Joural of Evirometal Protectio, 202, 3, 336-345 http://dx.doi.org/0.4236/jep.202.3052 Published Olie October 202 (http://www.scirp.org/joural/jep) No-Homogeeous Poisso Processes Applied to Cout Data: A Bayesia Approach Cosiderig Differet Prior Distributios Lorea Vicii, Luiz K. Hotta 2*, Jorge A. Achcar 3 Departmet of Statistics, Federal Uiversity of Sata Maria, Sata Maria, Brazil; 2 Departmet of Statistics, Uiversity of Campias, Campias, Brazil; 3 Departmet of Social Medicie, Medical School, Uiversity of São Paulo, Ribeirão Preto, Brazil. Email: * hotta@ime.uicamp,br Received July 22 d, 202; revised August 2 st, 202; accepted September 23 rd, 202 ABSRAC his article discusses the Bayesia approach for cout data usig o-homogeeous Poisso processes, cosiderig differet prior distributios for the model parameters. A Bayesia approach usig Marov Chai Mote Carlo (MCMC) simulatio methods for this model was first itroduced by [], taig ito accout software reliability data ad cosiderig o-iformative prior distributios for the parameters of the model. With the o-iformative prior distributios preseted by these authors, computatioal difficulties may occur whe usig MCMC methods. his article cosiders differet prior distributios for the parameters of the proposed model, ad studies the effect of such prior distributios o the covergece ad accuracy of the results. I order to illustrate the proposed methodology, two examples are cosidered: the first oe has simulated data, ad the secod has a set of data for pollutio issues at a regio i Mexico City. Keywords: No-Homogeeous Poisso Processes; Bayesia Aalysis; Marov Chai Mote Carlo Methods ad Simulatio; Prior Distributio. Itroductio he o-homogeeous Poisso model has bee applied to various situatios, such as the aalysis of software reliability data, air pollutio data ad medical cout data. Deote by M(t) the cumulative umber of evets i the time iterval (0, t] for t 0. M(t) is modelled by a o-homogeeous Poisso process with mea value fuctio m(t). he mea value fuctio m(t) chages accordig to the model, i.e., it assumes the shape of the distributio that is beig used. Oe ca also characterize the distributio by the itesity fuctio t. mt t If λ(t) is a costat, so that m(t) is liear, the M(t) is called a homogeeous Poisso process; otherwise the process is called a o-homogeeous Poisso process. May differet choices for the fuctio m(t) are cosidered i the literature, especially i software reliability []. Referece [2] cosidered that the expected umber of software failures for time t, give by the mea value fuctio m(t), is o-decreasig ad bouded above. * Correspodig author. Specifically, they have cosidered the mea fuctio m () t e t, () where θ, i our applicatio represets the expected maximum umber of days i which air quality stadard is violated by a particular pollutat ad β the rate at which evets occur. By (), we have that t mt e t. I [3] a geeralizatio of the model give by () was proposed, with the followig itesity fuctio 2 e t t t. (2) he correspodig mea fuctio is t m2 t e. (3) Note that () ad (3) ca be writte as special cases of the geeral form where the mea value fuctio is give as mt Ft, (4) where F(t) is a distributio fuctio. O the other had, for ay distributio fuctio F(t) we have a valid model. A widely used distributio fuctio, give its high flexibility of fit, is the Geeralized Gamma distributio. I this case the mea fuctio is
