Pith. sign in

REVIEW 2 major objections 6 minor 71 references

Conditional Distribution Estimation for Functional Responses with Random Forests

T0 review · 2 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Functional distributional random forests consistently estimate the conditional law of a curve-valued response, so a single fitted forest supports plug-in inference on any functional of that law.

desk verdict A genuine and useful extension of distributional random forests to functional responses, held back by a real gap between the consistency theory (exact MMD, infinite forest) and the implemented estimator (random-feature MMD, finite forest). read the letter →

arxiv 2608.08247 v1 pith:WMHS2QGQ submitted 2026-08-08 stat.ME stat.ML

Poorbita Kundu , Antonio R. Linero This is my paper · ORCID
classification stat.MEstat.ML MSC 62G0862G2062R10
keywords functionaldataanalysisrandomforestsconditionaldistributionestimationmaximummeandiscrepancykernelmethodsproperscoringrulesresponseMMDsplitcriterion
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces the functional distributional random forest (fDRF), which estimates the conditional law $F_x$ of a curve-valued response $Y$ given covariates $X$ as a weighted empirical distribution over the observed functions. The paper's central claim is that this estimator is consistent: under assumptions on the tree-growing process, the kernel, and the data, the estimated law $\hat{F}_x$ converges weakly-in-probability to $F_x$ as an element of $\mathcal{P}(\mathcal{Y})$, and with an additional tightness condition also in $\mathcal{P}(C([0,1]))$. This matters because a single fitted forest can then answer many scientific questions at once—conditional medians, quantiles, peak times, exceedance probabilities, and predictive dispersion—by plugging the estimated law into any functional of interest. The paper also proposes proper scoring rules for comparing functional distribution estimators and shows in simulations and NHANES accelerometer data that distributional changes missed by functional linear models are recovered.

What carries the argument

The central machinery is the MMD split criterion together with its random-feature approximation. For a positive definite kernel $\kappa$ on the function space, each candidate split is scored by the size-weighted empirical MMD between the response samples in the two child nodes, $\frac{n_{bL} n_{bR}}{n_b^2} D_{\mathrm{MMD}(\kappa)}$, which is CART applied to the RKHS feature map $\varphi(Y_i)$. To make this computable, the paper writes a Gaussian kernel on $L^2([0,1])$ as an expectation over cosine features of an isonormal Gaussian process $W(y)=\int_0^1 y(t)\,dB(t)$, then approximates it with $B$ random features that include discretized Brownian motion and are sampled afresh for each split; the same construction extends to other radial kernels through mixing variables. The theoretical guarantee is proved for the ideal infinite-forest estimator whose weights $w_i(x)$ are conditional expectations over tree randomness, under honesty, $\alpha$-regularity, subsampling, and uniform tightness assumptions.

What would settle it

Fit the implemented fDRF on data from a Gaussian process whose length scale varies with a one-dimensional covariate, at sample sizes 500 and 4000, and estimate the conditional distribution of $\sup_t Y(t)$ at a fixed covariate value. If the Wasserstein or Kolmogorov distance between the estimated and true laws of this supremum functional does not shrink as $N$ grows, the random-feature approximation of the MMD split criterion is not preserving the paper's weak-consistency guarantee.

Watch

Extended reading notes

Core claim

The central claim, stated on the paper's own terms, is that the conditional law $F_x = \mathcal{L}(Y \mid X = x)$ of a functional response can be estimated nonparametrically by a random forest whose splits maximize a maximum mean discrepancy (MMD) between the empirical distributions in the child nodes, with the estimated law $\hat{F}_x = \sum_{i=1}^N w_i(x)\delta_{Y_i}$. Theorem 2 proves $\hat{F}_x \to F_x$ weakly-in-probability as probability measures on $\mathcal{Y}$ under Assumptions 1–5, and Theorem 3 upgrades this to $\mathcal{P}(C([0,1]))$ under Assumption 6. The paper is careful that MMD convergence alone does not imply weak convergence in infinite-dimensional spaces, so the proof adds uniform tightness and uses Prohorov's theorem and a subsequence argument. Once consistency holds, the same $\hat{F}_x$ can be used for plug-in inference on functionals that are discontinuous in $L^2([0,1])$, such as point evaluations, suprema, peak times, and exceedance sets.

