An Environmentally-Adaptive Hawkes Process with An Application to COVID-19
AAn Environmentally-Adaptive Hawkes Process withAn Application to COVID-19
Tingnan Gong, Yu Chen and Weiping Zhang
Department of Statistics and FinanceUniversity of Science and Technology of ChinaHefei, Anhui, P.R. ChinaEmail: [email protected]
Abstract —We proposed a new generalized model based on theclassical Hawkes process with environmental multipliers, whichis called an environmentally-adaptive Hawkes (EAH) model.Compared to the classical self-exciting Hawkes process, theEAH model exhibits more flexibility in a macro environmentallytemporal sense, and can model more complex processes byusing dynamic branching matrix. We demonstrate the well-definedness of this EAH model. A more specified version of thisnew model is applied to model COVID-19 pandemic data throughan efficient EM-like algorithm. Consequently, the proposed modelconsistently outperforms the classical Hawkes process.
I. I
NTRODUCTION
The self-exciting point process has been proposed in variousfields such as epidemic forcasting (see [1], [2]). The Hawkesprocess is a classical self-exciting point process, in which,each event i occurs at time t i , and generates new eventswith stochastic intensity φ ( t − t i ) , where φ ( t ) is a generickernel. Thus Hawkes processes prove useful in modeling somereal-world complex systems with clusters (see [3], [4]). Thebranching matrix defined in a multivariate Hawkes processis responsible to describe the complex excitements betweendimensions (nodes) in a network. Particularly, the ( i, j ) entryin the branching matrix is non-negative, measuring the excite-ment from node j to node i . Whereas the branching matrixis static in a classical Hawkes process, the rigidity arises in atemporal sense.Many of current work utilized the Hawkes process to fit theCOVID-19 pandemic (see [5], [6]. [7]). With the self-excitingproperty, the Hawkes process is a seemingly proper tool tofit the COVID-19 spread. However, as shown in Section V,the self-exciting property tends to build an explosive Hawkesprocess and overestimate the deterioration. In many areas ofthe real world, indoor public spaces were closed and peoplewere required to wear a mask. Even many cities were lockeddown. Soon the spread of virus would slow down and exhibitnot like a typical Hawkes process anymore. In this paper,we expect to apply our proposed model to the COVID-19 pandemic to capture the time-varying epidemic control,which is brought by the alertness of the society and themacro countermove of the local governments. Thus decayingenvironmental multipliers are considered when constructingthe new model.We develop an environmentally-adaptive Hawkes (EAH)model with dynamic multipliers. The dynamic multiplier is called the environmental multiplier, which is a time-varyingfunction. By choosing proper environmental multipliers, themodel exhibits various temporal flexibility. The proposed EAHmodel is a novel and sophisticated version of the Hawkesmodel. We firstly provides a cluster representation of EAHmodel to justify its well-definedness (existence). From theanalysis of the existence, the sufficient condition ensuringthe non-explosiveness of an EAH model is given. Then westate the uniqueness of the EAH model. Also based on [8],an EM-like algorithm is developed to efficiently estimatesthe parameters in the environmentally-adaptive Hawkes modelwith decaying multipliers (EAHDM), a pandemic-specifiedversion of the EAH model.The rest of this paper is organized as follows. Section IIintroduces the background knowledge on the Hawkes process.In Section III, we formally construct the EAH model andclarify its existence and uniqueness. Several properties relatedto the probability generation functional (p.g.fl. see [10]) aregiven. Also, the multi-dimensional EAH and EAHDM modelare briefly introduced. In Section IV, we develop the EM-like algorithm. In Section V, we demonstrate the proposedmethodology via both simulations and the COVID-19 data.II. B ACKGROUND An M -dimensional Hawkes process is denoted by H ( t ) =( H ( t ) , · · · , H M ( t )) (cid:62) , which can be fully characterized byits intensity η = [ η ( t ) , η ( t ) , . . . , η M ( t )] (cid:62) as follows, η ( t ) = µ + (cid:90) t ( A (cid:12) Φ( t − τ )) d H ( τ ) , (1)where (cid:12) represents the Hadamard (elementwise) multiplica-tion and µ is the intensity vector of the immigrant process. Theconcept of the immigrant process (see [9]) will be further intro-duced in Section III. A = ( α ij ) M × M is the branching matrix. [Φ i,j ( t )] Mi,j =1 is the kernel function matrix, or fertility functionmatrix, which provides a useful tool to characterize the influ-ence of the history information