Exact Phase-Type Point Processes for Certified Measurement and Prediction of Event Streams

Sayak Dutta

cs.LGOct 10, 2026

Exact Phase-Type Point Processes for Certified Measurement and Prediction of Event Streams

Authors: Sayak Dutta

Abstract

Temporal point processes are used both to measure market mechanisms, for example how fast prices absorb news and how much of the reaction is the market responding to itself, and to predict the next event. We show that one algebraic object, the phase-type (Erlang-chain) state space, serves both purposes with exact likelihoods. For measurement, we build a multiscale phase-type Hawkes process whose parameters all enter linearly. Its log-likelihood is concave, and we compute the global maximum together with a Fenchel duality-gap certificate. A Markov embedding gives the response to an exogenous shock, echoes included, in closed form, and this response is stable if and only if the spectral radius of the branching matrix is below one. Applied to six markets around 670 scheduled macroeconomic releases (2022-2026) with a built-in placebo test, the model shows that the direct reaction to CPI, payrolls, PPI and the FOMC statement is absorbed in about a second, while the market's own echoes stretch half of the response to 16-137 s. For prediction, we introduce EPT-X, a neural point process that adds a history-dependent mixture-of-Erlang renewal channel to a linear Hawkes backbone. Its intensity and compensator remain in closed form, its states follow an affine recursion that trains with a parallel scan, and it can start from the certified fit of the measurement model. EPT-X has the best test likelihood on all five LOBSTER order-book streams and on Retweet and Taobao, and is level with the best published model on StackOverflow. We also find that the EasyTPP implementation of IntensityFree clamps short inter-event times before scoring, and that the gaps of the EasyTPP Amazon data are uniform on two fixed intervals.

Figures & tables

Panel of Figure 1 — Figure 1: EPT-X at one event. The GRU summary $\bm{h}_{n}$ sets the jump of the neural Erlang state, the hazard parameters and the renewal weights, while the Hawkes state is driven by the event mark alone. The grey boxes are the states carried from one event to the next. Until the next event, all four terms evolve in closed form. The dashed box is the optional residual mark law, which splits the total intensity across marks without changing it.

Classical estimators
	S1 exp	S2 power law	S3 hump	S4 multiscale	S5 network
Exp. kernel MLE	0.05 $\pm$ 0.05	26.05 $\pm$ 0.50	101.47 $\pm$ 1.81	711.28 $\pm$ 53.09	426.80 $\pm$ 368.57
Sum-of-exp. MLE	0.92 $\pm$ 0.13	0.18 $\pm$ 0.12	11.51 $\pm$ 0.46	–	–
ADM4	0.05 $\pm$ 0.05	26.05 $\pm$ 0.50	101.47 $\pm$ 1.81	3.13 $\pm$ 0.25	26.51 $\pm$ 0.29
Hawkes EM (nonparam.)	1.02 $\pm$ 0.07	1.73 $\pm$ 0.21	2.24 $\pm$ 0.12	4.55 $\pm$ 0.29	5.23 $\pm$ 0.25
Conditional law	2.84 $\pm$ 1.03	4.91 $\pm$ 0.59	10.66 $\pm$ 2.13	114.94 $\pm$ 1.20	78.64 $\pm$ 4.79

Table 1: Synthetic recovery: held-out log-likelihood deficit to the true model (

10^{-3}

nats per event, lower is better; mean

\pm

s.d. over five seeds). Bold: best; underlined: second best; shaded: our estimators.

Neural TPPs, published ( Chang et al., 2024 ; Boyd et al., 2025 )
	Amazon	Retweet	Taxi	Taobao	StackOverflow
RMTPP	$-$ 2.136	$-$ 7.098	0.346	1.003	$-$ 2.480
SAHP	$-$ 2.074	$-$ 6.708	0.298	1.168	$-$ 2.341
THP	$-$ 2.096	$-$ 6.659	0.372	0.790	$-$ 2.338
IntensityFree	0.496	$-$ 10.344	0.453	1.318	$-$ 2.233
MHP	$-$ 2.091	$-$ 6.564	0.370	0.636	$-$ 2.346

Table 2: Public benchmarks: test log-likelihood per event (higher is better). Published numbers are from Chang et al. (2024) and Boyd et al. (2025) . Our rows give the mean

\pm

s.d. over seeds (EPT-X: three to five seeds; other rows one seed), and the EPT-X configuration is selected per dataset on validation. Bold: best; underlined: second best; shaded: our models.

