ON A PROCESS DERIVED FROM A FILTERED POISSON PROCESS

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1 ON A PROCESS DERIVED FROM A FILTERED POISSON PROCESS MARIO LEFEBVRE and JEAN-LUC GUILBAULT A ontinuous-time and ontinuous-state stohasti proess, denoted by {Xt), t }, is defined from a proess known as a filtered Poisson proess. Various harateristis of the proess, suh as its mean and its variane at time t are alulated. We also use its onditional expetation, given the history of the proess in the interval [, t], as an estimator of Xt + δ). An appliation of this estimator to foreast the flow of a river is presented. AMS 2 Subjet Classifiation: Primary 6G55; Seondary 6G35. Key words: onditional expetation, speial funtions, estimation, hydrology, foreast. 1. INTRODUCTION Let {Nt), t } be a time-homogeneous Poisson proess with rate λ. The stohasti proess {Zt), t } defined by 1) Zt) = Nt) Y n t τ n ) k e t τn)/, Zt) = if Nt) =, where the random variables Y 1, Y 2,... are independent and identially distributed, and independent of Nt), the τ n s are the arrival times of the events of the Poisson proess, and k and > are onstants, is a partiular ase of proesses alled filtered Poisson proesses. The funtion wt, τ n, Y n ) := Y n t τ n ) k e t τn)/ is the response funtion of the proess. This type of proess is used in many appliations [see Parzen [5], p. 144)]. In partiular, the proess {Zt), t } seems appropriate to desribe the flow of a river see Lefebvre et al. [4] and Lefebvre and Guilbault [3]. In this appliation, the τ n s denote the times the days, in pratie) when intense preipitation events, either rain or snow, ourred and the Y n s are the inreases of the flow at the orresponding time instants τ n. It is often assumed that the random variables Y n have an exponential distribution with parameter µ. REV. ROUMAINE MATH. PURES APPL., 54 29), 2,

2 148 Mario Lefebvre and Jean-Lu Guilbault 2 In the ase when the times between the suessive events are long enough and the onstant is relatively small, we an neglet the events that ourred before the most reent one. It follows that Zt) Y Nt) t τ Nt) ) k e t τ Nt))/ if Nt) >. In this paper, we define the stohasti proess {Xt), t } by 2) Xt) = X) + wt, τ Nt), Y Nt) ), t, where wt, τ Nt), Y Nt) ) is a response funtion. Generally, a response funtion is non-negative. Here, we only assume that Note that, beause N) =, we have wt, τ Nt), Y Nt) ) = if Nt) =. w, τ N), Y N) ) = w, τ, Y ) =. To be more general, we an suppose that X) is a random variable [independent of Nt)]. The response funtion that will be used in an appliation of the model in hydrology is 3) wt, τ Nt), Y Nt) ) = Y Nt) t τ Nt) ) k e t τ Nt))/, t, with Y = τ =, so that w,, ) = if Nt) =. In Setion 2, we alulate, in partiular, the mean and the variane of the random variable Xt) for a general response funtion. Next, we onsider the onditional expetation E[Xt + δ) Ht)], whih gives us the mean value of the proess at time t + δ, given the history Ht) of the proess in the interval [, t]. We make use of this onditional expetation in Setion 3 in an appliation of the model in hydrology. The problem of estimating the various parameters of the model is onsidered in Setion 3 as well. Finally, we onlude this work with a few remarks in Setion THEORETICAL RESULTS First, we alulate the mean of the stohasti proess {Xt), t }. Proposition 2.1. For the stohasti proess defined in 2), E[Xt)] = E[X)] + λe λt E[wt, s, Y )]e λs ds. Proof. We ondition on Nt): E[Xt)] = E[Xt) Nt) = n]p [Nt) = n], n=

3 3 On a proess derived from a filtered Poisson proess 149 where λt λt)n P [Nt) = n] = e, n =, 1,.... In the ase when Nt) >, by linearity of the mathematial expetation, while E[Xt) Nt) = n] = E[X)] + E[wt, τ Nt), Y Nt) ) Nt) = n] = E[X)] + E[wt, τ n, Y n ) Nt) = n] E[Xt) Nt) = ] = E[X)]. Now, given that Nt) = n, the arrival time τ n of the last event in the interval [, t] is distributed like the maximum of n independent uniform U[, t] random variables see Lefebvre [2] or Ross [6]). That is, It follows that f τn s Nt) = n > )) = n t n sn 1, s t. E[wt, τ n, Y n ) Nt) = n] = E[wt, s, Y )] n t n sn 1 ds = n t n E[wt, s, Y )]s n 1 ds, where Y is a random variable distributed like the Y n s. Thus, E[Xt)] = E[X)]e λt + {E[X)] + nt n E[wt, s, Y )]s n 1 ds }e = E[X)] + { n t n = E[X)] + e λt } E[wt, s, Y )]s n 1 λt λt)n ds e λ n n 1)! E[wt, s, Y )]s n 1 ds. By the dominated onvergene theorem, E[Xt)] = E[X)] + e λt λ n n 1)! E[wt, s, Y )]sn 1 ds = E[X)] + λe λt E[wt, s, Y )] λs) m ds m! m= = E[X)] + λe λt E[wt, s, Y )]e λs ds. λt λt)n

