Likelihood Function for Multivariate Hawkes Processes
Yuanda Chen
January, 2016
Abstract
In this article we discuss the likelihood function for an M-variate Hawkes process and derive the
explicit formula for the case with exponential excitation kernels.
1
Definitions of Multivariate Point Processes
Definition 1.1. (Bremaud, 1981, pp.19-20) Let {Tk }k=1,2,... be a point process defined on (Ω, F, P), and
let {Zk }k=1,2,... be a sequence of {1, 2, . . . , M }-valued random variables, also defined on (Ω, F, P). Then the
double sequence {(Tk , Zk )}k=1,2,... is called an M -variate point process.
Remark 1.1. The univariate point process can be thought of as a special case of an M -variate point process
with M = 1, in which case it is also called univariate. In the case when M = 2, the M -variate point process
is often called bivariate. We use the phrase “multivariate” rather than “multidimensional” because there are
point processes where 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, 2003);
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, where the points locate on a one-dimensional
time line and represent the occurrence times of events .
For a realization {(tk , zk )}k=1,2,... of an M -variate point process, tk denotes the occurrence time of the
k-th event and zk indicates the type of it. It is said to be of type m ∈ {1, 2, . . . , M } if and only if zk = m.
We define the counting process associated with an M -variate point process as follows.
Definition 1.2. (Bremaud, 1981, p.19) For any m ∈ {1, 2, . . . , M } and t ≥ 0,
X
N m (t) =
1{Tk ≤t} · 1{Zk =m}
k=1,2,...
counts the number of occurrences of type-m events, up to and including time t, and the M -vector process
N(t) = N 1 (t), N 2 (t), . . . , N M (t)
is called the associated counting process.
N(t) = N 1 (t), N 2 (t), . . . , N M (t) is also called an M -variate point process.
Remark 1.2. It is convenient to denote the occurrence time of the k-th event among all the type-m events
as Tkm , so that {Tkm }k=1,2,... is then the point process that has N m (t) as its counting process. With this
m
notation, for any m ∈ {1, 2, . . . , M }, any positive integer k and any t such that tm
k ≤ t < tk+1 we have
m
m
N (t) = k. The point processes N for m ∈ {1, 2, . . . , 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.
Remark 1.3. (Cox and Lewis, 1972, p.404) 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 ∈ {1, 2, . . . , 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.
1
Figure 1: 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 Remark 1.2. The composited point process is shown
as the cross marks on the axis at the top, labeled with the double-sequence notation given in Definition 1.1.
Figure 1 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 1.1
while the ones at the bottom of each panel use the notation in Remark 1.2. For example, for this particular
realization ω, the fifth point {(T5 (ω), Z5 (ω))} is the second point in the third dimension, so the occurrence
time is denoted as T5 (ω) = t5 = t32 and Z5 (ω) = z5 = 3 indicating a type-3 event.
The natural filtration can be defined similarly as the univariate case as follows.
Definition 1.3. (Karr, 1991, p.54) For an M -variate point process, define
FtN = σ(N m (s) : 0 ≤ s ≤ t, m ∈ {1, 2, . . . , M })
to be the natural filtration.
The stochastic intensity functions can then be defined using the notation mentioned in Remark 1.2. More
precisely, the intensity along the m-th dimension can be defined by only considering the points that are of
type-m, or {Tkm }k=1,2,... .
Definition 1.4. (Cox and Lewis, 1972, p.414) Let N(t) = N 1 (t), N 2 (t), . . . , N M (t) be an M -variate point
process. The stochastic intensity functions for the process are defined as:
λm (t|FtN− ) = lim
h→0+
P {N m (t + h) − N m (t) > 0|FtN− }
h
for m ∈ {1, 2, . . . , M }, where FtN is the natural filtration of N containing the internal history of the process,
up to time t, along all dimensions.
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2
Likelihood Function
Theorem 2.1. (Daley and Vere-Jones, 2003, p.251, Proposition 7.3.III) Let {tk , zk }k=1,2,...,n be a realization
of an M -variate point process on the interval [0, T ], such that 0 < t1 < · · · < tn ≤ T and zk ∈ {1, 2, . . . , M }
for k = 1, 2, . . . , n. Denote this particular realization as ω and the set of parameters as θ, then the loglikelihood function satisfies
ln L(θ|ω) =
n
X
ln λzθk (tk |ω)
−
k=1
n Z
X
k=1
tk
T
Z
λθ (t|ω)dt −
λθ (t|ω)dt.
tk−1
(1)
tn
where λm (t|ω) is defined for the marginal point process N m , so that for any given ω, λm (t|ω) = λm (t|FtN− )(ω).
For a complete proof, please refer to (Daley and Vere-Jones, 2003).
By applying Theorem 2.1 to the multivariate Hawkes process, we have the following proposition about
the likelihood function. The theories in this section are the basis of the parameter calibration of Hawkes
processes.
Proposition 2.2. Given a particular realization ω that contains all points in each dimension {tm
k }k=1,2,...
for m = 1, 2, . . . , M on the interval [0, T ], the log-likelihood function for an M -variate Hawkes process is
M
X
=
ln Lm θ {tnk }n=1,2,...,M
ln L θ {tm
k }m=1,2,...,M
m=1
where
ln Lm θ {tnk }n=1,2,...,M
=
Z
−
T
λm
θ (t|ω)dt +
T
Z
0
m
ln λm
θ (t|ω)dN (t)
0
= −µm T −
M
X
αmn
β
n=1 mn
+
n
1 − e−βmn (T −tk )
i
{k:tn
k <T }
"
X
h
X
ln µm +
{k:tm
k <T }
M
X
#
αmn Rmn (k)
n=1
with Rmn (k) defined recursively as
m
m
X
Rmn (k) = e−βmn (tk −tk−1 ) Rmn (k − 1) +
m
n
e−βmn (tk −ti )
n
m
{i:tm
k−1 ≤ti <tk }
with initial condition:
Rmn (0) = 0.
Proof. We will show the derivation for a bivariate Hawkes process with exponential decays, where M = 2,
and the results naturally applies to the M -variate case for M ≥ 2. Denote the realization by ω and let
θ = {µ1×M , αM ×M , βM ×M }. By applying Theorem 2.1, we have
ln L θ t1k , t2k
= ln L1 θ t1k , t2k + ln L2 θ t1k , t2k
where
ln L1 θ t1k , t2k
Z
= −
T
λ1θ t t1k , t2k dt +
0
Z
0
3
T
ln λ1θ t t1k , t2k dN 1 (t)
X
= −Λ1θ (T |ω) +
ln λ1θ (t1k |ω)
{k:t1k <T }


