Efficient distributional regression trees learning algorithms for calibrated non-parametric probabilistic forecasts

Guillaume Obozinski; Quentin Duchemin

arxiv: 2502.05157 · v3 · pith:YHM3LB2Anew · submitted 2025-02-07 · 💻 cs.LG · cs.DS

Efficient distributional regression trees learning algorithms for calibrated non-parametric probabilistic forecasts

Quentin Duchemin , Guillaume Obozinski This is my paper

Pith reviewed 2026-05-23 03:14 UTC · model grok-4.3

classification 💻 cs.LG cs.DS

keywords probabilistic regression treesWIS lossCRPS lossdistributional regressionnon-parametric forecastsconformal predictioncalibrated uncertainty

0 comments

The pith

New algorithms learn probabilistic regression trees efficiently for WIS and CRPS losses by integrating min-max heaps, weight-balanced binary trees, and Fenwick trees.

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

The paper introduces novel algorithms for learning probabilistic regression trees optimized for the weighted interval score or continuous ranked probability score loss functions. These algorithms achieve computational efficiency through the use of data structures including min-max heaps, weight-balanced binary trees, and Fenwick trees. A sympathetic reader cares because this enables non-parametric models that estimate conditional distributions directly, leading to better calibration and coverage in uncertainty quantification compared to parametric alternatives. The methods also preserve the interpretability of decision trees and support applications like conformal prediction for group-conditional coverage.

Core claim

The paper presents algorithms that learn distributional regression trees by optimizing splits for WIS or CRPS, with efficiency from appropriate data structures, demonstrating competitive performance in numerical experiments and suitability for conformal prediction.

What carries the argument

Integration of min-max heaps, weight-balanced binary trees, and Fenwick trees into the tree learning process to compute optimal splits under WIS or CRPS losses.

If this is right

The learned trees produce non-parametric probabilistic forecasts with good calibration and coverage properties.
The trees maintain interpretability and explainability advantages of standard decision trees.
The methods achieve competitive performance with alternative approaches on benchmark tasks.
The trees are well-suited for conformal prediction to achieve group-conditional coverage guarantees.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

These algorithms could extend to other proper scoring rules beyond WIS and CRPS.
The data structures may enable scaling to much larger datasets than naive implementations allow.
The approach could combine with ensemble methods to further improve forecast reliability.

Load-bearing premise

The data structures integrate into the tree learning process without introducing hidden computational costs or approximation errors that compromise calibration or coverage.

What would settle it

A dataset where training the new algorithms takes significantly longer than expected or results in poorer CRPS scores and coverage compared to baseline methods.

Figures

Figures reproduced from arXiv: 2502.05157 by Guillaume Obozinski, Quentin Duchemin.

**Figure 1.** Figure 1: Fenwick tree for a list (ai)i∈{1,...,7} of length 7. By convention, a0 = 0. The nodes of the tree are indexed by the natural integers. 0 is at the root and the children of the root correspond to all integers of the form 2p with p ∈ N. More generally, nodes at depth k are the sum of k integers of the form 2p , i.e., they have k non-zero bits in their binary expansion. The parent of node i has the same binar… view at source ↗

**Figure 2.** Figure 2: Illustration of an update of the min-max heaps at a specific iteration of the algorithm to [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** Figure 3: Illustration of an update of the Fenwick tree at a specific iteration of the algorithm to perform [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

**Figure 4.** Figure 4: Estimation of the conditional CDF of y given x with CRPS-RF on a sample of size [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗

**Figure 5.** Figure 5: Time to compute the CRPS of the prefix sequences [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗

**Figure 6.** Figure 6: We train a single CRPS-RT and identify four distinct regions in the feature space, based [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗

