol}_{2}\) and set \(\widehat{\xi }_{i}=\widehat{\xi}_{i,m}\).

- (4)
If \(|\widehat{\xi}_{i}-\widehat{\xi}_{i-1}|> \mathrm{tol}_{3}\), set

*i*=*i*+1 and go to Step (2). Otherwise, the M-estimator is \(\widehat {\xi}=\widehat{\xi}_{i}\).

*ψ*function and the implementation of the iterative steps above are provided in the Appendix. In application, we encountered the same convergence problem outlined by Stefanski et al. (1986), and hence considered a one-step approximation which, under mild regularity conditions, is asymptotically equivalent to the convergent estimate (Stefanski et al. 1986). In the next two sections, we use a one-step M-estimate by carrying out steps 1–3 without the procedures Open image in new window and Open image in new window It remains to select an initial point

*ξ*

_{0}for the iteration procedure. As MLEs are easily affected by contamination, they are not good candidates for an initial point. Instead, we will determine the initial values of the M-estimate by a procedure somewhat analogous to a trimmed mean. Consider the observed point

*t*

_{i}. If the model (1) is correct, then the expected interval to the next point is \((\theta^{\beta}+ t_{i}^{\beta})^{1/\beta}-t_{i}\). Hence, we define

*t*

_{i}and

*t*

_{i+1}wherever

*z*

_{i}is larger than an upper cutoff. On the other hand, where

*z*

_{j}is less than a lower cutoff, we delete

*t*

_{j+1}. The cutoffs are selected as a certain percentile of the empirical distribution, as in the calculation of a trimmed mean. Equation (9) thus identifies those portions of the observed process where points are anomalously far apart or close together, and the perturbation of the data to find an initial point corrects for it. The initial point is taken as the MLE of the modified process.

## 5 Simulation Exercise

Before proceeding to our application, we will briefly report on the performance of our method on data simulated from the Weibull process, with the object of determining a ‘reasonable’ value of *b*. As discussed in Sect. 2, the reporting rate of volcanic eruptions decreases the further back in time we go. Initially, we will delete a number of points from the simulated sequence, with the proportion of missing events being higher in the early years than in later years. In particular, we divide the simulated events into four groups each containing a quarter of the occurrence times. The probability that a data point is deleted is assigned in the ratio 4:2:1 for the first, second, and third quarters, and no data is deleted from the fourth quarter. The MLEs and the one-step robust estimates for both the original and the missing data sets are then compared. Our second experiment is to add a Poisson process to the simulated sequences of events, again comparing the MLEs and the robust estimates for the original data and the data with additional points. Finally, we consider the combination of missing data and the addition of spurious points.

*β*=1.432 and

*θ*=6.415, as estimated for Etna by Ho (1991). For each sequence, we first obtained the MLEs, and the one-step robust estimates, the latter with a range of

*b*values. Then we deleted

*m*data points from each sequence as described above, recalculating the maximum likelihood and one-step robust estimates for various

*m*(Fig. 1). A similar procedure was followed for the addition of

*a*points to each sequence (Fig. 2) and combinations of missing and spurious points (Fig. 3). In all three cases, the robust estimates with smaller

*b*values are on average closer to the true values. However, there is a trade-off in that too small a value of

*b*results in higher variability in the estimates, as too much data can be down weighted. The estimator with

*b*=0.3 has the greatest observed variability in Figs. 1 to 3, and hence we will use

*b*=0.5 for our application in the next section. Like all situations involving trade-offs, this choice is necessarily subjective, and the results depend on both the amount of contamination and the degree to which the data are consistent with the underlying model.

## 6 Application: Flank Eruptions at Mount Etna

Mount Etna, in Sicily, is one of the world’s most intensively studied volcanoes. It has displayed a variety of eruption styles, but has predominately produced effusive eruptions from lateral vents on the volcano’s flanks, or from the central craters. The catalog of flank eruptions is considered complete since 1600 AD (Mulargia et al. 1985). However, the quasi-continuous nature of the summit activity, and poor weather and visibility conditions common in winter, mean that the summit eruption record was not complete as late as 1983 (Andronico and Lodato 2005). In contrast to the summit activity, flank eruptions occur at intervals of years, and are thus amenable to a point process analysis. Moreover, the mechanisms of the flank and summit eruptions appear to be different (Mulargia et al. 1992), and the two appear not to be related (Branca and Del Carlo 2005), justifying a separate analysis of flank eruptions, which are also more destructive (Salvi et al. 2006).

*t*=0 in (1) identified by some sort of change point procedure. The notable hiatus in the mid-eighteenth century (Fig. 4) usually features in such analysis. For example, Ho (1992) found departures from the trend (1) in 1702 and 1759, using a statistical control chart based on the Weibull process. Instead, we will here consider the mid-eighteenth century quiescence as possible catalog contamination, and use 1600 as the initial point for the current eruptive regime. Starting from the year 1600, we first fit the Weibull process to each of the five flank eruption catalogs and examine the residual process (Ogata 1988) \(\tau_{i} = \int_{0}^{t_{i}} \hat{\lambda}(t) \,\mathrm{d}t\) (Fig. 5). The residual processes for Catalogs 1, 2 and 4 are stationary with unit rate according to the Kolmogorov–Smirnov statistics. For Catalogs 3 and 5, the 95 % confidence limits are exceeded in the 1970s due to the fact (Fig. 4) that there are more events identified in the two catalogs during that time period than in the other three catalogs.

