Inference for Functional Data under Markov Constraints

Ulysse Naepels; Victor M. Panaretos

arxiv: 2604.18229 · v1 · submitted 2026-04-20 · 📊 stat.ME

Inference for Functional Data under Markov Constraints

Ulysse Naepels , Victor M. Panaretos This is my paper

Pith reviewed 2026-05-10 04:05 UTC · model grok-4.3

classification 📊 stat.ME

keywords functional data analysisMarkov propertycovariance estimationinverse covarianceshape constraintsGaussian processesprediction

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The pith

Markovianity on the covariance kernel provides a falsifiable alternative to smoothness for inference in functional data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper is trying to establish that Markovianity induces a usable shape constraint on the covariance kernel for Gaussian functional data, providing an alternative to smoothness for inference tasks that depend on the inverse covariance. It introduces a Markov transform of the empirical covariance and an adaptive estimator that enforces the Markov structure with no regularity beyond continuity. A sympathetic reader would care because this leads to improved prediction performance in simulations even under misspecification, and unlike smoothness, the Markov property is falsifiable via a new test.

Core claim

Focusing on the Gaussian case as a central motivating setting, we exploit the fact that Markovianity induces a shape constraint on the covariance kernel. Building on this observation, we introduce a Markov transform of the empirical covariance together with a corresponding estimator that enforces the Markov structure. The estimator is adaptive and requires no regularity of the underlying covariance beyond continuity. In simulation experiments, it is seen to improve prediction performance even under model misspecification. Unlike smoothness-based assumptions, Markovianity is falsifiable. To assess its validity, we further propose a novel and computationally efficient test for the Markov pro

What carries the argument

Markov transform of the empirical covariance, which enforces the Markov structure as a shape constraint on the estimator.

Load-bearing premise

That Markovianity induces a usable shape constraint on the covariance kernel in the Gaussian case, and that the proposed transform and estimator correctly enforce this without additional regularity assumptions.

What would settle it

Simulating Gaussian functional data from a known non-Markov covariance and checking whether the Markov estimator still improves prediction accuracy over the standard empirical covariance would settle whether the gains require the assumption to hold.

Figures

Figures reproduced from arXiv: 2604.18229 by Ulysse Naepels, Victor M. Panaretos.

**Figure 2.** Figure 2: Comparison of empirical kernel shapes for the Ornstein-Uhlenbeck process ( [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

**Figure 3.** Figure 3: Comparison of convergence in L2-norm as n grows for the Brownian motion (left) and the Kernel-Embedded Brownian motion with h = 0.1 (right) for p = 20. 6.2 Kriging and prediction A key advantage of the Markov approach is that it enforces sparsity at the level of the conditionalindependence structure, or equivalently, the precision matrix in finite dimension. Accordingly, we expect the principal advantages… view at source ↗

**Figure 4.** Figure 4: Kriging errors distributions for Brownian motion and kernel-embedded Brownian [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗

**Figure 5.** Figure 5: Power of the proposed test against local alternatives generated by kernel-embedded [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗

**Figure 6.** Figure 6: Example of a discrete (left) and a continuous (right) graphical model. In the discrete [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗

**Figure 7.** Figure 7: Graphical model with path (in dashed blue) going from [PITH_FULL_IMAGE:figures/full_fig_p030_7.png] view at source ↗

**Figure 8.** Figure 8: Samples of the kernel-embedded Brownian motion with [PITH_FULL_IMAGE:figures/full_fig_p033_8.png] view at source ↗

**Figure 9.** Figure 9: Comparison of convergence in L2-norm as n grows for the kernel-embedded Brownian motion with h = 0.07, 0.08, 0.09 [PITH_FULL_IMAGE:figures/full_fig_p034_9.png] view at source ↗

**Figure 10.** Figure 10: Comparison of kriging mean square errors for the Ornstein-Uhlenbeck process as [PITH_FULL_IMAGE:figures/full_fig_p034_10.png] view at source ↗

