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A Conformal Prediction Approach to Explore Functional Data
Jing Lei, Alessandro Rinaldo, Larry Wasserman
TL;DR
Functional data require exploratory tools that can identify typical, abnormal, and high-density curves despite their infinite-dimensional nature and the computational difficulty of ordinary conformal prediction. The paper combines inductive conformal prediction, basis projections, pseudo-densities, and conformity scores to construct simultaneous prediction bands and clustering trees. These prediction sets have distribution-free finite-sample guarantees and correspond to high-density regions, while author-described examples illustrate their use for detecting outliers and clusters.
Problem
Exploratory visualization of functional data is difficult because curves are infinite-dimensional, while ordinary conformal prediction is computationally costly for constructing functional prediction sets.
Method
The paper combines inductive conformal prediction with basis projections and selected conformity scores for prediction bands, and with pseudo-densities for conformal clustering trees.
Results
The resulting prediction bands and clustering-tree prediction sets provide distribution-free finite-sample coverage and reveal high-density structure in functional data.
Takeaways & Limitations
The methods provide tools for visualizing functional quantiles, detecting outliers, and identifying clusters or salient high-density regions.
Takeaways & Limitations
A standard conformity score based on −supt |X(t)| yields a valid prediction band that is usually too wide and therefore of limited use.
Abstract
from arXiv · showhide
This paper applies conformal prediction techniques to compute simultaneous prediction bands and clustering trees for functional data. These tools can be used to detect outliers and clusters. Both our prediction bands and clustering trees provide prediction sets for the underlying stochastic process with a guaranteed finite sample behavior, under no distributional assumptions. The prediction sets are also informative in that they correspond to the high density region of the underlying process. While ordinary conformal prediction has high computational cost for functional data, we use the inductive conformal predictor, together with several novel choices of conformity scores, to simplify the computation. Our methods are illustrated on some real data examples.
1 Introduction
The paper develops distribution-free exploratory tools for functional data, where curves are difficult to visualize and distinguish as typical or abnormal. It combines conformal prediction with inductive computation to construct simultaneous prediction bands and clustering trees with finite-sample interpretation.
- Functional data analysis studies functions rather than scalar or vector observations, supporting models for smooth processes in fields such as longitudinal data, genetics, and engineering.
- Visualization is challenging because functional data are infinite-dimensional, making typical and abnormal curves difficult to identify directly.
- The paper constructs simultaneous prediction bands that cover a random curve from the underlying process at a specified coverage level.
- Inductive conformal prediction and carefully chosen conformity scores make functional prediction bands computationally tractable, unlike ordinary conformal methods with infeasible prediction-set characterization.
- The paper constructs conformal clustering trees whose levels are estimated prediction sets with distribution-free finite-sample coverage, while higher tree levels correspond to higher pseudo-density regions.
- The methods are presented as the first prediction and visualization tools for functional data with finite-sample guarantees and are illustrated using real data examples.
2 Notation and Background
The section formulates distribution-free prediction for functional data and introduces inductive conformal prediction as a computationally efficient alternative. It also explains projection-based bands, conformity-score choices, and their finite-sample coverage and efficiency trade-offs.
- Prediction sets: The goal is to construct prediction sets for future functions under general distributions, with a less ambitious projected target when full-function coverage is unnecessary.Projection Π maps functions into a finite-dimensional space, allowing coverage for Π(Xn+1) rather than the entire function.
- Prediction bands: Distribution-free finite-sample prediction bands are desirable because existing provable bands commonly rely on Gaussianity.A band contains functions whose values lie within interval-valued sets Bn(t) at every t.
- Conformal prediction: Inductive conformal prediction randomly splits the data, fits a conformity function on one part, ranks held-out scores, and thresholds the resulting prediction set.The threshold is an order statistic of held-out conformity scores, and the method avoids refitting for every candidate function.
- Conformal prediction: For all sample sizes and distributions, Algorithm 1 guarantees P(Xn+1 ∈ Cn) ≥ 1 − α through exchangeability of the held-out scores.The guarantee follows because the future score has a uniformly distributed rank among the held-out and future scores.
- Method behavior: The modified data-splitting method can avoid the unbounded prediction sets caused by absurd candidate values in the standard conformal construction.Its prototypes and most conformity scores are not affected by such candidate values, whereas the omit-one approach is more computationally expensive.
