Thinning Algorithms for Simulating Point Processes

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1 Thinning Algorithms for Simulting Point Processes Yund Chen September, 2016 Abstrct In this tlk we will discuss the lgorithms for simulting point processes. The simplest point process is the (homogeneous) Poisson process, which hs n intensity function of constnt vlue λ. It cn be simulted by the sum of its interrrivl times. By llowing the intensity to vry, tking vlues given by deterministic function λ(t), we cn extend the Poisson process to the inhomogeneous cse. As proposed by (Lewis nd Shedler, 1979), inhomogeneous Poisson processes cn be simulted by thinning the points from the homogeneous versions. Further by llowing the intensity t time t to depend on ll informtion prior to t, we get the self-exciting processes such s the Hwkes processes. The simultion of such processes cn lso be performed by thinning, s given by (Ogt, 1981). 1 Introduction In this tlk we will discuss the simultion of point processes. A point process cn be thought of s being sequence of points on rel number line. The line represents time nd the points re the occurrence time of events. The ptterns how those points re locted gives us different point processes. The most commonly used nd simplest point process is the homogeneous Poisson process, which ssumes tht the points re evenly spred over ny intervl. It is perfectly resonble process to model the rrivl times of buses, ssuming you re stnding t bus stop. But it will not be good choice for modeling, sy erthqukes, becuse there re ftershocks fter the min ttck, or modeling occurrences of mrket events, which hs the clustering effect. The Hwkes process is developed for this purpose, nd our gol in this tlk is to introduced the simultion of Hwkes processes. Before we cn tlk bout point processes precisely, we need to define few concepts. 2 Fundmentls of Point Processes Definition 2.1. (Sigmn, 1995, p.1) Let (Ω, F, P) be probbility spce. A point process on R + is sequence of non-negtive rndom vribles {T k } k=1,2,..., such tht for ll k, T k T k+1. Here T k represents the occurrence times of events, nd such processes re clled temporl point processes. When the points re distributed in, sy two dimensionl spce, it is clled sptil point process. The elpsed time from n event to the next is clled the interrrivl time. Definition 2.2. (Ross, 1995, p.64) Consider point process {T k } k=1,2,.... Define W k = T k T k 1 with the convention T 0 = 0. The sequence {W k } k=1,2,... is clled the interrrivl times. Definition 2.3. (Sigmn, 1995, p.1, (1.2)) The process with right-continuous smple pths N(t) = k=1,2,... 1 {Tk t} is clled the counting process ssocited with the point process {T k } k=1,2,.... The counting process N(t) counts the number of occurrences up to nd including time t, nd N({t}) = N(t) N(t ) in the following. 1

2 Definition 2.4. (Dley nd Vere-Jones, 2003, p.47, Definition 3.3.II) A point process is clled simple when P {N({t}) = 0 or 1 for ll t} = 1. Definition 2.5. (Lst nd Brn, 1995, p.8, (1.5.1)) For point process {T k } k=1,2,... we define the point of explosion to be T = lim k T k nd the point process is clled non-explosive if T k lmost surely s k. We ssume the point processes discussed in this rticle re both simple nd non-explosive. Definition 2.6. (Grndell, 1997, p.6) A quntity which, for smll h, is of lower order thn h, is denoted s o(h), i.e. o(h) lim h 0 + h = 0. Definition 2.7. The intensity of point process N(t) is defined by Khinchin (1960) s P {N(t, t + h] > 0} λ(t) = lim. h 0 + h Intuitively λ(t) mesures the rte of occurrence of points becuse in most cses (under certin ssumptions) P {N(t, t + h] > 0} E[N(t, t + h)] lim = lim h 0 + h h 0 + h nd on the right hnd side we hve number over time, or rte. 3 Homogeneous Poisson Processes 3.1 Definition There re lterntive definitions for homogeneous Poisson processes. homogeneous Poisson process bsed on the intensity. The following definition defines Definition 3.1. (Ross, 2009, pp , Definition 5.3, Theorem 5.1) The point process N is (homogeneous) Poisson process with rte λ, λ > 0, if nd only if, for ll t 0 nd h 0 +, (i) N(0) = 0. (ii) The process hs independent increments. (iii) P {N(t + h) N(t) = 1} = λh + o(h). (iv) P {N(t + h) N(t) 2} = o(h). (i) sys the process strts from 0 t time 0. (ii) mens the increments re independent over disjoint bounded subintervls. (iii) simply sys tht λ is the intensity nd (iv) is clled orderly, which is the probbility distribution property nlogously to the smple pth property of simpleness in Definition

