Likelihood Function for Multivariate Hawkes Processes

Transcription

1 Lielihood Function for Multivariate Hawes Processes Yuanda Chen January, 6 Abstract In this article we discuss the lielihood function for an M-variate Hawes process and derive the explicit formula for the case with exponential excitation ernels. Definitions of Multivariate Point Processes Definition.. (Bremaud, 98, pp.9-) Let {T } =,,... be a point process defined on (Ω, F, P), and let {Z } =,,... be a sequence of {,,..., M}-valued random variables, also defined on (Ω, F, P). Then the double sequence {(T, Z )} =,,... is called an M-variate point process. Remar.. The univariate point process can be thought of as a special case of an M-variate point process with M =, in which case it is also called univariate. In the case when M =, the M-variate point process is often called bivariate. We use the phrase multivariate rather than multidimensional because there are point processes the points are themselves positions in a two- or three-dimensional region. They are often referred to as spatial point processes (for example, the subject covered in Møller and Waagepetersen, 3); while on the other hand, although we will inevitably use the word dimension, the point processes considered in this article are sometimes called temporal point processes, the points locate on a one-dimensional time line and represent the occurrence times of events. For a realization {(t, z )} =,,... of an M-variate point process, t denotes the occurrence time of the -th event and z indicates the type of it. It is said to be of type m {,,..., M} if and only if z = m. We define the counting process associated with an M-variate point process as follows. Definition.. (Bremaud, 98, p.9) For any m {,,..., M} and t, N m (t) = {T t} {Z =m} =,,... counts the number of occurrences of type-m events, up to and including time t, and the M-vector process N(t) = ( N (t), N (t),..., N M (t) ) is called the associated counting process. N(t) = ( N (t), N (t),..., N M (t) ) is also called an M-variate point process. Remar.. It is convenient to denote the occurrence time of the -th event among all the type-m events as T m, so that {T m} =,,... is then the point process that has N m (t) as its counting process. With this notation, for any m {,,..., M}, any positive integer and any t such that t m t < tm + we have N m (t) =. The point processes N m for m {,,..., M}, considered as univariate point processes, are often called the marginal processes of N, and on the other hand, N can regarded as the superposed marginal events. Remar.3. (Cox and Lewis, 97, p.44) By assuming a multivariate point process N is orderly, we exclude the possibility of multiple occurrence of events, both of the same type and of different types. More precisely, it means not only that N m is orderly for each m {,,..., M}, which is often called to be marginally orderly, but also that the superposed process N is orderly. Of course, this type of orderliness implies marginal orderliness.

2 Figure : A realization of a 3-variate point process. The counting processes for each dimension is shown in each panel, with the points labeled using notations in Remar.. The composited point process is shown as the cross mars on the axis at the top, labeled with the double-sequence notation given in Definition.. Figure is an illustrative example showing the notations based on the first 7 points in a realization of a 3-variate point process. The notation on the top uses the double sequence notation defined in Definition. while the ones at the bottom of each panel use the notation in Remar.. For example, for this particular realization ω, the fifth point {(T 5 (ω), Z 5 (ω))} is the second point in the third dimension, so the occurrence time is denoted as T 5 (ω) = t 5 = t 3 and Z 5 (ω) = z 5 = 3 indicating a type-3 event. The natural filtration can be defined similarly as the univariate case as follows. Definition.3. (Karr, 99, p.54) For an M-variate point process, define F N t = σ(n m (s) : s t, m {,,..., M}) to be the natural filtration. The stochastic intensity functions can then be defined using the notation mentioned in Remar.. More precisely, the intensity along the m-th dimension can be defined by only considering the points that are of type-m, or {T m } =,,... Definition.4. (Cox and Lewis, 97, p.44) Let N(t) = ( N (t), N (t),..., N M (t) ) be an M-variate point process. The stochastic intensity functions for the process are defined as: λ m (t Ft N P {N m (t + h) N m (t) > Ft N } ) = lim h + h for m {,,..., M}, Ft N is the natural filtration of N containing the internal history of the process, up to time t, along all dimensions.

