Hawkes Processes Turn Events Into Causes of Future Events

A Poisson process assumes that an event does not change the future event rate. A Hawkes process makes the opposite mechanism explicit: each event can create a temporary increase in intensity and generate descendants of its own.

Published
March 12, 2026
Author
Reading time
21 mins read
Difficulty
Intermediate

A homogeneous Poisson process makes a strong statement about event data. Conditional on a constant rate, one event does not alter the probability of another. The waiting time after an event has exactly the same distribution as the waiting time before it. Counts in disjoint intervals are independent. Clustering can occur by chance, but the event itself has no memory.

Many systems suggest a different mechanism. One earthquake is followed by aftershocks. A machine failure can leave a system in a temporarily fragile state. One social-media post can trigger replies and reposts. A network attack can produce secondary events. A purchase can prompt related transactions. A hospital admission can raise short-term readmission risk. In such settings an event is not merely an observation of an underlying rate. It can become part of the mechanism generating future events.

Hawkes processes provide one of the cleanest mathematical models for this kind of self-excitation. The central object is not a probability for a fixed time bin, but a conditional intensity that evolves continuously and jumps when events occur. Each event adds a temporary impulse to the future rate. The process can therefore produce bursts even when the exogenous baseline is constant.

The important quantity is not only how large each impulse is. It is the total expected number of events that one event ultimately creates through all generations of descendants. That quantity is controlled by the branching ratio. When the branching ratio is comfortably below one, event clusters die out. As it approaches one, clusters become larger and count variability increases sharply. At or above one, the stationary finite-intensity model breaks down.

This branching interpretation is what turns the Hawkes process from a convenient curve for fitting bursts into a structured stochastic model.

The conditional intensity is the model

Let

$$ N(t) $$

count events up to time $t$, with event times

$$ 0<t_1<t_2<\cdots. $$

For a simple point process, the conditional intensity is informally defined by

$$ \lambda(t\mid\mathcal H_t) = \lim_{\Delta t\downarrow0} \frac{ P\{ N(t+\Delta t)-N(t)=1 \mid \mathcal H_t \} }{ \Delta t }, $$

where

$$ \mathcal H_t $$

is the event history before time $t$.

A homogeneous Poisson process has

$$ \lambda(t\mid\mathcal H_t) = \mu, $$

a constant independent of history.

A linear Hawkes process replaces this with

$$ \lambda(t) = \mu + \sum_{t_i<t} \phi(t-t_i), $$

where

$$ \mu>0 $$

is the exogenous baseline intensity and

$$ \phi(s)\ge0 $$

is an excitation kernel.

Every past event contributes to the current rate. The contribution depends on its age.

A common choice is the exponential kernel,

$$ \phi(s) = \alpha e^{-\beta s} \mathbf 1\{s>0\}, $$

with

$$ \alpha>0, \qquad \beta>0. $$

The intensity becomes

$$ \lambda(t) = \mu + \sum_{t_i<t} \alpha e^{-\beta(t-t_i)}. $$

At an event time the intensity jumps upward by approximately

$$ \alpha. $$

It then decays exponentially back toward the baseline at rate

$$ \beta. $$

The decay half-life is

$$ t_{1/2} = \frac{ \log2 }{ \beta }. $$

The two kernel parameters therefore play different roles. The amplitude $\alpha$ sets the immediate jump. The decay $\beta$ sets how long that jump persists.

Their ratio controls something more fundamental.

The branching ratio controls endogenous amplification

The total expected direct excitation generated by one event is the integral of the kernel,

$$ n = \int_0^\infty \phi(s)\,ds. $$

For the exponential kernel,

$$ n = \int_0^\infty \alpha e^{-\beta s}\,ds = \frac{ \alpha }{ \beta }. $$

This quantity is called the branching ratio.

The reason for the name comes from the Poisson-cluster representation of a linear Hawkes process. Exogenous immigrant events arrive according to a Poisson process with rate

$$ \mu. $$

Each event independently generates a Poisson offspring process with mean number of direct children

$$ n. $$

Those children generate descendants according to the same mechanism.

