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From Jumps to Signatures: a Generative Method for Temporal Point Processes

This paper introduces sigTPP, a novel generative model for Temporal Point Processes that bridges the gap between discrete event sequences and continuous path analysis by using an interarrival embedding to extend signature methods, thereby enabling global trajectory-level training and mathematically justified distributional evaluation.

Original authors: Niels Cariou-Kotlarek, Vasileios Lampos

Published 2026-07-09

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Niels Cariou-Kotlarek, Vasileios Lampos

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Predicting the Unpredictable

Imagine you are trying to teach a computer to understand a sequence of events that happen at random times. These aren't just "events"; they are things like earthquakes, tweets, stock trades, or people checking into hotels. In math, we call these Temporal Point Processes (TPPs).

The problem is that these events are messy. They happen in "jumps" (suddenly, a tweet appears), not in a smooth, flowing line. Most current AI models try to learn these sequences by looking at one event at a time, like trying to understand a movie by only looking at individual frames. They often miss the bigger picture: how the whole story flows together.

This paper introduces a new way to teach AI to understand these event sequences by looking at the entire story at once, using a mathematical tool called Signatures.

The Problem: The "Staircase" vs. The "Slide"

To understand the paper's solution, you need to see the problem with the old way of doing things.

The Messy Staircase (The Data): Real-world event data looks like a staircase. It sits flat, then suddenly jumps up, then sits flat, then jumps up again. In math, these are called càdlàg paths (a fancy word for "jumpy").
The Smooth Slide (The Tool): The mathematical tool the authors want to use, called Signatures, is like a high-tech scanner that works best on smooth, sliding surfaces (continuous paths). It can't scan a staircase directly; it gets confused by the sharp corners.
The Old AI Approach: Previous AI models tried to learn the staircase by looking at each step individually. They asked, "Given the last step, what is the next one?" This is like trying to learn a song by only listening to one note at a time. You might get the notes right, but you miss the melody and the rhythm of the whole song.

The Solution: The "Interarrival Embedding"

The authors invented a clever trick to make the staircase compatible with the smooth scanner. They call this the Interarrival Embedding.

The Analogy: Turning a Staircase into a Ramp
Imagine you have a jagged staircase (the event data). You want to scan it with a device that only works on smooth ramps.

The Trick: Instead of scanning the stairs, you draw a smooth, straight line connecting the corners of the stairs.
The Result: You now have a smooth ramp (a continuous path) that perfectly represents the original staircase.
Why it works: This new ramp contains all the same information as the stairs (how many steps, how wide they are), but now it's smooth enough for the "Signature" scanner to read it.

This allows the authors to use the Signature tool on messy, jumpy event data for the first time.

The New Model: SIGTPP

Using this new "ramp" trick, they built a new AI model called SIGTPP.

Old Way (Local Learning): The old models were like a student memorizing flashcards. "If event A happens, then event B usually follows." They optimize the model one event at a time.
SIGTPP (Global Learning): SIGTPP is like a conductor listening to the whole orchestra. It looks at the entire sequence of events as one big picture. It uses the "Signature" of the smooth ramp to measure how similar the AI's generated story is to the real story.
The Goal: Instead of just getting the next note right, SIGTPP tries to make the AI generate a whole song that sounds exactly like the original, capturing the long-term rhythm and structure.

The Evaluation: How Do We Know It Works?

The authors realized that the usual way to test these models was flawed.

The "Pointwise" Trap:
Usually, people test these models by asking, "How close was your prediction to the actual time of the next event?"

The Flaw: Imagine a deterministic robot that always predicts the average time for an event. It might be very accurate on average (low error), but it's boring and wrong every single time because it never captures the variety or randomness of real life. It's like a weatherman who always predicts "70°F." He's often right, but he's not a good model of the weather.

The New Way (Distributional Metrics):
The authors created new tests that check if the AI captures the whole shape of the randomness.