4 15 Mario Lefebvre and Jean-Lu Guilbault 4 Corollary 2.1. For the response funtion given in 3), the mean of the stohasti proess at time t, when k > 1, is ) k+1 E[Xt)] = E[X)] + λe[y ] γ k + 1, λ + 1 ) ) t, λ + 1 where γ, ) is the inomplete gamma funtion. In partiular, when k =, 1,..., ) [ k+1 k E[Xt)] = E[X)]+λk!E[Y ] 1 e λ+ 1 )t λ + 1 ) ] m t m. λ + 1 m! m= That is, Proof. By Proposition 2.1 and by independene, ] E[Xt)] = E[X)] + λe λt E [Y t s) k e t s) e λs ds = E[X)] + λe λt E[Y ] E[Xt)] = E[X)] + λe λ+ 1 )t E[Y ] t s) k e t s) e λs ds. t s) k e λ+ 1 )s ds. Now, if k > 1 then [Gradshteyn and Ryzhik [1], p. 343)] ) I 1 := t s) k e λ+ 1 k+1 )s ds = e λ+ 1 )t γ λ + 1 k + 1, λ + 1 ) ) t. In partiular, when k =, 1,..., [Gradshteyn and Ryzhik [1], p. 89)], ) [ k+1 k I 1 = k! e λ+ 1 )t 1 e λ+ 1 )t λ + 1 ) ] m t m. λ + 1 m! m= Next, we alulate the variane of the proess {Xt), t }. Proposition 2.2. For the stohasti proess defined in 2), V [Xt)] = V [X)] + λe λt E[w 2 t, s, Y )]e λs ds Proof. By independene, { 2 λ 2 e 2λt E[wt, s, Y )]e ds} λs. V [Xt)] = V [X)] + V [wt, τ Nt), Y Nt) )].

5 5 On a proess derived from a filtered Poisson proess 151 Conditioning on Nt), we obtain E[w 2 t, τ Nt), Y Nt) )] = + E[w 2 λt λt)n t, τ Nt), Y Nt) ) Nt) = n]e = E[w 2 λt λt)n t, τ n, Y n ) Nt) = n]e. Now, Thus, E[w 2 t, τ n, Y n ) Nt) = n> )] = E[w 2 t, τ Nt), Y Nt) )] = { n t n = λe λt E[w 2 t, s, Y )] n t n sn 1 ds = n t n E[w 2 t, s, Y )]s n 1 ds. } E[w 2 t, s, Y )]s n 1 λt λt)n ds e { } λ E[w 2 t, s, Y )]s n 1 n 1 ds n 1)!. The result follows from the monotone onvergene theorem, by whih E[w 2 t, τ Nt), Y Nt) )] = λe λt E[w 2 λs) n 1 t, s, Y )] n 1)! ds as well as from Proposition 2.1. = λe λt E[w 2 t, s, Y )]e λs ds, Corollary 2.2. When the response funtion is that defined in 3), and k > 1 2, the variane of the stohasti proess at time t is given by ) 2k+1 V [Xt)] = V [X)] + λe[y 2 ] γ 2k + 1, λ + 2 ) ) t λ + 2 { ) k+1 λe[y ] γ k + 1, λ + 1 ) ) } 2 t. λ + 1 In partiular, if k =, 1 2, 1,..., γ 2k + 1, λ + 2 ) ) [ 2k t = 2k)! 1 e λ+ 2 )t m= λ + 2 ) ] m t m m!

