Quantile Regression (QR) provides a way to approximate a single conditional quantile. To have a more informative description of the conditional distribution, QR can be merged with deep learning techniques to simultaneously estimate multiple quantiles. However, the minimisation of the QR-loss function does not guarantee non-crossing quantiles, which affects the validity of such predictions and introduces a critical issue in certain scenarios. In this article, we propose a generic deep learning algorithm for predicting an arbitrary number of quantiles that ensures the quantile monotonicity constraint up to the machine precision and maintains its modelling performance with respect to alternative models. The presented method is evaluated over several real-world datasets obtaining state-of-the-art results as well as showing that it scales to large-size data sets.
University of British Columbia（不列颠哥伦比亚大学）；Carleton University（卡尔顿大学）
This paper proposes a new method to address the long-standing problem of lack of monotonicity in estimation of the conditional and structural quantile function, also known as quantile crossing problem. Quantile regression is a very powerful tool in data science in general and econometrics in particular. Unfortunately, the crossing problem has been confounding researchers and practitioners alike for over 4 decades. Numerous attempts have been made to find a simple and general solution. This paper describes a unique and elegant solution to the problem based on a flexible check function that is e…
The Hong Kong Polytechnic University（香港理工大学）；The Chinese University of Hong Kong（香港中文大学）
We propose a nonparametric quantile regression method using deep neural networks with a rectified linear unit penalty function to avoid quantile crossing. This penalty function is computationally feasible for enforcing non-crossing constraints in multi-dimensional nonparametric quantile regression. We establish non-asymptotic upper bounds for the excess risk of the proposed nonparametric quantile regression function estimators. Our error bounds achieve optimal minimax rate of convergence for the Holder class, and the prefactors of the error bounds depend polynomially on the dimension of the pr…
The Hong Kong Polytechnic University（香港理工大学）；Wuhan University（武汉大学）；The Chinese University of Hong Kong（香港中文大学）
We propose a penalized nonparametric approach to estimating the quantile regression process (QRP) in a nonseparable model using rectifier quadratic unit (ReQU) activated deep neural networks and introduce a novel penalty function to enforce non-crossing of quantile regression curves. We establish the non-asymptotic excess risk bounds for the estimated QRP and derive the mean integrated squared error for the estimated QRP under mild smoothness and regularity conditions. To establish these non-asymptotic risk and estimation error bounds, we also develop a new error bound for approximating $C^s$ …
Deep Distributional Learning with Non-crossing Quantile Network
The Hong Kong Polytechnic University（香港理工大学）；University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校）；ByteDance（字节跳动）；London School of Economics and Political Science（伦敦政治经济学院）
Melbourne Business School, the University of Melbourne（墨尔本商学院，墨尔本大学）；School of Political Science and Economics, Waseda University（早稻田大学政治经济学院）；Booth School of Business, the University of Chicago（芝加哥大学布斯商学院）
This paper proposes a method to address the longstanding problem of lack of monotonicity in estimation of conditional and structural quantile functions, also known as the quantile crossing problem. The method consists in sorting or monotone rearranging the original estimated non-monotone curve into a monotone rearranged curve. We show that the rearranged curve is closer to the true quantile curve in finite samples than the original curve, establish a functional delta method for rearrangement-related operators, and derive functional limit theory for the entire rearranged curve and its functiona…
Distributional reinforcement learning (DRL) extends the value-based approach by approximating the full distribution over future returns instead of the mean only, providing a richer signal that leads to improved performances. Quantile Regression (QR) based methods like QR-DQN project arbitrary distributions into a parametric subset of staircase distributions by minimizing the 1-Wasserstein distance. However, due to biases in the gradients, the quantile regression loss is used instead for training, guaranteeing the same minimizer and enjoying unbiased gradients. Non-crossing constraints on the q…
We investigate different methods for regularizing quantile regression when predicting either a subset of quantiles or the full inverse CDF. We show that minimizing an expected pinball loss over a continuous distribution of quantiles is a good regularizer even when only predicting a specific quantile. For predicting multiple quantiles, we propose achieving the classic goal of non-crossing quantiles by using deep lattice networks that treat the quantile as a monotonic input feature, and we discuss why monotonicity on other features is an apt regularizer for quantile regression. We show that latt…
Quantile regression relates the quantile of the response to a linear predictor. For a discrete response distributions, like the Poission, Binomial and the negative Binomial, this approach is not feasible as the quantile function is not bijective. We argue to use a continuous model-aware interpolation of the quantile function, allowing for proper quantile inference while retaining model interpretation. This approach allows for proper uncertainty quantification and mitigates the issue of quantile crossing. Our reanalysis of hospitalisation data considered in Congdon (2017) shows the advantages o…
Technical University of Denmark (DTU)（丹麦技术大学）；Massachusetts Institute of Technology (MIT)（麻省理工学院）
Spatio-temporal problems are ubiquitous and of vital importance in many research fields. Despite the potential already demonstrated by deep learning methods in modeling spatio-temporal data, typical approaches tend to focus solely on conditional expectations of the output variables being modeled. In this paper, we propose a multi-output multi-quantile deep learning approach for jointly modeling several conditional quantiles together with the conditional expectation as a way to provide a more complete "picture" of the predictive density in spatio-temporal problems. Using two large-scale dataset…