**Figure 7.** Figure 7: Estimation of the conditional CDF of y given x with PMQRF with 100 on a sample of size [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗

read the original abstract

The perspective of developing trustworthy AI for critical applications in science and engineering requires machine learning techniques that are capable of estimating their own uncertainty. In the context of regression, instead of estimating a conditional mean, this can be achieved by producing a predictive interval for the output, or to even learn a model of the conditional probability $p(y|x)$ of an output $y$ given input features $x$. While this can be done under parametric assumptions with, e.g. generalized linear model, these are typically too strong, and non-parametric models offer flexible alternatives. In particular, for scalar outputs, learning directly a model of the conditional cumulative distribution function of $y$ given $x$ can lead to more precise probabilistic estimates, and the use of proper scoring rules such as the weighted interval score (WIS) and the continuous ranked probability score (CRPS) lead to better coverage and calibration properties. This paper introduces novel algorithms for learning probabilistic regression trees for the WIS or CRPS loss functions. These algorithms are made computationally efficient thanks to an appropriate use of known data structures - namely min-max heaps, weight-balanced binary trees and Fenwick trees. Through numerical experiments, we demonstrate that the performance of our methods is competitive with alternative approaches. Additionally, our methods benefit from the inherent interpretability and explainability of trees. As a by-product, we show how our trees can be used in the context of conformal prediction and explain why they are particularly well-suited for achieving group-conditional coverage guarantees.

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

New algorithms for exact CRPS/WIS optimization in regression trees using heaps and Fenwick trees form the main contribution, with a side application to conformal prediction.

read the letter

This paper introduces new algorithms for learning distributional regression trees that optimize the CRPS or WIS loss directly. The efficiency comes from integrating min-max heaps, weight-balanced binary trees, and Fenwick trees to handle the order statistics and aggregates needed for split evaluation. That combination is the actual novelty, as most prior tree methods either rely on parametric forms or settle for approximations when dealing with full distributional outputs. The work also notes that the resulting trees lend themselves to conformal prediction setups that can deliver group-conditional coverage, which follows naturally from the structure rather than requiring extra machinery. Experiments are described as showing competitive performance against alternatives while preserving the interpretability of trees. These points are the parts that stand out as useful. The data-structure choices align with standard techniques for dynamic prefix sums and order maintenance, so the efficiency claim does not look obviously overstated on the surface. The focus on proper scoring rules that promote calibration is consistent with the stated goal of trustworthy uncertainty estimates. Soft spots are limited. The abstract gives little detail on the exact complexity bounds or on whether the data structures introduce any hidden constants that erode the gains in practice, and the conformal-prediction byproduct would be stronger with a short proof sketch or concrete coverage numbers. No load-bearing gaps in reasoning appear from the description, and there is no sign of circularity. The paper is aimed at people who already use tree models for probabilistic forecasting and want both calibration and some degree of explainability. A reader working on non-parametric methods in applied settings would find the algorithmic details worth examining. It deserves peer review because the core construction is specific enough to be checked and the efficiency angle is testable.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces novel algorithms for learning probabilistic regression trees optimized exactly for the weighted interval score (WIS) or continuous ranked probability score (CRPS). Computational efficiency during split evaluation is obtained by integrating min-max heaps, weight-balanced binary trees, and Fenwick trees for dynamic order statistics and prefix aggregates. Numerical experiments are reported to show competitive performance against alternative approaches; the tree structure is additionally shown to support conformal prediction with group-conditional coverage guarantees.

Significance. If the exactness and complexity claims hold, the work supplies an interpretable non-parametric method for distributional regression that directly optimizes proper scoring rules, thereby improving calibration and coverage relative to mean-focused trees. The explicit use of standard data structures for order-statistic queries is a positive feature that aligns with the requirements of split-score computation.

minor comments (3)

[§3] §3 (algorithm description): the integration of the three data structures is described at a high level; an explicit walk-through of one split-score update (including the sequence of heap/BST/Fenwick operations) would make the claimed O(log n) per candidate split verifiable.
[Table 2] Table 2 (or equivalent results table): the reported runtimes should include a column for the naïve O(n log n) baseline per split so that the practical speed-up factor is directly visible.
[§5.2] §5.2 (conformal prediction): the argument for group-conditional coverage relies on the tree partitioning; a short proof sketch or reference to the relevant property of the learned partitions would strengthen the claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central contribution consists of new algorithmic constructions for exact optimization of WIS/CRPS losses inside regression trees, achieved by integrating standard external data structures (min-max heaps, weight-balanced BSTs, Fenwick trees) for dynamic order statistics and prefix aggregates. These structures are invoked as known primitives whose properties are independent of the present work; the efficiency claims and the conformal-prediction by-product follow directly from the tree structure and the loss definitions without any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard properties of known data structures and loss functions from prior literature in machine learning and data structures; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

standard math Min-max heaps, weight-balanced binary trees, and Fenwick trees support the required update and query operations in logarithmic time as established in computer science literature.
Efficiency claims depend on these established data structure properties.