6 152 Mario Lefebvre and Jean-Lu Guilbault 6 while, if k =, 1,..., 4) γ k + 1, λ + 1 ) ) [ k t = k! 1 e λ+ 1 )t m= λ + 1 Proof. All is left to do is to alulate the integral ] I 2 := λe λt E [Y t s) k e t s) 2 e λs ds. By independene, and when k > 1 2 I 2 = λe λt E[Y 2 ] t s) 2k e 2t s) e λs ds, [Gradshteyn and Ryzhik [1], p. 343)], I 2 = λe λt E[Y 2 ]e 2t = λe[y 2 ] λ + 2 t s) 2k e λ+ 2 )s ds ) 2k+1 γ 2k + 1, λ + 2 ) ) t. ) ] m t m. m! In partiular, when k =, 1 2, 1,... [Gradshteyn and Ryzhik [1], p. 89)], ) [ 2k+1 2k I 2 = λ2k)!e[y 2 ] 1 e λ+ 2 )t λ + 2 ) ] m t m. λ + 2 m! We ontinue with the alulation of the onditional expetation E[Xt + δ) Ht)], where Ht) denotes the history of the proess in the interval [, t]. That is, if we know Ht), then we know the value of Xu) for any u in the interval [, t] as well as the arrival times τ n of the events and the value of the random variables Y n, n =, 1,.... m= Proposition 2.3. The onditional expetation of Xt + δ) given Ht) is E[Xt + δ) Ht)] = X) + wt + δ, τ Nt), Y Nt) )e λδ + λe λδ E[wt + δ, u + t, Y )]e λu du. Proof. Proeeding as in the proofs of the previous propositions, we write E[Xt + δ) Ht)] = = E[Xt + δ) Ht), Nt + δ) Nt) = n]p [Nt + δ) Nt) = n Ht)] = n= n= E[Xt + δ) Ht), Nt + δ) Nt) = n]e λδ λδ)n

7 7 On a proess derived from a filtered Poisson proess 153 beause the inrements of the Poisson proess are independent). Now, E[Xt + δ) Ht), Nt + δ) Nt) = ] = X) + wt + δ, τ Nt), Y Nt) ). Moreover, the knowledge of Ht), apart from the value of X), is not needed when Nt + δ) Nt) = n >. Therefore, E[Xt + δ) Ht), Nt + δ) Nt) = n> )] = E[Xt + δ) X), Nt + δ) Nt) = n> )] = E[X) + wt + δ, τ Nt+δ), Y Nt+δ) ) X), Nt + δ) Nt) = n> )] = X) + E[wt + δ, τ Nt+δ), Y Nt+δ) ) X), Nt + δ) Nt) = n> )] = X) + +δ t E[wt + δ, s, Y )] n δ n s t)n 1 ds = X) + n δ n E[wt + δ, u + t, Y )]u n 1 du. Finally, E[Xt + δ) Ht)] = {X) + wt + δ, τ Nt), Y Nt) )}e λδ + {X) + nδ n E[wt + δ, u + t, Y )]u n 1 λδ λδ)n du }e = X) + wt + δ, τ Nt), Y Nt) )e λδ { n δ + δ n E[wt + δ, u + t, Y )]u n 1 λδ λδ)n du }e = X) + wt + δ, τ Nt), Y Nt) )e λδ + λe λδ λ n 1 n 1)! = X) + wt + δ, τ Nt), Y Nt) )e λδ E[wt + δ, u + t, Y )]u n 1 du + λe λδ E[wt + δ, u + t, Y )]e λu du by dominated onvergene). From what preedes, we dedue the result below.

8 154 Mario Lefebvre and Jean-Lu Guilbault 8 Corollary 2.3. If the response funtion is that in 3), and k > 1, then the onditional expetation of Xt + δ) given Ht) is E[Xt + δ) Ht)] = X) + [Xt) X)]e λ+ 1 1 )δ δ + t τ Nt) ) k+1 + λe[y ] γ k + 1, λ + 1 ) ) δ. λ + 1 When k =, 1,..., the inomplete gamma funtion above is given by 4). Proof. First, wt + δ, τ Nt), Y Nt) ) = [Xt) X)]e δ 1 + δ t τ Nt) ) k. Furthermore, by independene [see Gradshteyn and Ryzhik [1], p. 343)], E[wt + δ, u + t, Y )]e λu du = = e δ E[Y ] δ u) k e λ+ 1 )u du E[Y δ u) k e δ u) ]e λu du ) = e δ k+1 E[Y ] e λ+ 1 )δ γ k + 1, λ + 1 ) ) δ. λ + 1 Finally, for formula 4) that gives the inomplete gamma funtion when k =, 1,..., see Gradshteyn and Ryzhik [1], p. 89). Remark. For the partiular response funtion onsidered, it is suffiient to know X), Xt) and τ Nt) to evaluate expliitly the onditional expetation E[Xt + δ) Ht)], and when k =, the value of τ Nt) is not required. Moreover, when k =, E[Xt+δ) Ht)]= [ X)+λE[Y ] If we assume that X) =, then λ+1 5) E[Xt + δ) Ht)] = e λ+ 1 )δ Xt) + λe[y ] ) k )] 1 e λ+ 1 )δ) +e λ+ 1 )δ Xt). ) 1 e λ+ 1 )δ). λ + 1 It is this formula, with δ = 1, that we will use in Setion 3 to foreast the flow of the Delaware River the day after the most reent observation of this flow. To onlude this setion, we alulate a mathematial expetation that ould be used, together with other results, to estimate the parameters of the