**Figure 11.** Figure 11: Comparison of kriging mean square errors for the kernel-embedded Brownian motion [PITH_FULL_IMAGE:figures/full_fig_p034_11.png] view at source ↗

**Figure 12.** Figure 12: Power of the proposed test against local alternatives generated by kernel-embedded [PITH_FULL_IMAGE:figures/full_fig_p035_12.png] view at source ↗

**Figure 13.** Figure 13: Power of the proposed test against local alternatives generated by kernel-embedded [PITH_FULL_IMAGE:figures/full_fig_p035_13.png] view at source ↗

**Figure 14.** Figure 14: Power of the proposed test against local alternatives generated by kernel-embedded [PITH_FULL_IMAGE:figures/full_fig_p035_14.png] view at source ↗

**Figure 15.** Figure 15: ROC curves for the kernel-embedded Brownian motion with [PITH_FULL_IMAGE:figures/full_fig_p036_15.png] view at source ↗

read the original abstract

Smoothness has long been the dominant form of parsimony in functional data analysis, to the point of occasionally being conflated with the very notion of functional data. However, many core inferential tasks depend on the inverse covariance, where sparsity--rather than smoothness--emerges as the more natural structural constraint. In this paper, we explore Markovianity as an alternative to smoothness. Focusing on the Gaussian case as a central motivating setting, we exploit the fact that Markovianity induces a shape constraint on the covariance kernel. Building on this observation, we introduce a Markov transform of the empirical covariance together with a corresponding estimator that enforces the Markov structure. The estimator is adaptive and requires no regularity of the underlying covariance beyond continuity. In simulation experiments, it is seen to improve prediction performance even under model misspecification. Unlike smoothness-based assumptions, Markovianity is falsifiable. To assess its validity, we further propose a novel and computationally efficient test for the Markov property based on a new characterization of continuous graphical structure.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper replaces smoothness with a Markov constraint on functional covariances via a new transform and adaptive estimator, plus a test for the property; the idea is fresh but the derivations need verification.

read the letter

The main thing here is that Naepels and Panaretos treat Markovianity as a usable shape constraint on the covariance kernel for Gaussian functional data, then build a transform of the empirical covariance and an estimator that enforces it. The estimator is adaptive and claims to need nothing beyond continuity. They also give a test for the continuous graphical Markov property. This is a clear departure from the smoothness default that has dominated functional data analysis for covariance work. The falsifiability angle is useful, and the simulations reportedly show prediction gains even under misspecification, which suggests the method has some practical buffer. The program is coherent on paper: Markovianity induces sparsity-like structure in the inverse covariance, and they exploit that directly instead of penalizing roughness. The adaptive part avoids forcing extra regularity assumptions that smoothness methods often carry. That said, the strength rests on the precise definition of the transform and the proof that it enforces the structure without hidden conditions. I would want to see the exact characterization of continuous graphical structure and how the test avoids the usual discretization pitfalls in functional settings. The simulation details also matter: how misspecification was generated, what the baselines were, and whether the gains hold across sample sizes and noise levels. Without those, the performance claim stays provisional. This is aimed at people doing covariance estimation or inverse problems in functional data, especially those already thinking in terms of conditional independence or graphical models on continuous domains. A reader looking for alternatives to kernel or spline penalties would get something concrete to try. It deserves peer review. The novelty is real and the setup is internally consistent enough that referees can check the technical claims and run their own simulations. If the math checks out, it opens a usable new direction.

Referee Report

0 major / 3 minor

Summary. The paper claims that Markovianity induces a usable shape constraint on the covariance kernel for Gaussian functional data, allowing a Markov transform of the empirical covariance to yield an adaptive estimator that enforces the structure under no regularity beyond continuity. Simulations indicate improved prediction even under misspecification. The work also develops a new computationally efficient test for the Markov property based on a characterization of continuous graphical structure, positioning Markovianity as a falsifiable alternative to smoothness assumptions in functional data analysis.