pith-pipeline@v0.9.0 · 5800 in / 1217 out tokens · 26648 ms · 2026-05-23T03:14:53.371517+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages · 1 internal anchor

[1]

Variational inference for nonparametric B ayesian quantile regression

Sachinthaka Abeywardana and Fabio Ramos. Variational inference for nonparametric B ayesian quantile regression. Proceedings of the AAAI Conference on Artificial Intelligence , 29(1), Feb. 2015

work page 2015
[2]

Breiman, J

L. Breiman, J. Friedman, C.J. Stone, and R.A. Olshen. Classification and Regression Trees . Taylor & Francis, 1984

work page 1984
[3]

Brehmer and Tilmann Gneiting

Jonas R. Brehmer and Tilmann Gneiting. Scoring interval forecasts: Equal-tailed, shortest, and modal interval. Bernoulli , 27(3), May 2021

work page 2021
[4]

Towards scalable quantile regression trees

Harish S Bhat, Nitesh Kumar, and Garnet J Vaz. Towards scalable quantile regression trees. In 2015 IEEE International Conference on Big Data (Big Data) , pages 53--60. IEEE, 2015

work page 2015
[5]

Classification and regression trees

Leo Breiman. Classification and regression trees . Routledge, 2017

work page 2017
[6]

Ray, Tilmann Gneiting, and Nicholas G

Johannes Bracher, Evan L. Ray, Tilmann Gneiting, and Nicholas G. Reich. Evaluating epidemic forecasts in an interval format. PLOS Computational Biology , 17(2):1--15, 02 2021

work page 2021
[7]

Fenchel- Y oung losses with skewed entropies for class-posterior probability estimation

Han Bao and Masashi Sugiyama. Fenchel- Y oung losses with skewed entropies for class-posterior probability estimation. In International Conference on Artificial Intelligence and Statistics , pages 1648--1656. PMLR, 2021

work page 2021
[8]

Alex J. Cannon. Quantile regression neural networks: Implementation in r and application to precipitation downscaling. Computers & Geosciences , 37(9):1277--1284, 2011

work page 2011
[9]

ACCRUE: A ccurate and reliable uncertainty estimate in deterministic models

Enrico Camporeale and Algo Car \`e . ACCRUE: A ccurate and reliable uncertainty estimate in deterministic models. International Journal for Uncertainty Quantification , 11(4), 2021

work page 2021
[10]

Distributional conformal prediction

Victor Chernozhukov, Kaspar W \"u thrich, and Yinchu Zhu. Distributional conformal prediction. Proceedings of the National Academy of Sciences , 118(48):e2107794118, 2021

work page 2021
[11]

M. H. DeGroot. Uncertainty, Information, and Sequential Experiments . The Annals of Mathematical Statistics , 33(2):404 -- 419, 1962

work page 1962
[12]

Bayesian density regression

David B Dunson, Natesh Pillai, and Ju-Hyun Park. Bayesian density regression. Journal of the Royal Statistical Society Series B: Statistical Methodology , 69(2):163--183, 2007

work page 2007
[13]

Robert F Engle and Simone Manganelli. Caviar. Journal of Business & Economic Statistics , 22(4):367--381, 2004

work page 2004
[14]

Gr \"u nwald and A

Peter D. Gr \"u nwald and A. Philip Dawid. Game theory, maximum entropy, minimum discrepancy and robust B ayesian decision theory . The Annals of Statistics , 32(4):1367 -- 1433, 2004

work page 2004
[15]

Nested conformal prediction and quantile out-of-bag ensemble methods

Chirag Gupta, Arun K Kuchibhotla, and Aaditya Ramdas. Nested conformal prediction and quantile out-of-bag ensemble methods. Pattern Recognition , 127:108496, 2022

work page 2022
[16]