Load-bearing premise

The load-bearing premise is uniform tightness of the conditional laws (Assumption 5), which is what upgrades MMD convergence to genuine weak convergence in infinite-dimensional function spaces; a second fragile point is that the theorem treats an idealized infinite forest with the exact MMD criterion, while the implemented algorithm uses a random-feature approximation whose error is not analyzed.

Editorial extensions

If this is right

  • One fitted fDRF yields plug-in estimates for any functional $\tau(F_x)$—conditional means, quantiles, exceedance probabilities, peak times, and predictive dispersion—without refitting.
  • Because $\hat{F}_x$ is supported on the observed curves, predictive draws preserve realistic geometry and automatically respect constraints such as nonnegativity or density normalization.
  • Under Assumptions 1–5 the estimated law converges weakly-in-probability in $\mathcal{P}(\mathcal{Y})$; adding Assumption 6 extends this to $\mathcal{P}(C([0,1]))$, justifying inference on point evaluations and suprema.
  • Energy and kernel scoring rules for functional responses allow comparison of full conditional-distribution estimators; Sobolev-based scores are more sensitive to roughness differences than $L^2$-based scores.
  • Simulations show the fDRF outperforms K-nearest neighbors and functional linear model baselines on heteroskedastic, mixture, warped, and variable-roughness Gaussian processes, with the weighted Sobolev kernel the strongest default.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: the consistency theorems cover an idealized infinite forest with the exact MMD, while the implemented algorithm uses $B$ random features with discretized Brownian motion resampled for each split; a uniform error bound for this approximation as $B,R\to\infty$ would be the natural step to make the applied estimator match the proven one.
  • Editorial extension: because the estimator is a weighted empirical law, the fDRF could be coupled with smoothed or calibrated weights to serve as a common-support device for causal or mediation estimators, a use the paper lists as future work.
  • Editorial extension: the same MMD-split-plus-tightness argument should extend to responses in other infinite-dimensional or non-Euclidean metric spaces—direction-valued, graph-valued, or manifold-valued data—whenever a characteristic kernel and uniform tightness are available.
  • Editorial extension: the little-bags intervals intentionally overstate variability because the within-bag correction is omitted; a testable refinement is to replace the normal quantile with a calibrated bootstrap quantile, especially for nonregular functionals like argmax peak times.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. This paper introduces the functional distributional random forest (fDRF), a random-forest method that estimates the conditional distribution of a functional response Y given covariates X by a weighted empirical distribution over the observed curves. Splits are chosen to maximize an empirical maximum mean discrepancy (MMD) between child-node response distributions, with kernels defined on function spaces such as L2 and Sobolev spaces. The paper establishes consistency of an idealized infinite-forest version of the estimator (Theorems 2 and 3), proposes proper scoring rules for evaluating conditional distribution estimators, reports simulation comparisons against K-nearest neighbors and functional linear models, and applies the method to NHANES accelerometer data to study how age and income affect activity profiles. The theoretical development explicitly addresses the fact that MMD convergence does not automatically imply weak convergence on infinite-dimensional spaces, using uniform tightness assumptions to upgrade the consistency statement.