Published ( Chang et al., 2024 ; Boyd et al., 2025 )
	Amazon	Retweet	Taxi	Taobao	StackOverflow
RMTPP	0.338	16488	0.283	0.126	1.049
SAHP	0.335	16102	0.290	0.126	1.031
THP	0.332	16268	0.285	0.125	1.033
IntensityFree	0.327	16625	0.362	0.125	1.340
MHP	0.329	16109	0.284	0.126	1.046

Table 3: Next-event prediction on the public benchmarks: time RMSE (top, lower is better; Retweet in seconds) and mark accuracy (bottom, %, higher is better). Published numbers from Chang et al. (2024) and Boyd et al. (2025) , Table 2(b,c) of each. EPT-X: mean

\pm

s.d. over seeds of the validation-selected configuration. Bold: best; underlined: second best; shaded: ours.

Classical Hawkes
	AAPL	AMZN	GOOG	INTC	MSFT
Exp. kernel MLE	1.961	–	$-$ 0.311	–	–
Hawkes EM (nonparam.)	3.370	2.243	2.851	5.771	6.258
ADM4	1.961	1.067	1.306	4.787	5.211
Conditional law	$-$ 1.018	$-$ 17.814	$-$ 32.784	0.379	$-$ 1.270
Neural TPPs (EasyTPP, our runs)

Table 4: LOBSTER order books: test log-likelihood per event (higher is better; mean

\pm

s.d. over seeds where several were run). Bold: best; underlined: second best; shaded: our models. ∗ As shipped, IntensityFree clamps gaps at

10^{-5}

s before scoring, so its score is not the likelihood of the observed data; it is greyed and not ranked. The IntensityFree row is retrained and scored on the actual gaps.

Data	Hawkes backbone	renewal	baseline	hazard terms	neural state
Amazon	0.4	99.6	0.0	0.0	0.0
Retweet	9.5	8.2	0.9	1.0	80.4
Taxi	13.2	80.9	0.2	5.3	0.4
Taobao	57.0	36.0	6.0	0.9	0.1
StackOverflow	49.4	3.0	0.5	44.9	2.2
LOBSTER AAPL	30.6	40.6	1.5	1.5	25.8

Table 5: Exact share (%) of the expected number of test events contributed by each channel of the selected EPT-X model (seed 0).

Data	Model	Compensator	Device	s / epoch	Parameters
Taxi	EPT-X, event loop	exact	GPU	9.1	146k
	EPT-X, parallel scan	exact	GPU	3.2	146k
	S2P2	Monte Carlo	GPU	7.7	35k
LOBSTER AAPL	EPT-X, event loop	exact	GPU	13.0	76k
	EPT-X, parallel scan	exact	GPU	4.5	76k
	S2P2	Monte Carlo	GPU	5.8	34k

Table 6: Training cost per epoch (including validation) and model size. EPT-X: one epoch of the same model timed with the event loop and with the parallel scan. EasyTPP baselines: average over their full epoch budget.

	Taxi	Taobao	StackOverflow	Amazon	INTC
Validation log-likelihood per event
EPT (no renewal channel)	0.392	1.200	$-$ 2.186	$-$ 1.193	5.360 $\pm$ 0.007
EPT-X	0.466 $\pm$ 0.004	1.205 $\pm$ 0.002	$-$ 2.161 $\pm$ 0.003	$-$ 0.487 $\pm$ 0.000	5.463 $\pm$ 0.003
+ sharp atoms (orders to 1024)	0.457	1.194	$-$ 2.173	0.604	5.516
+ sharp atoms, 48 scales	0.455	1.177	$-$ 2.182	0.643	5.511
+ deep encoder (3 layers)	0.396	1.142	$-$ 2.264	$-$ 0.507	5.461

Table 7: EPT-X ablations: validation log-likelihood per event (higher is better). The EPT and EPT-X rows are the final runs (mean

\pm

s.d. over seeds); each variant is one seed. Encoder and atom variants use a 100-epoch budget. The last three rows add one new component each, with the final budget on Taxi, Taobao and StackOverflow and 100 epochs on Amazon and INTC, where they are added to the sharp-atom configuration. Bold: best; underlined: second best.

Panel of Figure 2 — Figure 2: Measurement pipeline. Prices become $\delta$ -crossing events, which are grouped into release windows and matched placebo windows. A single certified MSX fit gives the duality gap and the placebo test, and its Markov embedding gives the response $\bm{m}(t)$ , the absorption times and the expected price drift in closed form.

Figure 3: Placebo test. Direct news-kernel mass, the expected number of extra events per release summed over all assets, for each release type (blue) and for its matched placebo at the same clock time on a release-free day (grey). A type passes if its mass exceeds the placebo mass by at least 50%.

