hawkes-process

Hawkes Process

The probability density function of hawkes process for the observation period $[0, T]$ is given by

$$ p_{[0,T]}(\boldsymbol{t}{n}) = \prod{i=1}^{n} \left[\mu + \sum_{j < i}g(t-t_j)\right] \times \exp\left[-\mu T - \sum_{i=1}^n\int_{t_i}^T g(s-t_i)ds \right] $$

where, $g(\tau)$ is a kernel function representing the influence from past events.

Maximum \mathcal{L}ikelihood Estimation of a hawkes process

Assume the following for kernel functions $$g(\tau) = ab \exp(-b\tau)$$

Maximum likelihood estimation of $\boldsymbol{\theta} = \{\mu, a, b\}$

The likelihood function $\mathcal{L}(\boldsymbol{\theta}|\boldsymbol{t}_n)$ is

$$ \mathcal{L}(\boldsymbol{\theta}|\boldsymbol{t}n) = p{[0,T]}(\boldsymbol{t}_n | \boldsymbol{\theta}) $$

$$\hat{\boldsymbol{\theta}} = \arg\min_{\boldsymbol{\theta}}-\log \mathcal{L}(\boldsymbol{\theta}|\boldsymbol{t}_n) $$

Estimate parameters by gradient descent method.

$$ \begin{aligned} \frac{\partial}{\partial\mu}\log \mathcal{L}(\boldsymbol{\theta} | \boldsymbol{t}n) &= \sum{i=1}^n \frac{1}{\lambda_i} - T \ \frac{\partial}{\partial a}\log \mathcal{L}(\boldsymbol{\theta} | \boldsymbol{t}n) &= \sum{i=1}^n \frac{1}{\lambda_i}\frac{\partial \lambda_i}{\partial a} - \sum_{i=1}^n [1 - \exp[-b(T-t_i)]] \ \frac{\partial}{\partial b}\log \mathcal{L}(\boldsymbol{\theta} | \boldsymbol{t}n) &= \sum{i=1}^n \frac{1}{\lambda_i}\frac{\partial \lambda_i}{\partial b} - \sum_{i=1}^n a(T-t_i) \exp[-b(T-t_i)] \end{aligned} $$ where, $$ \begin{aligned} \lambda_i &= \mu + \sum_{j<i}ab\exp[-b(t_i-t_j)] \ \frac{\partial \lambda_i}{\partial a} &= \sum_{j<i}b\exp[-b(t_i - t_j)] \ \frac{\partial \lambda_i}{\partial b} &= \sum_{j<i} a\exp[-b(t_i - t_j)][1-b(t_i-t_j)] \end{aligned} $$

Simple computation is $O(n^2)$. However, efficient computation is $O(n)$.

$$G_i = \sum_{j < i} ab\exp[-b(t_i - t)]$$

Introducing the above, we have the following asymptotic equation.

$$ \begin{aligned} G_{i+1} &= (G_{i} + ab)\exp[-b(t_{i+1} - t_i)] \\ \frac{\partial G_{i+1}}{\partial b} &= \left(\frac{\partial G_{i}}{\partial b} + a\right) \exp[-b(t_{i+1} - t_i)] - G_{i+1}(t_{i+1}-t_i) \end{aligned} $$

Using this, we can calculate $\lambda_i, \frac{\partial \lambda_i}{\partial a}, \frac{\partial \lambda_i}{\partial b}$

$$ \begin{aligned} \lambda_i = G_i + \mu, \quad \frac{\partial \lambda_i}{\partial a} = \frac{G_i}{a}, \quad \frac{\partial \lambda_i}{\partial b} &= \frac{\partial G_i}{\partial b} \end{aligned} $$

Usage

Start container

$ docker-compose up -d --build

Attach container

$ docker exec -it hawkes-process /bin/bash

Maximum likelihood estimation

$ python main.py

Reference

点過程の時系列解析