3 3.2 Properties Number of Occurrence The following property sttes tht the number of points from homogeneous Poisson process over ny intervl is Poisson distributed. Proposition 3.1. The number of events in ny intervl of length t is Poisson distributed with men λt. Tht is, for ll s, t 0, λt (λt)n P {N(t + s) N(s) = n} = e for n = 0, 1,.... Proof. Define p n (t) = P {N(t) = n} for t 0 nd integers n 0. Conditions (iii), (iv) of Definition 3.1 imply tht the process N is sttionry. Together with condition (ii) of Definition 3.1, it mens tht N hs both sttionry (not function of time t) nd independent increments, so we hve p 0 (t + h) = P {N(t + h) = 0} = P {N(t) N(0) = 0, N(t + h) N(t) = 0} [independent increments] = P {N(t) N(0) = 0} P {N(t + h) N(t) = 0} [sttionry] = P {N(t) = 0} P {N(h) = 0} = p 0 (t) (1 λh + o(h)) nd so, Let h 0 +, we hve the ODE or equivlently, with the generl solution By setting t = 0, we hve nd so, Now for n 1, p 0 (t + h) p 0 (t) h dp 0 (t) = p 0 (t)λ + o(h) h. = p 0 (t)λ dp 0 (t) p 0 (t) = λ p 0 (t) = Ce λt. 1 = P {N(0) = 0} = p 0 (0) = C p 0 (t) = e λt. (1) p n (t + h) = P {N(t + h) = n} = P {N(t) = n, N(t + h) N(t) = 0} + P {N(t) = n 1, N(t + h) N(t) = 1} +P {N(t + h) = n, N(t + h) N(t) 2} = P {N(t) = n} P {N(h) = 0} + P {N(t) = n 1} P {N(h) = 1} + o(h) = p n (t)(1 λh + o(h)) + p n 1 (t)(λh + o(h)) + o(h) = (1 λh)p n (t) + λhp n 1 (t) + o(h) (2) 3

4 In the derivtion bove we used the fct tht which is true, becuse, P {N(t + h) = n, N(t + h) N(t) 2} = o(h) P {N(t + h) = n, N(t + h) N(t) 2} P {N(t + h) N(t) 2} lim lim = 0. h 0 + h h 0 + h Rerrnging (2), we hve p n (t + h) p n (t) = λp n (t) + λp n 1 (t) + o(h) h h nd by letting h 0 + we hve dp n (t) = λp n (t) + λp n 1 (t) or equivlently, dp n (t) + λp n (t) = λp n 1 (t) (3) which is first order liner ODE nd by multiplying the integrting fctor e λt to both sides, we hve d ( e λt p n (t) ) = λe λt p n 1 (t) (4) We now use mthemticl induction to show tht (4) implies λt (λt)n p n (t) = e for ll integers n 1. When k = 1, (4) becomes d ( e λt p 1 (t) ) becuse of (1). Solving the ODE for p 1 (t) we get nd by setting t = 0 we hve so = λ e λt p 1 (t) = λt + C 0 = P {N(0) = 1} = p 1 (0) = C p 1 (t) = e λt λt nd (5) holds. Assuming (5) holds for k = n 1 1, then nd (4) becomes which yields or equivlently, By setting t = 0 we hve d ( e λt p n (t) ) λt (λt)n 1 p n 1 (t) = e (n 1)! = λe λt λt (λt)n 1 e (n 1)! = λ (λt)n 1 (n 1)! = λn (n 1)! tn 1 e λt p n (t) = λn (n 1)! tn n + C λt (λt)n p n (t) = e + Ce λt. 0 = P {N(0) = n} = p n (0) = C which proves tht (5) holds true for ll integers n 0, nd by sttionrity, for ny s, t 0, λt (λt)n P {N(t + s) N(s) = n} = P {N(t) N(0) = n} = P {N(t) = n} = e which completes the proof. 4 (5)

