Conformal Multivariate Prediction

The foundational engine that maps a vector-valued prediction and its true label to a scalar anomaly score. Unlike univariate cases, these measures must capture joint strangeness across all output dimensions simultaneously.

• Mahalanobis Distance: Scores a vector by its distance from a predicted mean, scaled by the inverse covariance matrix, naturally accounting for inter-target correlations.
• Copula-Based Scores: Transforms marginal nonconformity scores into a joint score using a copula, explicitly modeling the dependence structure between targets.
• Deep Set Encodings: Uses a neural network to learn a permutation-invariant representation of all target residuals, producing a single scalar nonconformity score for high-dimensional outputs.

The output of conformal multivariate prediction is a prediction region in ℝᵈ—a geometric set guaranteed to contain the true vector label with a user-specified probability. The shape of this region is determined by the chosen nonconformity measure.

• Ellipsoidal Regions: Produced by Mahalanobis-based scores, these are compact, convex, and interpretable, centered on the point prediction.
• Level-Set Regions: Generated by density-based scores, these can be non-convex and multi-modal, capturing complex, non-linear dependencies.
• Hyper-rectangular Regions: Result from applying a Bonferroni correction to independent marginal conformal sets, which is valid but often overly conservative and ignores inter-target correlation.

The core statistical property is the marginal coverage guarantee: P(Y_test ∈ Γ(X_test)) ≥ 1 - α. This holds distribution-free and with finite samples, provided the calibration and test data are exchangeable.

• Validity: The guarantee is exact, not asymptotic. It does not depend on the correctness of the underlying model or the distribution of the data.
• Limitation: The guarantee is marginal over the entire population. Achieving conditional coverage—validity for a specific X=x—is impossible without strong distributional assumptions. This is a critical trade-off to communicate to stakeholders.

The computationally efficient workflow for generating multivariate prediction regions without retraining the base model.

1. Partition: Split the training data into a proper training set and a disjoint calibration set.
2. Train: Fit the multivariate base model (e.g., a multi-output regressor) on the proper training set.
3. Calibrate: For each sample in the calibration set, compute the multivariate nonconformity score. Find the (1-α) empirical quantile of these scores, denoted as q̂.
4. Predict: For a new test point, the prediction region is the set of all vectors y for which the nonconformity score is ≤ q̂.

Extends Conformalized Quantile Regression (CQR) to the multivariate setting. A base model predicts a conditional quantile surface for each output dimension. A conformal calibration step then adjusts these surfaces to form a joint prediction region.

• Initial Region: The base model provides an initial, potentially miscalibrated, hyper-rectangular region from the marginal quantiles.
• Calibration Step: A multivariate nonconformity score measures how much this initial region must be expanded or contracted to cover the calibration points. The adjustment factor is applied isotropically or directionally to achieve the target coverage.

While validity is guaranteed, the efficiency—the size of the prediction region—is the key metric for comparing methods. A smaller region indicates a more informative, less uncertain prediction.

• Volume: The Lebesgue measure of the prediction region in ℝᵈ. Directly quantifies the total uncertainty.
• Sum of Marginal Widths: The sum of the lengths of the region's projections onto each axis. A simpler metric but ignores correlation.
• Empirical Coverage vs. Nominal Coverage: A well-calibrated method should have empirical coverage close to 1-α. A method with coverage far exceeding 1-α is likely producing inefficient, overly large regions.

The output of multivariate conformal prediction is a prediction region in the label space, not just independent intervals. The shape is determined by the nonconformity measure.

• Hyper-rectangles: Simple but often inefficient; assumes independence between outputs.
• Confidence Ellipsoids: Captures linear correlations using Mahalanobis distance.
• Arbitrary Level Sets: Non-parametric regions defined by the contours of a kernel density estimator on the calibration scores.

Guaranteeing validity while minimizing the region's volume is the central optimization challenge.

The standard marginal coverage guarantee relies on the joint exchangeability of the calibration and test tuples (X_i, Y_i). For multivariate outputs, this assumption implies that the entire vector-valued label is exchangeable.

• Temporal Violations: Strictly violated in time series; requires sliding window or sequential adaptations.
• Structured Outputs: For images or graphs, exchangeability must hold for the high-dimensional structured object, not just scalar pixels.

Violations require moving to weighted or adaptive conformal inference frameworks to maintain valid error control.

A specialized method that decouples the marginal behavior of each output dimension from their dependence structure. It fits a copula to the calibration residuals to model the joint tail probability.

• Marginal Calibration: Each output dimension is calibrated independently.
• Joint Correction: The copula captures non-linear dependencies (e.g., tail dependence) that a simple covariance matrix misses.

This is crucial for risk management applications where extreme joint co-movements of variables must be captured with valid coverage.