3 Lielihood Function Theorem.. (Daley and Vere-Jones, 3, p.5, Proposition 7.3.III) Let {t, z } =,,...,n be a realization of an M-variate point process on the interval [, T ], such that < t < < t n T and z {,,..., M} for =,,..., n. Denote this particular realization as ω and the set of parameters as θ, then the loglielihood function satisfies ln L(θ ω) = n = ln λ z θ (t ω) n = t t λ θ (t ω)dt t n λ θ (t ω)dt. () λ m (t ω) is defined for the marginal point process N m, so that for any given ω, λ m (t ω) = λ m (t F N t )(ω). For a complete proof, please refer to (Daley and Vere-Jones, 3). By applying Theorem. to the multivariate Hawes process, we have the following proposition about the lielihood function. The theories in this section are the basis of the parameter calibration of Hawes processes. Proposition.. Given a particular realization ω that contains all points in each dimension {t m } =,,... for m =,,..., M on the interval [, T ], the log-lielihood function for an M-variate Hawes process is ( ) ln L θ {t m } m=,,...,m = M ( ) ln L m θ {t n } n=,,...,m m= ( ) ln L m θ {t n } n=,,...,m = λ m θ (t ω)dt + ln λ m θ (t ω)dn m (t) = µ m T M n= + α mn β mn {:t m <T } ln {:t n <T } [ µ m + [ e βmn(t tn )] ] M α mn R mn () n= with R mn () defined recursively as R mn () = e βmn(tm tm ) R mn ( ) + {i:t m tn i <tm } with initial condition: R mn () =. Proof. We will show the derivation for a bivariate Hawes process with exponential decays, M =, and the results naturally applies to the M-variate case for M. Denote the realization by ω and let θ = {µ M, α M M, β M M }. By applying Theorem., we have ln L ( θ { t { }) }, t = ln L ( θ { t { }) }, t + ln L ( θ { t } { }) ln L ( θ { t } { }) = λ ( { } { }) T θ t t dt + ln λ ( { } { }) θ t t dn (t) 3

4 = Λ θ(t ω) + {:t <T } ln λ θ(t ω) = Λ θ(t ω) + ln µ + α () + α + ln µ + α + {:t <t } e β(t {:t <t } e β(t t ) + α t ) {:t <t } e β(t t ) = Λ θ(t ω) + ln [µ + α R () + α R ()] + log [µ + α R () + α R ()] + = Λ θ(t ω) + {:t <T } log [µ + α R () + α R ()] = µ T α β α β {:t <T } {:t <T } + [ e β(t t )] [ e β(t t )] {:t <T } log [µ + α R () + α R ()] and similarly, ln L ( θ { t } { }) = µ T α β {:t <T } [ e β(t t )] α β {:t <T } + [ e β(t t )] {:t <T } log [µ + α R () + α R ()] R mn () = {i:t n i <tm } = e βmn(tm tm ) R mn ( ) + 4 {i:t m tn i <tm }

5 with the convention R mn () =. References Bremaud, P. 98. Point Processes and Queues: Martingale Dynamics (Springer Series in Statistics). Springer. Cox, D. R. and P. A. W. Lewis 97. Multivariate point processes. In Proceedings of the Sixth Bereley Symposium on Mathematical Statistics and Probability, Volume 3: Probability Theory, Pp , Bereley, Calif. University of California Press. Daley, D. and D. Vere-Jones 3. An Introduction to the Theory of Point Processes, Volume. Springer. Karr, A. 99. Point Processes and Their Statistical Inference, Second Edition, (Probability: Pure and Applied). CRC / Marcel Deer, Inc. Møller, J. and R. P. Waagepetersen 3. Statistical Inference and Simulation for Spatial Point Processes (Chapman & Hall/CRC Monographs on Statistics & Applied Probability). Chapman and Hall/CRC. 5