No-Homogeeous Poisso Processes Applied to Cout Data: A Bayesia Approach Cosiderig Differet Prior Distributios 337 GG, m t I t (5) where I (s) is the itegral of the Gamma fuctio give by s x e d. 0 I s x x (6) From Equatio (6), we obtai the itesity fuctio give by GG ' GG t s m t t e. (7) his model is called Geeralized Gamma model. Whe is iteger we ca write F(t) as F t j0 j t t e. (8) j! he three models used i this article are described as follows: ) Model I: Here all the parameters are uow, ad λ GG (t) is give by (7); 2) Model II: with =, λ GG (t) is give by the fuctio λ 2 (t) give i (2) [3]; 3) Model III: with = ad α =, λ GG (t) reduces to λ (t) [2]. It is importat to poit out that other geeralized models could be cosidered i this case. A particular case is give by the expoetiated-weibull distributio (see, for example, [4-6]). his model permits bathtub forms for the itesity fuctio, besides the forms assumed by the Geeralized Gamma distributio. his case is beyod the goals of this paper. I the Bayesia aalysis of Models I, II ad III, we may have some difficulties i obtaiig the Bayesia ifereces usig MCMC simulatio methods. I order to do so, let us study the effect of differet prior distributios o the performace of sample simulatio algorithms for the posterior distributio i which we are iterested. he focus of this paper is to propose differet prior distributio specificatios for the parameters, because whe we cosider the models with all parameters beig free ad to be estimated, we foud some covergece problems i the MCMC algorithm for simulatig samples of the joit posterior distributio. herefore, a sesitivity aalysis of the estimatio of the parameters of the models is coducted which relates to the choice of prior distributios. he biggest cocer is with the parameter, a parameter of the Gamma distributio [5]. I this article we will preset two prior distributio proposals for the parameter of the Geeralized Gamma model. he first uses the prior distributios for the parameters used i []. Give the problems with this prior, especially of covergece of the Gibbs samplig algorithm, a secod prior distributio is suggested for the parameter, which is a trucated expoetial distributio. As we shall see, this distributio proves to be adequate. Other distributios were tested, but will ot be displayed. We chose to preset oly the two assumed prior distributios, because the first oe is suggested i the literature ad the secod oe has show good results. he covergece of the MCMC algorithm was aalysed by graphical methods ad by the Gelma-Rubi criterio. he Gelma-Rubi criterio [7] relies o parallel chais to test whether they all coverge to the same posterior distributio. I [8] the itroductio of a correctio factor was suggested. his corrected statistics was evaluated usig the CODA pacage i R. he closer this statistics is to.0, the more evidece we have that the chai is ear covergece. he limit value of.2 is sometimes used as a guidelie for approximate covergece [9]. his article is structured as follows. I Sectio 2 we preset the Bayesia iferece for the models discussed here. Sectio 3 itroduces a example of this process with simulated data, ad we use the two prior distributios metioed previously for the parameter. A applicatio to Mexico City s pollutio data is performed i Sectio 4. I Sectio 5 some cocludig remars are made. 2. Bayesia Iferece he data set is deoted by D = {; t, t 2,, t ; }, where is the umber of evets observed such that 0 t t 2 t, ad t i are the times of the evets observed durig the period of time (0, ]. he lielihood fuctio for the vector,,, cosiderig the trucated time of the model (see, for example, [0]) is give by L D t m i iexp. (9) I some cases it is ecessary to itroduce a latet variable; this latet variable may improve the covergece of the MCMC algorithm. We cosidered the itroductio of a latet variable N that has Poisso distributio with parameter F (see [] for details). Cosiderig the mea fuctio (5) with a Geeralized Gamma distributio, the lielihood fuctio for the parameters θ, β, α ad is expressed as L D t ti,,, i e i u exp u e d u, o
