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Manifold Constrained Conformal Prediction for Spatial Events

This paper introduces a novel manifold-constrained conformal prediction method that utilizes Wasserstein distance and flow-based sampling to generate calibrated, low-energy prediction sets for spatial events like tropical cyclones and earthquakes, outperforming existing baselines in both coverage accuracy and geometric fidelity.

Original authors: Collin Nill, Trevor Harris, Jason Adams

Published 2026-07-14

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Collin Nill, Trevor Harris, Jason Adams

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 where a storm will be born, or where the next earthquake will strike. You don't just want a single dot on a map; you want to know the whole "cloud" of possibilities. Will there be one big storm or ten tiny ones? Will they cluster near the equator or spread out?

For a long time, scientists have tried to draw a safety net around these predictions. They want to say, "We are 90% sure the real event will be inside this net." But here's the problem: the nets they've been using are often too loose. They include places where storms could mathematically happen but where nature would never actually send one. It's like drawing a safety net around a city that includes the moon because, technically, a rocket could go there.

This paper introduces a new way to tighten that net. The authors, Collin Nill, Trevor A. Harris, and Jason Adams, propose a method called Manifold Constrained Conformal Prediction.

The "Cloud" Problem

Think of a tropical cyclone season or an earthquake sequence not as a single point, but as a cloud of dots floating on a giant globe. Some clouds have 50 dots; others have 200. They can be scattered or clumped together.

Old methods tried to measure the distance between a predicted cloud and a real cloud by subtracting one from the other. But you can't subtract a cloud of 50 dots from a cloud of 200 dots easily. It's like trying to subtract a handful of marbles from a bucket of sand. The math gets messy, and the resulting "safety net" ends up being a giant, blurry ball that covers almost everything, including places that make no physical sense.

The New Trick: Measuring the "Shape"

The authors suggest a smarter way to measure the distance between these clouds. Instead of counting dots, they treat the clouds as distributions of weight. Imagine pouring sand onto a globe to represent where the storms might be.

They use a mathematical tool called the Spherical Sliced Wasserstein distance. Think of this like taking a giant, invisible slicer and cutting the globe into thin slices (like a loaf of bread). They compare how the "sand" is distributed on each slice. If two clouds look similar, their slices match up. If they are different, the slices look very different. This allows them to measure the distance between clouds of any size or shape without getting confused.

The "Manifold" Constraint: Staying on the Rails

Here is the real magic. Even with this better measuring tool, the "safety net" (the prediction set) they build is still too big. It includes weird, impossible shapes—like a storm cloud floating in the middle of the ocean where no ocean exists, or an earthquake happening in the middle of a solid mountain range.

In the real world, nature follows rules. Earthquakes happen along fault lines (cracks in the Earth's crust). Tropical cyclones are born in specific ocean basins with warm water. These rules form a hidden "track" or a manifold.

The authors' method adds a special "rail" to their prediction. They say: "We will build our safety net, but we will only allow the net to exist where the training data has shown us nature actually goes."

They do this by checking every possible prediction against the training data (the history of past storms and quakes). If a predicted cloud is too far away from any cloud seen in the past, they throw it out. This forces the prediction to stay "on the rails" of physical reality.

What They Found (and What They Didn't)

The authors tested this idea in three ways:

Fake Data: They created computer simulations of random dots, spirals, and self-exciting patterns (like earthquakes that trigger more earthquakes).
Real Storms: They looked at tropical cyclone genesis (where storms are born) using data from 1980 to 2025.
Real Quakes: They looked at earthquakes in California (magnitude 4.0 and above) from 1976 to 2025.

The Results:

Coverage: In their simulations, the method successfully caught the real events about 90% of the time (when they asked for a 90% guarantee). This matches the theory perfectly.
Better than the Rest: When they compared their method to other popular techniques (like "Highest Density Regions" or fancy AI models like Diffusion and VAEs), their method produced prediction sets that were much closer to the actual data.
- For the California earthquakes, their method had a "manifold distance" (how far off the fault lines the prediction was) of 0.19 (scaled by 100), while a standard flow-based method had 1.79. This means their method kept the predictions tightly hugging the fault lines, while others let them drift into impossible places.
- For tropical cyclones, their method kept the predictions concentrated in the known ocean basins, whereas other methods let the predictions spread out over land and cold water.

