Hawkes_Process_JAX

Hawkes Process MLE Inference Using JAX

• Run python hawkes.py

• Requires tick and JAX

• Hawkes Process model for event modelling https://en.wikipedia.org/wiki/Hawkes_process

$$\lambda_i (t) = \mu_i + \sum_j \sum_{t_{jl} < t} \alpha_{ij} \omega \exp{ \omega (t - t_{jl} ) }$$

Loglikelihood is defineds as,

$$\sum_{n=1}^N {\log(\lambda_{i_{t_n}} (t_n) )} - \sum_i \int_0^T \lambda_i (t) dt$$

• Before Optimization Negative Loglikelihood 268.909910517387
• After Optimization Negative Loglikelihood 219.8147895512949
• \mu True values [0.5 0.1 0.5]
• \mu Estimated Values [0.53409991 0.04881035 0.4799496 ]
• \alpha True values

$$[[0.1 0. 0. ] [0. 0.3 0. ] [0. 0. 0.5]]$$

• \alpha Estimated Values

$$[[0.01708269 0. 0.04558697] [0.04051707 0.37785955 0. ] [0.07921349 0.2555025 0.53284138]]$$

• \omega true value 0.3
• \omega estimate value 0.22339805387220335