Hawkes process - animation

Hawkes processes are a class of temporal point processes that model time occurrences of events. It extends the well-known class of Poisson processes in a way that enables simple and yet powerful description of self-excitation or inhibition. They are widely used in finance, neuroscience, social networks, and more, where the occurrence of one event influences the probability of future events. This post contains well-known facts about Hawkes processes, but also an interactive JavaScript simulation of a bivariate Hawkes process which is quite original up to my knowledge.

All temporal point processes below are denoted by $N$, i.e. $N$ is a random set of times.

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Introduction

Let us start with the simplest case, which is a univariate (linear) Hawkes process $N$. A univariate Hawkes process models a single type of events. Like any temporal point process model, it is characterized by its conditional intensity $\lambda_t$ which describes the instantaneous rate at which events occur, given the past history $\mathcal{F}_t$. If the history is empty at initial time $t=0$, the intensity reads as

\[\lambda_t = \mu + \sum_{T < t} h(t - T) = \mu + \int_0^{t-} h(t-s) N(\mathrm{d}s),\]

where:

• $\mu \geq 0$ is the baseline intensity (spontaneous event rate),
• $h: \mathbb{R} \to \mathbb{R}$ is the delay function, which models the influence of past events on future intensity,
• the sum runs across all times $T$ of the Hawkes process $N$ that occurred before time $t$.

The main restriction of this model is that the values of $h$ must be non negative. In turn, such a process cannot model inhibitory phenomena.

A multivariate Hawkes process generalizes this to multiple interacting event types. For a process with $n$ dimensions, the (random) sets are denoted by $N^1,\dots, N^n$ and the intensities read as

\[\lambda^i(t) = \mu_i + \sum_{j=1}^{n} \int_0^{t-} h_{i,j}(t-s) N^j(\mathrm{d}s).\]

Here, $h_{i,j}$ encodes how events of type $j$ excite events of type $i$. This framework is extremely useful when modeling networks of interacting neurons, trades in multiple assets, or social interactions.

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Delay Functions

When the delay functions are exponential, Hawkes processes enjoy a Markovian structure, therefore their simulation and theoretical study are way more efficient. For instance, if $n = 1$ and the delay function is given by $h(t) = \beta e^{−\alpha t}$, then

\[\Xi_t = \int_0^t \beta e^{−\alpha (t−s)} N(ds),\]

defines a Markov process which furthermore satisfies a simple stochastic differential equaiton with jumps. In this example, $\Xi$ follows an exponential decay to 0 at rate $\alpha$ with jumps of height $\beta$. The parameter $\beta$ describes the interaction strength.

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Nonlinear case

The Hawkes processes defined above are usually described as linear in contrast with the more generic case where the intensity (in the univariate case) reads as

\[\lambda_t = f\left( \int_0^{t-} h(t-s) N(\mathrm{d}s) \right),\]

where the activation function $f: \mathbb{R} \to \mathbb{R}_+$ ensures positivity and/or model saturation effects. Nonlinear Hawkes processes allow more flexible modeling, especially in networks with inhibition: delay functions $h$ with negative values are allowed here.

Remark: The linear case corresponds to $f(\xi) = \mu + \xi$ under the restriction that $h$ has non-negative values. It is commonly extended to the rectified-linear case, i.e. $f(\xi) = \max(0, \mu + \xi)$.

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JavaScript animation

Here is the link to an interactive simulation of a bivariate Hawkes process in JavaScript using Plotly.js. The intensities are given by: (the colors below correspond to the colors in the animation)

\[\begin{align*} {\color{darkblue} \lambda^1_t} &= \mu_1 + \int_0^t \beta_{11} e^{-\alpha(t-s)} {\color{darkblue} N^1}(\mathrm{ds}) + \int_0^t \beta_{12} e^{-\alpha(t-s)} {\color{orange} N^2}(\mathrm{ds}),\\ {\color{orange} \lambda^2_t} &= \mu_2 + \int_0^t \beta_{21} e^{-\alpha(t-s)} {\color{darkblue} N^1}(\mathrm{ds}) + \int_0^t \beta_{22} e^{-\alpha(t-s)} {\color{orange} N^2}(\mathrm{ds}). \end{align*}\]

The simulation algorithm uses the thinning of two underlying Poisson processes ${\color{darkblue}\Pi^1}$ and ${\color{orange}\Pi^2}$. That is the reason why most of the points are not modified when you modify the parameters. A good exercise is to wonder what would happen if the simulation algorithm used time change instead of thinning.

Here are the animation features:

• the parameters $\mu_1, \mu_2, \alpha, \beta_{11}, \beta_{12}, \beta_{21}, \beta_{22}$ can be modified via sliders;
• the dots on the x-axis represent the times of $N^1$ and $N^2$;
• the solid lines represent both conditional intensities $\lambda^1$ and $\lambda^2$;
• the dashed lines represent both baseline intensities $\mu_1$ and $\mu_2$;
• a button to re-simulate the underlying Poisson processes;
• a checkbox to toggle visibility of the underlying Poisson processes with cross markers.

The interaction strengths $\beta_{11}, \beta_{12}, \beta_{21}, \beta_{22}$ are set to 0 by default which corresponds to the case where $N^1$ and $N^2$ are independent Poisson processes.

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Try It Yourself

And feel free to reach me if you want the source code ot have some ideas for improvement.