Fitting finite mixtures of generalized linear regressions in R

Bettina Gr \"u n and Friedrich Leisch. Fitting finite mixtures of generalized linear regressions in R . Computational Statistics & Data Analysis , 51(11):5247--5252, 2007

work page 2007
[17]

Strictly proper scoring rules, prediction, and estimation

Tilmann Gneiting and Adrian E Raftery. Strictly proper scoring rules, prediction, and estimation. Journal of the American statistical Association , 102(477):359--378, 2007

work page 2007
[18]

Halim and F

S. Halim and F. Halim. Competitive P rogramming 3: T he New Lower Bound of Programming Contests . Number vol. 3. Lulu.com, 2013

work page 2013
[19]

Denoising diffusion probabilistic models

Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. Advances in neural information processing systems , 33:6840--6851, 2020

work page 2020
[20]

Hamilton and W Viscusi

James T. Hamilton and W Viscusi. Are risk regulators rational? evidence from hazardous waste cleanup decisions. American Economic Review , 89(4):1010--1027, 1999

work page 1999
[21]

Evaluating probabilistic forecasts with scoringRules

Alexander Jordan, Fabian Kr \"u ger, and Sebastian Lerch. Evaluating probabilistic forecasts with scoringRules . arXiv preprint arXiv:1709.04743 , 2017

work page internal anchor Pith review Pith/arXiv arXiv 2017
[22]

A multiple quantile regression approach to the wind, solar, and price tracks of GEFCom2014

Romain Juban, Henrik Ohlsson, Mehdi Maasoumy, Louis Poirier, and J Zico Kolter. A multiple quantile regression approach to the wind, solar, and price tracks of GEFCom2014 . International Journal of Forecasting , 32(3):1094--1102, 2016

work page 2016
[23]

Regression quantiles

Roger Koenker and Gilbert Bassett Jr. Regression quantiles. Econometrica: journal of the Econometric Society , pages 33--50, 1978

work page 1978
[24]

Normalizing flows: A n introduction and review of current methods

Ivan Kobyzev, Simon JD Prince, and Marcus A Brubaker. Normalizing flows: A n introduction and review of current methods. IEEE transactions on pattern analysis and machine intelligence , 43(11):3964--3979, 2020

work page 2020
[25]

Quantiles based personalized treatment selection for multivariate outcomes and multiple treatments

Karunarathna B Kulasekera and Chathura Siriwardhana. Quantiles based personalized treatment selection for multivariate outcomes and multiple treatments. Statistics in medicine , 41(15):2695--2710, 2022

work page 2022
[26]

Rage against the mean – a review of distributional regression approaches

Thomas Kneib, Alexander Silbersdorff, and Benjamin Säfken. Rage against the mean – a review of distributional regression approaches. Econometrics and Statistics , 26:99--123, 2023

work page 2023
[27]

Stepwise multiple quantile regression estimation using non-crossing constraints

Yufeng Liu and Yichao Wu. Stepwise multiple quantile regression estimation using non-crossing constraints. Statistics and its Interface , 2(3):299--310, 2009

work page 2009
[28]

Quantile regression forests

Nicolai Meinshausen and Greg Ridgeway. Quantile regression forests. Journal of machine learning research , 7(6), 2006

work page 2006
[29]

Handbook of data structures and applications

Dinesh P Mehta and Sartaj Sahni. Handbook of data structures and applications . Chapman and Hall/CRC, 2004

work page 2004
[30]

Generalized maximum entropy for supervised classification

Santiago Mazuelas, Yuan Shen, and Aritz Pérez. Generalized maximum entropy for supervised classification. IEEE Transactions on Information Theory , 68(4):2530--2550, 2022

work page 2022
[31]

Nievergelt and E

J. Nievergelt and E. M. Reingold. Binary search trees of bounded balance. SIAM Journal on Computing , 2(1):33--43, 1973

work page 1973
[32]

High-resolution image synthesis with latent diffusion models

Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Bj \"o rn Ommer. High-resolution image synthesis with latent diffusion models. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages 10684--10695, 2022

work page 2022
[33]

Handbook of quantile regression

Quantile Regression. Handbook of quantile regression . CRC Press: Boca Raton, FL, USA, 2017

work page 2017
[34]

