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 signals. Moreover we also show that if the true vector of quantiles is a discrete spline with a few polynomial pieces then both versions attain a near parametric rate of convergence. Corresponding results for the usual trend filtering estimators are known to hold only when the errors are sub-Gaussian. In contrast, our risk bounds are shown to hold under minimal assumptions on the error variables. In particular, no moment assumptions are needed and our results hold under heavy-tailed errors. Our proof techniques are general and thus can potentially be used to study other nonparametric quantile regression methods. To illustrate this generality we also employ our proof techniques to obtain new results for multivariate quantile total variation denoising and high dimensional quantile linear regression.
This paper considers an estimation of semiparametric functional (varying)-coefficient quantile regression with spatial data. A general robust framework is developed that treats quantile regression for spatial data in a natural semiparametric way. The local M-estimators of the unknown functional-coefficient functions are proposed by using local linear approximation, and their asymptotic distributions are then established under weak spatial mixing conditions allowing the data processes to be either stationary or nonstationary with spatial trends. Application to a soil data set is demonstrated wi…
Based on the novel concept of multivariate center-outward quantiles introduced recently in Chernozhukov et al. (2017) and Hallin et al. (2021), we are considering the problem of nonparametric multiple-output quantile regression. Our approach defines nested conditional center-outward quantile regression contours and regions with given conditional probability content irrespective of the underlying distribution; their graphs constitute nested center-outward quantile regression tubes. Empirical counterparts of these concepts are constructed, yielding interpretable empirical regions and contours wh…
We study trend filtering, a recently proposed tool of Kim et al. [SIAM Rev. 51 (2009) 339-360] for nonparametric regression. The trend filtering estimate is defined as the minimizer of a penalized least squares criterion, in which the penalty term sums the absolute $k$th order discrete derivatives over the input points. Perhaps not surprisingly, trend filtering estimates appear to have the structure of $k$th degree spline functions, with adaptively chosen knot points (we say ``appear'' here as trend filtering estimates are not really functions over continuous domains, and are only defined over…
Non-parametric and distribution-free two-sample tests have been the foundation of many change point detection algorithms. However, randomness in the test statistic as a function of time makes them susceptible to false positives and localization ambiguity. We address these issues by deriving and applying filters matched to the expected temporal signatures of a change for various sliding window, two-sample tests under IID assumptions on the data. These filters are derived asymptotically with respect to the window size for the Wasserstein quantile test, the Wasserstein-1 distance test, Maximum Me…
Cardiometabolic diseases have substantially increased in China in the past 20 years and blood pressure is a primary modifiable risk factor. Using data from the China Health and Nutrition Survey, we examine blood pressure trends in China from 1991 to 2009, with a concentration on age cohorts and urbanicity. Very large values of blood pressure are of interest, so we model the conditional quantile functions of systolic and diastolic blood pressure. This allows the covariate effects in the middle of the distribution to vary from those in the upper tail, the focal point of our analysis. We join the…
We present a simple quantile regression-based forecasting method that was applied in a probabilistic load forecasting framework of the Global Energy Forecasting Competition 2017 (GEFCom2017). The hourly load data is log transformed and split into a long-term trend component and a remainder term. The key forecasting element is the quantile regression approach for the remainder term that takes into account weekly and annual seasonalities such as their interactions. Temperature information is only used to stabilize the forecast of the long-term trend component. Public holidays information is igno…
Ground level Ozone is one of the six common air-pollutants on which the EPA has set national air quality standards. In order to capture the spatio-temporal trend of 1-hour and 8-hour average ozone concentration in the US, we develop a method for spatio-temporal simultaneous quantile regression. Unlike existing procedures, in the proposed method, smoothing across the sites is incorporated within modeling assumptions thus allowing borrowing of information across locations, an essential step when the number of samples in each location is low. The quantile function has been assumed to be linear in…
