A Bayesian Approach to Arrival Rate Forecasting for Inhomogeneous Poisson Processes for Mobile Calls

Transcription

1 A Bayesan Approach to Arrval Rate Forecastng for Inhomogeneous Posson Processes for Moble Calls Mchael N. Nawar Department of Computer Engneerng Caro Unversty Caro, Egypt Amr F. Atya Department of Computer Engneerng Caro Unversty Caro, Egypt Mohamed Saleh Department Operatons Research and Decson Support Faculty of Computers and Informaton Caro Unversty Caro, Egypt Khaled F. Elsayed Department of Electroncs & Communcatons Caro Unversty Caro, Egypt July 15, 2013

2 1 Introducton Moble call and short message arrvals can be characterzed as nhomogeneous Posson process (IHPP),.e. a Posson process wth tme-varyng arrval rate (see [4]). In ths paper we propose a Bayesan framework for the problem of estmatng ths underlyng varable arrval rate λ(t). Moreover, ths varable arrval rate s projected forward n tme to provde a forecast for the call arrval rate. There are several potental benefts to the problem of forecastng arrval rate: It can lead to better prcng of the dfferent servces, that takes nto account demand (n the form of forecasted or expected usage). Moreover, one can apply some form of dynamc prcng, that dynamcally adjusts the prce accordng to demand, n an attempt to optmze the revenue. One can dentfy peak hours, and therefore alert to congeston stuatons. One can obtan the seasonal varatons of usage. resources for the dfferent perods. Ths helps n plannng adequate For moble advertsng, ths can help n better targetng the users whle they are most actve n moble usage. Ths wll produce a better advertsng mpact. For these reasons forecastng the usage of the dfferent servces would be an mportant task. Estmatng the arrval rate for IHPP s has been an actve topc n the statstcs lterature. A few of these works have consdered the Bayesan methodology. For example [3] assumes the pror to be a Drchlet process. Also, [5] proposed a Bayesan model, whereby they utlze a transformaton that renders the arrvals as approxmately normal, and they mpose a smoothness pror n the form of a Wener process for the ntraday rate. Subsequently they use Markov Chan Monte Carlo (MCMC) to obtan the rate estmates. The work by [1] developed a general Bayesan estmaton model where the posteror s evaluated usng the MCMC algorthm. They used Metropols algorthm wth Gbbs samplng. Also, [2] propose another Bayesan estmaton model where the pror s unformly dstrbuted. The problem wth most exstng methods s that they need some Monte Carlo method, such as MCMC, to evaluate the ntegrals. It s well-known that MCMC s very hard to tune, and t often fals to mx. In ths work we propose an IHPP approach that models moble call arrval (or ntaton) rates. Moreover, the method takes nto account contnuty and seasonal relatons by ncorporatng them nto the pror n a Bayesan formulaton. The appeal of the Bayesan approach here stems from the fact that calls are sparse. The majorty of tme slots do not have any calls. So we need to enforce contnuty or smoothness relatons usng a Bayesan pror to avod a choppy estmate of the arrval rate curve. The man contrbuton s that we obtan an analytcal soluton for the forecasted rates, n the form of multple summatons. To our knowledge, ths s the only work that obtaned an analytcal soluton to the Bayesan IHPP problem. A Monte Carlo algorthm s also presented that allevates the large computatonal effort of the proposed method. In summary, we hope ths work wll shed lght nto understandng the characterstcs of the call and data servces processes. Ths could lead to more effcent utlzaton of resources, better prcng strateges, and better targetng of advertsng campagns.

3 2 The Model Let us partton the perod of each day nto a fxed number of ntervals of a certan sze (we took half-hour ntervals). We assume λ(t) to be constant wthn each nterval, leadng to a pecewse constant functon. Let λ tj be the arrval rate at perod j n day t, and let us arrange the λ tj s n the form of a vector λ. Moreover, let T be the sze (n days) of the avalable call records data, and let T 0 be the subsequent number of days to be forecasted. Let N tj be the number of calls ntated at perod j of day t. We propose a Bayesan framework, where the pror would enforce contnuty and seasonal relatons. For example two adjacent perods should have close values of arrval rates. Also, perods correspondng to the same hour of the day of two adjacent days have to be farly smlar. Seasonal patterns can be enforced by assumng that a perod of the day has a smlar arrval rate as the same perod of the day seven days from now (same day of the week). Fnally, we need to make sure that the arrvals are wthn the average range of arrvals. These four condtons can be enforced to varyng degrees by the followng Bayesan pror probablty densty: p(λ) exp [ T +T0 Mj=1 t=1 (λ tj L) 2 T +T0 t=1 T +T0 t=2 T +T0 t=w +1 2σ 2 0 Mj=1 (λ tj λ t,j1 ) 2 2σ 2 1 Mj=1 (λ tj λ t1,j ) 2 2σ 2 2 Mj=1 (λ tj λ tw,j ) 2 ] 2σ 2 3 where L represents the average arrval rate durng the hstorcal data, W = 7 represents the length of the week, and λ t0 λ tm. Ths pror functon s essentally a multvarate normal functon: p(λ) = N (λ; µ, Σ) (1) where N (λ; µ, Σ) s a normal densty functon wth mean vector µ, and covarance matrx Σ. It s easy to derve a smple recursve algorthm to obtan the Σ, based on the above formulaton of the pror. Moreover, one can prove that all entres of Σ are nonnegatve. For an nhomogeneous Posson process, t s well-known that the lkelhood functon of the observatons s gven by: p(n H λ) = [ λ(t ) ] T e = [ N >0 = [ N >0 λ N 0 λ(t)dt (2) ] e T M t=1 λ (3) ] λ N e e T λ where s the perod length (we use half an hour), and e s a vector of all ones. We gnore the fact that outgong calls have a certan call duraton, durng whch new calls cannot be ntated. Ths stuaton calls for the use of observaton censorng, and ths complcates (4)