Significance. The contribution is potentially significant: it extends distributional random forests from Euclidean responses to functional responses, provides a practical plug-in framework for inference on distributional features of curves, and gives a theoretically grounded treatment of weak convergence of the estimated conditional laws. The paper's handling of the non-local-compactness issue is a real technical contribution, and the proposed Sobolev-transform kernels and proper scoring rules are useful tools. The supplementary material includes detailed proofs, implementation algorithms, and NHANES analysis code, which supports reproducibility. The simulation study is reasonably extensive and the NHANES application illustrates genuinely distributional findings beyond conditional means. However, the consistency theorems apply to an idealized estimator using exact MMD and a fixed kernel, while the implemented algorithm uses a random-feature approximation and data-dependent kernel tuning; this gap must be addressed before the central plug-in claim can be regarded as fully established.

major comments (2)
  1. [Section 5 / Supplementary Algorithm 2] The consistency results in Section 5 are stated for an ideal infinite forest that uses the exact MMD split criterion (1) with D_MMD from (2), but Algorithm 2 of the Supplementary Material replaces D_MMD by the random-feature approximation in (4), with B random features, a finite-grid discretization of the Itô integral, and fresh randomness at each split. No theorem, uniform bound, or rate is provided connecting the approximate criterion to the exact one, and no condition on B or R_j is stated under which the approximation error vanishes as N→∞. Consequently, the estimator whose performance is reported in Sections 4 and 6 is not the estimator shown to be consistent in Theorems 2 and 3. This gap is load-bearing because the paper's central plug-in claim requires consistency of the fitted forest, not of a hypothetical exact-MMD forest. The Discussion (Section 7) acknowledges that discretely-sampled approximations are future work, which confirms the scope of the current theory.
  2. [Section 2.4 and Section 5 (Assumptions 2 and 4)] The theoretical results treat the kernel κ as fixed, whereas the implementation sets the length scale ℓ by the median-distance heuristic on the data and sets λ and the Sobolev normalizers (a,b) by data-dependent rules. Because Assumption 4 and the Lipschitz condition on x↦μ(F_x) in Assumption 2 are stated for a fixed kernel, the ported Theorem 1 and the proof of Theorem 2 do not directly cover the estimator actually fitted with a data-selected kernel. The paper should either state conditions under which the data-dependent kernel still satisfies the assumptions (e.g., a rate condition on ℓ̂), or clarify that the theory is for a pre-specified kernel and treat the heuristic as a practical implementation choice.
minor comments (6)
  1. [Abstract] The abstract claims the method 'supports inference on arbitrary functionals of the conditional distribution,' but Theorems 2 and 3 only establish weak convergence, which guarantees convergence for continuous bounded functionals (or continuous functionals on C[0,1] under Assumption 6). Recommend replacing 'arbitrary' with 'a broad class of' or 'continuous.'
  2. [Section 2.4] The sentence 'We see this in Section 2.4, where it is shown that the L2-kernel is outperformed when functions differ by smoothness' appears to refer to the simulation results, which are in Section 4, not Section 2.4.
  3. [Supplementary Material C.1] There are several typos in this section: 'a broad a broad class', 'kernels htat', and 'obtaiend' should be corrected.
  4. [Supplementary Algorithm 2] The notation q_b = min{max[Poisson(m_try),1],P} is unclear; please define m_try and explain whether Poisson(m_try) denotes a Poisson random variable.
  5. [Supplementary Lemma 2 proof] In the displayed chain of equalities in the proof of Lemma 2, the third line writes w_i where w_{m,i} appears in the neighboring lines; the intended conditioning argument is clear, but the notation should be made consistent.
  6. [Section 6 (NHANES)] The Discussion lists irregular and partially observed sampling as future work, but the NHANES application itself uses discretely sampled activity profiles; a sentence in Section 6 acknowledging that the theory is applied informally to discretely observed curves would help.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central consistency theorems are ported from an external source (Čevid et al., 2022), the estimators are evaluated on held-out data against external baselines, and the main implementation gap (random-feature MMD approximation) is an unproved approximation step rather than a circular reduction.