The result is a branching process embedded in continuous time.

When

$$ n<1, $$

the branching process is subcritical and each immigrant produces a finite cluster almost surely under the standard model.

The expected total cluster size, including the immigrant, is

$$ E[S] = 1+n+n^2+\cdots = \frac{ 1 }{ 1-n }. $$

If

$$ n=0.6, $$

then

$$ E[S] = \frac1{0.4} = 2.5. $$

One exogenous event creates, on average, a cluster containing 2.5 events in total.

The expected number of endogenous descendants is therefore

$$ 2.5-1 = 1.5. $$

In the stationary branching representation, the long-run fraction of events that are immigrants is

$$ 1-n, $$

and the endogenous fraction is

$$ n. $$

For

$$ n=0.6, $$

approximately 40% of stationary events are exogenous immigrants and 60% are descendants of earlier events under the model.

This interpretation is attractive and easy to overstate. The label endogenous refers to the fitted Hawkes branching structure. If the baseline is misspecified, events attributed to descendants can actually be responses to omitted common drivers.

The branching ratio is therefore a model-based decomposition of event generation, not a direct physical measurement.

Stationary mean rate is larger than the exogenous baseline

Take expectations in the linear Hawkes intensity.

Under stationarity,

$$ \bar\lambda = E[\lambda(t)] $$

is constant.

Each past event contributes an expected total kernel mass

$$ n, $$

so the mean intensity satisfies

$$ \bar\lambda = \mu + n\bar\lambda. $$

Therefore,

$$ \bar\lambda = \frac{ \mu }{ 1-n }, $$

provided

$$ n<1. $$

This equation separates exogenous input from endogenous amplification.

If

$$ \mu=2 $$

events per day and

$$ n=0.6, $$

then

$$ \bar\lambda = \frac2{0.4} = 5 $$

events per day.

A homogeneous Poisson process with rate five and this Hawkes process have the same long-run mean count.

Their temporal structure is very different.

This illustrates an identifiability point immediately. The mean rate alone identifies only the combination

$$ \frac{ \mu }{ 1-n }. $$

Many pairs

$$ (\mu,n) $$

produce the same mean.

For example,

$$ \mu=2, \qquad n=0.6, $$

and

$$ \mu=4, \qquad n=0.2 $$

both produce

$$ \bar\lambda=5. $$

Counts averaged over long periods cannot distinguish the two mechanisms from the mean alone.

The temporal dependence contains the missing information.

The same mean rate can hide radically different count variance

A Poisson process has the defining variance-to-mean property

$$ \operatorname{Var}[N(T)] = E[N(T)] $$

for any interval length $T$.

The Fano factor is therefore

$$ F(T) = \frac{ \operatorname{Var}[N(T)] }{ E[N(T)] } = 1. $$

A stationary Hawkes process is overdispersed because events arrive in clusters.

The branching representation makes the long-window variance transparent.

Let immigrants arrive as a Poisson process of rate

$$ \mu. $$

For a linear Hawkes process whose offspring count is Poisson with mean $n$, the total cluster size $S$ has

$$ E[S] = \frac1{1-n} $$

and

$$ \operatorname{Var}(S) = \frac{ n }{ (1-n)^3 }. $$

Therefore,

$$ E[S^2] = \operatorname{Var}(S) + E[S]^2 = \frac1{(1-n)^3}. $$

Ignoring boundary effects over a sufficiently long observation window, the total event count is approximately a compound Poisson sum of cluster sizes.

Its mean is

$$ E[N(T)] \approx \mu T E[S] = \frac{ \mu T }{ 1-n }, $$

and its variance is

$$ \operatorname{Var}[N(T)] \approx \mu T E[S^2] = \frac{ \mu T }{ (1-n)^3 }. $$

The asymptotic Fano factor is therefore

$$ F = \frac{ 1 }{ (1-n)^2 }. $$

For

$$ n=0.6, $$

we obtain

$$ F = \frac1{0.4^2} = 6.25. $$

Compare two processes with the same stationary mean rate of five events per day.

