International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

A Bayesian change point model to detect changes in event occurrence rates, with application to induced.. Gupta, Abhineet; Baker, Jack W. Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015A Bayesian Change Point Model To Detect Changes In EventOccurrence Rates, With Application To Induced SeismicityAbhineet GuptaGraduate Student, Dept. of Civil and Environmental Engineering, Stanford University,Stanford, CA 94305, USAJack W. BakerAssociate Professor, Dept. of Civil and Environmental Engineering, Stanford University,Stanford, CA 94305, USAABSTRACT: A significant increase in earthquake occurrence rates has been observed in recent years inparts of Central and Eastern US. There is a possibility that this increased seismicity is anthropogenicand is referred to as induced seismicity. In this paper, a Bayesian change point model is implementedto evaluate whether temporal features of observed earthquakes support the hypothesis that a change inseismicity rates has occurred. This model is then used to estimate when the change is likely to haveoccurred. The magnitude of change is also quantified by estimating the distributions of seismicity ratesbefore and after the change. These calculations are validated using a simulated data set with a knownchange point and event occurrence rates; and then applied to earthquake occurrence data for a site inOklahoma.1. INTRODUCTIONThe level of seismicity in the Central and EasternUS (CEUS) has increased markedly since approxi-mately 2009 (Ellsworth, 2013). For Oklahoma, thecumulative number of earthquakes with magnitude≥ 3 since 1974 is shown in Fig. 1. The figure showsa marked increase in seismicity rate in Oklahomaafter 2008. The magnitude 3 threshold was chosensince Coppersmith et al. (2012) described that thereis catalog completeness in CEUS at this magnitudelevel. (All earthquake data from Oklahoma usedin this paper has been obtained from catalogs de-veloped and maintained by Oklahoma GeologicalSurvey’s Leonard Geophysical Observatory.)There is a possibility that this increased seismic-ity is a result of underground wastewater injection(e.g., Ellsworth 2013; Keranen et al. 2013, 2014).Seismicity generated as a result of human activitiesis referred to as induced or triggered seismicity. In-duced seismicity in a region can alter its seismichazard and risk. One of the important componentsin calculating seismic hazard through Probabilis-tic Seismic Hazard Analysis (PSHA) is the activity1970 1980 1990 2000 2010 20200200400600800DateNumber of earthquakesFigure 1: Cumulative number of earthquakes in Okla-homa with magnitude ≥ 3 from 1974 to Sept 2014rate at a seismic source. For a given seismic source,we thus need to establish whether earthquake oc-currence data indicates that a change in rates hasoccurred, or whether the earthquake activity is con-sistent with normal features of a process having aconstant rate.112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015In this study, we describe a Bayesian change-point model that uses event occurrence data to indi-cate whether a change in event rates occurred. Weassume that the event occurrences belong to a Pois-son distribution. If a change in rates is detected,we also obtain the probability of change occurringat any given time, and the probability distributionsof rates before and after the change. Along with thedescription of the model, an algorithmic implemen-tation is provided. The model is validated throughits application to a simulated data set with knownproperties.The Bayesian model is then implemented on aregion in Oklahoma to evaluate whether change inrates has occurred in this region. Finally, the dis-tributions for seismicity rates before and after thechange are calculated. This detection could be usedto inform seismic hazard analysis and can serve asa decision support tool for operations that may belinked to induced seismicity.2. BAYESIAN MODEL FOR CHANGE POINT DE-TECTIONChange point models are used to detect changesin occurrence rates of events. A Bayesian modelfor change point