and the effect from node j tonode i . d H ( τ ) = (cid:2) d H ( τ ) , d H ( τ ) , . . . , d H M ( τ ) (cid:3) (cid:62) is theRadon-Nykodym differential vector of the Hawkes process,which is also a counting measure vector. The intensity ofa point process can be intuitively understood as the rateof the arrivals. From (1), given a fixed timing τ , the in-tensity of a Hawkes process is stimulated by an increment a r X i v : . [ s t a t . A P ] J a n (cid:12) Φ( t − τ )d H ( τ ) . Now that the rate is increased due to thenew arrival, the next arrival comes sooner in the probabilitysense. This is the so-called "self-exciting" property and thereason that the Hawkes process is commonly used to simulatethe time series with clusters.The branching matrix A commonly can be considered as adefault setting of the whole network. With the growth of M ,the Hawkes process can describe rather a complex network ina space sense. However, in a temporal perspective, a constantmatrix A is too rigid to capture the effects from the changingfactors of the macro outer environment, for example, theepidemic control policies from local governments. Thus wedevelop an environmentally adaptive Hawkes (EAH) modelwith a time-varying multiplier A ( t ) .III. E NVIRONMENTALLY A DAPTIVE H AWKES MODEL
In this section we develop the EAH model along with itsexistence, uniqueness and some interesting properties relatedto the p.g.fl.. Our theoretical discussion is restrained to one-dimension for convenience.
A. Existence and Uniqueness of an EAH Model
Definition 1 (An EAH process) . An univariate point process { N ( t ) , t ≥ } is an EAH process, if it satisfies N (0) = 0 . The intensity λ ( t ) is given by the Stieltjes integral λ ( t ) = µ + α ( t ) × φ ( t ) ∗ d N ( t ) , (2) where µ > and the environmental multiplier α ( t ) ≥ , ∗ means the convolution operation and φ ( t ) ≥ is thekernel function. In Definition 1, α ( t ) serves as an characterization of thetemporal evolution of the macro outer environment. The kenel φ ( t ) is of the similar role as in a Hawkes process. Let α ( t ) = 1 , the EAH model reduces to a Hawkes process, whichcan be considered as a cluster process defined in Section 3.2in [10]. The process of cluster centers N c ( t ) , is a Poissonprocess with rate µ . The arrivals of N c ( t ) are often calledimmigrants. Each immigrant produces a subsidiary process,which is also known as a cluster. Actually, in one cluster,any individual reproduces the next generation, including theimmigrant. Thus the cluster is formed by the arrivals of all-generation descendants of an immigrant. For an event arrives at τ , the process of its direct descendants is constructed as a non-homogenous Poisson process with the intensity φ ( t − τ ) . Thesummation of the intensities of the process of cluster centersand all clusters gives the intensity of a Hawkes process.Clusters in the Hawkes process are identical processes afterdifferent temporal translations. Here "identical" means thateach cluster exhibits the same statistical structure leadingto same properties, for example, the mean size of clusters.Define s = (cid:82) ∞ φ ( τ )d τ , which is the mean size of a non-homogenous Poisson process with intensity φ ( t ) , namely themean size of the direct descendants of any event, includingthe immigrants. Focus on a cluster and s equals to the mean size of the first generation which is formed by the directdescendants of the immigrant. Iteratively, s the mean size ofthe second generation and so on. The mean size of a clusteris the summation of all generations, namely (cid:80) ∞ n =0 s n , whichconverges with ≤ s < .By the same method as in the Hawkes process, we willconstruct a cluster process which possesses the intensity (2) tojustify the existence of EAH model. In a cluster representationof the EAH model, the "identical" property in a Hawkesprocess does not exist anymore. The mean size of a directdescendants process is related to the specific arrival time of thecorresponding parent. Then the mean size of each generationdepends on the arrivals time in the former generation itera-tively. Such a high dependence on the history results in hugetrouble in direct calculation of the mean cluster size. Hencewe almost surely bound the mean cluster sizes to ensure thenon-explosiveness, namely that the clusters are almost surelyfinite. Denote m ( u ) = (cid:90) ∞−∞ α ( t ) φ ( t − u ) dt, which represents the mean size of the first generation if theancestor of the cluster arrives at u . Theorem 1 (Existence) . Consider the intensity λ ( t ) definedin (2), if there exists an m ( < m < ) such that m ( u ) ≤ m for all u > , then there exists a non-explosive cluster processwith λ ( t ) satisfying λ ( t ) ≤ µ − m a.s.. Proof.
