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Arbitrarily Conditioned Hierarchical Flows for Spatiotemporal Events

This paper introduces Arbitrarily Conditioned Hierarchical Flows (ARCH), a hierarchical flow matching framework that overcomes the limitations of existing autoregressive point process models by enabling flexible conditioning on arbitrary observed events to unify and improve performance across spatiotemporal forecasting, inverse inference, and trajectory recovery tasks.

Original authors: Keyan Chen, Qiwei Yuan, Zhitong Xu, Bin Shen, Shandian Zhe

Published 2026-05-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Keyan Chen, Qiwei Yuan, Zhitong Xu, Bin Shen, Shandian Zhe

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

Imagine you are trying to predict the future of a chaotic city. You see people moving, cars stopping, and rain falling at random times and places. Your goal is to understand the hidden rules that govern these events so you can answer questions like: "When will the next accident happen?", "Where did that first earthquake start?", or "What happened in the middle of this storm that we missed?"

For a long time, computer scientists have tried to build models to do this, but they usually had to make a big compromise: they could either make the model very simple (and easy to calculate) or very complex (to capture reality), but rarely both. Also, most models were like a strict teacher who only lets you ask questions in order: "What happens next?" They couldn't easily answer, "What happened before?" or "Fill in the missing pieces."

This paper introduces a new tool called ARCH (Arbitrarily Conditioned Hierarchical Flows). Think of ARCH as a super-smart, flexible detective that can solve these puzzles in any direction you ask.

Here is how it works, broken down into simple concepts:

1. The Two-Step Dance (Hierarchical Flows)

Most models try to guess the "When" and the "Where" of an event all at once, like trying to catch a butterfly with one hand while guessing its color with the other. It's messy and hard to get right.

ARCH splits the job into a clean, two-step dance:

Step 1 (The Time): First, it figures out when the event will happen.
Step 2 (The Place): Once it knows the time, it figures out where it will happen.

The Analogy: Imagine you are planning a surprise party.

Old models try to guess the guest list, the time, and the location all in one giant, confusing guess.
ARCH first asks, "What time is the party?" (It gets a solid answer). Then, it asks, "Given that time, where is the best spot?" Because it separates the questions, it can be much more precise and can also calculate exactly how "risky" or likely an event is at any specific moment.

2. The Magic Mask (Arbitrarily Conditioned)

This is the paper's biggest superpower. Imagine you have a movie script, but some pages are torn out, some are blank, and some are already written.

Standard Models: They can only read the script from the beginning to the end. If a page is missing in the middle, they get stuck. They can only predict the next scene.
ARCH: ARCH wears a magic mask. You can tell it, "Ignore the first 10 pages, I want to guess what happens in the middle," or "I know the middle, but tell me how the movie started."

You can give ARCH any mix of information (some past events, some future events, some missing pieces), and it will use its training to fill in the blanks. It doesn't need to be retrained for every new type of question. It's like a Swiss Army knife for event prediction.

3. The "Risk Meter" (Conditional Intensities)

In the world of earthquakes, crime, or disease, knowing when something might happen isn't enough; you need to know the instantaneous risk.

Think of a weather forecast that says, "It might rain tomorrow." That's vague.
ARCH provides a "Risk Meter" that says, "Right now, at this exact second, the chance of rain is 15%."

Because of its two-step design, ARCH can calculate this risk meter accurately and quickly. Other complex models (like the "diffusion" models mentioned in the paper) are great at making up stories, but they are often "black boxes" that can't tell you the specific risk level at a specific moment. ARCH keeps the door open so you can see the math behind the risk.

4. What Did They Test?

The authors tested this detective on three real-world scenarios:

Earthquakes: Predicting where and when the next tremor hits.
COVID-19: Tracking how the virus spread through time and space.
Citibike: Predicting where people in a city will rent bikes next.

They also tested it on fake data designed to be tricky.

The Results:

Better Accuracy: ARCH was better at predicting the next event than almost all other models.
Better Risk Calculation: On the fake data, it was much better at figuring out the "Risk Meter" (the conditional intensity) than models that tried to force the data into simple shapes.
The "Fill-in-the-Blank" Champion: When asked to guess missing events (like a torn-out page in a script) or guess the very first event based on later ones, ARCH crushed the competition. It was far better than standard "guessing by looking at neighbors" methods.

The Bottom Line

The paper claims that ARCH is a new way to model events in time and space that is flexible, accurate, and transparent. It breaks the old rule that you have to choose between a simple model and a smart one. By separating "time" and "place" and using a flexible masking system, it can answer almost any question about a sequence of events, whether that question is about the future, the past, or the missing pieces in between.

