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The Multivariate Hawkes Process in High Dimensions: Beyond Mutual Excitation

Shizhe Chen, Ali Shojaie, Eric Shea-Brown, and Daniela Witten

June 18th, 2019

Abstract

The Hawkes process is a class of point processes whose future depends on their own his- tory. Previous theoretical work on the Hawkes process is limited to a special case in which a past event can only increase the occurrence of future events, and the link function is linear. However, in neuronal networks and other real-world applications, inhibitory relationships may be present, and the link function may be non-linear. In this paper, we develop a new approach for investigating the properties of the Hawkes process without the restriction to mutual exci- tation or linear link functions. To this end, we employ a thinning process representation and a coupling construction to bound the dependence coefficient of the Hawkes process. Using recent developments on weakly dependent sequences, we establish a concentration inequality for second-order statistics of the Hawkes process. We apply this concentration inequality to cross-covariance analysis in the high-dimensional regime, and we verify the theoretical claims with simulation studies.

Keywords: Hawkes process; thinning process; weak dependence.

1 Introduction Hawkes (1971) proposed a class of point process models in which a past event can affect the probability of future events. The Hawkes process and its variants have been widely applied to model recurrent events in many fields, with notable applications to earthquakes (Ogata, 1988), crimes (Mohler et al., 2011), interactions in social networks (Simma and Jordan, 2010; Perry and Wolfe, 2013), financial events (Chavez-Demoulin et al., 2005; Bowsher, 2007; Aı̈t-Sahalia et al., 2015), and spiking histories of neurons (see e.g., Brillinger, 1988; Okatan et al., 2005; Pillow et al., 2008).

There is currently a significant gap between applications and statistical theory for the Hawkes process. Hawkes (1971) considered the mutually-exciting Hawkes process, in which an event excites the process, i.e. one event may trigger future events. Later, Hawkes and Oakes (1974) de- veloped a cluster process representation for the mutually-exciting Hawkes process, which is an es- sential tool for subsequent theoretical developments (Reynaud-Bouret and Schbath, 2010; Hansen et al., 2015; Bacry et al., 2015). The cluster process representation requires two key assumptions: (i) the process is mutually-exciting; and (ii) the link function is linear, implying that the effects of

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past events on the future firing probabilities are additive. In many applications, however, one might wish to allow for inhibitory events and non-additive aggregation of effects from past events. For instance, it is well-known that a spike of one neuron may inhibit the activities of other neurons (see e.g., Purves et al., 2001), meaning that it decreases the probability that other neurons will spike. Furthermore, non-linear link functions are often used when analyzing spike trains (Paninski et al., 2007; Pillow et al., 2008). In these cases, many existing theoretical results do not apply, since Hawkes and Oakes’s cluster process representation is no longer viable.

In this paper, we propose a new analytical tool for the Hawkes process that applies beyond the mutually-exciting and linear setting. We employ a new representation of the Hawkes process to replace the cluster process representation. To demonstrate the application of this new analytical tool, we establish a concentration inequality for second-order statistics of the Hawkes process, without restricting the process to be mutually-exciting or to have a linear link function. We apply this tool to study smoothing estimators of cross-covariance functions of the Hawkes process.

While this paper was under revision, it came to our attention that Costa et al. (2018) have concurrently studied the Hawkes process with inhibitions in the one-dimensional case. By contrast, our work allows for multiple dimensions and considers the high-dimensional setting. We will provide some additional remarks on their proposal in Section 3.3.

The paper is organized as follows. We introduce the Hawkes process and review the existing literature in Section 2. In Section 3, we present the construction of a coupling process, and derive a new concentration inequality for the Hawkes process. In Section 4, we study the theoretical properties of smoothing estimators of the cross-covariance functions of the Hawkes process, and corroborate our findings with numerical experiments. Proofs of the main results are in Section 5. We conclude with a discussion in Section 6. Technical proofs are provided in the Appendix.

2 Background on the Hawkes Process In this section, we provide a very brief review of point processes in general, and the Hawkes process in particular. We refer interested readers to the monograph by Daley and Vere-Jones (2003) for a comprehensive discussion of point processes.

We use the notation f ∗ g(t) ≡ ∫∞ −∞ f(∆)g(t − ∆)d∆ to denote the convolution of two

functions, f and g. We use ‖a‖2 to denote the `2-norm of a vector a ∈ Rp. Furthermore, ‖f‖2,[l,u] ≡

{ ∫ u l f 2(t)dt

}1/2 will denote the `2-norm of a function f on the interval [l, u], and ‖f‖∞ ≡ supx |f(x)| will denote the maximum of f . We use Γmax(A) for the maximum eigen- value of a square matrixA. We use J to denote a p-vector of ones. The notation 1[C] is an indicator variable for the event C.