Significance. If the central claims hold, the paper offers a substantive alternative to smoothness-based parsimony in functional data analysis by leveraging falsifiable Markov constraints, which naturally induce sparsity-like structure on the inverse covariance. The adaptive estimator requiring only continuity and the efficient test for graphical structure represent potential advances for prediction and inference in settings such as longitudinal functional data or time-series of curves. Explicit credit is due for the emphasis on minimal assumptions and the simulation evidence of robustness to misspecification.

minor comments (3)

[Abstract and §1] The abstract and introduction would benefit from a brief explicit statement of the functional domain (e.g., [0,1] or general interval) and the precise definition of the Markov property for the covariance kernel to aid readers unfamiliar with the continuous graphical model literature.
[Simulation section] In the simulation experiments, the description of how the misspecification scenarios are constructed (e.g., the specific non-Markov covariances used) should be expanded with a table or pseudocode to allow reproducibility and clearer assessment of the reported performance gains.
[Methods] Notation for the empirical covariance, its Markov transform, and the resulting estimator could be consolidated into a single table or displayed equation block early in the methods section to reduce cross-referencing.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our work on inference for functional data under Markov constraints. We appreciate the recognition of the Markov transform estimator, its adaptivity under minimal assumptions, the simulation evidence of robustness, and the new test for the Markov property as a falsifiable alternative to smoothness. The recommendation for minor revision is noted, and we will incorporate any editorial adjustments in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper derives a Markov transform and estimator from a new characterization of the shape constraint that Markovianity imposes on the covariance kernel for Gaussian functional data. This is presented as an alternative to smoothness, with the estimator being adaptive and requiring only continuity. The test for the Markov property rests on a novel characterization of continuous graphical structure. No load-bearing step reduces by construction to a fitted input, self-citation chain, or renamed known result; the central claims introduce independent content and are falsifiable. This matches the absence of any quoted reduction in the provided material.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract provides limited detail; central claim rests on the domain assumption that Markovianity imposes a usable shape constraint on the covariance kernel.

axioms (1)

domain assumption Markovianity induces a shape constraint on the covariance kernel in the Gaussian case
This is the key premise enabling the Markov transform and estimator.

pith-pipeline@v0.9.0 · 5471 in / 1127 out tokens · 25937 ms · 2026-05-10T04:05:23.772739+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

Al-Ghattas and D

Omar Al-Ghattas and Daniel Sanz-Alonso. Covariance operator estimation via adaptive threshold- ing. arXiv preprint arXiv:2405.18562 ,

work page arXiv
[2]

The Annals of Statistics , author =

doi: 10.1214/009053607000000758. URL https: //doi.org/10.1214/009053607000000758. Stamatis Cambanis. On some continuity and differentiability properties of paths of gaussian pro- cesses. Journal of Multivariate Analysis , 3(4):420–434,

work page doi:10.1214/009053607000000758
[3]

URL https://doi.org/10.1080/ 10485250215323

doi: 10.1080/10485250215323. URL https://doi.org/10.1080/ 10485250215323. Joseph L. Doob. Stochastic Processes. Wiley publications in statistics. Wiley,

work page doi:10.1080/10485250215323
[4]

URL http://www.jstor.org/stable/25662231

ISSN 00905364, 21688966. URL http://www.jstor.org/stable/25662231. Yu Liu and Zhao Ren. Minimax estimation of large precision matrices with bandable Cholesky factor. The Annals of Statistics , 48(4):2428 – 2454,

work page arXiv
[5]