What They Rule Out:
The paper explicitly argues against using standard "Highest Density Region" (HDR) methods or unconstrained generative models for this specific job. They found that while some of those models (like Diffusion) might look statistically accurate on paper, they often generate "physically implausible" results—like predicting a hurricane forming in the middle of the Sahara Desert. The authors show that without their "manifold constraint," these models fail to respect the physical rules of the world.

How Sure Are They?

The authors are very confident in the math behind the coverage guarantee (it's a proven lower bound), but they are careful to say their performance results come from experiments and simulations.

They don't claim this is the "final solution" for all natural disasters.
They note that in their simulations, the method works best when they have enough historical data to define the "rails" (the manifold).
They found that if they set the "tolerance" for the rails too tight, the method might miss the real event. But they showed that by choosing a tolerance that covers all the training data, they get nearly perfect coverage.

The Bottom Line

This paper doesn't just say "we have a better calculator." It says, "We have a way to force our predictions to respect the laws of physics."

By combining a new way to measure the distance between event clouds with a rule that says "stay close to where nature has been before," they created a prediction tool that is both statistically reliable (it catches the truth 90% of the time) and physically sensible (it doesn't predict earthquakes in the sky). It's a way to make sure that when we look at the future of natural hazards, our safety nets are tight enough to be useful, but not so loose that they include the impossible.

Technical Summary: Manifold Constrained Conformal Prediction for Spatial Events

Problem Statement

Natural hazards such as tropical cyclones, earthquakes, and wildfires manifest as discrete, spatially organized configurations of events (point clouds) rather than continuous fields. The number, positions, and spatial arrangement of these events are critical for risk assessment, particularly for compound and extreme hazards. However, forecasting these phenomena presents unique challenges:

Variable Cardinality: The number of events in a configuration varies, making standard residual-based conformity scores (e.g., $y - \hat{y}$ ) ill-defined.
Lack of Calibration: Modern forecasting systems (climate models, deep learning ensembles) produce distributions over configurations but are not guaranteed to be calibrated; a 90th percentile ensemble does not inherently imply 90% coverage of the true event cloud.
Physical Implausibility: Standard conformal prediction regions, when defined over the space of all possible distributions, often include physically impossible configurations (e.g., earthquakes occurring in the middle of the ocean or cyclones forming in non-tropical latitudes) because they lack constraints based on the underlying data-generating manifold.

Existing methods, such as Highest Density Region (HDR) estimators or Bayesian posteriors, either fail to provide post-hoc coverage guarantees for arbitrary models or require specific parametric assumptions that may not hold.

Methodology

The authors propose a Manifold Constrained Conformal Prediction framework that operates on the space of probability measures to generate calibrated prediction sets for spatial point clouds.

1. Representation and Scoring

Empirical Measures: Point clouds $Y$ are represented as empirical measures $\mu_Y = \frac{1}{|Y|}\sum \delta_y$ , embedding them into the Wasserstein space $P_2(S^2)$ . This allows for the comparison of clouds with different cardinalities.
Conformity Score: The discrepancy between an observed target and a model prediction is measured using the Spherical Sliced Wasserstein (SSW) distance, $SSW_2(\mu_Y, f(X))$ . This metric respects the geodesic geometry of the sphere and is computationally tractable ( $O(n \log n)$ ) via Monte Carlo projection onto great circles.
Ambient Region: A standard split conformal prediction region is constructed as an SSW ball of radius $\tau_\alpha$ around the prediction: $C^{amb}_\alpha = \{Y : S(Y, f(X)) \leq \tau_\alpha\}$ . This guarantees coverage $P(Y_{n+1} \in C^{amb}_\alpha) \geq 1-\alpha$ under exchangeability.

2. Manifold Restriction

To eliminate physically implausible configurations, the method restricts the prediction set to a subset of the space supported near the training data manifold $\mathcal{M}$ .