2.1 A Brief Review of Point Processes Let B(R) denote the Borel σ-field of the real line, and let {ti}i∈Z be a sequence of real-valued random variables such that ti+1 > ti and 0 ≤ t1. Here, time t = 0 is a reference point in time, e.g., the start of an experiment. We define a simple point process N on R as a family {N(A)}A∈B(R) that takes on non-negative integer values such that the sequence {ti}i∈Z consists of event times of the process N , i.e., N(A) =

∑ i 1[ti∈A] for A ∈ B(R). We write N

( [t, t + dt)

) as dNj(t), where

dt denotes an arbitrarily small increment of t.

2

Now suppose thatN is a marked point process, in the sense that each event time ti is associated with a mark mi ∈ {1, . . . , p} (see e.g., Definition 6.4.I. in Daley and Vere-Jones, 2003). With a slight abuse of notation, we can then view N as a multivariate point process, N ≡

( Nj ) j=1,...,p

, for which the jth component process, Nj , satisfies Nj(A) =

∑ i 1[ti∈A,mi=j] for A ∈ B(R). To

simplify the notation, in what follows, we will let {tj,1, tj,2, . . .} denote the event times of Nj . Let Ht denote the history of N up to time t. The intensity process λ(t) =

( λ1(t), . . . , λp(t)

)T is a p-variateHt-predictable process, defined as

λj(t)dt = P(dNj(t) = 1 | Ht), j = 1, . . . , p. (1)

2.2 A Brief Overview of the Hawkes Process For the Hawkes process (Hawkes, 1971), the intensity function (1) takes the form

λj(t) = φj

{ µj +

p∑ k=1

( ωk,j ∗ dNk

) (t)

} , j = 1, . . . , p, (2)

where ( ωk,j ∗ dNk

) (t) =

∫ ∞ 0

ωk,j(∆)dNk(t−∆) = ∑ i:tk,i≤t

ωk,j(t− tk,i).

We refer to µj ∈ R as the background intensity, and ωk,j(·) : R+ 7→ R as the transfer function. If the link function φj on the right-hand side of (2) is non-linear, then λj(t) is the intensity of a non-linear Hawkes process (Brémaud and Massoulié, 1996). We will refer to the class of Hawkes processes that allows for non-linear link functions and negative transfer functions as the general- ized Hawkes process.

In this paper, we assume that we observe the event times of a stationary process dN on [0, T ], whose intensities follow (2). The existence of a stationary process is guaranteed by the following assumption.

Assumption 1 We assume that φj(·) is αj-Lipschitz for j = 1, . . . , p. Let Ω be a p × p matrix whose entries are Ωj,k = αj

∫∞ 0 |ωk,j(∆)| d∆ for 1 ≤ j, k ≤ p. We assume that there exists a

generic constant γΩ such that Γmax(Ω) ≤ γΩ < 1.

Note that in Assumption 1, the constant γΩ does not depend on the dimension p. For any fixed p, Brémaud and Massoulié (1996) establish that the intensity process of the form (2) is stable in distribution, and thus a stationary process dN exists given Assumption 1. We refer interested readers to Brémaud and Massoulié (1996) for a rigorous discussion of stability for the Hawkes process.

We define the mean intensity Λ = (Λ1, . . . ,Λp)T ∈ Rp as

Λj ≡ E[dNj(t)]/dt, j = 1, . . . , p. (3)

Following Equation 5 of Hawkes (1971), we define the (infinitesimal) cross-covariance V (·) =( Vk,j(·)

) p×p : R 7→ R

p×p as

Vk,j(∆) ≡

{ E[dNj(t)dNk(t−∆)]/{dtd(t−∆)} − ΛjΛk j 6= k E[dNk(t)dNk(t−∆)]/{dtd(t−∆)} − Λ2k − Λkδ(∆) j = k

, (4)

3

for any ∆ ∈ R, and 1 ≤ j, k ≤ p. Here δ(·) is the Dirac delta function, which satisfies δ(x) = 0 for x 6= 0 and

∫∞ −∞ δ(x)dx = 1.

Example 1 (Linear Hawkes processes) Suppose that φ(x) = x and ωk,j(∆) ≥ 0 for all ∆ ∈ R+ and for all j, k ∈ {1, . . . , p}. The intensity in (2) takes the form

λj(t) = µj +

p∑ k=1

( ωk,j ∗ dNk

) (t), j = 1, . . . , p. (5)

This is known as the l