Energy Distance & Wasserstein Distance: Think of this as comparing two clouds of data points. Do the AI's generated events look like the same "cloud" as the real events? Do they have the same spread, the same clusters, and the same long-range patterns?
The Signature Distance: This measures if the "fingerprint" of the AI's story matches the fingerprint of the real story.

The Results: A New Champion

The authors tested SIGTPP on 9 different datasets (4 made-up synthetic ones and 5 real-world ones like earthquakes, taxi rides, and Stack Overflow questions).

The Winner: SIGTPP consistently beat the other top models (like Diffusion models, VAEs, and GANs).
The Score: On average, SIGTPP was the best or very close to the best in 64% of the tests.
The Improvement: When they looked at how much better it was, SIGTPP improved upon the previous best models by at least 19% on average.
The Key Insight: While other models were good at guessing the next event, SIGTPP was much better at capturing the entire structure of the sequence. It didn't just get the notes right; it got the melody right.

Summary

This paper solves a math problem where "jumpy" data didn't fit "smooth" tools.

They invented a way to turn jumps into smooth ramps (Interarrival Embedding).
They used this to build an AI (SIGTPP) that learns the whole story at once, rather than just the next chapter.
They proved that this new AI creates more realistic, structured sequences of events than any previous method, especially when looking at the big picture rather than just individual moments.

The paper concludes that to truly understand event sequences, we need to stop looking at single events and start looking at the shape of the entire journey.

Technical Summary: From Jumps to Signatures: a Generative Method for Temporal Point Processes

Problem Statement

Temporal Point Processes (TPPs) model discrete events occurring in continuous time, underpinning applications in finance, seismology, and health. While Neural TPPs (NTPPs) have advanced the field using expressive neural networks, they face two primary limitations:

Training Objectives: Prevailing methods optimize local, per-event losses (e.g., log-likelihood maximization). These decompose into sums of conditional losses, failing to provide a global signal for structural properties like autocorrelation, long-range dependencies, and joint timing distributions.
Evaluation Metrics: Standard evaluation relies on Negative Log-Likelihood (NLL) or pointwise metrics (e.g., Mean Absolute Error). These are insensitive to global sequence structure, can mask mode collapse, and often favor underdispersed predictions. Furthermore, existing distributional discrepancy measures (like MMD or Wasserstein distances) lack theoretical justification on the space of counting paths (càdlàg step functions).

A significant theoretical gap exists: Rough path signatures, which offer universal feature maps and determinacy guarantees for continuous paths of bounded variation, cannot be directly applied to TPPs because TPP sample paths are discontinuous (càdlàg), violating the continuity assumptions required for signature-based generative modeling.

Methodology: SIGTPP

The paper proposes SIGTPP, a signature-based generative framework that bridges the gap between discrete event sequences and continuous path theory. The methodology consists of four core components:

1. The Interarrival Embedding ( $\Phi$ )

To apply signature methods to discrete jumps, the authors introduce a stable, injective lift from the space of counting paths ( $\mathcal{N}$ ) to the space of continuous paths of bounded variation ( $\mathcal{C}$ ).

Mechanism: For a counting path $\eta$ with event times $t_k$ , the embedding constructs a piecewise-linear continuous function $x = \Phi(\eta)$ . The value of the function at each event time $t_k$ is set to the interarrival time $\tau_k = t_k - t_{k-1}$ .
Time Augmentation: The signature is computed on the time-augmented curve $t \mapsto (t, \Phi(\eta)_t)$ in $\mathbb{R}^2$ . This encodes both absolute event times and interarrival durations, removing tree-like components to ensure the signature is injective.
Theoretical Guarantee: The paper proves $\Phi$ is Lipschitz continuous and injective. Under mild separation assumptions (minimum interarrival time $\delta > 0$ ), the inverse map is Hölder continuous, ensuring that discrepancies in the embedded space reflect genuine discrepancies in the original counting-path space.

2. Global Signature Loss

Instead of optimizing per-event conditional densities, SIGTPP optimizes a global loss on complete trajectories.