detection is implemented here toquantify changes in seismicity rates. The unknownsin our problem are the date of change, event ratebefore the change and rate after the change. Un-like parameter estimation techniques where a singlevalue of a parameter is obtained, using a Bayesianmodel yields a probability distribution for the pa-rameters. The probability distribution for rates be-fore and after the change is helpful in seismic haz-ard calculation as it allows a more rigorous account-ing of uncertainties.For our model, it is assumed that occurrence ofearthquakes is a Poisson process. Declustering anearthquake catalog (i.e. removing dependent af-tershocks and foreshocks and only preserving in-dependent mainshocks) accomplishes the require-ments of this assumption as described by Gardnerand Knopoff (1974) and van Stiphout et al. (2012).Furthermore, it is assumed that the seismicity ratesbefore and after the change, as well as the date ofchange are mutually independent. Using Bayesiananalysis to detect change point in a Poisson processhas been described by Raftery and Akman (1986).2.1. Marginal posterior distributions of time ofchange and event ratesWe define data in an observation period [0, T] asa vector of inter-event times (i.e. the time betweensuccessive events) t . The first event occurs at time0 and the n+1th event occurs at time T .t = {t1, t2, . . . , tn} s.t. ∑iti = T (1)The variables τ , λ1, and λ2 define the date ofchange, event occurrence rate before the change,and occurrence rate after the change, respectively.Since the events belong to a Poisson process, theinter-event times are exponentially distributed.f Tλ (s)(t) = λ (s)e−λ (s) t (2)such thatλ (s) ={λ1, 0≤ s≤ τλ2, τ < s≤ T(3)A Gamma distribution with parameters k j and θ jis used to define the conjugate prior of λ j.pi(λ j) ∝ λk j−1j e−λ j/θ j (4)The prior distribution for the time of change, τ isassumed to be uniformly distributed over the obser-vation period. This implies that change is equallylikely to occur at any time during the observationperiod.pi(τ) = 1T, 0≤ τ ≤ T (5)The likelihood function for the unknown param-eters can be written asL (τ,λ1,λ2 | t)=tτ∏t=t1λ1e−λ1 ttn∏t=tτ+1λ2e−λ2 t= λN(τ)1 e−λ1 τ .λN(T )−N(τ)2 e−λ2 (T−τ) (6)where N(t) represents the number of events be-tween [0, t]. Using the fact that all parameters are212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015mutually independent, the posterior density is cal-culated aspi(τ,λ1,λ2 | t)∝L (τ,λ1,λ2 | t)pi(λ1)pi(λ2)pi(τ) (7)Then the marginal distributions for each of τ , λ1and λ2 can be obtained by integrating the poste-rior density over the remaining two variables. Themarginal posterior distribution of τ is calculated aspi(τ | t) ∝∫ ∞−∞∫ ∞0pi(λ1,λ2,τ | t)dλ1 dλ2= pi(τ).∫ ∞0λN(τ)+k1−11 e−λ1(τ+ 1θ1)dλ1.∫ ∞0λN(T )−N(τ)+k2−12.e−λ2(T−τ+ 1θ2)dλ2=1T.Γ(r1(τ))Γ(r2(τ))S1(τ)r1(τ)S2(τ)r2(τ)(8)wherer1(τ) = N(τ)+ k1S1(τ) = τ + 1θ1r2(τ) = N(T )−N(τ)+ k2S2(τ) = T − τ + 1θ2(9)The marginal posterior distribution of λ1 is cal-culated as shown below. A closed form solutionfor this double integration does not exist. Hence, toevaluate the probability distribution, the time rangeis discretized on a per day basis, and summed up toapproximate the marginal distribution.pi(λ1 | t) ∝∫ T0∫ ∞0pi(λ1,λ2,τ | t)dλ2 dτ=∫ T0∫ ∞0λ r2(τ)−12 e−λ2S2(τ) dλ2.pi(τ)λ r1(τ)−11 e−λ1S1(τ) dτ≈T∑τ=0[1T.λ r1(τ)−11 e−λ1S1(τ). Γ(r2(τ))S2(τ)r2(τ)](10)Similarly, the marginal posterior distribution ofλ2 can be calculated aspi(λ2 | t) ∝T∑τ=0[1T.λ r2(τ)−12 e−λ2 S2(τ). Γ(r1(τ))S1(τ)r1(τ)](11)Another quantity of interest is the ratio of pre-change event rate to post-change event rate (or vice-versa), defined as β = λ1/λ2. Lindley (1965) de-scribes the following function of β conditional onτ to follow the F-distribution with d.o.f. p1 and p2,where pk = 2rk(τ).S1(τ)r2(τ)S2(τ)r1(τ)β ∼ Fp1,p2 (12)The above equation can be used to compute theprobability distribution of β conditional on τ .p(β | τ, t) = 1B(r1(τ),r2(τ))(r1(τ)r2(τ))r1(τ).(S2(τ)+S1(τ)βS2(τ))−(r1(τ)+r2(τ)).(S1(τ)r2(τ)S2(τ)r1(τ)β)r1(τ)−1.S1(τ)r2(τ)S2(τ)r1(τ)(13)The marginal distribution of β is then calculatedasp(β | t) =∫ T0p(β | τ, t)pi(τ | t)dτ∝∫ T0pi(τ)β r1(τ)−1.