The Poisson process has

$$ F=1. $$

The Hawkes process with

$$ \mu=2, \qquad n=0.6 $$

has long-window Fano factor

$$ 6.25. $$

Over thirty days, both have expected count

$$ 150. $$

The Poisson variance is

$$ 150, $$

with standard deviation approximately

$$ 12.25. $$

The Hawkes long-window variance is approximately

$$ 150(6.25) = 937.5, $$

with standard deviation approximately

$$ 30.62. $$

The mean count is identical.

The uncertainty in the count is not.

This is why aggregation into average rates can hide the entire mechanism of clustering.

The branching ratio and the decay rate answer different temporal questions

The branching ratio determines the total expected number of descendants.

It does not determine when those descendants occur.

Consider two exponential kernels:

$$ \phi_1(t) = 0.6e^{-t}, $$

and

$$ \phi_2(t) = 6e^{-10t}. $$

Both have

$$ n = \frac{ \alpha }{ \beta } = 0.6. $$

They therefore have the same expected cluster size,

$$ 2.5, $$

the same stationary endogenous fraction,

$$ 0.6, $$

and, for the same baseline $\mu$, the same stationary mean intensity and asymptotic long-window Fano factor.

Their temporal clustering is different.

For the first kernel,

$$ \beta=1 $$

per day and the excitation half-life is

$$ \frac{ \log2 }{ 1 } \approx 0.693 $$

days.

For the second,

$$ \beta=10 $$

per day and the half-life is

$$ \frac{ \log2 }{ 10 } \approx 0.0693 $$

days, about 1.66 hours.

One process creates broad clusters spread over days.

The other creates tight bursts concentrated within hours.

This distinction matters scientifically.

If only daily counts are observed, the two models can become hard to distinguish because the timestamp resolution has erased the timescale on which excitation decays.

Fine temporal resolution can identify a short kernel that coarse aggregation cannot.

The observation process therefore contributes directly to parameter identifiability.

Likelihood fitting uses the event times and the compensator

For a point process with conditional intensity

$$ \lambda(t), $$

observed over

$$ [0,T], $$

the log likelihood, up to conventions at boundaries, is

$$ \ell = \sum_{i:t_i\le T} \log\lambda(t_i) - \int_0^T \lambda(t)\,dt. $$

The first term rewards high intensity at observed event times.

The second penalizes models that maintain high intensity over intervals where no events occur.

For the exponential Hawkes model,

$$ \lambda(t) = \mu + \sum_{t_i<t} \alpha e^{-\beta(t-t_i)}, $$

the integrated intensity has a closed form:

$$ \int_0^T \lambda(t)\,dt = \mu T + \frac{ \alpha }{ \beta } \sum_{t_i<T} \left[ 1- e^{-\beta(T-t_i)} \right]. $$

The log likelihood is therefore

$$ \ell(\mu,\alpha,\beta) = \sum_{i=1}^{N(T)} \log \left[ \mu + \sum_{j<i} \alpha e^{-\beta(t_i-t_j)} \right] - \mu T - \frac{ \alpha }{ \beta } \sum_{i=1}^{N(T)} \left[ 1- e^{-\beta(T-t_i)} \right]. $$

This uses exact event times rather than counts in arbitrary bins.

Maximum likelihood is a natural estimator, but the likelihood geometry can be difficult.

A high baseline and weak excitation can mimic a lower baseline with stronger excitation if the observation window is short.

Amplitude and decay can trade off through

$$ n=\alpha/\beta. $$

Events near the beginning of the observation window may have unobserved parents before time zero.

Events near the end can have descendants after time $T$ that are never observed.

These edge effects matter especially for slowly decaying kernels.

The optimization problem should therefore be inspected rather than treated as a black box. Profile likelihoods, multiple initializations and bootstrap uncertainty can reveal weak identification that one numerical optimum hides.

The branching representation gives another route to estimation

The cluster interpretation suggests an expectation-maximization view.

Every observed event is either an immigrant generated by the baseline or a descendant triggered by one earlier event.