Figure 4: Response of each asset to a CPI release, in excess events per second (both axes logarithmic). Orange: direct news kernel. Blue: echo-corrected response

\bm{m}(t)

, which adds the events triggered by earlier events. Dotted lines and panel titles give

t_{50}

, direct / echo-corrected.

Release	EURUSD	USDJPY	Gold	S&P 500	BTC	ETH
CPI	64 / 9.1	126 / 11.0	19 / 4.8	137 / 11.0	17 / 6.1	33 / 7.8
NFP	94 / 10.1	109 / 10.7	16 / 5.2	80 / 8.8	27 / 7.5	24 / 7.3
PPI	120 / 10.8	94 / 10.3	40 / 6.4	79 / 8.6	23 / 7.2	39 / 8.7
FOMC	89 / 10.1	76 / 9.8	82 / 9.0	77 / 8.9	41 / 9.2	36 / 9.1
Retail	59 / 9.1	77 / 9.8	87 / 9.2	114 / 10.4	26 / 7.6	26 / 7.8
GDP ∗	95 / 10.0	81 / 9.6	26 / 5.1	145 / 11.2	27 / 7.3	27 / 7.5

Table 8: Echo-corrected absorption times from the pooled model:

t_{50}

in seconds /

t_{90}

in minutes after the release. ∗ Release types whose news-kernel mass does not exceed their placebo by 50% (Figure 3 ).

Figure 5: Expected cumulative price change, in basis points, after a CPI surprise of

+1

(blue) or

-1

(orange) standard deviation, from the pooled model. Time axis logarithmic.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

$\ell_{1}$ level	held-out LL (rel.)	mean release mass	mean placebo mass
0.2	$-$ 93.7	10.79	6.23
0.4	$-$ 27.1	5.20	2.89
0.8	0.0	1.81	1.03
1.6	$-$ 20.5	0.44	0.37

Appendix

Table 9: Choice of the

\ell_{1}

penalty on the news kernels. Held-out log-likelihood on the 2024 windows relative to the best level (bold), and the mean news-kernel mass of release and placebo windows.

	RMSE of $\log(1+N)$			QLIKE of realised variance
Method	1 min	5 min	15 min	1 min	5 min	15 min
Climatology	1.440	1.308	1.199	1.263	0.916	0.701
Event-study OLS	1.140	0.925	0.776	2.552	0.580	0.372
HAR	1.217	0.985	0.820	1.467	0.762	0.468
MSX closed form	1.322	1.049	0.955	2.913	1.036	0.518

Appendix

Table 10: Post-release activity forecasts on the 163 test releases of 2025–2026, from models fitted on 2022–2024.

N

is the number of

\delta

-crossings after the release; QLIKE scores the forecast of realised variance. Mean over assets; lower is better. Bold: best; underlined: second best; shaded: our model.

S1: exponential
	$G$ error	$\rho$ error	kernel $L^{1}$	dLL ( $10^{-3}$ )	edge AUC
Exp. kernel MLE	0.018 $\pm$ 0.007	0.005 $\pm$ 0.003	0.018 $\pm$ 0.008	0.05 $\pm$ 0.05	n/a
Sum-of-exp. MLE	0.065 $\pm$ 0.022	0.030 $\pm$ 0.013	0.165 $\pm$ 0.023	0.92 $\pm$ 0.13	n/a
ADM4	0.018 $\pm$ 0.007	0.005 $\pm$ 0.003	0.018 $\pm$ 0.008	0.05 $\pm$ 0.05	n/a
Hawkes EM (nonparam.)	0.035 $\pm$ 0.015	0.017 $\pm$ 0.007	0.173 $\pm$ 0.010	1.02 $\pm$ 0.07	n/a
NPHC (cumulants)	0.247 $\pm$ 0.087	0.032 $\pm$ 0.024	–	–	n/a

Appendix

Table 11: Synthetic recovery, all metrics (mean

\pm

s.d. over five seeds): relative error of the branching matrix and of its spectral radius, kernel

L^{1}

error and held-out deficit dLL (

10^{-3}

nats/event), all lower is better, and edge AUC (higher is better). AUC needs absent true edges, so it is defined only for S4 and S5 (n/a for the complete graphs S1–S3). “–”: not produced by the method; NPHC estimates only the branching matrix. Missing rows are fits that did not converge. Bold: best; underlined: second best within each scenario; shaded: ours.