full rationale

The derivation chain is not circular. Theorem 1 is explicitly 'Theorem 2 of Čevid et al. (2022)' and is taken as an external starting point; Theorems 2–3 add genuine functional-analytic work (weak convergence from MMD consistency under uniform tightness assumptions). Assumptions 5–6 are conditions on the true conditional laws, not fitted targets, and the proofs do not assume the conclusion. The MMD split criterion (Eqs. 1–2), the plug-in estimator (Eq. 5), and the downstream functionals are all defined independently of the quantities being predicted. Empirical evaluation is conducted on held-out test data and compares against KNN and FLM baselines using proper scoring rules imported from Gneiting and Raftery (2007) and Bülte et al. (2025); no fitted parameter is renamed as a prediction. The only self-citation (Kundu et al., 2025, for the exceedance function) is not load-bearing: the exceedance function is a simple threshold-occupancy summary used as a downstream target, and its consistency follows from the same plug-in logic as other functionals. The most serious weakness is a theory–implementation gap, not circularity: Algorithm 2 replaces the exact D_MMD in Eq. (1) with the random-feature approximation of Eq. (4), and the Discussion concedes that 'our theoretical results are derived for continuous-time curves, while extending these results to account for discretely-sampled approximations remains an important direction for future research.' This means the implemented estimator is not formally covered by Theorems 1–3, but that is an unproved approximation step, not an equivalence between inputs and outputs. No circular step can be exhibited from the paper's own equations or citations.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The central claim rests on standard forest structural assumptions, smoothness and tightness of the conditional laws, and characteristic kernels on function spaces. The only entity introduced is the fDRF estimator, which is a method, not a new postulated object. The main unproven premise is that the random-feature MMD approximation inherits the consistency of the exact MMD criterion.

free parameters (5)
  • Kernel choice kappa = Gaussian, Sobolev, or FPCA kernels
    The splitting criterion and downstream sensitivity depend on which functional kernel is selected; the paper gives no data-driven rule for choosing among kernel families and lists kernel choice as a limitation.
  • Sobolev mixing weight lambda = 0.5 in illustrations
    Sets equal weight on function and derivative in the Sobolev norm; chosen by hand as a default.
  • Kernel length scale ell = median{||T(Y_i)-T(Y_j)||_G : i<j}
    Set by the standard median distance heuristic; data-dependent but not fitted to target functionals.
  • Sobolev scale normalizers (a,b) = tuned to equalize scale between ||Y-Y'||^2 and ||DY-DY'||^2
    Hand-tuned normalization so the Sobolev components contribute equally; not derived from first principles.
  • Random feature count B = 200
    Computational choice used in simulations and NHANES; larger B reduces approximation error at increased cost.
assumptions (7)
  • standard math Assumption 1 (P1-P5): subsampling, honesty, alpha-regularity, symmetry, random-split.
    Taken verbatim from Cevid et al. (2022) and Wager and Athey (2018); used in Theorem 1 and all subsequent results.
  • domain assumption Assumption 2: the kernel mean embedding x -> mu(F_x) is Lipschitz.
    Requires the conditional law to change smoothly in the covariate; used to bound split localization.
  • domain assumption Assumption 3: covariates have a continuous density bounded away from zero on [0,1]^P.
    Standard condition for forest consistency; ensures every leaf eventually localizes.
  • domain assumption Assumption 4: kernel is bounded, continuous, and characteristic.
    Needed to make MMD a metric and to ensure the limiting measure is identified; the Gaussian and Sobolev kernels are assumed characteristic on the function space.
  • domain assumption Assumption 5: uniform tightness of the conditional distributions.
    Crucial to upgrading MMD convergence to weak convergence; requires that most probability mass of F_x lies in compact sets uniformly over neighborhoods of x.
  • domain assumption Assumption 6: smoothness embedding to get convergence in C[0,1].
    Requires either C continuously embedded into Y or Y into C; used for inference on non-L2-continuous functionals like point evaluation and supremum.
  • ad hoc to paper Random-feature approximation validity.
    The algorithm uses a finite random-feature approximation of the MMD (Equation 4) but the theory applies to the exact MMD; no theorem connects the two.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Conditional Distribution Estimation for Functional Responses with Random Forests." pith.science (2026). https://pith.science/paper/WMHS2QGQ