Unconditional quantile regression (UQR) attracts attention in various fields to investigate the impacts of explanatory variables on quantiles of the marginal distribution of an explained variable. This study attempts to introduce spatial dependence into the UQR within the framework of random effects eigenvector spatial filtering, resulting in the model that we term the spatially filtered UQR (SF-UQR). We then develop a computationally efficient approach for SF-UQR estimation. Finally, the performance of the SF-UQR is tested with a hedonic land price model for the Tokyo metropolitan area. SF-UQ…
We focus on the problem estimating a monotone trend function under additive and dependent noise. New point-wise confidence interval estimators under both short- and long-range dependent errors are introduced and studied. These intervals are obtained via the method of inversion of certain discrepancy statistics arising in hypothesis testing problems. The advantage of this approach is that it avoids the estimation of nuisance parameters such as the derivative of the unknown function, which existing methods are forced to deal with. While the methodology is motivated by earlier work in the indepen…
We study additive models built with trend filtering, i.e., additive models whose components are each regularized by the (discrete) total variation of their $k$th (discrete) derivative, for a chosen integer $k \geq 0$. This results in $k$th degree piecewise polynomial components, (e.g., $k=0$ gives piecewise constant components, $k=1$ gives piecewise linear, $k=2$ gives piecewise quadratic, etc.). Analogous to its advantages in the univariate case, additive trend filtering has favorable theoretical and computational properties, thanks in large part to the localized nature of the (discrete) tota…
Climate change is commonly associated with an overall increase in mean temperature in a defined past time period. Many studies consider temperature trends at the global scale, but the literature is lacking in in-depth analysis of the temperature trends across Australia in recent decades. In addition to heterogeneity in mean and median values, daily Australia temperature data suffers from quasi-periodic heterogeneity in variance. However, this issue has barely been overlooked in climate research. A contribution of this article is that we propose a joint model of quantile regression and variabil…
We propose a novel approach for density estimation called histogram trend filtering. Our estimator arises from looking at surrogate Poisson model for counts of observations in a partition of the support of the data. We begin by showing consistency for a variational estimator for this density estimation problem. We then study a discrete estimator that can be efficiently found via convex optimization. We show that the estimator enjoys strong statistical guarantees, yet is much more practical and computationally efficient than other estimators that enjoy similar guarantees. Finally, in our simula…
While many approaches have been proposed for discovering abrupt changes in piecewise constant signals, few methods are available to capture these changes in piecewise polynomial signals. In this paper, we propose a change point detection method, PRUTF, based on trend filtering. By providing a comprehensive dual solution path for trend filtering, PRUTF allows us to discover change points of the underlying signal for either a given value of the regularization parameter or a specific number of steps of the algorithm. We demonstrate that the dual solution path constitutes a Gaussian bridge process…
There are many research works and methods about change point detection in the literature. However, there are only a few that provide inference for such change points after being estimated. This work mainly focuses on a statistical analysis of change points estimated by the PRUTF algorithm, which incorporates trend filtering to determine change points in piecewise polynomial signals. This paper develops a methodology to perform statistical inference, such as computing p-values and constructing confidence intervals in the newly developed post-selection inference framework. Our work concerns both…
We develop an air mobility index and use the newly developed Apple's driving trend index to evaluate the impact of COVID-19 on the crude oil price. We use quantile regression and stationary and non-stationary extreme value models to study the impact. We find that both the \textit{air mobility index} and \textit{driving trend index} significantly influence lower and upper quantiles as well as the median of the WTI crude oil price. The extreme value model suggests that an event like COVID-19 may push oil prices to a negative territory again as the air mobility decreases drastically during such p…
Several studies have focused on the Realized Range Volatility, an estimator of the quadratic variation of financial prices, taking into account the impact of microstructure noise and jumps. However, none has considered direct modeling and forecasting of the Realized Range conditional quantiles. This study carries out a quantile regression analysis to fill this gap. The proposed model takes into account as quantile predictors both the lagged values of the estimated volatility and some key macroeconomic and financial variables, which provide important information about the overall market trend a…