Hawkes Processes

See all 14 papers

Sep 18, 2026

Probabilistic Forecasting of Business Process Executions with Neural Temporal Point Processes

Operators of service-based systems act on forecasts of how a running execution will continue, and such a forecast is actionable only if its reliability is known. Mainstream deep-learning models for this task are discriminative and deterministic: they emit a single next activity and a single remaining-time estimate, without a distribution to reason over. We instead cast the problem as generative sequence modelling with marked temporal point processes, which define a joint density over the next mark and its inter-event time and therefore deliver predictive distributions by construction. Real event logs violate the simple-point-process assumption these models rest on, since consecutive events frequently carry identical timestamps; we handle such ties explicitly and combine a transformer encoder with a mixture decoder over inter-event times, trained by exact log-likelihood. On ten public logs, the resulting model matches discriminative baselines on point accuracy, dominates them on the calibration and sharpness of remaining-time distributions, and is the cheapest at inference, since a full predictive distribution is obtained in a single forward pass without sampling.

Jiaxin Yuan, Daniela Grigori, Han van der Aa

Paris Dauphine University-PSL, Pl. du Maréchal de Lattre de Tassigny, 75016 Paris, France · University of Vienna, Universitätsring 1, 1010 Wien, Austria

Probabilistic Forecasting Predictive Process Monitoring Next-Event Prediction Temporal Point Processes Process Mining

Ask PDF

Jul 13, 2026

NeuroMem-FHP: A Likelihood-Free Deep Learning Framework for Parameter Estimation of Fractional Hawkes Process

In this paper, we propose deep learning based NeuroMem-FHP framework for estimating the parameters of the fractional Hawkes process (FHP), a self-exciting point process that captures long-range dependence through a fractional Mittag-Leffler excitation kernel. Two neural architectures, namely a Long Short-Term Memory (LSTM) network and a Transformer, are developed to estimate the model parameters

(μ,γ,α,β)

directly from sequences of inter-arrival times without requiring computationally intensive likelihood optimization. Experiments on synthetic data that both neural models significantly outperform the classical Maximum Likelihood Estimation (MLE) method, with the Transformer achieving the highest estimation accuracy (MSE =

0.1634

), followed by the LSTM (MSE =

0.1752

), compared to MLE (MSE =

2.8032

). An ablation study further examines the effects of key hyperparameters on model performance. The proposed framework is also on two real-world high-frequency datasets, namely AAPL NBBO transaction data and Montgomery County 911 emergency call records. Using a predictive validation approach, event sequences simulated from the estimated parameters closely reproduce the empirical distribution, tail behavior, and temporal dependence structure of the observed data. These results demonstrate that Transformer-based parameter estimation provides an accurate and efficient alternative to conventional estimation techniques for FHP and offers a promising framework for modeling event-driven systems with long-memory dynamics.

Neha Gupta, Aditya Maheshwari

Operations Management and Quantitative Techniques Area, Indian Institute of Management Indore, Indore 453556, India.

Temporal Point Processes Parameter Estimation Hawkes Processes

Ask PDF

May 17, 2026

Structured Neural Marked Point Processes for Interpretable Event Interaction Modeling

Multi-class event streams arise in numerous real-world applications, where uncovering structured, interpretable inter-event relationships, together with accurate prediction, remains a central challenge. Existing neural point process models are highly expressive but encode event interactions in a black-box manner, preventing explicit discovery of structured dependencies. In this paper, we propose a structured neural marked point process (SNMPP) that achieves high modeling flexibility while enabling explicit event-wise and class-wise relationship discovery from data. Our model constructs a product-form neural influence kernel composed of a signed interaction network over event types and a delay-aware monotonic temporal network. This design enables explicit characterization of inter-class influence topology -- including excitation, inhibition, and neutrality -- while flexibly capturing diverse temporal decay patterns and potential influence delays. For efficient learning, we develop a stratified Monte Carlo estimator for stochastic training. Extensive experiments on synthetic and real-world benchmark datasets validate the ability of our approach to uncover structured relationships and deliver strong predictive performance.

Zhitong Xu, Qiwei Yuan, Yinghao Chen +2

Kahlert School of Computing, University of Utah · Celonis AI

Temporal Point Processes Neural Network Interpretability Sequence Modeling

Ask PDF

Abstract

Hawkes Processes

Explore similar work

Probabilistic Forecasting of Business Process Executions with Neural Temporal Point Processes

NeuroMem-FHP: A Likelihood-Free Deep Learning Framework for Parameter Estimation of Fractional Hawkes Process

Structured Neural Marked Point Processes for Interpretable Event Interaction Modeling