See Appendix.In real applications, it is natural to have a beginning point ofthe process. So we clarify the uniqueness of EAH model withan arbitrarily fixed beginning point, which is straight supportedby the first lemma in [9].
Theorem 2.
Given an arbitrarily fixed start time, there existsat most one simple point process, whose complete intensity isgiven by (2).B. Other properties
Now we describe the p.g.fl. of the EAH model, based onwhich we further derive the distribution of the residual timeof an EAH model and the length of a particular cluster in anEAH process.
Theorem 3.
The p.g.fl. of the EAHs has the form G ( z ( · )) = exp (cid:26)(cid:90) ∞−∞ µ [ F ( z ( · ) | t ) −
1] d t (cid:27) , (3) where F ( z ( · ) | t ) is the p.g.fl. of a cluster generated by animmigrant arriving at time t , and including that immigrant.The functional F ( z ( · ) | t ) satisfies the functional equation F ( z ( · ) | t ) = z ( t ) exp (cid:26)(cid:90) ∞ t [ F ( z ( · ) | τ ) − α ( τ ) φ ( τ − t )d τ (cid:27) . (4) roof. See Appendix.Now we can have an iteration formula of the forwardrecurrence time, also known as the residual time of an EAHmodel.
Theorem 4.
The forward recurrence time, which is observedfrom time y , L y of an EAH model has survivor function R L y ( l ) = P ( L y > l ) given by R L ( l ) = exp (cid:26) µ (cid:90) y −∞ [ γ ( y, l | t ) − dt − µl (cid:27) , where γ ( y, l | t ) satisfies the equation γ ( y, l | t ) = exp (cid:110)(cid:82) y + ly [ γ ( y, l | τ ) − α ( τ ) φ ( τ − t ) dτ (cid:111) , y ≥ t ;1 , y ≤ t − l ;0 , elsewhere . Proof.
See Appendix.Also, the p.g.fl. of an EAH determines the length of acluster, i.e. the time between the immigrant and the lastindividual in this cluster. If the immigrant arrives at t , wedenote the length of the cluster as J t . Theorem 5.
The distribution function D J t ( y ) = P ( J t ≤ y ) of the length of a cluster, whose ancestor arrives at time t ,satisfies, for any y ≥ , D J t ( y ) =exp {− m ( t ) + (cid:90) t + yt D J τ ( y + t − τ ) α ( τ ) φ ( t − τ )d τ } . (5) Proof.
See Appendix.