338 No-Homogeeous Poisso Processes Applied to Cout Data: A Bayesia Approach Cosiderig Differet Prior Distributios such that, L,,, D t i i 2.. Prior Distributios exp ti I. i ) N~Poisso I ; (0) he first set of prior distributios for the geeral Model I are the distributios suggested by []: 2) ~Gamma ab, ; ab, ow; 3) ~Gamma cd, ; cd, ow; 4) ~ where to 0 ; 5) ~ 2 where 2 to 0, where P(λ) deotes the Poisso distributio with parameter λ. Observe that the prior distributios for ad α are improper priors. With this choice of prior distributio we caot guaratee that for all data sets the posterior distributio is proper; i other words, we could have covergece problems i the simulatio algorithm for samples of the joit posterior distributio. o overcome this problem, we assume a trucated expoetial prior for, but other priors could be used as the Gamma distributio. his is a subject of a future wor. hus, i the secod set, also for the geeral Model I, we assume other prior specificatios for the parameters α ad, i.e., give by 4) ~ Gamma e, f ; e, f ow; 5) ~ trucated expoetial a, g ; a, g ow, where a, b, c, d, e, f ad a are the ow hyperparameters of the gamma distributio where Γ(a, b) deotes a gamma distributio with mea a b ad variace a 2 b. Further, let us assume prior idepedece amog the parameters. he values of the hyperparameters are give i the applicatios, followig expert opiio or eve from some iformatio i the data (use of empirical Bayesia methods). As we have covergece problems i the chais of the parameter, the effects of other priors for this parameter are studied i the applicatios. hese priors distributios are give i Sectio 3. I the restricted Models II ad III the same prior distributios are used for the free parameters. 2.2. Posterior Distributios he iferece will be coducted based o iformatio supplied by the posterior distributio of the parameters. From the Bayes formula, the joit posterior distributio is obtaied by combiig the prior ad the lielihood fuctio. he lielihood fuctio for homogeeous or o-homogeeous Poisso processes is give by, L D ti ti exp xdx exp u d, t u i t i L D t exp u d u. 0 i or i hus, we have L D t m i iexp. () Iitially we will cosider the case where the prior distributio of is that suggested by []. I this case, the posterior distributio is give by P D t N! Na c i i 2 I e ( ). N b d ti i (2) Because the posterior distributio i (2) has o closed form, we resort to MCMC methods to geerate samples for the joit posterior distributio. he algorithm used to obtai posterior distributio samples i (2) is give as follows. For the subset of parameters whose full coditioal desity is ow, we simulate samples directly from these distributios, usig the Gibbs Samplig algorithm; for the set of parameters whose coditioal desities are ot ow stadard desities, the samples are tae usig the steps of the Metropolis-Hastigs algorithm [,2]. Recommeded refereces for a review of the MCMC methods are give i [3]. he coditioal posterior distributios required for MCMC methods are give by ) N,,,, D ~ ; P I 2) N,,,, D ~ an, b ; 3) N,,,, D e xi d c i N I ;
No-Homogeeous Poisso Processes Applied to Cout Data: A Bayesia Approach Cosiderig Differet Prior Distributios 339 ti i 4) N,,,, D ti e i N I 5), N,,, D t i i N I 2 he distributio for the parameter θ has a closed form distributio, so the posterior distributios of θ are obtaied through the Gibbs samplig method. Sice the posterior distributios of α, β ad do ot have a closed form, we resort to MCMC methods to simulate samples of these radom quatities. he algorithm used to obtai posterior distributio samples of the parameters α, β ad, whose full coditioal desities are ot ow, is the Metropolis-Hastigs algorithm. Whe we adopt the prior specificatio of the secod set, proposed i Model I, we have that the posterior distributio is give by P D N! Na e c N ti I (3) i e b d ti fa i I 0, g. ; ad the oly chages i the margial posterior desities occur i 4) ad 5), which are replaced by t i e f i i i 4) N,,,, D t e I 5), N,,, D t N 3. A Simulated Study i i N a I e I0, g I this sectio we use a set of simulated data with a sample size of 300 observatios with θ = 300, β = 0.02, α = ad =. For both sets ad 2 of the prior specificatios of Model I we foud problems of covergece of the Gibbs samplig algorithm for all parameters. We used a bur-i of 5,000 iteratios to elimiate the effects of the iitial ;,. values, ad after the bur-i period aother 