@misc{pith2026260808247,
  author       = {Pith},
  title        = {Pith review of: Conditional Distribution Estimation for Functional Responses with Random Forests},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WMHS2QGQ}},
  note         = {Machine review of arXiv:2608.08247}
}
read the original abstract

Many functional data analyses reduce random functions to scalar summaries or conditional mean curves. This is limiting when we wish to understand how covariates affect the distribution of entire functional responses, including their shape, timing, or variability. We study the problem of estimating conditional laws of functional outcomes and show that these objects can be estimated and evaluated in a practical nonparametric framework. To do this, we introduce functional distributional random forests, which estimate each conditional law as a covariate-dependent distribution over sampled functions by training a random forest to minimize a kernel-based maximum mean discrepancy within the leaf nodes of the decision tree. This supports inference on arbitrary functionals of the conditional distribution while keeping predictive samples tied to realistic curves. We consider a variety of kernels defined on function spaces, including Sobolev and operator-induced kernels. We also provide conditions for consistency of our estimator and develop scoring rules for comparing it to baseline estimators. In simulations, our method recovers distributional changes that are missed by baseline methods. In an application to NHANES accelerometer data, it identifies interesting covariate-associated changes in both median activity profiles and predictive dispersion.

Figures

Figures reproduced from arXiv: 2608.08247 by the authors.

Figure 1
Figure 1. Data-generating mechanisms used in the simulation study. Each row shows one [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Samples from the conditional predictive distributions of density-valued functional [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Top: pairwise comparison heatmap of different methods under the different simula [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Predictive distributions for NHANES hourly activity profiles. Panels vary age and [PITH_FULL_IMAGE:figures/full_fig_p025_4.png]
Figure 5
Figure 5. Figure 5: Pointwise differences in median day-averaged log activity trajectories relative to the [PITH_FULL_IMAGE:figures/full_fig_p027_5.png]
Figure 6
Figure 6. Figure 6: Inference for the circular mean and circular standard deviation of peak time and [PITH_FULL_IMAGE:figures/full_fig_p028_6.png]
Figure 7
Figure 7. Figure 7: Median exceedance functions for weekly NHANES activity curves, with pointwise [PITH_FULL_IMAGE:figures/full_fig_p029_7.png]
Figure 8
Figure 8. Figure 8: Pointwise differences in median exceedance functions relative to the 25-year-old [PITH_FULL_IMAGE:figures/full_fig_p030_8.png]
Figure 9
Figure 9. Figure 9: Predictive distribution of the exceedance function in different strata as a function [PITH_FULL_IMAGE:figures/full_fig_p045_9.png]
Figure 10
Figure 10. Figure 10: Estimated conditional distributions from the [PITH_FULL_IMAGE:figures/full_fig_p049_10.png]
Figure 11
Figure 11. Figure 11: Simulation results using L2-based scoring rules. Top: pairwise comparison heatmaps under the L2 energy score (left) and the Gaussian kernel score constructed from the L2 norm (right). Blue indicates that the row method performed better, while red indicates that the co…
Figure 12
Figure 12. Figure 12: Predictive distribution of the peak time (left) and maximal effort (right) across [PITH_FULL_IMAGE:figures/full_fig_p052_12.png]
Figure 13
Figure 13. Figure 13: Pointwise median day-averaged log activity trajectories with pointwise 95% con [PITH_FULL_IMAGE:figures/full_fig_p053_13.png]
Figure 14
Figure 14. Figure 14: Empirical coverage of nominal 95% confidence intervals under the settings de [PITH_FULL_IMAGE:figures/full_fig_p055_14.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

71 extracted references · 59 canonical work pages

  1. [1]

    Electronic Journal of Statistics , year =

    Peng, Wei and Coleman, Tim and Mentch, Lucas , title =. Electronic Journal of Statistics , year =

    work page

  2. [2]

    Journal of Machine Learning Research , year =

    Mentch, Lucas and Hooker, Giles , title =. Journal of Machine Learning Research , year =

    work page

  3. [3]