C. Multivariate EAH and EAHDM models
Similarly to the multivariate Hawkes process, we character-ize a multivariate EAH model by its intensity defined as λ ( t ) = µ + (cid:90) t α ( t ) (cid:12) Φ( t − τ )d N ( τ ) , t ≥ , (6)where α ( t ) = ( α ij ( t )) M × M is the environmental multipli-ers matrix. And the kernel function matrix, [Φ i,j ( t )] Mi,j =1 ischosen to be an exponential kernel matrix, namely Φ i,j ( t ) =exp {− β i,j t } . For an EAH model applied on COVID-19, wereduce our EAH to an EAHDM. Consider an M -dimensionalmodel, λ ( t ) = µ + d ( t ) (cid:90) t A (cid:12) Φ( t − τ )d N ( τ ) , t ≥ , (7)where A ∈ R M × M with all entries being non-negative, and d ( t ) : R → R is a non-negative and decreasing function. InCOVID-19 pandemic, the environment turned adverse to thespread of the virus due to the alertness of the society and thecontrol of the government. Then this characteristic is revealedby the user-specified decaying function d ( t ) . IV. EM- LIKE A LGORITHM
We develop the EM-like algorithm for the EAHDM modelbased on the algorithm in [8]. Considering a simple pointprocess, all arrivals are distinct ordered by { t , t , . . . , t n } .The vector { u , u , . . . , u n } with u j ∈ { , , . . . , M } rep-resents the corresponding dimensions (nodes) each arrivalbelongs to. Define auxiliary probabilities [ p ij ] n × n satisfying (cid:80) j ≤ i p ij = 1 , ∀ i ∈ { , , . . . , n } .Similarly to [11], let L ( A , d ( t ) , β ) be the log likelihood ofthe multivariate EAH model. By the Jensen inequality in theconcave case, we have L ( A , d ( t ) , β ) ≥ (cid:88) t i In this section, we investigate the performance of ourmethod via numerical studies. In simulation, we will generatedata from an EAHDM model and estimate A by the EM-likealgorithm. In real data experiments, we will train a Hawkesprocess model and an EAHDM model with the COVID-19pandemic data from Jan. 16 to Feb. 11 in 2020. And wewill exhibit the one-step prediction with the Hawkes processmodel and the EAHDM model, which proves that the EAHDMmodel outperforms the Hawkes process. Two common casesexist where the EAHDM model will degenerate. Note that noexogenous factor steadily infects people. Hence we can assumethat no immigrant process exists in the spread of COVID-19 pandemic. Only the several individuals in the beginningcan be acknowledged as immigrants , the confirmed casesafterwards were all descendants in those clusters generatedfrom immigrants . So µ = in both simulations and real dataexperiments. A. Simulation In the simulation, a -dimension network is constructed.More specifically, we fix β = 0 . and choose three A s inwhich A , ∈ { . , . , . } , A , ∈ { . , . , . } , A , ∈{ . , . , . } and the left entries are all . Based on thesame method in [12], we generate the data for 8 days. Since µ = , the data on the first day is manually given to triggerthe whole process. Hence three nodes are equally provided arrivals on the first day, on am and pm, respectively.Then we will descritize the data with binsize ∆ = 0 . toease the further computation. We estimate A by using EM-like algorithm in Section IV with the known Hawkes skeleton(see [13]) imbedded. The results are shown in Table I.TABLE I: Estimates of A . A , A , A , Simulation 1 True Parameter 1.500 1.500 1.500Simulation 1 Estimate 1.581 1.500 1.514Simulation 2 True Parameter 1.800 1.500 1.200Simulation 2 Estimate 1.725 1.500 1.206Simulation 3 True Parameter 2.000 1.500 1.000Simulation 3 Estimate 1.979 1.500 