5 3.2.2 Interrrivl Times Immeditely we hve the following property bout the interrrivl times. Proposition 3.2. (Ross, 1995, p.64, Proposition 2.2.1) The interrrivl times {W k } k=1,2,... of (homogeneous) Poisson process with rte λ > 0, re independent identiclly distributed exponentil rndom vribles hving men 1/λ. Proof. From Proposition 3.1, P {W 1 > t} = P {N(t) = 0} = e λt so W 1 is exponentilly distributed with prmeter λ. For W 2, by independent increments, we hve nd P {W 2 > t W 1 = s} = P {N(t + s) N(s) = 0 N(s) N(s ) = 1} = P {N(t + s) N(s) = 0} = P {N(t) = 0} = e λt P {W 2 > t} = E [P {W 2 > t W 1 }] = e λt so W 2 is exponentilly distributed with prmeter λ. Also the fct tht P {W 2 > t} = P {W 2 > t W 1 = s} implies the independence between W 2 nd W 1. Repeting the sme rgument proves the proposition. Nturlly we hve the following lgorithm for generting the points t k in homogeneous Poisson process by generting the interrrivl times w k nd tking the sum t k = k i=1 w i. This is sometimes clled the interrrivl scheduling. Assuming there is rndom number genertor uniform(0,1) tht genertes uniformly distributed rndom vribles on (0, 1). Algorithm 1: Simultion of Homogeneous Poisson Process with Rte λ, on [0, T ]. Input: λ, T 1 Initilize n = 0, t 0 = 0; 2 while True do 3 Generte u uniform(0,1); 4 Let w = ln u/λ; // so tht w exponentil(λ) 5 Set t n+1 = t n + w; 6 if t n+1 > T then 7 return {t k } k=1,2,...,n 8 else 9 Set n = n + 1; 10 end 11 end Joint Density From Proposition 3.1 we cn esily derive the joint density of the occurrence times. Theorem 3.3. The joint density tht exctly n events occur in the intervl [0, T ] for Poisson process with rte λ tking vlues 0 < t 1 < t 2 < < t n T is f T1,...,T n (t 1,..., t n ; T n+1 > T ) = λ n e λt. 5

6 Proof. For k = 1, 2,..., n + 1, let W k = T k T k 1 be the interrrivl times nd w k = t k t k 1 be the reliztion, with the convention T 0 = t 0 = 0. By Proposition 3.2, {W k } k=1,2,...,n+1 re independent identiclly distributed exponentil rndom vribles with prmeter λ nd density nd til CDF f W (w) = λe λw P {W > t} = F (t) = 1 F (t) = e λt so by the chnge of vribles formul (see Csell nd Berger, 2002, p.158, (4.3.2) nd p.185, (4.6.7)), the joint density function of T 1,..., T n is f T1,...,T n (t 1,..., t n ; T n+1 > T ) = f W1,...,W n (w 1,..., w n ) J 1 P {W n+1 > T t n } = n k=1 λ(t tn) f W (t k t k 1 ) e where n = λ n e λ k t k 1 ) k=1(t e λ(t t n) = λ n e λt J ij = dt i dw j nd J = 1 becuse J is lower tringulr mtrix with ll non-zero elements being 1. 4 Inhomogeneous Poisson Processes 4.1 Definition By llowing the intensity λ in Definition 3.1 to vry ccording to deterministic function of t, we hve the following definition for inhomogeneous Poisson process. Definition 4.1. (Ross, 2009, p.339, Definition 5.4) The point process N is sid to be n inhomogeneous Poisson process with intensity function λ(t) 0, t 0, if (i) N(0) = 0. (ii) The process hs independent increments. (iii) P {N(t + h) N(h) = 1} = λ(t)h + o(h). (iv) P {N(t + h) N(h) 2} = o(h). Anlogously, we hve similr property for inhomogeneous Poisson processes. Proposition 4.1. Let the point process N be n inhomogeneous Poisson process with intensity function λ(t), then N(t) follows Poisson distribution with prmeter t λ(s)ds, i.e. 0 P {N(t) = n} = e t 0 λ(s)ds ( t 0 λ(s)ds ) n Proof. The proof is similr to the second prt of the proof for Proposition 3.1. Anlogously to (1) for the homogeneous cse, we now hve p 0 (t) = e t 0 λ(s)ds (6) 6