Loss Function: The model minimizes the Signature-Wasserstein-1 (Sig-W1) distance between the expected truncated signatures of the generated paths and the observed data paths.
Optimization: The loss is defined as the $\ell_2$ distance between the expected truncated signatures:
$\text{Sig-W}^{(M)}_1(\mu_\theta, \nu) := \left\| \mathbb{E}_{N_\theta \sim \mu_\theta} [S_M(\Phi(N_\theta))] - \mathbb{E}_{N \sim \nu} [S_M(\Phi(N))] \right\|_2^2$
This replaces the adversarial discriminator of GANs with a direct metric on the tensor algebra, avoiding training instability.

3. Evaluation Metrics

The authors derive three theoretically grounded distributional discrepancies for evaluating generative TPPs, all based on the pathwise metric $d_N(\eta, \xi) = \int_0^{T_{max}} |\eta_t - \xi_t| dt$ :

Energy Distance (E): A metric induced by the negative type property of the $L_1$ space.
Wasserstein-1 Distance (W1): Lifts the pathwise metric to a distributional metric.
Signature-Wasserstein-1 (Sig-W1): Uses the expected signatures as a discrepancy measure, valid due to the injectivity of $\Phi$ .

4. Model Architecture

SIGTPP is implemented as a 1-step autoregressive generator using a single-layer LSTM. It predicts log-interarrival times conditioned on the history state, current cumulative time, and exogenous noise.

Key Contributions

Interarrival Embedding: A novel, stable, and injective mapping that allows signature-based methods to be applied to discrete event sequences (càdlàg paths).
SIGTPP Model: The first signature-based generative model for TPPs, trained via a global path-level loss rather than local conditional losses.
Theoretical Framework: Proofs establishing the Lipschitz continuity and bijectivity of the embedding, and the derivation of mathematically valid distributional discrepancies (E, W1, Sig-W1) on the space of counting paths.
Evaluation Analysis: A demonstration that pointwise metrics (MAE/MSE) fail to assess generative quality, often favoring deterministic regressors that ignore distributional variation.

Experimental Results

The model was evaluated on 4 synthetic datasets (Poisson, Inhomogeneous Poisson, Univariate Hawkes, Multivariate Hawkes) and 5 real-world datasets (Earthquake, Stack Overflow, Taobao, Taxi, Yelp).

Performance: SIGTPP achieved the best average rank across 8 complementary metrics. It outperformed or matched the strongest baselines (VAE, DDPM, WGAN) in 64% of dataset-metric pairs.
Relative Improvement: According to a relative score metric, SIGTPP improved against every baseline by at least 19% on average.
Distributional Fidelity: SIGTPP significantly outperformed baselines on path-level distributional metrics (E, W1, Sig-W1), showing superior alignment with the joint temporal law.
Trade-offs: While SIGTPP excelled at global structure, it slightly underperformed on the Continuous Ranked Probability Score (CRPS) compared to diffusion-based models (DDPM) and VAEs. The authors attribute this to the fact that path-level objectives prioritize global distributional alignment over local conditional calibration.
Pointwise Metrics: The deterministic baseline (DETER) achieved the best MAE/MSE scores, confirming that pointwise errors do not measure generative quality or distributional fidelity.

Significance and Claims

The paper claims to provide a "common pathwise framework" that unifies the generation and evaluation of TPPs. Its primary significance lies in:

Extending Signatures to Discrete Events: Successfully adapting rough path theory to càdlàg processes via the interarrival embedding.
Global vs. Local: Demonstrating that optimizing a global signature loss yields better structural fidelity (autocorrelation, long-range dependencies) than traditional per-event likelihood maximization.
Evaluation Rigor: Establishing that standard pointwise metrics are insufficient for generative TPPs and providing a suite of theoretically justified distributional discrepancies (E, W1, Sig-W1) as the standard for evaluation.

The authors acknowledge limitations, including the computational cost of high-degree signatures, the artificial nature of the linear interpolation lift, and the preliminary nature of the theoretical bounds. However, they assert that the framework offers a robust alternative for modeling complex temporal dependencies where global structure is critical.