0.971 The EM-like algorithm commonly converges in severaliterations, which is rather efficient. The estimators are accuratein a mutual relationship sense but not in a absolute sense. Inother words, the ratios among the estimators are robust thoughtheir values are not. Actually whether the estimates are largeor small is determined by the binsize. If the binsize is chosensmall, then estimates turns large. If the bins are crude, thenvice versa. In Table I, we calibrate the estimates through equal A , , then estimates of A , and A , are also close to thetruth since the mutual relationship is estimated nicely. In realdata application, we calibrate A by minimizing some metricbetween the real data and our predictions. B. COVID-19 data We now utilize the Hawkes processes and the EAHDMmodel to fit the COVID-19 data in China and then make one- step predictions. Our data is obtained from [14] and we choosethe data source National Health Commission of Chinese. Theperiod is from Jan. 16 to Feb. 11 in 2020 containing addedconfirm cases in four provinces. Those four places are Hubei,Guangdong, Zhejiang and Henan, whose pandemic were oncethe most severe. In many implementation work involved theHawkes process, β is set manually (for example in [8]). In ourexperiments, we let β ∈ { . , . } . In the EAHDM model, d ( t ) = (cid:40) / max(7 , t ) , ≤ t ≤ / ( t . − . , t > . The results shown in Figures 1 and 2 indicate that the staticbranching matrix A in the Hawkes process makes it hardto capture the evolution of the environment temporally. Thusthe one-step predictions in the Hawkes process overestimatethe severity after about Feb. 3. In the EAHDM model, thedrawback of the Hawkes process is obviously fixed. Also, as β becomes smaller (from . to . ), the overestimates in theHawkes process get worse, while the EAHDM model behavesmore robust. It is interesting that if β = 1 , the Hawkes processwill ease its overestimates, but still not as good as the EAHDMmodel. Additionally, it is not recommended to choose a β evenlarger than . . Compared to d ( t ) which reflects the evolutionof outer environment, β means the decay rate of the influencefrom history events which can be interpreted as the infectivityof COVID-19 virus. If β = 1 , then exp {− βt } will be toosmall to capture the influence of the history only after days,which is not reasonable according to our common perceptionto the COVID-19 virus.VI. C ONCLUSION The Hawkes process is a classical self-exciting processwhich is frequently used to model the data with clusters.Considering the Hawkes process lacks of the flexibility ina environmentally temporal sense, we develop a novel andsophisticated EAH model and clarify its existence, uniquenessand the non-explosive condition. In implementation stage, wereduced the EAH model to the EAHDM model, which is stillan extension of the Hawkes process. In simulations, the EM-like algorithm requires a few iterations and provides nice esti-mates of the mutual relationship between nodes in a network.In the real data experiments, a proper decay function d ( t ) displays that the EAHDM outperforms the Hawkes processwhen predicting the evolution of the COVID-19 pandemicin China. Also, the EAHDM model is more robust with themanually chosen parameter β than the Hawkes process.A CKNOWLEDGMENT We are indebted to the National Science Foundation ofChina (Nos. 71771203, 11671374). We are grateful to Prof.Yao Xie for helpful discussion.R EFERENCES[1] S. Meyer, J. Elias, and M. Höhle, “A