URL https://doi.org/10.1214/19-AOS1893

doi: 10.1214/19-AOS1893. URL https://doi.org/10.1214/19-AOS1893. Cyrus B. Mehr and James A. McFadden. Certain Properties of Gaussian Processes and Their First-Passage Times. Journal of the Royal Statistical Society: Series B (Methodological) , 27(3): 505–522,

work page doi:10.1214/19-aos1893
[6]

and B¨ uhlmann, P

doi: 10.1214/009053606000000281. URL https://doi.org/10.1214/009053606000000281. Neda Mohammadi and Victor M. Panaretos. Functional data analysis with rough sample paths? Journal of Nonparametric Statistics , 36(1):4–22,

work page doi:10.1214/009053606000000281
[7]

and Peters, Jonas , year=

doi: 10.1214/19-AOS1857. Kartik G. Waghmare and Victor M. Panaretos. The completion of covariance kernels. The Annals of Statistics, 50(6):3281–3306,

work page doi:10.1214/19-aos1857
[8]

Their structure can be encoded by a graph, which provides a concise representation of the interactions among the variables

Appendix A Graphical models Conditional dependencies and independencies characterize how a collection of random variables relate to one another, both directly and indirectly. Their structure can be encoded by a graph, which provides a concise representation of the interactions among the variables. Moreover, the conditional independence relation and its as...

work page 2019
[9]

  X (j,j′)∈Ji(k)×Ji(k+1) var[ϵij,ij′]   . Moreover, var[ϵij,ij′] = K(tij′, tij′) − K(tij, tij′)2 K(tij, tij) = 1 K(tij, tij) K(tij, tij)K(tij′, tij′) − K(tij, tij)K(tij, tij′) + K(tij, tij′)K(tij, tij) − K(tij, tij′)2 = K(tij′, tij′) − K(tij, tij′) + K(tij, tij′) K(tij, tij) K(tij, tij) − K(tij, tij′) ≤ w(|tij − tij′|) 1 + K(tij, tij′) K(tij, tij) 27 ...

work page 1972
[10]

It is clear that if X is Markov then Equation (8) holds

Proof of Theorem 5.3. It is clear that if X is Markov then Equation (8) holds. For the converse, we adopt the perspective of continuously indexed graphical models [Waghmare and Panaretos, 2025] and we show that if X is not Markov, then Equation (8) cannot hold. An illustration of the proof is given in Figure

work page 2025
[11]

Then, there exist s < t and a closed interval U included in ( s, t) such that Xs ̸ ⊥ ⊥Xt | XU

Assume that X does not satisfy the Markov property. Then, there exist s < t and a closed interval U included in ( s, t) such that Xs ̸ ⊥ ⊥Xt | XU. Let Ω be a graphical model for X. By definition of Ω (see Remark A.4), s and t cannot be separated by U in Ω. Therefore, we can find a path p(s, t) in Ω from s to t such that this path does not cross U. Further...

work page 1973
[12]

We can then characterize the Markov property of X as follows

We say that the coordinates of X are locally dependent if Xj ̸ ⊥ ⊥Xj+1 | X{j,j+1}c for all j ∈ [p − 1]. We can then characterize the Markov property of X as follows. Theorem C.6. Let X = (X1, . . . , Xp) ∈ Rp be a non-degenerate Gaussian vector, and assume that the coordinates of X are locally dependent. Then, X satisfies the Markov property if and only i...

work page 2018

[1] [1]

Al-Ghattas and D

Omar Al-Ghattas and Daniel Sanz-Alonso. Covariance operator estimation via adaptive threshold- ing. arXiv preprint arXiv:2405.18562 ,

work page arXiv

[2] [2]

The Annals of Statistics , author =

doi: 10.1214/009053607000000758. URL https: //doi.org/10.1214/009053607000000758. Stamatis Cambanis. On some continuity and differentiability properties of paths of gaussian pro- cesses. Journal of Multivariate Analysis , 3(4):420–434,

work page doi:10.1214/009053607000000758

[3] [3]

URL https://doi.org/10.1080/ 10485250215323

doi: 10.1080/10485250215323. URL https://doi.org/10.1080/ 10485250215323. Joseph L. Doob. Stochastic Processes. Wiley publications in statistics. Wiley,

work page doi:10.1080/10485250215323

[4] [4]