(S2(τ)+S1(τ)β )−(r1(τ)+r2(τ)) dτ≈T∑τ=01T.β r1(τ)−1.(S2(τ)+S1(τ)β )−(r1(τ)+r2(τ)) (14)It is noted after the evaluation of eq. 14 that thereis some difference in the equation terms comparedto those obtained by Raftery and Akman (1986).We are able to verify and validate our equation withthe same data as Raftery and Akman (1986), andbelieve this formulation to be correct.312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20152.2. Bayes factorThe change-point model described in the previoussection does not indicate whether the data supportsthe presence of a change in the observed time range.When the model is applied to observed data, itassumes that there is a change and calculates theprobability of change on any given date. To indi-cate whether data favors a change-point model, theBayes factor is used.The Bayes factor is a Bayesian alternative to hy-pothesis testing. In this case, it is defined as theratio of the likelihood function for a constant ratemodel (H0) to that for a change model (H1). Hence,it is used to compare which model better describesthe data.B01(t) =L (H0 | t)L (H1 | t)(15)The constant rate model is characterized by a sin-gle unknown parameter - the rate of occurrence, λ0.A gamma distribution with parameters k0 and θ0,similar to eq. 4, is used as its prior distribution.ThenL (H0 | t) =∫ ∞0L (λ0 | t)pi(λ0)dλ0=Γ(N(T )+ k0)Γ(k0)(1θ0)k0.(1θ0+T)−(N(T )+k0)(16)andL (H1 | t) =∫ T0∫ ∞0∫ ∞0L (τ,λ1,λ2 | t).pi(λ1)pi(λ2)pi(τ)dλ1 dλ2 dτ≈(1θ1)k1( 1θ2)k2 1Γ(k1)Γ(k2).T∑τ=0[pi(τ)Γ(r1(τ))Γ(r2(τ))S1(τ)r1(τ)S2(τ)r2(τ)](17)If the value of parameters for gamma conjugatepriors are k j = 0.5 and θ j→ ∞ f or j = 0,1,2, thenit is shown by Raftery and Akman (1986) that theequation for Bayes factor can be simplified toB01(t) = 4√pi T−nΓ(n+1/2).[T∑τ=0Γ(r1(τ))Γ(r2(τ)). S1(τ)−r1(τ)S2(τ)−r2(τ)]−1(18)Smaller values of Bayes factor (less than 1) im-ply that the change model is more strongly sup-ported by the data. Bayes factor values of less than0.01 (or greater than 100) are typically used to indi-cate decisive preference for one model or the other(Kass and Raftery, 1995). In this study, a Bayes fac-tor of smaller than 1×10−3 is used to indicate thatdata favors the change model compared to the con-stant rate model (i.e if the Bayes factor for an ob-served period is calculated to be less than 1×10−3,it indicates that a change has occurred within thisobserved period).3. ALGORITHMIC IMPLEMENTATION OF THEMODELSince all the unknown variables in the model de-scribed above (τ , λ1, and λ2) are continuous andclosed form solutions are not possible for marginaldistributions of λ1, and λ2, some approximationsare required to implement the algorithm for thechange-point model. Additionally, overflow condi-tions are encountered in the algorithm, for instancewhen computing Γ(x) function for big values. Theapproximations and inputs required for the imple-mentation of our algorithm are described below.Later, the verification and validation processes forthe algorithm are described.3.1. Approximations and prior parameter inputsfor the algorithmUnder the first approximation, the continuous vari-ables are discretized. Since it is not computation-ally possible to calculate the marginal distributionof τ over a continuous range, the algorithm is im-plemented on a per-day basis. Hence, probability ofchange happening at time τ is calculated for everyday of the observation period. The proportionalityin eq. 8 is converted to a probability by dividingthe values for each day by the sum of values for all412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015days in the observation period, such that the sum ofprobabilities equals one.To compute the marginal distributions for λ1, λ2,and β , the integral over τ is discretized to a sum-mation by again dividing the observation periodon a per-day basis. Then the marginal distribu-tions are approximated as shown in the last step ofeqs. 