7 by replcing λ with λ(t) in the derivtion. Also (3) becomes dp n (t) + λ(t)p n (t) = λ(t)p n 1 (t) nd by multiplying e t 0 λ(s)ds to both sides, we now hve the ODE ( d e ) t 0 λ(s)ds p n (t) We gin use mthemticl induction by ssuming p n 1 (t) = = λ(t)e t 0 λ(s)ds p n 1 (t). e t 0 λ(s)ds ( t 0 λ(s)ds ) n 1 (n 1)! so tht solving which yields ( d e ) t 0 λ(s)ds p n (t) = λ(t) ( t 0 λ(s)ds ) n 1 (n 1)! ( ) t n p n (t) = e t 0 λ(s)ds 0 λ(s)ds. b Apprently, the number of points in the intervl [, b] follows Poisson distribution with prmeter λ(s)ds, or more precisely, P {N(b) N() = n} = e b λ(s)ds ( b λ(s)ds ) n Letting n = 0 we get the probbility tht there is no point in the intervl [, b] to be P {N(, b] = 0} = e b λ(s)ds (7) which determines the lw of occurrence for the next point. For simple point process, points occur one by one nd by the checking this condition we cn mke sure whether the point process is n inhomogeneous Poisson process. 4.2 Simultion To simulte the points in n inhomogeneous Poisson process, we cn simulte the points sequentilly. We only need to gurntee tht (7) is stisfied nd the following is one such lgorithm, sometimes clled the Lewis thinning lgorithm. Figure 1 shows n illustrtive exmple of simulted inhomogeneous Poisson process on the intervl [0, 2π], with the intensity function λ(t) = 1 + sin(t) shown s the solid curve. The rte of the homogeneous Poisson process which domintes λ(t) is set to be λ = sup 0 t 2π λ(t) = 2 nd is shown s the horizontl dshed line. The simulted points for the homogeneous Poisson process re shown s circles, while those ccepted s the points for the inhomogeneous Poisson process re lbeled with cross mrks. Verticlly bove ech point on the x-xis, there is nother point plotted with height of D λ, where D is the uniform rndom vrible used in line 7 of Algorithm 2. The points bove λ(t) re those rejected nd re mrked with circles; the ones below ccepted nd lbeled with cross mrks. The following theorem describes the lgorithm mthemticlly nd it mkes sure the resulting point process stisfies (7). 7.