*b*=0.5) of the parameters in (1) are given in Table 1. The MLEs differ considerably. While Catalog 2, with few pre-1700 events has the largest \(\widehat{\beta}\), the reverse occurs for Catalog 1, with its relatively fewer later events. The estimates (2) then determine \(\widehat{\theta}\) from \(\widehat{\beta}\) and the observed number of events, resulting in the large values of \(\widehat{\theta}\) in Catalogs 2, 3 and 4 from high values of \(\widehat{\beta}\) and, in the case of Catalog 3, a larger number of events. The discrepancies are much reduced in the one-step M-estimates: The shape parameter

*β*is moved towards 1.465 or thereabouts, and the scale parameter

*θ*toward a value around 25 for all the catalogs. The estimated errors in the MLEs are calculated from the Hessian of the log likelihood function, while those of the robust estimates are obtained via 200 simulations of a parametric bootstrap. The latter is performed by simulating a sequence of points from the robust estimate of the intensity \(\tilde{\lambda}(t)\), and then accepting points with probability \(\min\{\bar{\lambda}(t)/\tilde{\lambda }(t),1\}\), using the kernel density estimate where

*h*is the bandwidth and

*α*being between 1 and 1.5 (Vere-Jones 1992). The boundary effects are dealt with by weighting the terms in Eq. (10) according to how much of the kernel is within the time interval (Diggle 1985). Finally, potential spurious points are added by simulation from the intensity

Maximum likelihood and one-step M-estimates (M) for the parameters of the Weibull processes fitted to the five catalogs, Catalog 1: Seibert and Simkin (2002); Catalog 2: Mulargia et al. (1985) + Sandri et al. (2005); Catalog 3: Branca and Del Carlo (2004); Catalog 4: Tanguy et al. (2007); Catalog 5: Behncke et al. (2005)

Catalog 1 | Catalog 2 | Catalog 3 | Catalog 4 | Catalog 5 | ||
---|---|---|---|---|---|---|

MLE | \(\widehat{\beta}\) | 1.35 | 1.67 | 1.57 | 1.51 | 1.41 |

s.e. | 0.18 | 0.22 | 0.19 | 0.20 | 0.18 | |

\(\widehat{\theta}\) | 20.29 | 35.76 | 27.41 | 27.99 | 21.80 | |

s.e. | 8.31 | 11.77 | 9.23 | 10.33 | 8.35 | |

M | \(\widehat{\beta}\) | 1.37 | 1.51 | 1.53 | 1.47 | 1.46 |

s.e. | 0.26 | 0.40 | 0.34 | 0.35 | 0.31 | |

\(\widehat{\theta}\) | 21.02 | 26.92 | 25.46 | 25.97 | 24.23 | |

s.e. | 12.69 | 21.25 | 16.82 | 18.24 | 15.68 |

*T*] into subintervals

*I*

_{i}with equal number of expected events

*N*

_{E}(

*I*

_{i})=5, using the robust estimate \(\widehat{\xi}\), and count the number of observed events in each interval

*N*(

*I*

_{i}). The standardized residual, defined as

## 7 Discussion

We have applied Assunção and Guttorp (1999)’s robust estimator for a nonhomogeneous Poisson process to the Weibull process, in order to investigate volcanic eruption records of Mt. Etna. The robust estimator in this work may not be the best possible, as extensive analysis is needed to find the best robustifying function according to desired criteria. Volcanic records suffer from missing and spurious data. Missing data result in a lower observed intensity in the neighborhood from where the data are missing. Unlike the usual missing data problem in statistics, a missing datum in a point process not only results in one less observation, but also a larger interval between points. Robust methods can only detect this when the missing data lead to an interval too inconsistent with that expected from the bulk of the data. Such large intervals are comparable to outliers in general statistical terms, which explains the rationale for a robust method, which will help reduce the influence of such outliers on the parameter estimates. As we have seen with the examples of Catalogs 3 and 5 above, the method is also able to detect and ameliorate the effects of possibly spurious data, which result in anomalously short intervals. As demonstrated via simulation, the variance of the robust estimator is generally larger than that of the MLE. This is inevitable and is the reason why the MLE is used to estimate parameters when there is no reason to suspect contamination of the data. However, when the data are heavily contaminated, there is a trade-off in accuracy for the greater (spurious) precision, and the less efficient robust estimator should be favored because the MLEs are sensitive to the influence of even one contaminated observation. In theory, the asymptotic variance of the robust estimator can be calculated from Theorem 2 in Assunção and Guttorp (1999). Given the complexity of the result and the fact that our application is far from the asymptotic regime, we have confined our treatment to that in the simulation section.

## Notes

### Acknowledgements

This work was carried out while the first author was supported as a Massey University postdoctoral fellow by the Natural Hazards Research Platform and the Earthquake Commission. We thank an anonymous reviewer and the editor for helpful suggestions on the original manuscript.

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