10, 11 and 14. The results are calculated atdiscrete values of λ1, λ2, and β over a certain range[Xlow,Xhigh], and the proportionality is converted toprobability again by dividing each result by the sumof results over the complete range. It is ensured thatthe selected range [Xlow,Xhigh] for each of the vari-ables is wide enough such that probabilities at theextreme points are essentially zero.Secondly, to address the overflow problem, theresults are calculated in the log domain and thenconverted back to the original domain. One of thereasons for the overflow problem is computation ofΓ(x) which grows very rapidly with increasing val-ues of x. In the log domain, algorithms are availableto calculate log[Γ(x)] directly without first needingto calculate Γ(x). Another overflow problem is en-countered when the results obtained in log domainat each discrete value are converted back to origi-nal space and summed together. Their sum couldbe very large and outside the floating point rangeof the computer. Hence, to calculate probability ateach discrete value, the log results are scaled suchthat the sum is within the range of computation.As an example of the implementation of approx-imations and modifications described above, the al-gorithm for calculating the probability of changehappening on any given day is written as:(a) Discretize the observation period on a per-daybasis, τ = {τi}Ti=1.(b) At each τi, calculateprobi =− log(T )+ log[Γ(r1(τi))]+ log[Γ(r2(τi))]− r1(τi) log[S1(τi)]− r2(τi) log[S2(τi)].(c) Find a scale such that∑ieprobi−scale ≤ realMaxwhile preserving as many smallest values aspossible. Here realMax is the largest finitefloating-point number in IEEE double preci-sion. Update probi = probi− scale.(d) Normalize to obtain probability,probi =eprobi∑i eprobi.The final step in the implementation of the algo-rithm is selecting the values for the prior parameters(k j and θ j) for the gamma distribution. The valuesare selected as the same ones that are used in thedevelopment of the formulation for Bayes factor ineq. 18 (i.e. k j = 0.5 and θ j → ∞ f or j = 0,1,2).Brief analysis was performed to assess the sensitiv-ity of the prior parameter values on results, and itwas found that marginal distributions of τ , λ1, λ2,and β did not vary significantly with different priorparameters.3.2. Verification and validationThe algorithm developed above for the change-point model is verified by comparing the resultswith those presented in Raftery and Akman (1986).(The data used for comparison is the coal-miningdisasters data described by Jarrett (1979) and ex-tended per Raftery and Akman (1986)) . The re-sults are in very good agreement. The algorithmsare further validated through application on simu-lated data.To obtain the simulated data, inter-event timesare randomly generated from an exponential dis-tribution with different rates and a known changepoint. The change-point model is then implementedon this data and the results for τ , λ1, and λ2 arecompared with the inputs. Here, we describe onesimulation case for validation.For this validation case, 100 inter-event durationare generated at a rate of 0.005 events/day and an-other 50 are generated at a rate of 0.015 events/day.A plot of cumulative number of events is shownin Fig. 2. The first event is assumed to occur on2000-01-01. Since the event rate reduces startingfrom the 101st event, a change occurs between the100th and the 101st event. Without any additional512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015information, we cannot determine a specific date ofchange. Hence, it is assumed that change occurredthe day before the 101st event which is known fromthe data as 2052-06-01.2000 2025 2050050100150DateNumber of events  Known dateof hangeFigure 2: Cumulative number of events for simulateddata with known change on 2052-06-02The change-point model yields a Bayes factor of6.3×10−7 for this data, implying that data stronglysupports the change-point model. The probabilityof change happening on a given date is plotted onFig. 