Quantile regression has received increased attention in the statistics community in recent years. This article adapts an auxiliary variable method, commonly used in Bayesian variable selection for mean regression models, to the fitting of quantile regression curves. We focus on the fitting of regression splines, with unknown number and location of knots. We provide an efficient algorithm with Metropolis-Hastings updates whose tuning is fully automated. The method is tested on simulated and real examples and its extension to additive models is described. Finally we propose a simple postprocessi…
We consider the problem of nonparametric quantile regression for twice censored data. Two new estimates are presented, which are constructed by applying concepts of monotone rearrangements to estimates of the conditional distribution function. The proposed methods avoid the problem of crossing quantile curves. Weak uniform consistency and weak convergence is established for both estimates and their finite sample properties are investigated by means of a simulation study. As a by-product, we obtain a new result regarding the weak convergence of the Beran estimator for right censored data on the…
Quantile regression (QR) is a principal regression method for analyzing the impact of covariates on outcomes. The impact is described by the conditional quantile function and its functionals. In this paper we develop the nonparametric QR-series framework, covering many regressors as a special case, for performing inference on the entire conditional quantile function and its linear functionals. In this framework, we approximate the entire conditional quantile function by a linear combination of series terms with quantile-specific coefficients and estimate the function-valued coefficients from t…
Estimating the structures at high or low quantiles has become an important subject and attracted increasing attention across numerous fields. However, due to data sparsity at tails, it usually is a challenging task to obtain reliable estimation, especially for high-dimensional data. This paper suggests a flexible parametric structure to tails, and this enables us to conduct the estimation at quantile levels with rich observations and then to extrapolate the fitted structures to far tails. The proposed model depends on some quantile indices and hence is called the quantile index regression. Mor…
Quantile regression, that is the prediction of conditional quantiles, has steadily gained importance in statistical modeling and financial applications. The authors introduce a new semiparametric quantile regression method based on sequentially fitting a likelihood optimal D-vine copula to given data resulting in highly flexible models with easily extractable conditional quantiles. As a subclass of regular vine copulas, D-vines enable the modeling of multivariate copulas in terms of bivariate building blocks, a so-called pair-copula construction (PCC). The proposed algorithm works fast and acc…
The R package quantreg.nonpar implements nonparametric quantile regression methods to estimate and make inference on partially linear quantile models. quantreg.nonpar obtains point estimates of the conditional quantile function and its derivatives based on series approximations to the nonparametric part of the model. It also provides pointwise and uniform confidence intervals over a region of covariate values and/or quantile indices for the same functions using analytical and resampling methods. This paper serves as an introduction to the package and displays basic functionality of the functio…
Quantile regression, the prediction of conditional quantiles, finds applications in various fields. Often, some or all of the variables are discrete. The authors propose two new quantile regression approaches to handle such mixed discrete-continuous data. Both of them generalize the continuous D-vine quantile regression, where the dependence between the response and the covariates is modeled by a parametric D-vine. D-vine quantile regression provides very flexible models, that enable accurate and fast predictions. Moreover, it automatically takes care of major issues of classical quantile regr…
A collection of quantile curves provides a complete picture of conditional distributions. Properly centered and scaled versions of estimated curves at various quantile levels give rise to the so-called quantile regression process (QRP). In this paper, we establish weak convergence of QRP in a general series approximation framework, which includes linear models with increasing dimension, nonparametric models and partial linear models. An interesting consequence is obtained in the last class of models, where parametric and non-parametric estimators are shown to be asymptotically independent. App…
We develop a general approach to estimating the derivative of a function-valued parameter $θ_o(u)$ that is identified for every value of $u$ as the solution to a moment condition. This setup in particular covers many interesting models for conditional distributions, such as quantile regression or distribution regression. Exploiting that $θ_o(u)$ solves a moment condition, we obtain an explicit expression for its derivative from the Implicit Function Theorem, and estimate the components of this expression by suitable sample analogues, which requires the use of (local linear) smoothing. Our esti…
Quantile regression is a field with steadily growing importance in statistical modeling. It is a complementary method to linear regression, since computing a range of conditional quantile functions provides a more accurate modelling of the stochastic relationship among variables, especially in the tails. We introduce a non-restrictive and highly flexible nonparametric quantile regression approach based on C- and D-vine copulas. Vine copulas allow for separate modeling of marginal distributions and the dependence structure in the data, and can be expressed through a graph theoretical model give…
Measuring interdependence between probabilities of default (PDs) in different industry sectors of an economy plays a crucial role in financial stress testing. Thereby, regression approaches may be employed to model the impact of stressed industry sectors as covariates on other response sectors. We identify vine copula based quantile regression as an eligible tool for conducting such stress tests as this method has good robustness properties, takes into account potential nonlinearities of conditional quantile functions and ensures that no quantile crossing effects occur. We illustrate its perfo…