900,000 iteratios were simulated. aig every 500th step, which gives a fial sample of size 800 we obtaied Mote Carlo estimates for the radom quatities of iterest. We also obtaied posterior estimates of the restricted models: Model II, = ad Model III, =, α =. We used a bur-i of 50,000 iteratios, ad after this we obtaied 5000 samples of the posterior distributios, a sample at each teth iteratio. I Model I, assumig set of the priors of [], we have, N ~ P I ; ~ 0.00, 0.00 ; ~ 0.00,0.00 ; ~, 0; ~, 0. Let us call the model with this prior specificatio as Model I(). I Model I with the secod set of prior specificatios, we assume a Gamma prior distributio for α give by Γ(0.00, 0.00). For the parameter we coducted a study of the sesitivity of differet priors assumig differet hyperparameter values. his was doe because, whe this parameter is estimated, it is difficulty to obtai covergece of the algorithm used to simulate samples for the parameters. he prior distributios tested for this parameter iclude: Uiform (0, ), Γ(0.00, 0.00) ad Γ(0,000), besides other values for the hyperparameters of such distributios. he trasformatio log() was also tested, where log(k) ~ N(a, b ), with may values for the hyperparameters a, b, icludig N(0, 0.0), N(0, 0) ad N(0, ). I most tests the covergece of the Marov chai simulatios was ot satisfactory. he best result was obtaied for a trucated expoetial distributio with mea parameter equal to 0.95 ad trucated i 3. Let us call the model with this prior distributio Model I(2). able shows summaries of the posterior the estimates of Models I, II ad III. For Model I() which uses a o-iformative prior distributio specificatio for parameter, the graphs of the MCMC chais are show i Figure (letters (a), (c), (e) ad (g)). Oly for parameter θ (letter (a)) do we appear to have clear covergece, while for the parameter β (letter (c)) there is clear ocovergece, ad for parameter (letter (g)) there is possible ocovergece. Figure 2 (letters (a), (c), (e) ad (g)) presets the posterior distributios of the two chais. Oly for the parameter θ (letter
340 No-Homogeeous Poisso Processes Applied to Cout Data: A Bayesia Approach Cosiderig Differet Prior Distributios able. Posterior summaries of the estimates of the parameters for Models I, II ad III. rue model: θ = 300, β = 0.02, α = ad =. Sample size of 300. Priors: for the parameters θ, β ad α the prior distributios are the oes i Sectio 2. For the parameter i Model I: () No-iformative prior distributio proposed i []; (2) Expoetial prior distributio with mea parameter 0.95 ad trucated i 3. For Model I, we have: st row of each parameter correspods to a estimate of chai, startig from a give value; 2d row correspods to a estimate of chai 2, startig from the true values. Models Priors Posterior Estimates ad Priors Parameters Mea S.D. Mea Media S.D. CI (95%) θ 3. 3.62 33.5 32.97 8.63 (282.44, 344.89) 3.62 33.49 33.04 8.35 (284.53, 343.25) β 3.62 0.220 * 0.27 * 0.224 * (0.03 *, 0.6855 * ) I() 3.62 0.334 * 0.084 * 0.525 * (0.009 *,.502 * ) α - - 0.709 * 0.687 * 0.96 * (0.444 *,.070 * ) - - 0.744 * 0.757 * 0.255 * (0.348 *,.27 * ) - - 2.026 *.787 * 0.889 * (0.90 *, 3.6825 * ) - - 2.202 *.559 *.5665 * (0.89 *, 5.588 * ) θ 3.62 30.3 309.60 7.40 (282.58, 339.62) 3.62 30.39 30.35 7.63 (28.83, 339.85) β 3.62 0.036 0.023 0.04 (0.0030, 0.3) I(2) 3.62 0.043 0.024 0.054 (0.0026, 0.66) α 0 0.984 0.967 0.95 (0.696,.333) 0 0.983 0.962 0.255 (0.644,.355) 0.95 0.57.05.058 0.336 (0.653,.736) 0.95 0.57.33.058 0.403 (0.627,.97) θ 3.62 30.9 30.04 7.87 (28.45, 340.5) II β 3.62 0.09 0.09 0.004 (0.04, 0.027) α - - 0.999 0.999 0.049 (0.97,.078) θ 3.62 30.09 309.44 7.38 (282.47, 339.) III β 3.62 0.09 0.09 0.00 (0.07, 0.02) * here is o covergece for these parameters. (a) (b) (c) (d) (e) (f) (g) (h) Figure. Covergece of chais ad 2, for the parameters of Model I(), for which the graphs are labeled (a), (c), (e) ad (g), ad Model I(2), for which the graphs are labeled (b), (d), (f) ad (h).