- Conformity scores: Conformity-score choice affects both computational usefulness and statistical efficiency, and an optimal score for functional data remains an open question.The supremum-based score produces valid but often overly wide bands, while projection densities and pseudo-densities provide alternative constructions.
3 Prediction Bands Based on Projections
The paper constructs simultaneous functional prediction bands by projecting curves into finite-dimensional spaces, applying inductive conformal prediction, and approximating coefficient-level prediction sets with Gaussian mixtures. The resulting bands retain finite-sample coverage while enabling tractable computation and are illustrated on neuron and phoneme data.
- Projection-based prediction bands: Projection into a finite-dimensional function space enables simultaneous prediction bands with finite-sample coverage for projected future curves.The projection may use fixed or data-driven bases, including empirical eigenfunctions.
- Projection-based prediction bands: Algorithm 2 splits the data, computes basis projection coefficients, evaluates conformity scores on held-out observations, and ranks those scores to form a prediction set.The output is a time-indexed band Bn(t) for all t in [0,1].
- Gaussian mixture approximation: Gaussian mixture density scores make the coefficient-level prediction set easier to characterize and project into functional bands.The method uses the first p empirical covariance eigenfunctions and estimated mixture parameters.
- Coverage guarantees: The band retains correct coverage without assuming the mixture model is correct and for all choices of projection dimension and mixture-component count.The paper also states that the refined band is a union of K bands.
- Gaussian mixture approximation: Each mixture component yields an ellipsoidal coefficient set whose pointwise projections form intervals, and the final band is their union.These projections are available in closed form, simplifying computation.
- Approximation refinements: The Gaussian-mixture level-set approximation is usually conservative, although refinements are possible when component overlap makes the band unnecessarily wide.The paper motivates a more refined approximation for separated components and an alternative score for overlapping components.
- Data illustrations: For neuron data, using p = 2 and K = 3 produced empirical coverage of 916 out of 1000 curves.The band was plotted with projected sample curves.
- Data illustrations: For phoneme data, overlapping mixture components made the rough approximation too wide, motivating a different conformity score; the resulting band had 90.5% empirical coverage.The refined score is particularly relevant to the non-Gaussian “dcl” cluster modeled by two mixture components.
4 Methods Based on Pseudo-Densities
The paper uses pseudo-density estimators within conformal prediction to construct finite-sample prediction sets, summarize functional data, and visualize hierarchical clusters. These tools identify anomalies, typical and high-density curves, prototypes, and cluster structure.
- Pseudo-density construction: Pseudo-density estimators support clustering curves despite the absence of a σ-finite dominating measure, so they are not ordinary density functions.They can nevertheless be used for tasks such as clustering functional observations.
- Conformal prediction sets: Conformal level sets provide distribution-free, finite-sample prediction sets with coverage at least 1 − α.The result holds for all distributions P and sample sizes n.
- Functional-data summaries: The method summarizes functional data through anomalies, median sets, high-density curves, and mean-shift prototypes.For neuron data, the median set captures common shapes, high-density curves occur in two larger groups, and anomalies are mostly irregular.
- Conformal cluster tree: The conformal tree is formed from connected components of a distance graph as α varies, with tree height indexed by α.Its leaves correspond to local pseudo-density modes, while isolated clusters smaller than 10 are ignored in the plotted trees.
- Neuron-data example: For neuron data, the conformal tree suggests three mixture components, with prominent splits near α = 0.08 and α = 0.24 and a subtler split at α = 0.79.The tree obtained with h = ϵ = 1000 differs but still strongly suggests the three-component structure.
5 Conclusion and Future Directions
The paper extends conformal methods to functional data, producing distribution-free simultaneous prediction bands and conformal clustering structures. It also identifies open questions involving conformity scores, tuning parameters, and distance choices.
- Contributions: Inductive conformal prediction with basis projections yields distribution-free simultaneous prediction bands with finite-sample guarantees.The bands can also be interpreted as quantile sets corresponding to high-density regions of the underlying process.
- Contributions: The resulting conformal prediction sets reveal hierarchical and salient data features through pseudo-density-based clustering.The paper proposes methods for extracting information from and visualizing these sets.
- Future directions: Open questions include extending conformity scores to high-dimensional problems, selecting tuning parameters, and choosing distances for pseudo-densities.The paper notes that L2 distance induces unbounded bands, while the analytic distance is more sensitive to curve shape.