Quantile regression is an effective technique to quantify uncertainty, fit challenging underlying distributions, and often provide full probabilistic predictions through joint learnings over multiple quantile levels. A common drawback of these joint quantile regressions, however, is \textit{quantile crossing}, which violates the desirable monotone property of the conditional quantile function. In this work, we propose the Incremental (Spline) Quantile Functions I(S)QF, a flexible and efficient distribution-free quantile estimation framework that resolves quantile crossing with a simple neural …
In this paper, we develop uniform inference methods for the conditional mode based on quantile regression. Specifically, we propose to estimate the conditional mode by minimizing the derivative of the estimated conditional quantile function defined by smoothing the linear quantile regression estimator, and develop two bootstrap methods, a novel pivotal bootstrap and the nonparametric bootstrap, for our conditional mode estimator. Building on high-dimensional Gaussian approximation techniques, we establish the validity of simultaneous confidence rectangles constructed from the two bootstrap met…
We study quantile trend filtering, a recently proposed method for nonparametric quantile regression with the goal of generalizing existing risk bounds known for the usual trend filtering estimators which perform mean regression. We study both the penalized and the constrained version (of order $r \geq 1$) of univariate quantile trend filtering. Our results show that both the constrained and the penalized version (of order $r \geq 1$) attain the minimax rate up to log factors, when the $(r-1)$th discrete derivative of the true vector of quantiles belongs to the class of bounded variation signal…
We derive the joint asymptotic distribution of empirical quantiles and expected shortfalls under general conditions on the distribution of the underlying observations. In particular, we do not assume that the distribution function is differentiable at the quantile with strictly positive derivative. Hence the rate of convergence and the asymptotic distribution for the quantile can be non-standard, but our results show that the expected shortfall remains asymptotically normal with a $\sqrt{n}$-rate, and we even give the joint distribution in such non-standard cases. In the derivation we use the …
This paper proposes a new approach to estimating the distribution of a response variable conditioned on observing some factors. The proposed approach possesses desirable properties of flexibility, interpretability, tractability and extendability. The conditional quantile function is modeled by a mixture (weighted sum) of basis quantile functions, with the weights depending on factors. The calibration problem is formulated as a convex optimization problem. It can be viewed as conducting quantile regressions for all confidence levels simultaneously while avoiding quantile crossing by definition.…
Beyond Straightness: Non-Crossing Flow Matching via Quantile AlignTree Coupling
Institute of Statistics and Big Data, Renmin University of China（中国人民大学统计与大数据研究院）；Department of Statistics and Data Science, Tsinghua University（清华大学统计与数据科学系）
The chain rule for derivatives of a function of a function is extended to a function of a statistical functional, and applied to obtain approximations to the cumulants, distribution and quantiles of functions of sample moments, and so to obtain third order confidence intervals and estimates of reduced bias for functions of moments. As an example we give the distribution of the standardized skewness for a normal sample to magnitude $O(n^{-2})$, where $n$ is the sample size.
The Chinese University of Hong Kong（香港中文大学）；Tencent AI Lab（腾讯人工智能实验室）；City University of Hong Kong（香港城市大学）
We propose a parsimonious quantile regression framework to learn the dynamic tail behaviors of financial asset returns. Our model captures well both the time-varying characteristic and the asymmetrical heavy-tail property of financial time series. It combines the merits of a popular sequential neural network model, i.e., LSTM, with a novel parametric quantile function that we construct to represent the conditional distribution of asset returns. Our model also captures individually the serial dependences of higher moments, rather than just the volatility. Across a wide range of asset classes, t…
Shared mobility services require accurate demand models for effective service planning. On the one hand, modeling the full probability distribution of demand is advantageous because the entire uncertainty structure preserves valuable information for decision-making. On the other hand, demand is often observed through the usage of the service itself, so that the observations are censored, as they are inherently limited by available supply. Since the 1980s, various works on Censored Quantile Regression models have performed well under such conditions. Further, in the last two decades, several pa…
Harvard University（哈佛大学）；Massachusetts Institute of Technology（麻省理工学院）
Molecular photoswitches are the foundation of light-activated drugs. A key photoswitch is azobenzene, which exhibits trans-cis isomerism in response to light. The thermal half-life of the cis isomer is of crucial importance, since it controls the duration of the light-induced biological effect. Here we introduce a computational tool for predicting the thermal half-lives of azobenzene derivatives. Our automated approach uses a fast and accurate machine learning potential trained on quantum chemistry data. Building on well-established earlier evidence, we argue that thermal isomerization proceed…
Shanghai Jiao Tong University（上海交通大学）；Technical University of Denmark（丹麦技术大学）
We present a data-driven approach for probabilistic wind power forecasting based on conditional normalizing flow (CNF). In contrast with the existing, this approach is distribution-free (as for non-parametric and quantile-based approaches) and can directly yield continuous probability densities, hence avoiding quantile crossing. It relies on a base distribution and a set of bijective mappings. Both the shape parameters of the base distribution and the bijective mappings are approximated with neural networks. Spline-based conditional normalizing flow is considered owing to its non-affine charac…