8 Algorithm 2: (Lewis nd Shedler, 1979, p.7, Algorithm 1) Simultion of n Inhomogeneous Poisson Process with Bounded Intensity Function λ(t), on [0, T ]. Input: λ(t), T 1 Initilize n = m = 0, t 0 = s 0 = 0, λ = sup 0 t T λ(t); 2 while s m < T do 3 Generte u uniform(0,1); 4 Let w = ln u/ λ; // so tht w exponentil( λ) 5 Set s m+1 = s m + w; // {s m } re points in the homo. Poisson process 6 Generte D uniform(0,1); 7 if D λ(s m+1 )/ λ then // ccepting with probbility λ(s m+1 )/ λ 8 t n+1 = s m+1 ; // {t n } re points in the inhomo. Poisson process 9 n = n + 1; // updting n to the index of lst point in {t n } 10 end 11 m = m + 1; // updting m to the index of lst point in {s m } 12 end 13 if t n T then 14 return {t k } k=1,2,...,n 15 else 16 return {t k } k=1,2,...,n 1 17 end Theorem 4.2. Consider homogeneous Poisson process N(t) with intensity function λ. Let t 1, t 2,..., t N(T ) be the points of the process in the intervl (0, T ]. Suppose tht for 0 t T, 0 λ(t) λ. For k = 1, 2,..., N(T ), delete the point t k with probbility 1 λ( t k )/ λ; then the remining points form point process N(t) stisfying (7). Proof. Consider the probbility where there re n points in the point process N(t) nd ech of them is deleted nd denote it s p n. From Theorem 3.3 the joint density of n points tking vlues t 1 < < t n b is λ n e λ(b ). Consider the unordered vlues s 1,..., s n such tht {s 1,..., s n } = {t 1,..., t n }, where ech s k tkes vlue in [, b]. The probbility p n = b b t 1 = e λ(b ) = e λ(b ) = e λ(b ) b t n 1 λn e λ(b ) b b b n k=1 ( 1 λ(t ) k) 1 2 n λ n ( λ λ(tk ) ) ds 1 ds 2 ds n k=1 ( b ( λ λ(tk ) ) ds ( λ(b ) b ) n λ(s)ds Now the number of points in the point process N(t), or n, cn be ny non-negtive integer nd the sum ( n b p n = e λ(b ) λ(b ) λ(s)ds) / n=0 n=0 ) n = e λ(b ) e λ(b ) b λ(s)ds = e b λ(s)ds equls the probbility tht there is no point in the intervl [, b], nd (7) is stisfied. 8

9 Figure 1: An illustrtive exmple of simulting n inhomogeneous Poisson process with intensity function λ(t) = 1 + sin(t) on the intervl [0, 2π] using Algorithm 2. The intensity functions for the homogeneous nd inhomogeneous Poisson processes re shown s the horizontl dshed line nd the solid curve, respectively. On the x-xis, the simulted points for the homogeneous Poisson process re mrked by circles, nd the points lbeled s cross mrks, re the simulted points for the inhomogeneous Poisson process. Verticlly bove ech point on the x-xis, point with height of D λ is lso plotted, with D being the uniform rndom vrible generted during the simultion. The points bove λ(t) re rejected nd mrked with circles, while the points below re ccepted s the points for the inhomogeneous Poisson process nd re mrked with crosses. Remrk 4.1. Notice tht in Theorem 4.2 the only constrint on the constnt λ is tht λ(t) λ for ll t in the domin of interest, [0, T ]. Becuse λ is the intensity of the homogeneous Poisson process N, so lrger vlue of λ will led to more generted points in N. At the mentime, λ determines the probbility t which point is ccepted, nd lrger λ corresponds to smller probbility of point cceptnce. As result, smller vlue of λ is preferred in prctice when using Algorithm 2, for higher efficiency (less itertions of the while-loop) nd hence λ = sup t [0,T ] λ(t) is often the choice. Becuse Poisson process hs independent increments, i.e. P {N(s, t] = 0 N(0, s] = 0} = P {N(0, t] = 0} P {N(0, s] = 0} t = e 0 λ(u)du e s 0 λ(u)du = e t s λ(u)du = P {N(s, t] = 0} 9