4 the analyss consderably. However, censorng s not requred f we are consderng short messages or ncomng calls where we lump together receved calls and calls that found the recever to be busy. Let us partton the λ vector nto two components: the hstorcal (or the tranng) component λ H, and the component to be forecasted λ F. Moreover, let N H be the vector of number of calls per tme slot correspondng to the hstorcal perod. Let us partton the mean vecotr and the covarance matrx along the lnes of hstorcal and forecasted parts: µ = ( ) µh µ F ( ) ΣHH Σ Σ = HF Σ T HF Moreover, let H be the subset of the hstorcal component that corresponds to nonzero entres of the N H vector,.e. the perods where calls have actually beng made. Ths means that λ H, N H, and Σ H H are respectvely the subvector and submatrx of the hstorcal component that corresponds to nonzero entres of the N H vector. Usng Bayesan analyss, the posteror estmate of the densty of λ F can be derved as: Σ F F p(λ F N H ) = p(n H λ H )p(λ F ) (7) p(n H ) p(nh λ H, λ F )p(λ H λ F )p(λ F )dλ H = (8) p(nh λ H )p(λ H )dλ H (5) (6) = p(nh λ H )p(λ H, λ F )dλ H p(nh λ H )p(λ H )dλ H I = I 1 I 2 (9) These nnocent -lookng ntegrals are n actualty very formdable to evaluate, because they are very hgh dmensonal. We have derved a formula that wll be descrbed below. For smplcty of notaton, let Σ H H R. Defne the covarance matrx Q, and the vectors γ and β, as Q = Σ H H Σ HF Σ 1 F F Σ T HF (10) γ = µ H Σ HH e (11) β = µ H Σ HH e + Σ HF Σ 1 F F (λ F + Σ T HF e µ F ) (12) where e s a vector of all ones. Assume that the followng product of terms can be expanded as: (λ H + β ) N H λ Hl + c 2 λ Hl + + β N H (13) l l S 1 l l S 2 where λ H and N H are the th component of respectvely λ H and N H. Also S 1 s the set of ndces who are represented n the hghest power of the product of λ s, where that set can have repeated ndces to account for the exponents N H (therefore the cardnalty of S 1 s the total number of calls n the hstorcal perod). A smlar defnton apples for S 2, S 3,

5 etc, except that t apples for the subsequent lower powers of the λ H products. Let K be the cardnalty of S. We defne a smlar formula, but wth the γ vector n place of β: (λ H + γ ) N H λ Hl + g 2 λ Hl + + γ N H (14) l l S 1 l l S 2 The posteror probablty p(λ F N H ) of Eq. (9) s derved as: I = N (λ F ; µ F Σ T HF e, Σ F F ) [ J 1 Q j Q kl... Q uv + c 2 J 2 Q j Q kl... Q uv β N ] H J 1 R j R kl... R uv + g 2 J 2 R j R kl... R uv γ N H (15) where the summatons J k are taken over all allocatons of the elements n the set S k nto dstnct pars (.e., j, k, l, u, v are dstnct elements of S k ). Only terms where K s even exst n ths formula, as odd terms are just excluded. 3 Smulatons We tested the developed approach by applyng t to the problem of forecastng moble call arrvals. Specfcally, we have a hstorcal data set consstng of 28 days. Each day s parttoned nto 48 half-hour ntervals. It s requred to forecast the arrval rates of each of next day s ntervals. We consdered data representng some users call profle. As a short glmpse of the result, Fgure 1 shows the densty of the forecast of the frst nterval n forecast horzon for a specfc user. Fgure 2 shows the densty of the forecast of the ffth nterval n forecast horzon for the same user. A thorough analyss and more smulatons results wll be presented as well. In summary, the proposed approach leads to effectve estmaton and forecastng. Fgure 1: Densty of the forecast of the frst nterval References [1] J. E. R. Cd and J. A. Achcar. Bayesan nference for nonhomogeneous Posson processes n software relablty models assumng nonmonotonc ntensty functons. Computatonal statstcs & data analyss 32.2 (1999):

6 Fgure 2: Densty of the forecast of the ffth nterval [2] M. Guda, R. Calabra, and G. Pulcn. Bayes nference for a non-homogeneous Posson process wth power ntensty law [relablty]. Relablty, IEEE Transactons on 38.5 (1989): [3] A. Kottas. A nonparametrc Bayesan approach to nference for non-homogeneous Posson processes. [4] L. Leems. Nonparametrc estmaton of the cumulatve ntensty functon for a nonhomogeneous Posson process. Management Scence 37.7 (1991): [5] J. Wenberg, L. Brown, and J. Stroud. Bayesan forecastng of an nhomogeneous Posson process wth applcatons to call center data. Journal of the Amercan Statstcal Assocaton (2007):