space–time conditional intensitymodel for invasive meningococcal disease occurrence,” Biometrics ,vol. 68, no. 2, pp. 607–616, 2012. an 20 Jan 27 Feb 03 Feb 10 time_exp_add1 h i s _and_e s t[ i , ] HubeiHubei_est (a) Jan 20 Jan 27 Feb 03 Feb 10 time_exp_add1 h i s _and_e s t[ i , ] GuangdongGuangdong_est (b) Jan 20 Jan 27 Feb 03 Feb 10 time_exp_add1 h i s _and_e s t[ i , ] HenanHenan_est (c) Jan 20 Jan 27 Feb 03 Feb 10 time_exp_add1 h i s _and_e s t[ i , ] ZhejiangZhejiang_est (d) Jan 20 Jan 27 Feb 03 Feb 10 Date A dded c on f i r m c a s e s HubeiHubei_est (e) Jan 20 Jan 27 Feb 03 Feb 10 Date A dded c on f i r m c a s e s GuangdongGuangdong_est (f) Jan 20 Jan 27 Feb 03 Feb 10 Date A dded c on f i r m c a s e s HenanHenan_est (g) Jan 20 Jan 27 Feb 03 Feb 10 Date A dded c on f i r m c a s e s ZhejiangZhejiang_est (h) Fig. 1: One-step estimate on the network formed by Hubei,Guangdong, Zhejiang and Henan. β = 0 . . The black linesrepresent the real data and the red dotted lines represent theone-step predictions. Subfigures (a-d) are from the Hawkesprocess. Subfigures (e-h) are from the EAHDM. Jan 20 Jan 27 Feb 03 Feb 10 time_exp_add1 h i s _and_e s t[ i , ] HubeiHubei_est (a) Jan 20 Jan 27 Feb 03 Feb 10 time_exp_add1 h i s _and_e s t[ i , ] GuangdongGuangdong_est (b) Jan 20 Jan 27 Feb 03 Feb 10 time_exp_add1 h i s _and_e s t[ i , ] HenanHenan_est (c) Jan 20 Jan 27 Feb 03 Feb 10 time_exp_add1 h i s _and_e s t[ i , ] ZhejiangZhejiang_est (d) Jan 20 Jan 27 Feb 03 Feb 10 Date A dded c on f i r m c a s e s HubeiHubei_est (e) Jan 20 Jan 27 Feb 03 Feb 10 Date A dded c on f i r m c a s e s GuangdongGuangdong_est (f) Jan 20 Jan 27 Feb 03 Feb 10 Date A dded c on f i r m c a s e s HenanHenan_est (g) Jan 20 Jan 27 Feb 03 Feb 10 Date A dded c on f i r m c a s e s ZhejiangZhejiang_est (h) Fig. 2: One-step estimate on the network formed by Hubei,Guangdong, Zhejiang and Henan. β = 0 . . The black linesrepresent the real data and the red dotted lines represent theone-step predictions. Subfigures (a-d) are from the Hawkesprocess. Subfigures (e-h) are from the EAHDM. 2] S. Meyer, L. Held et al. , “Power-law models for infectious diseasespread,” Annals of Applied Statistics , vol. 8, no. 3, pp. 1612–1639, 2014.[3] E. Bacry, I. Mastromatteo, and J.-F. Muzy, “Hawkes processes in fi-nance,” Market Microstructure and Liquidity , vol. 1, no. 01, p. 1550005,2015.[4] A. G. Hawkes, “Hawkes processes and their applications to finance: areview,” Quantitative Finance , vol. 18, no. 2, pp. 193–198, 2018.[5] W.-H. Chiang, X. Liu, and G. Mohler, “Hawkes process modelingof covid-19 with mobility leading indicators and spatial covariates,” medRxiv , 2020.[6] J. V. Escobar, “A hawkes process model for the propagation of covid-19:Simple analytical results,” EPL (Europhysics Letters) , vol. 131, no. 6,p. 68005, 2020.[7] L. Lesage, “A hawkes process to make aware people of the severity ofcovid-19 outbreak: application to cases in france,” Ph.D. dissertation,Université de Lorraine; University of Luxembourg, 2020.[8] S. Li, Y. Xie, M. Farajtabar, A. Verma, and L. Song, “Detecting changesin dynamic events over networks,” IEEE Transactions on Signal andInformation Processing over Networks , vol. 3, no. 2, pp. 346–359, 2017.[9] A. G. Hawkes and D. Oakes, “A cluster process representation of a self-exciting process,” Journal of Applied Probability , pp. 493–503, 1974.[10] D. Vere-Jones, “Stochastic models for earthquake occurrence,” Journalof the Royal Statistical Society: Series B (Methodological) , vol. 32, no. 1,pp. 1–45, 1970.