10 for ny 0 < s < t T, nd the genertion of points is sequentil (the vrible s m is incresing through the itertions of the while-loop), the choice of λ cn be modified s λ = sup t [sm,t ] λ(t), to llow djustment of λ when generting the next point in N. 5 Hwkes Processes 5.1 Definition Definition 5.1. (Donld L. Snyder, 1991, p.287) A point process is clled self-exciting if the intensity λ( ) depends not only on time t but lso the entire pst of the point process. The entire pst, or history of point process, is defined mthemticlly by the definition given below. Definition 5.2. (Grndell, 1997, p.51) For ny process N(t), the nturl filtrtion F N = ( Ft N ; t 0 ) is defined by Ft N = σ {N(s); s t}. In other words, Ft N up to time t. is the σ-lgebr generted by N up to time t, nd represents the internl history of N With this definition, we cn define the intensity λ( ) for self-exciting point process nlogously s (2.7). However, more ppropritely here, it should be clled the intensity process, becuse λ( ) is itself rndom process whose smple pth depends on the reliztion of N( ). Definition 5.3. (Lst nd Brn, 1995, p.10, (1.6.4)) Let N(t) be point process with nturl filtrtion Ft N. The left-continuous process defined by λ(t Ft N P {N(t + h) N(t) > 0 Ft N } ) = lim h 0 + h is clled the stochstic intensity function of the point process. The reson for using left continuity is connected with predictbility: if the conditionl intensity hs discontinuity t point of the process, then its vlue t tht point should be defined by the history before tht point, not by wht hppens t the point itself (Dley nd Vere-Jones, 2003, p.232). The Hwkes processes introduced below is n exmple of self-exciting point processes. Definition 5.4. (Hwkes, 1971, p.84, (7) nd (8)) A univrite simple point process N(t) stisfying (i) N(t) = 0. (ii) λ(t) is left-continuous stochstic process given by the Stieltjes integrl λ(t) = µ + t αe β(t s) dn(s) = µ + αe β(t t k) 0 {k:t k <t} (8) where µ > 0 nd 0 < α < β. (iii) λ(t) is the stochstic intensity of the point process P {N(t + h) N(t) = 1 Ft N } = λ(t)h + o(h) (iv) The point process is orderly P {N(t + h) N(t) 2 Ft N } = o(h) is clled univrite Hwkes process with exponentil decy on [0, ). 10

11 From Definition 5.4, it is esy to see tht for given reliztion, λ(t) is piece-wisely non-incresing with jumps of size α t occurrences of points. More properties re shown in Figure 2 which illustrtes the smple pths of the conditionl intensity λ(t) nd the ssocited counting process N(t) for given univrite Hwkes process with exponentil decy tht hs prmeters µ = 1.2, α = 0.6 nd β = 0.8. The left-continuous conditionl intensity, which strts from nd keeps t the bse intensity µ = 1.2 until the occurrence of the first point, is shown in the top pnel. After ech occurrence, the intensity jumps with size of α = 0.6 nd then immeditely strts decying t rte determined by β = 0.8. On the other hnd, the ssocited right-continuous counting process, which is step function tht jump by 1 t ech occurrence, is shown in the bottom pnel. 5.2 Simultion From (8) it is obvious tht given the informtion of the first k points, t 1,..., t k, the intensity λ(t) is deterministic on [t k, t k+1 ], where t k+1 is the loction of the next point which is stochstic. As result, the genertion of the next point in Hwkes process cn be considered s generting the first point in n inhomogeneous Poisson process. Ogt modified Algorithm 2 to simulte Hwkes processes. This is sometimes clled Ogt s modified thinning lgorithm. In his modifiction, λ is llowed to be djusted s described in Remrk 4.1. Algorithm 3: (Ogt, 1981, p.25, Algorithm 2) Simultion of Univrite Hwkes Poisson with Exponentil Kernel γ(u) = αe βu, on [0, T ]. Input: µ, α, β, T 1 Initilize T = Ø, s = 0, n = 0; 2 while s < T do 3 Set λ = λ(s + ) = µ + τ T αe β(s τ) ; 4 Generte u uniform(0,1); 5 Let w = ln u/ λ; // so tht w exponentil( λ) 6 Set s = s + w; // so tht s is the next cndidte point 7 Generte D uniform(0,1); 8 if D λ λ(s) = µ + τ T αe β(s τ) then // ccepting with prob. λ(s)/ λ 9 n = n + 1; // updting the number of points ccepted 10 t n = s; // nming it t n 11 T = T {t n }; // dding t n to the ordered set T 12 end 13 end 14 if t n T then 15 return {t k } k=1,2,...,n 16 else 17 return {t k } k=1,2,...,n 1 18 end Figure 3 illustrtes the use of Algorithm 3 for univrite Hwkes process with exponentil decy. The prmeters re µ = 1.2, α = 0.6 nd β = 0.8. All the cndidte points re mrked on the x-xis where the verticl dotted lines re, mong which the ones ccepted to be the points for the Hwkes process re lbel with t k where k = 1, 2,..., 8. The conditionl intensity λ(t) is plotted s the solid curve. The left-continuous piece-wisely constnt λ used in ech itertion inside the while loop t line 2 of Algorithm 3 is plotted s the dshed lines. On ech verticl dotted line, there re two points plotted. The lower ones re mrked with tringles nd represent the vlues of the uniform rndom vrible D (rnges from 0 to 1) generted t line 7 of Algorithm 3; the upper ones represent the vlues of D λ (rnges from 0 to λ) t line 8 nd re either lbeled with cross mrks for those ccepted (when D λ λ( )) or with circles for those rejected. 11