Those parent labels are latent.

For event $i$, the probability that it is an immigrant is proportional to

$$ \mu, $$

while the probability that event $j<i$ is its parent is proportional to

$$ \phi(t_i-t_j). $$

Normalizing by the total intensity gives responsibilities such as

$$ P( i\text{ is immigrant} \mid \mathcal H_{t_i} ) = \frac{ \mu }{ \lambda(t_i) }, $$

and

$$ P( j\to i \mid \mathcal H_{t_i} ) = \frac{ \phi(t_i-t_j) }{ \lambda(t_i) }. $$

An EM algorithm can alternate between estimating these latent branching probabilities and updating the baseline and kernel parameters.

This is useful because it produces a probabilistic decomposition of observed events into exogenous and triggered contributions.

It does not remove model dependence.

If an unmodelled external shock causes many events close together, the Hawkes model may assign them parent-child links because that is the only mechanism available to explain the cluster.

Posterior branching probabilities are therefore conditional explanations under the fitted model, not observed genealogies.

Time rescaling gives a strong goodness-of-fit diagnostic

A fitted point-process model should not be validated only by comparing observed and predicted event counts.

The time-rescaling theorem provides a much sharper diagnostic.

Let consecutive event times be

$$ t_{i-1} $$

and

$$ t_i. $$

Define the integrated conditional intensity between them,

$$ Z_i = \int_{t_{i-1}}^{t_i} \hat\lambda(s)\,ds. $$

If the fitted conditional intensity is correct under suitable regularity conditions, then the transformed gaps should behave as independent unit-rate exponential variables,

$$ Z_i \sim \operatorname{Exp}(1). $$

Equivalently,

$$ U_i = 1-e^{-Z_i} $$

should be approximately iid

$$ U(0,1). $$

This turns a complicated history-dependent event process into a simple residual sequence.

One can examine a Q-Q plot of the

$$ Z_i $$

against the exponential distribution, test the

$$ U_i $$

for uniformity and inspect serial dependence.

Different residual patterns suggest different failures.

Too many small transformed gaps indicate underprediction of short-term clustering.

Long serial runs can indicate omitted seasonality or dependence.

A good count fit can fail the rescaling diagnostic badly because total counts do not test event timing.

Point-process models should be validated on the timescale at which they claim to model events.

Omitted seasonality can look like self-excitation

Self-excitation is not the only mechanism that produces clusters.

Suppose events are generated by an inhomogeneous Poisson process with deterministic baseline

$$ \mu(t) $$

that is high during working hours and low overnight.

Events will cluster during the high-rate periods even though no event causes another.

If we fit a Hawkes process with a constant baseline,

$$ \mu(t)=\mu, $$

the model has only one way to explain the bursts: increase the excitation kernel.

The fitted branching ratio can therefore be positive even when the true data-generating process has no endogenous event triggering.

The same confounding occurs with common shocks.

A news announcement can produce many market orders.

A storm can generate many infrastructure failures.

A software deployment can trigger many error events.

If the common cause is omitted, the event sequence can resemble mutual or self-excitation.

This is the Hawkes version of model discrepancy.

The baseline should contain known exogenous structure before the residual clustering is interpreted as endogenous.

A richer model can use

$$ \lambda(t) = \mu(t) + \sum_{t_i<t} \phi(t-t_i), $$

where

$$ \mu(t) $$

contains calendar effects, covariates or another parametric or nonparametric trend.

The Hawkes kernel then models clustering beyond what the exogenous baseline explains.

This separation is essential if the branching ratio is going to receive a scientific interpretation.

Near-critical estimates require particular skepticism

As

$$ n\uparrow1, $$

several quantities diverge.

The stationary mean amplification is

$$ \frac1{1-n}. $$

The expected cluster size is

$$ \frac1{1-n}. $$

The long-window Fano factor is

$$ \frac1{(1-n)^2}. $$

Small errors in the estimated branching ratio therefore produce large changes in implied behaviour near one.