Conformalized quantile regression

Yaniv Romano, Evan Patterson, and Emmanuel Candes. Conformalized quantile regression. Advances in neural information processing systems , 32, 2019

work page 2019
[35]

R. A. Rigby and D. M. Stasinopoulos. Generalized Additive Models for Location, Scale and Shape . Journal of the Royal Statistical Society Series C: Applied Statistics , 54(3):507--554, 04 2005

work page 2005
[36]

A comparison of some conformal quantile regression methods

Matteo Sesia and Emmanuel J Cand \`e s. A comparison of some conformal quantile regression methods. Stat , 9(1):e261, 2020

work page 2020
[37]

Engression: Extrapolation through the lens of distributional regression, 2024

Xinwei Shen and Nicolai Meinshausen. Engression: Extrapolation through the lens of distributional regression, 2024

work page 2024
[38]

Bayesian nonparametric quantile regression using splines

Paul Thompson, Yuzhi Cai, Rana Moyeed, Dominic Reeve, and Julian Stander. Bayesian nonparametric quantile regression using splines. Computational Statistics & Data Analysis , 54(4):1138--1150, 2010

work page 2010
[39]

Le, Timothy D

Ichiro Takeuchi, Quoc V. Le, Timothy D. Sears, and Alexander J. Smola. Nonparametric quantile estimation. Journal of Machine Learning Research , 7:1231 – 1264, 2006. Cited by: 319

work page 2006
[40]

A review on quantile regression for stochastic computer experiments

L \'e onard Torossian, Victor Picheny, Robert Faivre, and Aur \'e lien Garivier. A review on quantile regression for stochastic computer experiments. Reliability Engineering & System Safety , 201:106858, 2020

work page 2020
[41]

Continuity of generalized entropy

Aolin Xu. Continuity of generalized entropy. In 2020 IEEE International Symposium on Information Theory (ISIT) , pages 2246--2251, 2020

work page 2020
[42]

Regularized simultaneous model selection in multiple quantiles regression

Hui Zou and Ming Yuan. Regularized simultaneous model selection in multiple quantiles regression. Computational Statistics & Data Analysis , 52(12):5296--5304, 2008

work page 2008

[1] [1]

Variational inference for nonparametric B ayesian quantile regression

Sachinthaka Abeywardana and Fabio Ramos. Variational inference for nonparametric B ayesian quantile regression. Proceedings of the AAAI Conference on Artificial Intelligence , 29(1), Feb. 2015

work page 2015

[2] [2]

Breiman, J

L. Breiman, J. Friedman, C.J. Stone, and R.A. Olshen. Classification and Regression Trees . Taylor & Francis, 1984

work page 1984

[3] [3]

Brehmer and Tilmann Gneiting

Jonas R. Brehmer and Tilmann Gneiting. Scoring interval forecasts: Equal-tailed, shortest, and modal interval. Bernoulli , 27(3), May 2021

work page 2021

[4] [4]

Towards scalable quantile regression trees

Harish S Bhat, Nitesh Kumar, and Garnet J Vaz. Towards scalable quantile regression trees. In 2015 IEEE International Conference on Big Data (Big Data) , pages 53--60. IEEE, 2015

work page 2015

[5] [5]

Classification and regression trees

Leo Breiman. Classification and regression trees . Routledge, 2017

work page 2017

[6] [6]

Ray, Tilmann Gneiting, and Nicholas G

Johannes Bracher, Evan L. Ray, Tilmann Gneiting, and Nicholas G. Reich. Evaluating epidemic forecasts in an interval format. PLOS Computational Biology , 17(2):1--15, 02 2021

work page 2021

[7] [7]

Fenchel- Y oung losses with skewed entropies for class-posterior probability estimation

Han Bao and Masashi Sugiyama. Fenchel- Y oung losses with skewed entropies for class-posterior probability estimation. In International Conference on Artificial Intelligence and Statistics , pages 1648--1656. PMLR, 2021

work page 2021

[8] [8]

Alex J. Cannon. Quantile regression neural networks: Implementation in r and application to precipitation downscaling. Computers & Geosciences , 37(9):1277--1284, 2011

work page 2011

[9] [9]