No-Homogeeous Poisso Processes Applied to Cout Data: A Bayesia Approach Cosiderig Differet Prior Distributios 34 (a) (b) (c) (d) (e) (f) (g) Figure 2. Posterior distributio of the parameters for chais ad 2 of Model I, where graphs (a), (c), (e) ad (g) correspod to the first set of priors, ad graphs (b), (d), (f) ad (h) correspod to the secod set of priors. he true value of each parameter is represeted by a vertical lie. (h) (a)) is there ay idicatio of covergece. hese results are cofirmed by the Gelma-Rubi s statistics which are equal to.25 ad.5, for the parameters β ad respectively. hat is, we did ot get covergece for the MCMC chais with these prior distributios. I Model I(2) we use, as the ew proposed prior distributio for the parameter, a expoetial distributio with mea parameter 0.95 ad trucated i 3. he chais are preseted i Figure (letters (b), (d), (f) ad (h)) ad there is a idicatio of covergece. Figure (letters (b), (d), (f) ad (h)) shows the posterior distributio of the parameters for chais ad 2, ad there is also idicatio of covergece. he covergece is cofirmed by the Gelma-Rubi s statistics, with the largest value equal to.0. he mai coclusio reached i this sesitivity aalysis is that i the case of the proposed Model 2, this is close to the process that geerates the actual data; i other words, it is possible to have a good approximatio of the posterior distributio of the parameters. Eve usig prior distributios for parameters α, β ad θ which are ot very iformative, Model I(2) ca estimate the true value of these parameters. his does ot hold true for Model I().
342 No-Homogeeous Poisso Processes Applied to Cout Data: A Bayesia Approach Cosiderig Differet Prior Distributios 4. A Applicatio to Pollutio Data his sectio applies Models I() ad I(2) to fit data correspodig to the maximum daily mea measuremets of ozoe gas, based o data from the ortheast (NE) regio of Mexico City. he sample has 98 observatios, which correspod to times whe a certai threshold established for the air quality stadard is violated i the iterval time (0,] hese data are available at www.sma.df.gob.mx/simat, ad we too eightee years of observatios, from Jauary st, 990 to December 3 st, 2008 [4]. For these models ad the assumed data set we used a bur-i of 5,000 iteratios, ad after this we obtaied 7000 samples of the posterior distributios, with a sample at each oe-hudredth iteratios. able 2 shows the posterior estimates of the parameters for Models I() ad I( 2). I Figure 3, we show the trace plots of chais ad 2, respectively, for the parameters of the models, where graphs (a), (c), (e) ad (g) correspod to Model I(), ad graphs (b), (d), (f) ad (h) correspod to Model I(2). We ca observe i Figure 3, that i graphs (a), (c), (e) ad (g), related to Model I(), oly parameter θ appears to have covergece (graph (a)), ad there is evidece that there is o covergece for the other parameters. I this model we used a o-iformative prior distributio specificatio for the parameter. I Model I(2), graphs (b), (d), (f) ad (h), appear to have covergece. I this model we proposed as the prior distributio for the parameter a expoetial distributio with mea parameter 0.99 ad trucated i 6. he largest value of the Gelma-Rubi s statistics for the parameters of the geeral Model I(2) is equal to.00 idicatig covergece of the MCMC chais. I Model I() the Gelma-Rubi s statistics are equal to.22,.23 ad.9, for the parameters α, β ad respectively. So we did ot get the covergece of the MCMC chais usig the o iformative prior for the parameter. I Figure 4, we have the posterior distributio of the parameters for chais ad 2 for Models I() ad I(2). he graphs labelled (a), (c), (e) ad (g) correspod to the first set of priors; ad the graphs t labelled (b), (d), (f) able 2. Posterior estimates of parameters, usig a sample of real data correspodig to the NE regio of Mexico City. Priors: for parameters θ, β ad α the prior distributios are the oes i Sectio 2. Priors for parameter : Model I() o-iformative prior distributio proposed i []; Model I(2) expoetial prior distributio with mea parameter 0.99 trucated i 6. I the st row, for each parameter, we preset the estimates correspodig to chai, ad i the 2d row, we preset estimates correspodig to chai 2. Models Priors Posterior Estimates ad Priors Parameters Mea S.D. Mea Media S.D. IC (95%) θ 3.62 995.64 995.7 32.50 (942.78, 049.44) 3.62 999.3 998.33 34.84 (944.7, 057.79) β 3.62 3.7 0 4* 0.6 0 5* 2.4 0 3* (0.8 0 6*,.3 0 4* ) I() 3.62 2.9 0 3*. 