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Conjugate Bayesian Two-step Change Point Detection for Hawkes Process
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Proposes CoBay-CPD, a conjugate Bayesian two-step change point detector for nonlinear Hawkes processes that uses Pólya-Gamma and marked Poisson data augmentation to obtain an analytical Gibbs sampler, evaluated on synthetic, WannaCry, and NYC collision data.
Problem
Bayesian two-step change point detection for Hawkes processes requires non-conjugate inference (MCMC, variational) that lacks analytical form, making it computationally slow and unsuitable for timely CPD.
Approach
Augment the sigmoid-link nonlinear Hawkes likelihood with Pólya-Gamma variables and a marked Poisson process so it becomes conditionally conjugate to a Gaussian prior on basis weights and a gamma-type prior on the intensity upper bound, enabling closed-form Gibbs sampling of posterior parameters, followed by thinning-based predictive intervals for detection.
Key results
- On synthetic data CoBay-CPD achieves FNR 0.13 and FPR 0.46% with MSE 0.05 and runtime 4.62 min, versus SMCPD 0.38/0.76%/5.50 min and SVCPD+Inhi 0.33/0.60%/23.09 min.
- On WannaCry, CoBay-CPD reaches FNR 0.21 and runtime 6.24 min versus 9.72 min for SVCPD and 29.76 min for SVCPD+Inhibition.
- On NYC Vehicle Collisions, CoBay-CPD reaches FNR 0.13, FPR 0.89%, and runtime 8.70 min, lower than all baselines (SMCPD runtime 24.67 min).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use CoBay-CPD instead of SMCPD | Hawkes change point detection · Synthetic data | FNR: 0.38 $\pm$ 0.41 -> 0.13 $\pm$ 0.22 | Helped |
| use CoBay-CPD instead of SMCPD | Hawkes change point detection · Synthetic data | RT (minute): 5.50 $\pm$ 0.31 -> 4.62 $\pm$ 0.10 | Helped |
| use CoBay-CPD instead of SVCPD | Hawkes change point detection · WannaCry | FNR: 0.34 $\pm$ 0.12 -> 0.21 $\pm$ 0.04 | Helped |
| use CoBay-CPD instead of SVCPD+Inhi | Hawkes change point detection · WannaCry | FPR: 0.00 $\pm$ 0.00 -> 0.05 $\pm$ 0.02 | Hurt |
| use CoBay-CPD instead of SVCPD+Inhi | Hawkes change point detection · NYC Vehicle Collisions | FNR: 0.22 $\pm$ 0.16 -> 0.13 $\pm$ 0.16 | Helped |
| increase the number of basis functions from 1 to 3 (ablation) | CoBay-CPD · Synthetic data | FNR: 0.38 $\pm$ 0.41 -> 0.13 $\pm$ 0.22 | Helped |
| increase the number of basis functions from 1 to 3 (ablation) | CoBay-CPD · Synthetic data | RT (minute): 1.57 $\pm$ 0.03 -> 3.62 $\pm$ 0.10 | Hurt |
| use a 95% rather than 90% confidence interval (ablation) | CoBay-CPD · Synthetic data | FNR: 0.13 $\pm$ 0.22 -> 0.50 $\pm$ 0.00 | Hurt |
| use an 85% rather than 90% confidence interval (ablation) | CoBay-CPD · Synthetic data | FPR: 0.46 $\pm$ 0.26 -> 1.83 $\pm$ 0.43 | Hurt |
| increase prior covariance from σ²=0.5 to σ²=10 (ablation) | CoBay-CPD · Synthetic data | FNR: 0.13 $\pm$ 0.22 -> 0.50 $\pm$ 0.00 | Hurt |
| decrease prior covariance from σ²=0.5 to σ²=0.01 (ablation) | CoBay-CPD · Synthetic data | FPR: 0.46 $\pm$ 0.26 -> 0.76 $\pm$ 0.26 | Hurt |
Surprising findings
- CoBay-CPD has a higher FPR than SVCPD+Inhibition on WannaCry (0.05 vs 0.00), trading false positives for lower FNR.
- Excessively small differences in intensity upper bound (delta lambda = 0.1) cause CoBay-CPD to miss the change point (FNR 1.00).
Limitations
- Extending to multivariate Hawkes processes is challenging because change points must occur at specific event locations.
- Hyperparameters (prior covariance, confidence interval, basis count) must be tuned by cross-validation and are sensitive.
- Performance degrades when adjacent intensity parameters are very similar (delta lambda = 0.1 gives FNR 1.00).
Experimental setup
Datasets: synthetic concatenated Hawkes segments WannaCry Cyber Attack traffic logs (208 observations) NYC Vehicle Collisions (Oct 14 2017, 477 records)
Benchmarks: synthetic concatenated Hawkes dataset WannaCry Cyber Attack NYC Vehicle Collisions
Metrics: FNR FPR MSE running time
Baselines: SMCPD SVCPD SVCPD+Inhibition
Builds on: Hawkes process Bayesian two-step change point detection Pólya-Gamma data augmentation for point processes SMCPD SVCPD
Closest papers by meaning
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