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Extending JumpProcess.jl for fast point process simulation with time-varying intensities

Guilherme Augusto Zagatti, Samuel A. Isaacson, Christopher Rackauckas, Vasily Ilin, See-Kiong Ng, Stéphane Bressan

TL;DR

JumpProcesses.jl is extended with a new simulation algorithm, Coevolve, that enables the rapid simulation of processes with locally-bounded variable intensity rates and significantly improves the computational performance of JumpProcesses.jl when simulating such processes.

Abstract

Point processes model the occurrence of a countable number of random points over some support. They can model diverse phenomena, such as chemical reactions, stock market transactions and social interactions. We show that JumpProcesses.jl is a fast, general-purpose library for simulating point processes. JumpProcesses.jl was first developed for simulating jump processes via stochastic simulation algorithms (SSAs) (including Doob's method, Gillespie's methods, and Kinetic Monte Carlo methods). Historically, jump processes have been developed in the context of dynamical systems to describe dynamics with discrete jumps. In contrast, the development of point processes has been more focused on describing the occurrence of random events. In this paper, we bridge the gap between the treatment of point and jump process simulation. The algorithms previously included in JumpProcesses.jl can be mapped to three general methods developed in statistics for simulating evolutionary point processes. Our comparative exercise revealed that the library initially lacked an efficient algorithm for simulating processes with variable intensity rates. We, therefore, extended JumpProcesses.jl with a new simulation algorithm, Coevolve, that enables the rapid simulation of processes with locally-bounded variable intensity rates. It is now possible to efficiently simulate any point process on the real line with a non-negative, left-continuous, history-adapted and locally bounded intensity rate coupled or not with differential equations. This extension significantly improves the computational performance of JumpProcesses.jl when simulating such processes, enabling it to become one of the few readily available, fast, general-purpose libraries for simulating evolutionary point processes.

Extending JumpProcess.jl for fast point process simulation with time-varying intensities

TL;DR

JumpProcesses.jl is extended with a new simulation algorithm, Coevolve, that enables the rapid simulation of processes with locally-bounded variable intensity rates and significantly improves the computational performance of JumpProcesses.jl when simulating such processes.

Abstract

Point processes model the occurrence of a countable number of random points over some support. They can model diverse phenomena, such as chemical reactions, stock market transactions and social interactions. We show that JumpProcesses.jl is a fast, general-purpose library for simulating point processes. JumpProcesses.jl was first developed for simulating jump processes via stochastic simulation algorithms (SSAs) (including Doob's method, Gillespie's methods, and Kinetic Monte Carlo methods). Historically, jump processes have been developed in the context of dynamical systems to describe dynamics with discrete jumps. In contrast, the development of point processes has been more focused on describing the occurrence of random events. In this paper, we bridge the gap between the treatment of point and jump process simulation. The algorithms previously included in JumpProcesses.jl can be mapped to three general methods developed in statistics for simulating evolutionary point processes. Our comparative exercise revealed that the library initially lacked an efficient algorithm for simulating processes with variable intensity rates. We, therefore, extended JumpProcesses.jl with a new simulation algorithm, Coevolve, that enables the rapid simulation of processes with locally-bounded variable intensity rates. It is now possible to efficiently simulate any point process on the real line with a non-negative, left-continuous, history-adapted and locally bounded intensity rate coupled or not with differential equations. This extension significantly improves the computational performance of JumpProcesses.jl when simulating such processes, enabling it to become one of the few readily available, fast, general-purpose libraries for simulating evolutionary point processes.
Paper Structure (13 sections, 16 equations, 1 figure, 3 tables, 5 algorithms)

This paper contains 13 sections, 16 equations, 1 figure, 3 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. The evolutionary point process
  3. The homogeneous process
  4. Exact simulation methods
  5. Inverse methods
  6. Thinning methods
  7. Queuing methods
  8. Implementation
  9. Empirical evaluation
  10. Statistical analysis of Coevolve
  11. Benchmarks
  12. Conclusion
  13. Acknowledgements

Figures (1)

  • Figure 1: Simulations of 10-nodes compound Hawkes process with parameters $\lambda = 0.5 , \alpha = 0.1 , \beta = 2.0$ for $200$ units of time. (a) and (b) sampled trajectory and intensity rate for a single simulation for the three selected nodes in (c) for the first $20$ units of time. (c) underlying 10-nodes network with three random nodes selected. (d) Q-Q plot of transformed inter-event time for 250 simulations colored by node.