    Electronic Journal of Statistics , year =

    Song, Yanglei and Chen, Xiaohui and Kato, Kengo , title =. Electronic Journal of Statistics , year =

    work page

  4. [4]

    Order-Explicit Linearization of High-Dimensional

    Ritzwoller, David M and Syrgkanis, Vasilis , journal=. Order-Explicit Linearization of High-Dimensional

    work page

  5. [5]

    The Annals of Mathematical Statistics , year =

    Hoeffding, Wassily , title =. The Annals of Mathematical Statistics , year =

  6. [6]

    The Annals of Statistics , volume=

    The Jackknife Estimate of Variance , author=. The Annals of Statistics , volume=. 1981 , publisher=

    work page 1981

  7. [7]

    Pacific-Asia conference on knowledge discovery and data mining , pages=

    Evaluating the replicability of significance tests for comparing learning algorithms , author=. Pacific-Asia conference on knowledge discovery and data mining , pages=. 2004 , organization=

    work page 2004

  8. [8]

    Advances in neural information processing systems , volume=

    Inference for the generalization error , author=. Advances in neural information processing systems , volume=

Show all 71 references
  1. [9]

    Computational Statistics & Data Analysis , volume=

    Standard errors for bagged and random forest estimators , author=. Computational Statistics & Data Analysis , volume=. 2009 , publisher=

    2009

  2. [10]

    2009 , publisher=

    Malliavin calculus and its applications , author=. 2009 , publisher=

    2009

  3. [11]

    Journal of the American Statistical Association , volume=

    Cauchy combination test: a powerful test with analytic p-value calculation under arbitrary dependency structures , author=. Journal of the American Statistical Association , volume=. 2020 , publisher=

    2020

  4. [12]

    arXiv preprint arXiv:2504.02296 , year=

    Exceedance and force of centrality for functional data , author=. arXiv preprint arXiv:2504.02296 , year=

    arXiv

  5. [13]

    Journal of Machine Learning Research , volume=

    Confidence and uncertainty assessment for distributional random forests , author=. Journal of Machine Learning Research , volume=

  6. [14]

    Wiley Interdisciplinary Reviews: Computational Statistics , volume=

    Circular data , author=. Wiley Interdisciplinary Reviews: Computational Statistics , volume=. 2010 , publisher=

    2010

  7. [15]

    Machine Learning , volume=

    Random Forests , author=. Machine Learning , volume=. 2001 , doi=

    2001

  8. [16]

    Computational Statistics & Data Analysis , volume=

    Grouped Variable Importance with Random Forests and Application to Multiple Functional Data Analysis , author=. Computational Statistics & Data Analysis , volume=. 2015 , doi=

    2015

  9. [17]

    Journal of Chemometrics , volume=

    Random Forests for Functional Covariates , author=. Journal of Chemometrics , volume=. 2016 , doi=

    2016

  10. [18]

    Scientific Reports , volume=

    Functional Random Forest with Applications in Dose-Response Predictions , author=. Scientific Reports , volume=. 2019 , doi=

    2019

  11. [19]

    Scientific Reports , volume=

    Functional Random Forests for Curve Response , author=. Scientific Reports , volume=. 2021 , doi=

    2021

  12. [20]

    Functional Data Analysis , author=

  13. [21]

    Annual Review of Statistics and Its Application , volume=

    Functional Regression , author=. Annual Review of Statistics and Its Application , volume=. 2015 , doi=

    2015

  14. [22]

    Statistical Modelling , volume=

    A General Framework for Functional Regression Modelling , author=. Statistical Modelling , volume=. 2017 , doi=

    2017

  15. [23]

    Petersen, Alexander and M. Fr. The Annals of Statistics , volume=. 2019 , doi=

    2019

  16. [24]

    Random Forest Weighted Local Fr

    Qiu, Rui and Yu, Zhou and Zhu, Ruoqing , journal=. Random Forest Weighted Local Fr

  17. [25]

    Journal of Machine Learning Research , volume=

    Quantile Regression Forests , author=. Journal of Machine Learning Research , volume=