Technical Summary: Arbitrarily Conditioned Hierarchical Flows for Spatiotemporal Events

Problem Statement

Spatiotemporal events, ubiquitous in domains such as weather, epidemics, and crime, exhibit complex distributions that are difficult to model, particularly given sparse observations. Existing approaches face two primary limitations:

Structural Rigidity vs. Expressiveness: Traditional point process models (e.g., Poisson, Hawkes) rely on strong structural assumptions (e.g., independent increments, additive interactions) that limit their ability to capture complex distributions. Conversely, recent diffusion-based generative models offer high expressiveness but often lose the ability to compute tractable and well-calibrated conditional intensities, a critical quantity for quantifying instantaneous event risk.
Inference Limitations: Most existing methods are restricted to autoregressive, event-by-event prediction. They struggle to support broader inference tasks such as inverse inference (recovering initial events), reconstructing missing event locations, or recovering partial event trajectories.

Methodology: ARCH

The authors propose Arbitrarily Conditioned Hierarchical Flows (ARCH), a hierarchical flow matching framework designed to balance expressive power with tractable intensity computation and flexible inference.

1. Hierarchical Flow-Matching Framework

Instead of generating event time ( $s$ ) and spatial location ( $x$ ) simultaneously via a single flow, ARCH decomposes the generation process into two sequential flows:

Flow $G_1$ : Generates the event time $s$ conditioned on the history $H$ .
Flow $G_2$ : Generates the spatial location $x$ conditioned on both the history $H$ and the generated time $s$ .

This decomposition, $p(s, x | H) = p(s | H)p(x | s, H)$ , enables the efficient computation of the conditional intensity (instantaneous event risk), $\lambda(s, x | H)$ .

Temporal Intensity: The marginal temporal intensity $\lambda(s | H)$ is computed using the Picard–Lindelöf theorem and the order-preserving property of one-dimensional ODEs. This allows the Conditional Cumulative Distribution Function (CDF) of the event time to be derived from the CDF of the initial noise variable without expensive numerical integration over the spatiotemporal domain.
Spatial Density: The conditional spatial density $p(x | s, H)$ is computed by solving the companion log-probability ODE backward from the target state to the noise prior.
Final Intensity: The full conditional intensity is factored as $\lambda(s, x | H) = \lambda(s | H) p(x | s, H)$ .

2. Arbitrarily Conditioned Training and Inference

To support tasks beyond standard autoregressive prediction, ARCH employs a hybrid masking strategy within a Transformer-based history encoder and generative decoder architecture. During training, masks are sampled to create diverse conditioning scenarios:

Autoregressive Mask: Supports standard next-event prediction where each token attends only to past events.
Random Condition Mask: Supports the generation of an arbitrary subset of events conditioned on the remaining observed events (e.g., imputing missing events).
Consecutive Mask: Supports the generation of a contiguous subsequence of events (e.g., trajectory recovery) conditioned on events outside the block.

This unified training objective allows a single model to perform forecasting, inverse inference, and trajectory recovery without task-specific retraining.

Key Contributions

Model Architecture: A hierarchical flow matching framework that decouples temporal and spatial generation, retaining the flexibility to capture complex distributions while enabling tractable and accurate computation of conditional intensities.
Unified Inference Framework: The introduction of a hybrid masking strategy that unifies forecasting, inverse inference, missing-event imputation, and partial trajectory recovery within a single model.
Theoretical Guarantees: Leveraging ODE properties (Picard–Lindelöf theorem) to compute marginal CDFs and conditional intensities efficiently, avoiding the computational intractability of high-dimensional integration required by joint flow models.

Experimental Results

The authors evaluated ARCH on synthetic datasets (Spatiotemporal Hawkes Process and Spatiotemporal Self-Correcting Process) and three real-world benchmarks (Earthquake, COVID-19, Citibike).

Intensity Recovery: On synthetic datasets, ARCH achieved significantly lower relative $L_2$ errors in recovering conditional intensities compared to structured baselines (DeepSTPP, AutoSTPP, NSTPP). Notably, ARCH successfully captured both excitation-dominated and inhibition-driven dynamics, whereas Hawkes-based baselines failed on the self-correcting process due to structural misspecification.
Next-Event Prediction: ARCH achieved state-of-the-art or comparable performance in spatial and temporal prediction across all real-world datasets, often outperforming diffusion-based baselines (DSTPP) while providing the added benefit of calibrated risk estimates.
Inverse Inference & Imputation: In tasks involving the recovery of initial events or missing events, ARCH substantially outperformed KNNImpute, demonstrating the efficacy of the arbitrarily conditioned formulation.
Trajectory Recovery & Forecasting: ARCH effectively reconstructed missing event trajectories and performed joint multi-step forecasting. It consistently outperformed autoregressive variants and diffusion baselines in long-horizon forecasting, suggesting superior capability in capturing long-range dependencies.

Significance and Claims

The paper claims that ARCH addresses a critical gap in spatiotemporal event modeling by providing a framework that is both expressive enough to model complex, non-linear distributions and tractable enough to compute instantaneous event risks. By moving beyond the standard autoregressive paradigm, ARCH enables a unified approach to diverse inference tasks (forecasting, reconstruction, imputation) that were previously ill-suited for existing point process models. The authors posit that this flexibility substantially broadens the scope of spatiotemporal modeling, making it applicable to risk monitoring, survival analysis, and intervention planning where accurate conditional intensities are essential.