ACCRUE: A ccurate and reliable uncertainty estimate in deterministic models

Enrico Camporeale and Algo Car \`e . ACCRUE: A ccurate and reliable uncertainty estimate in deterministic models. International Journal for Uncertainty Quantification , 11(4), 2021

work page 2021

[10] [10]

Distributional conformal prediction

Victor Chernozhukov, Kaspar W \"u thrich, and Yinchu Zhu. Distributional conformal prediction. Proceedings of the National Academy of Sciences , 118(48):e2107794118, 2021

work page 2021

[11] [11]

M. H. DeGroot. Uncertainty, Information, and Sequential Experiments . The Annals of Mathematical Statistics , 33(2):404 -- 419, 1962

work page 1962

[12] [12]

Bayesian density regression

David B Dunson, Natesh Pillai, and Ju-Hyun Park. Bayesian density regression. Journal of the Royal Statistical Society Series B: Statistical Methodology , 69(2):163--183, 2007

work page 2007

[13] [13]

Robert F Engle and Simone Manganelli. Caviar. Journal of Business & Economic Statistics , 22(4):367--381, 2004

work page 2004

[14] [14]

Gr \"u nwald and A

Peter D. Gr \"u nwald and A. Philip Dawid. Game theory, maximum entropy, minimum discrepancy and robust B ayesian decision theory . The Annals of Statistics , 32(4):1367 -- 1433, 2004

work page 2004

[15] [15]

Nested conformal prediction and quantile out-of-bag ensemble methods

Chirag Gupta, Arun K Kuchibhotla, and Aaditya Ramdas. Nested conformal prediction and quantile out-of-bag ensemble methods. Pattern Recognition , 127:108496, 2022

work page 2022

[16] [16]

Fitting finite mixtures of generalized linear regressions in R

Bettina Gr \"u n and Friedrich Leisch. Fitting finite mixtures of generalized linear regressions in R . Computational Statistics & Data Analysis , 51(11):5247--5252, 2007

work page 2007

[17] [17]

Strictly proper scoring rules, prediction, and estimation

Tilmann Gneiting and Adrian E Raftery. Strictly proper scoring rules, prediction, and estimation. Journal of the American statistical Association , 102(477):359--378, 2007

work page 2007

[18] [18]

Halim and F

S. Halim and F. Halim. Competitive P rogramming 3: T he New Lower Bound of Programming Contests . Number vol. 3. Lulu.com, 2013

work page 2013

[19] [19]

Denoising diffusion probabilistic models

Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. Advances in neural information processing systems , 33:6840--6851, 2020

work page 2020

[20] [20]

Hamilton and W Viscusi

James T. Hamilton and W Viscusi. Are risk regulators rational? evidence from hazardous waste cleanup decisions. American Economic Review , 89(4):1010--1027, 1999

work page 1999

[21] [21]

Evaluating probabilistic forecasts with scoringRules

Alexander Jordan, Fabian Kr \"u ger, and Sebastian Lerch. Evaluating probabilistic forecasts with scoringRules . arXiv preprint arXiv:1709.04743 , 2017

work page internal anchor Pith review Pith/arXiv arXiv 2017

[22] [22]

A multiple quantile regression approach to the wind, solar, and price tracks of GEFCom2014

Romain Juban, Henrik Ohlsson, Mehdi Maasoumy, Louis Poirier, and J Zico Kolter. A multiple quantile regression approach to the wind, solar, and price tracks of GEFCom2014 . International Journal of Forecasting , 32(3):1094--1102, 2016

work page 2016

[23] [23]

Regression quantiles

Roger Koenker and Gilbert Bassett Jr. Regression quantiles. Econometrica: journal of the Econometric Society , pages 33--50, 1978

work page 1978

[24] [24]

Normalizing flows: A n introduction and review of current methods

Ivan Kobyzev, Simon JD Prince, and Marcus A Brubaker. Normalizing flows: A n introduction and review of current methods. IEEE transactions on pattern analysis and machine intelligence , 43(11):3964--3979, 2020

work page 2020

[25] [25]

Quantiles based personalized treatment selection for multivariate outcomes and multiple treatments