  18. [26]

    The Annals of Applied Statistics , volume=

    Random Survival Forests , author=. The Annals of Applied Statistics , volume=. 2008 , doi=

    2008

  19. [27]

    The Annals of Applied Statistics , volume=

    Distributional Regression Forests for Probabilistic Precipitation Forecasting in Complex Terrain , author=. The Annals of Applied Statistics , volume=. 2019 , doi=

    2019

  20. [28]

    Pospisil, Taylor and Lee, Ann B. , year=

  21. [29]

    Proceedings of the 24th International Conference on Artificial Intelligence and Statistics , series=

    Wasserstein Random Forests and Applications in Heterogeneous Treatment Effects , author=. Proceedings of the 24th International Conference on Artificial Intelligence and Statistics , series=. 2021 , publisher=

    2021

  22. [30]

    The Annals of Statistics , volume=

    Generalized Random Forests , author=. The Annals of Statistics , volume=. 2019 , doi=

    2019

  23. [31]

    Journal of Machine Learning Research , volume=

    Distributional Random Forests: Heterogeneity Adjustment and Multivariate Distributional Regression , author=. Journal of Machine Learning Research , volume=

  24. [32]

    Journal of Machine Learning Research , volume=

    A Kernel Two-Sample Test , author=. Journal of Machine Learning Research , volume=

  25. [33]

    Journal of Machine Learning Research , volume=

    A Kernel Two-Sample Test for Functional Data , author=. Journal of Machine Learning Research , volume=

  26. [34]

    Gaussian Processes for Machine Learning , author=

  27. [35]

    Annals of Mathematics , volume=

    Metric Spaces and Completely Monotone Functions , author=. Annals of Mathematics , volume=. 1938 , doi=

    1938

  28. [36]

    Advances in Neural Information Processing Systems , volume=

    Random Features for Large-Scale Kernel Machines , author=. Advances in Neural Information Processing Systems , volume=

  29. [37]

    arXiv preprint arXiv:2411.02770 , year=

    A Spectral Mixture Representation of Isotropic Kernels with Application to Random Fourier Features , author=. arXiv preprint arXiv:2411.02770 , year=

  30. [38]

    Computational Statistics & Data Analysis , volume=

    Generative Models for Functional Data Using Phase and Amplitude Separation , author=. Computational Statistics & Data Analysis , volume=. 2013 , doi=

    2013

  31. [39]

    Advances in Neural Information Processing Systems , volume=

    Time-Series Generative Adversarial Networks , author=. Advances in Neural Information Processing Systems , volume=

  32. [40]

    Proceedings of the 40th International Conference on Machine Learning , series=

    Variational Autoencoding Neural Operators , author=. Proceedings of the 40th International Conference on Machine Learning , series=. 2023 , publisher=

    2023

  33. [41]

    Journal of Machine Learning Research , volume=

    Autoencoders in Function Space , author=. Journal of Machine Learning Research , volume=

  34. [42]

    arXiv preprint arXiv:2502.12902 , year=

    Probabilistic neural operators for functional uncertainty quantification , author=. arXiv preprint arXiv:2502.12902 , year=

    arXiv

  35. [43]

    Journal of the American statistical Association , volume=

    Strictly proper scoring rules, prediction, and estimation , author=. Journal of the American statistical Association , volume=. 2007 , publisher=

    2007

  36. [44]

    Annual Review of Statistics and Its Application , volume=

    Proper scoring rules for estimation and forecast evaluation , author=. Annual Review of Statistics and Its Application , volume=. 2025 , publisher=

    2025

  37. [45]

    Wiley Series in Probability and Statistics , year=

    Convergence of Probability Measures , author=. Wiley Series in Probability and Statistics , year=

  38. [46]

    Hitchhiker's guide to the fractional

    Di Nezza, Eleonora and Palatucci, Giampiero and Valdinoci, Enrico , journal=. Hitchhiker's guide to the fractional. 2012 , publisher=

    2012

  39. [47]