If

$$ n=0.8, $$

the expected cluster size is

$$ 5. $$

If

$$ n=0.9, $$

it is

$$ 10. $$

If

$$ n=0.95, $$

it is

$$ 20. $$

A confidence interval from 0.85 to 0.97 is not a minor parameter uncertainty. It spans radically different cluster regimes.

Finite windows can bias branching-ratio estimates near criticality. Slow baseline variation can masquerade as long-memory self-excitation. Kernel misspecification can push mass into the branching ratio. Missing or duplicated events can change the apparent degree of clustering.

A claim that a system is "near critical" should therefore be supported by more than one unconstrained maximum-likelihood estimate.

Profile likelihood, alternative kernel families, baseline sensitivity and simulation from the fitted model are all relevant.

Criticality is a strong mechanistic claim.

The data must distinguish it from ordinary heterogeneity.

Multivariate Hawkes processes turn excitation into a directed network

Suppose there are

$$ d $$

event types.

Let

$$ N_i(t) $$

count type-$i$ events.

A multivariate linear Hawkes process has intensities

$$ \lambda_i(t) = \mu_i + \sum_{j=1}^d \int_0^t \phi_{ij}(t-s) \,dN_j(s). $$

An event of type $j$ can increase future intensity of type $i$ through kernel

$$ \phi_{ij}. $$

Define the branching matrix

$$ G $$

with entries

$$ G_{ij} = \int_0^\infty \phi_{ij}(s)\,ds. $$

The entry

$$ G_{ij} $$

is the expected number of direct type-$i$ offspring produced by one type-$j$ event under the branching representation.

The stationarity condition generalizes from a scalar inequality to a spectral one:

$$ \rho(G) < 1, $$

where

$$ \rho(G) $$

is the spectral radius.

The stationary mean intensity vector satisfies

$$ \bar\lambda = \mu + G\bar\lambda. $$

Therefore,

$$ \bar\lambda = (I-G)^{-1} \mu. $$

This is a network amplification equation.

Indirect paths matter.

A type-$1$ event can excite type 2, which excites type 3, which feeds back into type 1.

The matrix inverse expands formally as

$$ (I-G)^{-1} = I + G + G^2 + G^3 + \cdots $$

when

$$ \rho(G)<1. $$

The powers of

$$ G $$

represent successive generations of event propagation.

This connects Hawkes processes naturally to network science.

The weighted directed graph encoded by

$$ G $$

describes predictive excitation among event types.

It should not automatically be read as a causal intervention network.

An omitted common driver can create apparent cross-excitation.

Predictive excitation is not automatically causal influence

Suppose events of type A consistently occur shortly before events of type B.

A multivariate Hawkes model may estimate

$$ \phi_{BA}(t)>0. $$

This means that, conditional on the modelled history, type-A events improve prediction of future type-B intensity.

This resembles Granger causality in continuous time.

It is not automatically equivalent to the intervention statement:

forcing an A event would cause more B events.

A hidden process can generate both.

A calendar effect can produce sequential patterns.

Measurement latency can make one event type appear to precede another systematically.

Selection or logging rules can create artificial ordering.

Causal interpretation therefore requires assumptions beyond the Hawkes likelihood.

The model is strongest as a description of temporal predictive dependence.

When used causally, the baseline, covariates and observation mechanism must be rich enough that common causes have been addressed credibly.

This is the same principle that applies to regression and time series more broadly.

Temporal order helps.

It does not identify interventions by itself.

Linear Hawkes models only excite

The classical linear Hawkes intensity is additive:

$$ \lambda(t) = \mu + \sum_i \phi(t-t_i). $$

If

$$ \phi(t)\ge0, $$

events can only increase future intensity.

Many systems contain inhibition.

A refractory period after a biological event can temporarily reduce the probability of another.

A maintenance repair can reduce immediate failure risk.

A purchase can temporarily reduce the chance of another purchase.

Signed kernels in a linear model can drive the intensity negative, which is invalid.

One solution is a nonlinear Hawkes model,

$$ \lambda(t) = g \left[ \mu + \sum_i h(t-t_i) \right], $$

where

$$ h $$

may take positive and negative values and

$$ g $$

maps the linear predictor to a nonnegative intensity.