[11] D. J. Daley and D. Vere-Jones, “An introduction to the theory of pointprocesses, volume 1: Elementary theory and methods,” Verlag New YorkBerlin Heidelberg: Springer , 2003.[12] Y. Ogata, “On lewis’ simulation method for point processes,” IEEEtransactions on information theory , vol. 27, no. 1, pp. 23–31, 1981.[13] P. Embrechts and M. Kirchner, “Hawkes graphs,” Theory of Probability& Its Applications , vol. 62, no. 1, pp. 132–156, 2018.[14] T. Wu, X. Ge, G. Yu, and E. Hu, “Open-source analytics tools forstudying the covid-19 coronavirus outbreak,” medRxiv , 2020.[15] T. E. Harris, “The theory of branching process,” 1964.[16] M. Westcott, “On existence and mixing results for cluster point pro-cesses,” Journal of the Royal Statistical Society: Series B (Methodolog-ical) , vol. 33, no. 2, pp. 290–300, 1971. A PPENDIX Proof of Theorem 1: Proof. The first step is to construct a cluster process whoseintensity is exactly the form in (2). The process of clustercenters N c ( t ) is set to be a Poisson process of rate µ . Foreach individual arriving at time τ , the process of the arrivalsof its direct descendants is constructed as a non-homogenousPoisson process with the intensity α ( t ) φ ( t − τ ) .So far we construct the cluster process we want intuitively.If the existence of such a cluster process is ensured, based onthe fact that the cluster process is the superposition of everyclusters and the process of cluster centers, it is straightforwardto see the constructed process possesses exactly the intensityin (2).So in the second step we try to ensure the existence ofconstructed process and control its rate. Now that the processof cluster centers is a homogenous Poisson process, which isrelatively easy to study with, we focus on the size of eachcluster. From Chapter 1 in [15], particularly Theorem 5.1, wemay firstly bound the size of first generation and the size ofother generations can be controled by a simple iteration.WLOG we consider a immigrant generated by N c ( · ) at time u . As said above, the first generation directly generated bythe immigrant follows a non-stationary α ( t ) φ ( t − u ) , t > u Poisson process. In Section 4.1 of [10], we recall the p.g.fl ofnon-homogeneous Poisson process as G ( ξ ( · ) | u ) = exp (cid:26)(cid:90) ( ξ ( t ) − α ( t ) φ ( t − u ) dt (cid:27) . (9)Now we truncate the cluster process at the first generation(including the first generation). Since p.g.fl. provides sufficientcharacterization of a point process, Section 4.2 of [10] indi-cates that the process of cluster centers with its first generationdescendents is equivalent to a Neyman-Scott cluster process.Actually, G ( ξ ( · ) | u ) = exp (cid:26) m ( u ) (cid:90) α ( t ) φ ( t − u ) m ( u ) ( ξ ( t ) − dt (cid:27) . The cluster size in such a Neyman-Scott process, namely thesize of first generation of a cluster, whose ancestor arrivedat time u , in the formerly constructed process, has a Poissondistribution with mean m ( u ) . Namely, the size of the firstgeneration of a cluster whose ancestor arriving at time u obeys Poi( m ( u )) .Now the cluster in the constructed process contains allgenerations, by Theorem 5.1 in [15], we can calculate theupper bound of mean cluster size with probability one as, ∞ (cid:88) n =0 m n = 11 − m . Meanwhile, by Theorem 6.1 in [15], when m < , thecluster is a.s. finite. As a result, Corollary 3.2 along withTheorem 3 in [16] ensures the existence of this non-explosivecluster