12 12 Figure 2: An illustrtive exmple of the left-continuous conditionl intensity λ(t) (top pnel) nd the ssocited right-continuous counting process N(t) (bottom pnel) for univrite Hwkes process with exponentil decy tht hs prmeters µ = 1.2, α = 0.6 nd β = 0.8. The conditionl intensity strts from nd keeps t the bse intensity µ = 1.2 until the occurrence of the first point. After ech occurrence, the intensity jumps with size of α = 0.6 nd then immeditely strts decying t rte determined by β = 0.8. On the other hnd, the ssocited counting process is step function tht jump by 1 t ech occurrence.

13 Figure 3: An illustrtive exmple of simulting univrite Hwkes process with exponentil kernel γ(u) = αe βu where α = 0.6, β = 0.8 nd bse intensity u = 1.2, using Algorithm 3. All the cndidte points re mrked on the x-xis where the verticl dotted lines re, mong which the ones ccepted to be the points for the Hwkes process re lbel with t k where k = 1, 2,..., 8. The conditionl intensity λ(t) is plotted s the solid curve. The left-continuous piece-wisely constnt λ used in ech itertion inside the while loop t line 2 of Algorithm 3 is plotted s the dshed lines. On ech verticl dotted line, there re two points plotted. The lower ones re mrked with tringles nd represent the vlues of the uniform rndom vrible D (rnges from 0 to 1) generted t line 7 of Algorithm 3; the upper ones represent the vlues of D λ (rnges from 0 to λ) t line 8 nd re either lbeled with cross mrks for those ccepted (when D λ λ( )) or with circles for those rejected. References Csell, G. nd R. Berger Sttisticl Inference, Duxbury dvnced series in sttistics nd decision sciences. Thomson Lerning. Dley, D. nd D. Vere-Jones An Introduction to the Theory of Point Processes, Volume 1. Springer. Donld L. Snyder, M. I. M Rndom Point Processes in Time nd Spce, Springer Texts in Electricl Engineering, 2 edition. Springer-Verlg New York. Grndell, J Mixed Poisson Processes (Chpmn & Hll/CRC Monogrphs on Sttistics & Applied Probbility). Chpmn nd Hll/CRC. 13

14 Hwkes, A. G Spectr of some self-exciting nd mutully exciting point processes. Biometrik, 58(1): Khinchin, A Mthemticl Methods in the Theory of Queueing, Griffin s sttisticl monogrphs & courses no. 7. Hfner. Lst, G. nd A. Brn Mrked Point Processes on the Rel Line: The Dynmicl Approch (Probbility nd Its Applictions). Springer. Lewis, P. A. W. nd G. S. Shedler Simultion of nonhomogeneous Poisson processes by thinning. Nvl Res. Logistics Qurt, 26: Ogt, Y On lewis simultion method for point processes. IEEE Trns. Inform. Theory, 27(1): Ross, S. M Stochstic Processes. Wiley. Ross, S. M Introduction to Probbility Models, Tenth Edition. Acdemic Press. Sigmn, K Sttionry Mrked Point Processes: An Intuitive Approch (Stochstic Modeling Series). Chpmn nd Hll/CRC. 14