3. A 95% credible interval for dates of changeis determined between 2050-03-07 and 2054-05-28. Hence, the actual date of change is within the95% credible interval. Additionally, the date ofchange with highest associated probability is foundto be 2052-06-01. This date exactly matches ourinput date of change. Thus, the Bayesian changepoint model correctly estimates the date of changeon simulated data.The rates of event occurrence are also estimatedusing the model. The results of probability distribu-tions of rates before and after the change are shownin Fig. 4. The modes of posterior distributions forrates are 0.0054 and 0.0157 respectively, which arein good agreement with the input rates.4. CHANGE POINT MODEL IMPLEMENTATIONIN OKLAHOMAThe change-point model described in previous sec-tions, is used to quantify the change in seismicityrates in a local region of Oklahoma. The local re-gion that is used in this paper is a 25 km radius2050 2052 205490100110120Number of events01234x 10−3DateProbability of change  Known dateof change95% cr dibleintervalNumber ofeventsFigure 3: Probability of change on simulated data rep-resented by the thick solid line, with its 95% credibleinterval between 2050-03-07 and 2054-05-280 0.01 0.02 0.03 0.0400.λ (events/day)Probability of rate  Pre−changeratePost−changerateInput ratesFigure 4: Probability of pre-change and post-changeratesarea centered around the location of "Well 1" de-scribed in Keranen et al. (2013). The center of thisregion is located 10 km north-west of Prague, OKat (35.56◦N, 96.75◦W).The cumulative number of events within the cho-sen region from the declustered catalog are shownwith the dashed line in Fig. 5. No events are ob-served between 1974 and 2009 in this region. ABayes factor of 9.6× 10−28 is calculated for thisdata, implying that the data almost certainly sup-ports the change-point model over a constant ratemodel. The probability of change occurring on anygiven day is then calculated and is shown in Fig. 5.612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015The highest probability for change is observed on2009-06-13. A 95% credible interval is calculatedto be between 2008-12-11 and 2010-02-24. Thisimplies that there is a high chance that change inseismicity rates in this local region occurs betweenlate 2008 and early 2010.2010 2012 2015010203040DateNumber of earthquakes00.0020.0040.0060.0080.01Probability of change on date  95% credibleintervalCumulativenumber ofearthquakesFigure 5: Probability of change on any given day forthe local region, represented by the solid lineThe probabilities of seismicity rates before andafter the change are also calculated and are shownin Fig. 6. Since the pre-change rate is governed bythe period of no seismicity between 1974 and 2009,its distribution is left-tailed. The modes for pre-change and post-change rates are 6.6× 10−5 and1.8×10−2, respectively.10−8 10−6 10−4 10−2 10000. RateProbability of rate  Pre−change ratePost−change rateFigure 6: Probability of activity rates for earthquakesin the local regionThe ratio of pre-change to post-change rate, β isshown in Fig. 7. The highest probability is calcu-lated at a ratio of 3.4×10−3. This implies that thepost-change rate is about 300 times greater than thepre-change rate.10−10 10−5 10000.0050.010.0150.020.025β = λ1/λ2ProbabilityFigure 7: Probability of ratio of pre-change to post-change rates in the local region5. CONCLUSIONSAn algorithm for change-point detection in eventrates using Bayesian statistics was developed in thispaper. The Bayes factor was defined in the model todetermine whether event occurrence data supportedthat a change had occurred in event rates. If thedata supported a change model, the probability ofchange happening on a given day could be com-puted. The model could also be used to estimate theprobability distributions of event rates before andafter the change. The model was validated throughits application on a simulated data set with knownproperties.After validating the model on simulated data, itwas implemented on a region in Oklahoma. Themodel detected that a change in seismicity rates oc-curred sometime between late 2008 and mid 2010.This period agreed well with the time of changeexpected through a visual inspection of the data.The post-change rate was estimated to be