The statistical theory of extremes is extended to observations that are non-stationary and not independent. The non-stationarity over time and space is controlled via the scedasis (tail scale) in the marginal distributions. Spatial dependence stems from multivariate extreme value theory. We establish asymptotic theory for both the weighted sequential tail empirical process and the weighted tail quantile process based on all observations, taken over time and space. The results yield two statistical tests for homoscedasticity in the tail, one in space and one in time. Further, we show that the c…
University of Patras（帕特雷大学）；Roma Tre University（罗马第三大学）；National Technical University of Athens（雅典国立技术大学）
Machine and statistical learning algorithms can be reliably automated and applied at scale. Therefore, they can constitute a considerable asset for designing practical forecasting systems, such as those related to urban water demand. Quantile regression algorithms are statistical and machine learning algorithms that can provide probabilistic forecasts in a straightforward way, and have not been applied so far for urban water demand forecasting. In this work, we aim to fill this gap by automating and extensively comparing several quantile-regression-based practical systems for probabilistic one…
The paper considers nonparametric specification tests of quantile curves for a general class of nonstationary processes. Using Bahadur representation and Gaussian approximation results for nonstationary time series, simultaneous confidence bands and integrated squared difference tests are proposed to test various parametric forms of the quantile curves with asymptotically correct type I error rates. A wild bootstrap procedure is implemented to alleviate the problem of slow convergence of the asymptotic results. In particular, our results can be used to test the trends of extremes of climate va…
We study trend filtering, a relatively recent method for univariate nonparametric regression. For a given positive integer $r$, the $r$-th order trend filtering estimator is defined as the minimizer of the sum of squared errors when we constrain (or penalize) the sum of the absolute $r$-th order discrete derivatives of the fitted function at the design points. For $r=1$, the estimator reduces to total variation regularization which has received much attention in the statistics and image processing literature. In this paper, we study the performance of the trend filtering estimator for every po…
The vast majority of the neural network literature focuses on predicting point values for a given set of response variables, conditioned on a feature vector. In many cases we need to model the full joint conditional distribution over the response variables rather than simply making point predictions. In this paper, we present two novel approaches to such conditional density estimation (CDE): Multiscale Nets (MSNs) and CDE Trend Filtering. Multiscale nets transform the CDE regression task into a hierarchical classification task by decomposing the density into a series of half-spaces and learnin…
We extend the notion of trend filtering to tensors by considering the $k^{\rm th}$-order Vitali variation, a discretized version of the integral of the absolute value of the $k^{\rm th}$-order total derivative. We prove adaptive $\ell^0$-rates and not-so-slow $\ell^1$-rates for tensor denoising with trend filtering. For $k=\{1,2,3,4\}$ we prove that the $d$-dimensional margin of a $d$-dimensional tensor can be estimated at the $\ell^0$-rate $n^{-1}$, up to logarithmic terms, if the underlying tensor is a product of $(k-1)^{\rm th}$-order polynomials on a constant number of hyperrectangles. For…
This paper reviews a class of univariate piecewise polynomial functions known as discrete splines, which share properties analogous to the better-known class of spline functions, but where continuity in derivatives is replaced by (a suitable notion of) continuity in divided differences. As it happens, discrete splines bear connections to a wide array of developments in applied mathematics and statistics, from divided differences and Newton interpolation (dating back to over 300 years ago) to trend filtering (from the last 15 years). We survey these connections, and contribute some new perspect…
We propose a novel approach to sentiment data filtering for a portfolio of assets. In our framework, a dynamic factor model drives the evolution of the observed sentiment and allows to identify two distinct components: a long-term component, modeled as a random walk, and a short-term component driven by a stationary VAR(1) process. Our model encompasses alternative approaches available in literature and can be readily estimated by means of Kalman filtering and expectation maximization. This feature makes it convenient when the cross-sectional dimension of the portfolio increases. By applying t…