Karunarathna B Kulasekera and Chathura Siriwardhana. Quantiles based personalized treatment selection for multivariate outcomes and multiple treatments. Statistics in medicine , 41(15):2695--2710, 2022

work page 2022

[26] [26]

Rage against the mean – a review of distributional regression approaches

Thomas Kneib, Alexander Silbersdorff, and Benjamin Säfken. Rage against the mean – a review of distributional regression approaches. Econometrics and Statistics , 26:99--123, 2023

work page 2023

[27] [27]

Stepwise multiple quantile regression estimation using non-crossing constraints

Yufeng Liu and Yichao Wu. Stepwise multiple quantile regression estimation using non-crossing constraints. Statistics and its Interface , 2(3):299--310, 2009

work page 2009

[28] [28]

Quantile regression forests

Nicolai Meinshausen and Greg Ridgeway. Quantile regression forests. Journal of machine learning research , 7(6), 2006

work page 2006

[29] [29]

Handbook of data structures and applications

Dinesh P Mehta and Sartaj Sahni. Handbook of data structures and applications . Chapman and Hall/CRC, 2004

work page 2004

[30] [30]

Generalized maximum entropy for supervised classification

Santiago Mazuelas, Yuan Shen, and Aritz Pérez. Generalized maximum entropy for supervised classification. IEEE Transactions on Information Theory , 68(4):2530--2550, 2022

work page 2022

[31] [31]

Nievergelt and E

J. Nievergelt and E. M. Reingold. Binary search trees of bounded balance. SIAM Journal on Computing , 2(1):33--43, 1973

work page 1973

[32] [32]

High-resolution image synthesis with latent diffusion models

Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Bj \"o rn Ommer. High-resolution image synthesis with latent diffusion models. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages 10684--10695, 2022

work page 2022

[33] [33]

Handbook of quantile regression

Quantile Regression. Handbook of quantile regression . CRC Press: Boca Raton, FL, USA, 2017

work page 2017

[34] [34]

Conformalized quantile regression

Yaniv Romano, Evan Patterson, and Emmanuel Candes. Conformalized quantile regression. Advances in neural information processing systems , 32, 2019

work page 2019

[35] [35]

R. A. Rigby and D. M. Stasinopoulos. Generalized Additive Models for Location, Scale and Shape . Journal of the Royal Statistical Society Series C: Applied Statistics , 54(3):507--554, 04 2005

work page 2005

[36] [36]

A comparison of some conformal quantile regression methods

Matteo Sesia and Emmanuel J Cand \`e s. A comparison of some conformal quantile regression methods. Stat , 9(1):e261, 2020

work page 2020

[37] [37]

Engression: Extrapolation through the lens of distributional regression, 2024

Xinwei Shen and Nicolai Meinshausen. Engression: Extrapolation through the lens of distributional regression, 2024

work page 2024

[38] [38]

Bayesian nonparametric quantile regression using splines

Paul Thompson, Yuzhi Cai, Rana Moyeed, Dominic Reeve, and Julian Stander. Bayesian nonparametric quantile regression using splines. Computational Statistics & Data Analysis , 54(4):1138--1150, 2010

work page 2010

[39] [39]

Le, Timothy D

Ichiro Takeuchi, Quoc V. Le, Timothy D. Sears, and Alexander J. Smola. Nonparametric quantile estimation. Journal of Machine Learning Research , 7:1231 – 1264, 2006. Cited by: 319

work page 2006

[40] [40]

A review on quantile regression for stochastic computer experiments

L \'e onard Torossian, Victor Picheny, Robert Faivre, and Aur \'e lien Garivier. A review on quantile regression for stochastic computer experiments. Reliability Engineering & System Safety , 201:106858, 2020

work page 2020

[41] [41]

Continuity of generalized entropy

Aolin Xu. Continuity of generalized entropy. In 2020 IEEE International Symposium on Information Theory (ISIT) , pages 2246--2251, 2020

work page 2020

[42] [42]

Regularized simultaneous model selection in multiple quantiles regression

Hui Zou and Ming Yuan. Regularized simultaneous model selection in multiple quantiles regression. Computational Statistics & Data Analysis , 52(12):5296--5304, 2008

work page 2008