process. Now ends the proof. Proof of Theorem 3 Proof. By the first formula in Section 3.2 of [10], we have G ( z ( · )) = G ( F ( z ( · ) | t )) , where G ( z ( · )) is the p.g.fl. of a homogenous Poisson processwith intensity µ . By the first formula in Section 4.1 of [10], G ( z ( · )) is rather clear to be G ( z ( · )) = exp (cid:26)(cid:90) ∞−∞ µ ( z ( t ) − t (cid:27) . We only need to figure out F ( z ( · ) | t ) . We define F n ( z ( · ) | t ) to be the p.g.fl of the cluster which consists of all birthsin all generations up to and including the n -th generationdescendants with the immigrant arriving at t .Denote H ( z ( · ) | t ) is the p.g.fl. of a non-homogenous Poissonprocess, by which an individual arrives at time t generates itsdescendants, in a EAH model. So actually H ( z ( · ) | t ) = exp (cid:26)(cid:90) ∞ t ( z ( τ ) − α ( τ ) φ ( t − τ )d τ (cid:27) . Then again, similarly with G ( z ( · )) = G ( F ( z ( · ) | t )) , bythe smoothing formula of p.g.fl., F n ( z ( · ) | t )= z ( t ) H ( F n − ( z ( · ) | τ ) | t )= z ( t ) exp (cid:26)(cid:90) ∞ t ( F n − ( z ( · ) | τ ) − α ( τ ) φ ( τ − t )d τ (cid:27) . ere F ( z ( · ) | t ) = E ( z ( t ) | t ) = z ( t ) represents the ancestor ofthe cluster, which is also the immigrant arriving at time t . Let n → ∞ and the proof ends. Proof of Theorem 4 Proof. If we take z ( x ) = (cid:26) z y ≤ x ≤ y + l elsewhere , (10)then F ( z ( · ) | t ) = E (cid:16) e (cid:82) log( z ( u )) dN s ( u | t ) (cid:17) = E (cid:16) e (cid:82) y + ly zdN s ( u | t ) (cid:17) = E (cid:16) z N s (( y,y + l ) | t ) (cid:17) =: π ( y, l, z | t ) , (11)say is the p.g.f. for the number of events in the interval ( y, y + l ) for a cluster whose originating event occurs at time t . It iseasy to see π ( y, l, z | t ) = E (cid:0) z N s (( y,y + l ) | t ) (cid:1) t ≤ y E (cid:0) z N s (( t,y + l ) | t ) (cid:1) y < t < y + l t ≥ y + l . Again, based on (4), π ( y, l, z | t ) = exp (cid:110)(cid:82) y + ly ( π ( y, l, z | τ ) − α ( τ ) φ ( τ − t )d τ (cid:111) t ≤ y t ≥ y + lz exp (cid:110)(cid:82) y + lt ( π ( y, l, z | τ ) − α ( τ ) φ ( τ − t )d τ (cid:111) elsewhere(12)Then G ( z ( · )) = exp (cid:26)(cid:90) ∞−∞ µ ( F ( z ( · ) | t ) − dt (cid:27) = exp (cid:26)(cid:90) ∞−∞ µ ( π ( y, l, z | t ) − dt (cid:27) = exp (cid:40)(cid:90) y + l −∞ µ ( π ( y, l, z | t ) − dt (cid:41) =: Q ( y, l, z ) . (13)Now we look at the distribution of the residual time of a EAH, P ( L y > l )= P ( Start observing from y, no events happen in [ t, t + l ))= E (0 N ( y,y + l ) )= Q ( y, l, (cid:40)(cid:90) y + l −∞ µ [ π ( y, l, | t ) − dt (cid:41) . Since π ( y, l, | t ) = exp (cid:110)(cid:82) y + ly ( π ( y, l, | τ ) − α ( τ ) φ ( τ − t )d τ (cid:111) t ≤ y t ≥ y + l elsewhere . (14) Then P ( L y > l ) = exp (cid:40)(cid:90) y + l −∞ µ [ π ( y, l, | t ) − dt (cid:41) = exp (cid:26)(cid:90) y −∞ µ [ π ( y, l, | t ) − dt − µl (cid:27) . We define γ ( y, l | t ) = π ( y, l, | t ) and the proof ends. Proof of Theorem 5 Proof. Let z ( x ) = (cid:26) x ≤ y x > y . Then F ( z ( · ) | t ) = E (cid:18) exp (cid:26)(cid:90) ∞−∞ log z ( τ ) dN s ( τ | t ) (cid:27)(cid:19) = E (0 N s (( y, ∞ ) | t ) ) = P ( N s (( y, ∞ ) | t ) = 0) . Also we find that D J t ( y ) = P ( J t ≤ y ) = P ( N s (( t + y, ∞ ) | t ) = 0) . Based on (4), for y ≥ , D J t ( y ) = exp (cid:26)(cid:90) ∞ t ( D J τ ( y + t − τ ) − α ( τ ) φ ( τ − t )d τ (cid:27) = exp (cid:26) − m ( t ) + (cid:90) t + yt D J τ ( y + t − τ ) α ( τ ) φ ( τ − t )d τ (cid:27)(cid:27)