For example,

$$ g(x) = e^x $$

guarantees positivity.

The price is that the clean Poisson branching representation is generally lost.

The linear nonnegative Hawkes model is mathematically elegant precisely because excitation and branching coincide.

Inhibition requires a different probabilistic structure.

The model class should follow the mechanism rather than stretching a convenient branching interpretation beyond where it applies.

Simulation reveals what the fitted model actually implies

For a subcritical linear Hawkes process, there are two natural simulation views.

The branching method first simulates immigrant events from a Poisson process with rate

$$ \mu. $$

Each event generates offspring according to an inhomogeneous Poisson process with intensity

$$ \phi(t). $$

Those offspring generate further descendants until clusters die out.

This simulation mirrors the cluster interpretation directly.

Ogata's thinning method works from the conditional intensity. It constructs an upper bound on the current rate, proposes candidate event times from a dominating Poisson process and accepts each candidate with probability equal to the ratio of actual intensity to the bound.

Thinning is flexible and extends beyond cases with a simple cluster representation.

Simulation is not merely a way to create pretty event plots.

A fitted Hawkes model implies distributions of cluster sizes, inter-event times, count variance, burst duration and extreme event density.

Simulating those quantities and comparing them with observed counterparts is a posterior-predictive or parametric-bootstrap style model check.

If the fitted model matches the likelihood but fails to reproduce cluster-duration or count-variance behaviour, the kernel family is probably missing relevant structure.

The model should be judged by the event patterns it claims to explain.

Hawkes processes model memory in event time

The Poisson process says that once the current rate is known, the event history itself is irrelevant.

The Hawkes process says that history matters through a weighted memory of past events.

For the exponential kernel, that memory decays exponentially.

For power-law kernels, memory can persist much longer.

For multivariate kernels, one event type can alter the future intensity of another.

The branching ratio summarizes total endogenous amplification.

The kernel shape controls the timing of that amplification.

The baseline represents exogenous opportunity.

These components answer different questions and should not be collapsed into one measure of burstiness.

The worked example makes this separation concrete.

A Poisson process and a Hawkes process can both average five events per day.

The Poisson Fano factor remains one.

A Hawkes process with branching ratio 0.6 has long-window Fano factor 6.25.

Two Hawkes models can then share that same mean and branching ratio while having excitation half-lives of 16.6 hours and 1.66 hours.

Mean rate, total amplification and clustering timescale are different statistical objects.

That is why modelling the event times themselves can reveal structure that binned counts hide.

A Hawkes process is useful when events plausibly alter the future event rate.

It is misleading when clustering is really exogenous seasonality, hidden common shocks or measurement artifacts.

The distinction is not solved by the model name.

It is solved by how the intensity is specified, how the model is validated and how cautiously the excitation parameters are interpreted.

References

Bacry, E., Mastromatteo, I., & Muzy, J.-F. (2015). Hawkes processes in finance. Market Microstructure and Liquidity, 1(1), 1550005.

Daley, D. J., & Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Volume I (2nd ed.). Springer.

Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika, 58(1), 83–90.

Hawkes, A. G., & Oakes, D. (1974). A cluster process representation of a self-exciting process. Journal of Applied Probability, 11(3), 493–503.

Ogata, Y. (1981). On Lewis' simulation method for point processes. IEEE Transactions on Information Theory, 27(1), 23–31.

Ogata, Y. (1988). Statistical models for earthquake occurrences and residual analysis for point processes. Journal of the American Statistical Association, 83(401), 9–27.

Rasmussen, J. G. (2013). Bayesian inference for Hawkes processes. Methodology and Computing in Applied Probability, 15, 623–642.

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Diogo Ribeiro (2026). Hawkes Processes Turn Events Into Causes of Future Events. Faculty of Media Arts and Design, Technical University of Porto. https://diogoribeiro7.github.io/mathematics/hawkes_processes_turn_events_into_causes_of_future_events/.

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