Empirical Manifold: Since the true manifold is unknown, it is approximated by the training targets $D_0$ . A "manifold discrepancy" $d_M(Y)$ is defined as the minimum SSW distance from a candidate cloud $Y$ to any training cloud in $D_0$ .
Tube Constraint: A tolerance $\epsilon$ is introduced to define an empirical tube. The final prediction set is the intersection of the ambient conformal ball and this tube:
$C_\alpha(X) = \{Y : S(Y, f(X)) \leq \tau_\alpha, \ d_M(Y) \leq \epsilon\}$
Coverage Guarantee: The authors derive a lower bound for the coverage of this intersected set. While the intersection reduces the set size, the coverage gap is shown to be small and exponentially decaying with the size of the training set, provided $\epsilon$ is chosen to adequately cover the training data.

3. Constrained Flow-Based Sampling

Because the resulting set $C_\alpha$ is not analytically tractable, the authors introduce a constrained gradient flow to sample from its boundary.

Velocity Field: A time-indexed candidate cloud $Y(t)$ is evolved via an ODE. The velocity field $v(y)$ is derived to simultaneously drive the conformity score $S(y)$ toward the quantile $\tau_\alpha$ and the manifold discrepancy $d_M(y)$ toward the boundary $\epsilon$ (if outside).
Optimization: The velocity is the minimum-norm solution to a linear system defined by the gradients of the score and the manifold discrepancy, ensuring the trajectory converges to the boundary of the manifold-restricted region without introducing unnecessary drift.
Ensemble Generation: By marginalizing over a sequence of these samplers, the method produces a calibrated predictive distribution represented as an ensemble of geophysically coherent event clouds.

Key Contributions

Novel Conformal Framework: The first conformal prediction method specifically designed for spatial point clouds on a sphere, utilizing optimal transport (SSW) for scoring.
Manifold Constraint: The introduction of an empirical manifold constraint that restricts prediction sets to physically plausible regions, addressing the issue of "off-manifold" samples common in unconstrained distributional prediction.
Theoretical Guarantees: Derivation of a coverage lower bound for the manifold-constrained sets, quantifying the trade-off between the constraint tolerance $\epsilon$ and the coverage probability.
Sampling Algorithm: A modified flow-based sampling procedure that generates ensembles satisfying both the conformal coverage guarantee and the manifold constraint.

Results

The method was evaluated on synthetic data (Homogeneous Poisson, Spiral, Inhomogeneous Poisson, Hawkes processes) and real-world geophysical data (Tropical Cyclone genesis and California Earthquakes).

Coverage: The method achieves near-nominal coverage (e.g., ~90% for $\alpha=0.1$ ) across all synthetic and real-world settings. The empirical coverage is shown to be robust to the number of SSW slices and calibration size, provided $\epsilon$ is set conservatively to cover the training data.
Forecast Skill (Energy Distance): The constrained velocity field (CVF) consistently achieves lower Energy Distance (ED) compared to unconstrained flow-based methods (VF) and standard baselines (HDR, Diffusion, Normalizing Flows, VAE).
Manifold Adherence: CVF significantly reduces the Manifold Distance (MD) compared to unconstrained methods.
- In Tropical Cyclone forecasting, CVF concentrates predictions in observed basins, whereas unconstrained methods (like Diffusion) often spread mass into invalid regions (e.g., mid-Atlantic) to minimize ED, resulting in high MD.
- In Earthquake forecasting, where events are strictly confined to fault lines, CVF maintains low MD (0.19) compared to unconstrained flows (1.79), ensuring predictions remain geophysically valid.
Comparison to Baselines: While generative models (Diffusion) sometimes achieve competitive ED, they often fail to respect the physical manifold. HDR methods provide valid regions but lack the flexibility and calibration guarantees of the proposed conformal approach.

Significance

The paper claims that quantifying uncertainty for spatial events requires moving beyond scalar summaries or unconstrained distributional predictions. The proposed method provides a post-hoc calibration procedure that works with any forecasting model, guarantees coverage under exchangeability, and ensures that the resulting prediction sets are geophysically coherent. By constraining predictions to the data manifold, the method produces uncertainty sets that are not only statistically valid but also actionable for risk assessment, avoiding the inclusion of physically impossible scenarios that plague unconstrained generative models. The approach bridges the gap between rigorous statistical coverage guarantees and the physical constraints inherent in natural hazard modeling.