approx-imately 300 times the pre-change rate. This highincrease in activity rates can substantially increasethe seismic hazard. Assuming that this local region712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015is a seismic source and that the magnitude distri-bution is held constant from pre-change to post-change, this would imply an increase of about 300times in rate of exceeding any ground motion atthis site from this source. However, more researchneeds to be carried out before calculating hazardfrom this increased seismicity. The magnitude dis-tribution of induced earthquakes could be differentthan tectonic earthquakes due to difference in b-values (in a Gutenberg-Richter relation) or due toan upper bound on earthquake magnitudes. Addi-tionally, there could be a difference in ground mo-tions from induced earthquakes compared to natu-ral earthquakes (Hough, 2014).The change point model does not make any asso-ciation with the causes of rate change. Hence, afterdetermining that a change has occurred, it should belinked with a physical phenomenon that might havecaused this change. In the case of induced seismic-ity in Oklahoma, some of these physical phenom-ena could be change in number of injection wells,cumulative injection volume or basement pore pres-sure. The information about dates of change pro-vided by the change-point model can be used toidentify the causes of induced seismicity. This iden-tification can assist in decision-making for opera-tions potentially linked with induced seismicity andcan thus be used as a tool for risk mitigation.Although the change-point model described herewas applied for the case of induced seismicity, thisis a versatile model with other potential applica-tions. Some of the other applications of this modelcould be to detect change in storm occurrence ratesto inform decisions about climate change, or todetect change in population migration rates to in-form decisions about developing urban infrastruc-ture. Thus the change-point model described in thispaper can serve as a decision-support tool for a va-riety of applications involving occurrences of po-tentially non-stationary events.6. REFERENCESCoppersmith, K. J., Salomone, L. A., Fuller, C. W.,Glaser, L. L., Hanson, K. L., Hartleb, R. D., Lettis,W. R., Lindvall, S. C., McDuffie, S. M., McGuire,R. K., and others (2012). “Central and Eastern UnitedStates (CEUS) seismic source characterization (SSC)for nuclear facilities project.” Report no., ElectricPower Research Institute (EPRI).Ellsworth, W. L. (2013). “Injection-induced earth-quakes.” Science, 341(6142), 1225942.Gardner, J. K. and Knopoff, L. (1974). “Is the se-quence of earthquakes in southern California, with af-tershocks removed, Poissonian?.” Bulletin of the Seis-mological Society of America, 64(5), 1363–1367.Hough, S. E. (2014). “Shaking from injection-inducedearthquakes in the Central and Eastern United States.”Bulletin of the Seismological Society of America,104(5), 2619–2626.Jarrett, R. G. (1979). “A note on the intervals betweencoal-mining disasters.” Biometrika, 66(1), 191–193.Kass, R. E. and Raftery, A. E. (1995). “Bayes fac-tors.” Journal of the American Statistical Association,90(430), 773–795.Keranen, K. M., Savage, H. M., Abers, G. A., andCochran, E. S. (2013). “Potentially induced earth-quakes in Oklahoma, USA: Links between wastew-ater injection and the 2011 Mw 5.7 earthquake se-quence.” Geology, 41(6), 699–702.Keranen, K. M., Weingarten, M., Abers, G. A., Bekins,B. A., and Ge, S. (2014). “Sharp increase in centralOklahoma seismicity since 2008 induced by massivewastewater injection.” Science, 345(6195), 448–451.Lindley, D. V. (1965). Introduction to probability andstatistics from Bayesian viewpoint. part 2 inference,Vol. 2. Cambridge University Press Archive.Raftery, A. and Akman, V. (1986). “Bayesian analysis ofa Poisson process with a change-point..” Biometrika,73(1), 85–89.van Stiphout, T., Zhuang, J., and Marsan, D. (2012).“Seismicity declustering.” Community Online Re-source for Statistical Seismicity Analysis, 10.8


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