9 9 On a proess derived from a filtered Poisson proess 155 model with response funtion 3). Suppose that k = and that X) =. We then have Hene, Xt) = Y Nt) e t τnt))/ ln Xt) = ln Y Nt) t τ Nt). E[ln Xt)] = E[ln Y Nt) ] 1 E[t τ Nt)]. The quantity At) := t τ Nt) is alled the age of the Poisson proess, whih is a speial renewal proess. The average value of At) over a long period an be alulated relatively easily for any renewal proess, see Lefebvre [2], p. 318). In the ase of the Poisson proess, using the fat that the time between the suessive events follows an exponential distribution with parameter λ and that this distribution possesses the memoryless property, we dedue that in the limit the average age of the proess is given by lim E[At)] = E[Expλ)] = 1 t λ. If we now assume that the random variables Y n are exponentially distributed with parameter µ, we must finally alulate We find that E[ln Y Nt) ] = ln y) µe µy dy. ln)y µe µy dy = ln µ γ, where γ,5772 is the Euler-Masheroni onstant. Thus, 6) lim E[ln Xt)] = ln µ γ 1 t λ. In pratie, if we assume that the proess onsidered is in equilibrium, or in stationary regime, we an estimate lim E[ln Xt)] by the arithmeti mean t of the natural logarithms of the observations over a long period of time. Sine we an estimate the parameter λ by the average number of events by time unit during the period onsidered and, theoretially, the parameter µ by the inverse of the arithmeti mean of the observations of the random variables Y n, we an use 6) to obtain a point estimate of. However, in fat, it is not always easy to determine the value of the Y n s, so that the problem of estimating the parameters λ, µ and in the model is not trivial. Furthermore, in the general ase, the onstant k must also be determined. In the next setion, we will see that formula 5) enables us to obtain more) preise foreasts of the flow of the Delaware River for the next day.

10 156 Mario Lefebvre and Jean-Lu Guilbault 1 3. FORECAST OF THE FLOW OF THE DELAWARE RIVER The authors onsidered in [3] the problem of modeling and foreasting the flow of the Delaware River, loated in the North-Eastern part of the United States. To do so, they proposed as model a filtered Poisson proess, suh as that defined by 1). They found that the hoie k = 1 2 gave a very good fit of the model to the real data. However, for the foreasts of the flow one day ahead, it is almost neessary to limit ourselves to the basi ase, namely that when k =. Otherwise, in pratie, the formulas to estimate Zt + 1) beome inonvenient. Based on model 1), with k =, the estimator of the flow Zt + 1) of the river on day t + 1, given the observed flow Zt), is the onditional expetation 7) E[Zt + 1) Zt)] = e 1/ Zt) + λ µ 1 e 1/ ), where µ 1 = E[Y ]. The observations of the flow for the period from Otober 1st, 22 to September 3th, 23 were used to estimate the parameters λ, µ and ; then, use was made of the model to foreast the daily flow of the river during a 11-day period in 24. We found that the mean of the absolute values of the foreasting errors was 838 ft 3 /s. Now, we dedue from 5) that ) 8) E[Xt + 1) Ht)] = e λ+ 1 ) Xt) + λe[y ] 1 e λ+ 1 )). λ + 1 If we use the point estimates of the parameters λ, µ and omputed in Lefebvre and Guilbault [3], we find that the mean of the absolute values of the foreasting errors obtained with the above formula is about 752 ft 3 /s, whih is very good if we ompare with results obtained with other estimators. Moreover, the foreast of the flow on the next day is better, 67 times out of 11, than that given by equation 7). Next, if we onsider the average of the foreasts provided by 7) and 8) as estimator of the flow on the next day, we find that the absolute values of the foreasting errors is redued to 686 ft 3 /s, whih is even better than the previous results. Remark. As mentioned in the introdution, the model proposed in this paper originates from a filtered Poisson proess and an serve as an approximate model when the time between the suessive events is suffiiently long. Consequently, if δ is small, for example if we are interested in the onditional expetation of Xt + 1) that is, the flow on the next day, as above), we may