URL http://www.jstor.org/stable/25662231

ISSN 00905364, 21688966. URL http://www.jstor.org/stable/25662231. Yu Liu and Zhao Ren. Minimax estimation of large precision matrices with bandable Cholesky factor. The Annals of Statistics , 48(4):2428 – 2454,

work page arXiv

[5] [5]

URL https://doi.org/10.1214/19-AOS1893

doi: 10.1214/19-AOS1893. URL https://doi.org/10.1214/19-AOS1893. Cyrus B. Mehr and James A. McFadden. Certain Properties of Gaussian Processes and Their First-Passage Times. Journal of the Royal Statistical Society: Series B (Methodological) , 27(3): 505–522,

work page doi:10.1214/19-aos1893

[6] [6]

and B¨ uhlmann, P

doi: 10.1214/009053606000000281. URL https://doi.org/10.1214/009053606000000281. Neda Mohammadi and Victor M. Panaretos. Functional data analysis with rough sample paths? Journal of Nonparametric Statistics , 36(1):4–22,

work page doi:10.1214/009053606000000281

[7] [7]

and Peters, Jonas , year=

doi: 10.1214/19-AOS1857. Kartik G. Waghmare and Victor M. Panaretos. The completion of covariance kernels. The Annals of Statistics, 50(6):3281–3306,

work page doi:10.1214/19-aos1857

[8] [8]

Their structure can be encoded by a graph, which provides a concise representation of the interactions among the variables

Appendix A Graphical models Conditional dependencies and independencies characterize how a collection of random variables relate to one another, both directly and indirectly. Their structure can be encoded by a graph, which provides a concise representation of the interactions among the variables. Moreover, the conditional independence relation and its as...

work page 2019

[9] [9]

  X (j,j′)∈Ji(k)×Ji(k+1) var[ϵij,ij′]   . Moreover, var[ϵij,ij′] = K(tij′, tij′) − K(tij, tij′)2 K(tij, tij) = 1 K(tij, tij) K(tij, tij)K(tij′, tij′) − K(tij, tij)K(tij, tij′) + K(tij, tij′)K(tij, tij) − K(tij, tij′)2 = K(tij′, tij′) − K(tij, tij′) + K(tij, tij′) K(tij, tij) K(tij, tij) − K(tij, tij′) ≤ w(|tij − tij′|) 1 + K(tij, tij′) K(tij, tij) 27 ...

work page 1972

[10] [10]

It is clear that if X is Markov then Equation (8) holds

Proof of Theorem 5.3. It is clear that if X is Markov then Equation (8) holds. For the converse, we adopt the perspective of continuously indexed graphical models [Waghmare and Panaretos, 2025] and we show that if X is not Markov, then Equation (8) cannot hold. An illustration of the proof is given in Figure

work page 2025

[11] [11]

Then, there exist s < t and a closed interval U included in ( s, t) such that Xs ̸ ⊥ ⊥Xt | XU

Assume that X does not satisfy the Markov property. Then, there exist s < t and a closed interval U included in ( s, t) such that Xs ̸ ⊥ ⊥Xt | XU. Let Ω be a graphical model for X. By definition of Ω (see Remark A.4), s and t cannot be separated by U in Ω. Therefore, we can find a path p(s, t) in Ω from s to t such that this path does not cross U. Further...

work page 1973

[12] [12]

We can then characterize the Markov property of X as follows

We say that the coordinates of X are locally dependent if Xj ̸ ⊥ ⊥Xj+1 | X{j,j+1}c for all j ∈ [p − 1]. We can then characterize the Markov property of X as follows. Theorem C.6. Let X = (X1, . . . , Xp) ∈ Rp be a non-degenerate Gaussian vector, and assume that the coordinates of X are locally dependent. Then, X satisfies the Markov property if and only i...

work page 2018