X
= −Λ1θ (T |ω) + ln µ1 + α11 (0) + α12
e
−β12 (t11 −t2k )

{k:t2k <t11 }


X
+ ln µ1 + α11
e
−β11 (t12 −t1k )
X
+ α12
{k:t1k <t12 }
e
−β12 (t12 −t2k )
{k:t2k <t12 }
+···
= −Λ1θ (T |ω) + ln [µ1 + α11 R11 (1) + α12 R12 (1)]
+ log [µ1 + α11 R11 (2) + α12 R12 (2)] + · · ·
X
= −Λ1θ (T |ω) +
log [µ1 + α11 R11 (k) + α12 R12 (k)]
{k:t1k <T }
= −µ1 T −
−
α11
β11
α12
β12
h
X
1
1 − e−β11 (T −tk )
i
{k:t1k <T }
h
X
2
1 − e−β12 (T −tk )
i
{k:t2k <T }
X
+
log [µ1 + α11 R11 (k) + α12 R12 (k)]
{k:t1k <T }
and similarly,
ln L2 θ t1k , t2k
=
−µ2 T −
−
α21
β21
α22
β22
h
i
1
1 − e−β21 (T −tk )
X
{k:t1k <T }
X
h
i
2
1 − e−β22 (T −tk )
{k:t2k <T }
X
+
log [µ2 + α21 R21 (k) + α22 R22 (k)]
{k:t2k <T }
where
Rmn (k)
=
m
X
n
e−βmn (tk −ti )
m
{i:tn
i <tk }
=
m
m
e−βmn (tk −tk−1 ) Rmn (k − 1) +
X
n
m
{i:tm
k−1 ≤ti <tk }
4
m
n
e−βmn (tk −ti )

with the convention Rmn (0) = 0.
References
Bremaud, P.
1981. Point Processes and Queues: Martingale Dynamics (Springer Series in Statistics). Springer.
Cox, D. R. and P. A. W. Lewis
1972. Multivariate point processes. In Proceedings of the Sixth Berkeley Symposium on Mathematical
Statistics and Probability, Volume 3: Probability Theory, Pp. 401–448, Berkeley, Calif. University of
California Press.
Daley, D. and D. Vere-Jones
2003. An Introduction to the Theory of Point Processes, Volume 1. Springer.
Karr, A.
1991. Point Processes and Their Statistical Inference, Second Edition, (Probability: Pure and Applied).
CRC / Marcel Dekker, Inc.
Møller, J. and R. P. Waagepetersen
2003. Statistical Inference and Simulation for Spatial Point Processes (Chapman & Hall/CRC Monographs
on Statistics & Applied Probability). Chapman and Hall/CRC.
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