11 11 On a proess derived from a filtered Poisson proess 157 assert that the probability of more than one event in the interval t, t + δ] is negligible. Sine E[Xt + δ) Ht), Nt + δ) Nt) = 1] k= = X) + E[Y ] e δ/ δ we obtain E[Xt + δ) Ht)] k= = E[Xt + δ) X), Xt)] = X) + E[Y ] δ 1 e δ/ ), e u/ du {X) + [Xt) X)]e δ/ }e λδ + {X) + E[Y ] δ 1 e δ/ )}λδe λδ = e λδ{ X)1 + λδ) + [Xt) X)]e δ/ + λe[y ]1 e δ/ ) }. If we assume that X) =, then so that E[Xt + δ) Ht)] k= = E[Xt + δ) Xt)] 9) E[Xt + 1) Xt)] e λδ {Xt)e δ/ + λe[y ]1 e δ/ )}, { e 1/ Xt) + λ } µ 1 e 1/ ) e λ. If we make again use of the point estimates of the parameters λ, µ and omputed in Lefebvre and Guilbault [3], then the foreast 7) of the flow on the next day is simply multiplied by the fator e λ. Sine ˆλ,1458, we multiply the foreasts of the flow by about,8643. The mean of the absolute values of the foreasting errors obtained with this estimator is approximately equal to 771 ft 3 /s, and the foreast of the flow on the next day is better, 66 times out of 11, than that given by equation 7). Finally, if we onsider as estimator of the flow on the next day the mean of the foreasts provided by 7), 8) and 9), we now find that the mean of the absolute values of the foreasting errors is redued to 669 ft 3 /s. 4. CONCLUSION We studied a stohasti proess derived from a filtered Poisson proess that an serve as an approximate model or as an almost) exat model in ertain pratial situations. We saw how the parameters that appear in the proess an be estimated in order to be able to use the model in real appliations. In Setion 3, we saw that the estimator of the value of the random variable Xt + 1), given the most reent observation of the proess [that is,

12 158 Mario Lefebvre and Jean-Lu Guilbault 12 Xt)], enabled us to improve the auray of the foreasts in an appliation in hydrology. Beause we assumed that X) = and k =, the estimator in question only requires the value of Xt). If k >, the formula used beomes more ompliated and requires the knowledge of the arrival times τ n of the events of the Poisson proess. In the appliation in hydrology, we assume that the system has been in operation long enough for the proess to be in equilibrium. If the time between two onseutive events is very long, the value of Xt) dereases to zero. In pratie, the flow of the Delaware River never goes below a ertain threshold. This threshold, whih is the minimal flow over a long period, ould be the deterministi) value of X) in the more general model. It would be suffiient then to define X t) = Xt) X) to retrieve the proess for whih X) =. Finally, another model that would be worth studying is the filtered Poisson proess defined by where Zt) = Nt) Y n t τ n ) k e t τn)/ gt, τ n ), Zt) = if Nt) =, { 1 if t τn [, s], gt, τ n ) = otherwise. Thus, we would keep all the events that ourred at most s time units before time t, rather than keeping all the events in the interval [, t] or only the most reent event. The quantity s would be another parameter in the model that should be hosen in an appropriate way depending on the appliation onsidered. Aknowledgements. This work was supported by the Natural Sienes and Engineering Researh Counil of Canada. REFERENCES [1] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Produts, 6th Ed. Aademi Press, San Diego, 2. [2] M. Lefebvre, Proessus stohastiques appliqués. Hermann, Paris, 25. [3] M. Lefebvre and J.L. Guilbault, Using filtered Poisson proesses to model a river flow. Appl. Math. Model ), [4] M. Lefebvre, J. Ribeiro, J. Rousselle, O. Seidou and N. Lauzon, Probabilisti predition of peak flood disharges. In: A. Der Kiureghian, S. Madanat and J.M. Pestana Eds.), Pro. ICASP 9 Conf., pp Millpress, Rotterdam, 23.

13 13 On a proess derived from a filtered Poisson proess 159 [5] E. Parzen, Stohasti Proesses. Holden-Day, San Franiso, [6] S.M. Ross, Introdution to Probability Models, 8th Ed. Aademi Press, San Diego, 23. Reeived 2 September 28 Éole Polytehnique de Montréal Département de mathématiques et de génie industriel C.P. 679, Suursale Centre-ville Montréal, Québe H3C 3A7, Canada mario.lefebvre@polymtl.a, jean-lu.guilbault@polymtl.a