0 5*.2 0 2* (2.7 0 6*, 2.5 0 2* ) α - -.530 *.569 * 0.235 * (0.966 *,.808 * ) - -.425 *.507 * 0.269 * (0.663 *,.663 * ) - -.4 *.002 * 0.409 * (0.87 *, 2.007 * ) - -.332 *.067 * 0.823 * (0.95 *, 3.78 * ) θ 3.62 043.54 043.38 35.39 (986.58, 02.4) 3.62 044.7 044.5 35.79 (985.53, 03.67) β 0, 0,00 0.055 0.052 0.022 (0.025, 0.095) I(2) 0, 0,00 0.056 0.053 0.02 (0.026, 0.094) * here is o covergece for these parameters. α 0 0.592 0.589 0.042 (0.530, 0.662) 0 0.588 0.586 0.038 (0.529, 0.656) 0.99 0.95 4.457 4.445 0.52 (3.660, 5.328) 0.99 0.95 4.486 4.467 0.49 (3.697, 5.324)
No-Homogeeous Poisso Processes Applied to Cout Data: A Bayesia Approach Cosiderig Differet Prior Distributios 343 (a) (b) (c) (d) (e) (f) (g) Figure 3. Covergece of chais ad 2, for the parameters of Model I(), i which the graphs are labelled (a), (c), (e) ad (g), ad Model I(2), i which the graphs are labelled (b), (d), (f) ad (h). (h) ad (h) correspod to the secod set of priors. he results cofirm the covergece i the Model I(2) ad ocovergece i Model I() 5. Coclusios I this article we proposed a sesitivity aalysis with various specificatios of prior distributios for the model previously itroduced i [], which was developed usig the Bayesia approach. We have coducted a study of the effect of prior distributios o the covergece ad accuracy of the results, ad i this way we have bee able to propose a prior distributio for the parameter that gives covergece of the samples simulatio algo- rithm. Observe that usig improper prior distributios for the parameter could ot guaratee that the posterior distributio was proper, as this depeded o the data set (as was observed i our applicatio with real data). After tryig several prior distributio specificatios for this parameter, it was possible to propose a prior distributio that would cosiderably improve the covergece of the chais. Such improvemets ca be oted i credible itervals, i which the rage of the iterval is smaller usig a trucated expoetial prior distributio; we may also observed these best results through graphical aalysis whe usig a trucated expoetial prior distributio for the parameter, we obtaied covergece of the chais. It is importat to poit out that other iformative priors
344 No-Homogeeous Poisso Processes Applied to Cout Data: A Bayesia Approach Cosiderig Differet Prior Distributios (a) (b) (c) (d) (e) (f) (g) Figure 4. Posterior distributios of parameters for chais ad 2, respectively for Model I, where graphs (a), (c), (e) ad (g), correspod to the first set of priors ad graphs (b), (d), (f) ad (h) correspod to the secod set of priors. (h) for the parameter could be cosidered, such as a Gamma distributio. his possibility is a goal for a future study. 6. Acowledgemets his wor was partially supported by grats from Capes, CNPq ad FAPESP. We would lie to tha oe referee ad Laboratório Epifisma. REFERENCES [] J. A. Achcar, K. D. Dey ad M. Niverthi, A Bayesia Approach Usig Nohomogeeous Poisso Process for Software Reliability Models, I: A. S. Basu, S. K. Basu ad S. Muhopadhyay, Eds., Frotiers i Reliability, World Scietific Publishig Co., Sigapore City, 998, pp. -8. doi:0.42/97898286580_000 [2] A. L. Goel ad K. Oumoto, A Aalysis of Recurret Software Errors i a Real-ime Cotrol System, Proceedigs of the 978 Aual Coferece, ACM 78, Washigto DC, 4-6 December 978, pp. 496-50. doi:0.45/80027.80460 [3] A. L. Goel, A Guideboo for Software Reliability Assessmet, echical Report, Syracuse Uiversity, Syracse, 983. [4] G. S. Muldholar, D. K. Srivastava ad M. Friemer, he Expoetiated-Weibull Family: A Reaalysis of the Bus- Motor Failure Data, echometrics, Vol. 37, No. 4, 995, pp. 436-445. doi:0.080/0040706.995.0484376 [5] V. G. Cacho, H. Bolfarie ad J. A. Achcar, A Bayesia Aalysis of the Expoetiated-Weibull Distributio, Joural of Applied Statistical Sciece, Vol. 8, No. 4, 999, pp. 227-242.
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