    Journal of the American Statistical Association , volume=

    Estimation and inference of heterogeneous treatment effects using random forests , author=. Journal of the American Statistical Association , volume=. 2018 , publisher=

    2018

  40. [48]

    Ramsay, James O and Silverman, Bernard W , year=

  41. [49]

    Annual Review of Statistics and its application , volume=

    Wang, Jane-Ling and Chiou, Jeng-Min and M. Annual Review of Statistics and its application , volume=. 2016 , publisher=

    2016

  42. [50]

    Journal of the American statistical association , volume=

    Yao, Fang and M. Journal of the American statistical association , volume=. 2005 , publisher=

    2005

  43. [51]

    The Annals of Statistics , volume=

    Yao, Fang and M. The Annals of Statistics , volume=

  44. [52]

    The Annals of Statistics , volume=

    Hall, Peter and M. The Annals of Statistics , volume=

  45. [53]

    Biometrika , volume=

    Tang, Rong and M. Biometrika , volume=. 2008 , publisher=

    2008

  46. [54]

    The Annals of Statistics , volume=

    Dai, Xiongtao and M. The Annals of Statistics , volume=

  47. [55]

    Marron, J. S. and Ramsay, J. O. and Sangalli, L. M. and Srivastava, A. , journal=

  48. [56]

    2006 , publisher=

    Ferraty, Fr. 2006 , publisher=

    2006

  49. [57]

    The Annals of Statistics , volume=

    Chen, Yaqing and M. The Annals of Statistics , volume=. 2022 , publisher=

    2022

  50. [58]

    2015 , publisher=

    Pya, Natalya and Wood, Simon N , journal=. 2015 , publisher=

    2015

  51. [59]

    The Annals of Statistics , volume=

    Functional data analysis for density functions by transformation to a Hilbert space , author=. The Annals of Statistics , volume=

  52. [60]

    Journal of the American Statistical Association , volume=

    Additive functional regression for densities as responses , author=. Journal of the American Statistical Association , volume=

  53. [61]

    The Annals of Statistics , volume=

    Total variation regularized Fr\'echet regression for metric-space valued data , author=. The Annals of Statistics , volume=

  54. [62]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

    Modelling function-valued stochastic processes, with applications to fertility dynamics , author=. Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

  55. [63]

    Journal of the American Statistical Association , volume=

    Modeling time-varying random objects and dynamic networks , author=. Journal of the American Statistical Association , volume=

  56. [64]

    The Annals of Statistics , volume=

    Metric statistics: Exploration and inference for random objects with distance profiles , author=. The Annals of Statistics , volume=

  57. [65]

    Exponential increases in high-temperature extremes in

    Davariashtiyani, Ali and Taherkhani, Masoud and Fattahpour, Sahar and Vitousek, Sean , journal=. Exponential increases in high-temperature extremes in

  58. [66]

    Quantifying the varying predictive value of physical activity measures obtained from wearable accelerometers on all-cause mortality over short to medium time horizons in

    Tabacu, Lucia and Ledbetter, Mark and Leroux, Andrew and Crainiceanu, Ciprian and Smirnova, Ekaterina , journal=. Quantifying the varying predictive value of physical activity measures obtained from wearable accelerometers on all-cause mortality over short to medium time horizons in

  59. [67]

    Computational Statistics & Data Analysis , volume=

    Shape-constrained estimation in functional regression with Bernstein polynomials , author=. Computational Statistics & Data Analysis , volume=

  60. [68]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

    Generalized additive and index models with shape constraints , author=. Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

  61. [69]

    arXiv preprint arXiv:2406.12817 , year=

    Decomposition-based intrinsic modeling of shape-constrained functional data , author=. arXiv preprint arXiv:2406.12817 , year=

    arXiv

  62. [70]

    Biometrika , volume=

    Fr\'echet analysis of variance for random objects , author=. Biometrika , volume=

  63. [71]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

    Empirical likelihood-based inference for functional means with application to wearable device data , author=. Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.