# Non parametric Bayesian estimation for Hawkes processes

Sophie Donnet, Judith Rousseau, Vincent Rivoirard

► To cite this version:

Sophie Donnet, Judith Rousseau, Vincent Rivoirard. Non parametric Bayesian estimation for Hawkes processes. International Society for Bayesian Analysis World Meeting, ISBA 2014, Jul 2014, Cancun, Mexico. pp.1. hal-01565527

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# Non-parametric Bayesian estimation for intensity function in Hawkes processes

<figure><img src="image_2.png" /></figure>

Sophie Donnet, Vincent Rivoirard, Judith Rousseau

UMR AgroParisTech/INRA 518. Team MORSE. INRA. Paris France

sophie.donnet@agroparistech.fr

## Introduction

Hawkes process is the most basic self-exciting counting process. It as been widely used to modelize earthquakes [1-3], financial or economical data but also in neurobiology. The Bayesian estimation of the conditional intensity function has been tackled by [4] in a parametric context. In this work, we propose to consider a non-parametric Bayesian estimation

## Hawkes processes

* The Hawkes process $(N_t)_{t \in \mathbb{R}}$ is defined by its conditional intensity

$$ \lambda(t) = \nu + \int_{-\infty}^{t^{-}} h(t-u)dN_{u} = \nu + \sum_{T \in N, T < t} h(t-T) \quad (1) $$

where

* $\nu$ is a positive parameter,

* $h$ a nonnegative function with support on $\mathbb{R}^+$ and $\int h < 1$

* $dN_u$ is the point measure associated to the process.

* Interpretation : $\lambda(t)dt = \mathbb{P}(\text{point } t; t+dt]$ conditionally to the past before t). So from (1) : constant rate $\nu$ and also all the previous occurrences condition the apparition of an occurrence at t.

* Stationarity :

* We assume that $h$ has a bounded support $[0, s_{max}]$ such that $\int_0^{s_{max}} h(u)du < 1$

* This condition guarantees the existence of a stationary version of the process

* And the number of events on $[T^*, T^* + \Delta]$:

$$ E[N^*] = E[N_{[T^*, T^* + \Delta]}] = \Delta^* \frac{\nu}{1 - \int_0^{s_{max}} h(u)du} $$

## Likelihood and prior distributions

* Parameters of interest : $h$ and $\nu$

* Observations and likelihood : computation of the likelihood on $[T^* - s_{max}, T^* + \Delta]$

$$ \log \ell(N; h, \nu) = \sum_{T_i \in [T^*, T^*+\Delta]} \log \lambda(T_i) - \int_{T^*}^{T^*+\Delta} \lambda(t)dt $$

* Prior distributions

* $h$ step function. $h(t) = \sum_{k=0}^{K-1} \alpha_k \mathbb{1}_{[s_k, s_{k+1}]}(t) = I_h \sum_{k=0}^{K-1} \bar{\alpha}_k \mathbb{1}_{[s_k, s_{k+1}]}(t)$ with

* $s_K = s_{max}$ known and $s_0 = 0$

* $I_h = \int_0^{s_{max}} h(t)dt < 1$

* $\sum_{k=0}^{K-1} \bar{\alpha}_k(s_{k+1} - s_k) = 1$

* $I_h \sim \mathcal{B}(a_h, b_h)$

* Prior on $s$ : $\left( \frac{s_0}{s_{max}} \cdots \frac{s_K}{s_{max}} \right) \sim \text{Dir}(a_1, \dots, a_K)$ (2)

* Prior on $\bar{\alpha}_1, \dots, \bar{\alpha}_K : \bar{\alpha}_0(s_1 - s_0), \dots, \bar{\alpha}_K(s_K - s_{K-1}) \sim \text{Dir}(b_1, \dots, b_K)$

* $\nu \sim \Gamma(a_\nu, b_\nu)$

## Sampling of the posterior distribution : MCMC

At iteration (l),

1. Sampling of $\nu$ with a classical random walk on $\log \nu$ to guaranty the positivity

2. Sampling of $I_h$ with a random walk on $[0, 1]$

$$ I_h^c = \Phi^{-1}(\Phi((I_h)^{l-1}) + \varepsilon), \quad \varepsilon \sim 0.5\mathcal{N}(0, \rho_1) + 0.5\mathcal{N}(0, \rho_2), \quad \rho_1 < \rho_2 $$

3. Sampling of $\bar{\alpha}_k$

* $k \sim U\{0, \dots, K-1\}$ and $\bar{\alpha}_k^c = \bar{\alpha}_k^{l-1} + \xi$, with $\xi \sim \sum_{r=1}^4 0.25\mathcal{N}(0, \rho_r)$

* Correction of the others $\bar{\alpha}$ to keep $\sum_{k=0}^K \bar{\alpha}_k(s_{k+1} - s_k) = 1$ :

$$ \Delta_k = \bar{\alpha}_k^c(s_{k+1} - s_k) - \bar{\alpha}_k^{l-1}(s_{k+1} - s_k) $$

$$ \bar{\alpha}_j^c(s_{j+1} - s_j) = \bar{\alpha}_j^{l-1}(s_{j+1} - s_j) - \frac{\Delta_k}{K-1} \quad \forall j \neq k. $$

4. Sampling of $s$

* $k \sim U\{1, \dots, K-1\}$ and $s_k^c \sim \text{Triang}(s_{k-1}^{l-1}, sl - 1, s_{k+1}^{l-1})$

* Correction of the $\bar{\alpha}_{k-1}^l$ and $\bar{\alpha}_k^l$ to keep the integral equal to 1:

$$ \bar{\alpha}_{k-1}^c = \bar{\alpha}_{k-1}^l \frac{s_k^{l-1} - s_{k-1}^{l-1}}{s_k^c - s_{k-1}^{l-1}} \quad \text{and} \quad \bar{\alpha}_k^c = \bar{\alpha}_{k-1}^l \frac{s_{k+1}^{l-1} - s_k^{l-1}}{s_{k+1}^{l-1} - s_k^c} $$

## First numerical results

$$ T^* = 50 \quad \Delta = 50, \quad I_h = 0.5, \quad \nu = 10 $$

* Data

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* MCMC - 200 000 iterations

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

Work in progress

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

## Perspectives of work

* Unknown $K$ : reversible jump

* $\pi(\bar{\alpha}_k = 0) > 0$

* $\pi(\bar{\alpha}_k < 0) > 0$ to modelize inhibition

* Multi-dimensional Hawkes processes to modelize interactions

## Bibliography

[1] Hawkes AG, Adamopoulos L (1973) Cluster models for earthquakes: regional comparisons. Bull Int Statist Inst 45:454-461

[2] Ogata Y (1988) Statistical models for earthquake occurrences and residual analysis for point processes. J Am Stat Assoc 83(401):9-27

[3] Ogata Y (1998) Space-time point-process models for earthquake occurrences. Ann Inst Stat Math 50(2):379-402

[4] Rasmussen J. (2013) Bayesian Inference for Hawkes Processes. Methodol Comput Appl Probab (2013) 15:623D642

[5] Daley, D. J. and Vere-Jones, D. (2005). An Introduction to the Theory of Point Processes. Springer Series in Statistics I. Springer, New York. MR0950166

[6] Reynaud Bouret P. and Schbath, S. (2010). Adaptive estimation for Hawkes processes; Application to genome analysis. The Annals of Statistics 2010, Vol. 38, No. 5, 2781D2822