Pinball Loss Function Research Articles

On Normality Assumption in Residual Simulation for Probabilistic Load Forecasting

Grid modernization has brought in various types of active demand, and intermittent and distributed generation resources to challenge the traditional power system planning and operation practices. As a result, more and more decision making processes rely on probabilistic forecasts as an input. While residual simulation has been recognized as one way to generate probabilistic load forecasts, the research on the application side of probabilistic load forecasting has been heavily relying on unverified distributions of load forecasting residuals, such as normal distribution. In this paper, we study the normality assumption from a different angle. Instead of trying to prove or disprove its validity via hypothesis tests, we attempt to understand whether applying the normality assumption helps improve the quality of probabilistic load forecasts. We apply a proper scoring rule, the pinball loss function, to evaluate a set of probabilistic load forecasts developed from different underlying linear and nonlinear models. To ensure the solidity of our conclusion, we conduct two case studies, one based on data from a large generation and transmission cooperative in the U.S., and the other based on data from the probabilistic load forecasting track of the Global Energy Forecasting Competition 2014.

Asymmetric ν-twin support vector regression

Twin support vector regression (TSVR) aims at finding 𝜖-insensitive up- and down-bound functions for the training points by solving a pair of smaller-sized quadratic programming problems (QPPs) rather than a single large one as in the conventional SVR. So TSVR works faster than SVR in theory. However, TSVR gives equal emphasis to the points above the up-bound and below the down-bound, which leads to the same influences on the regression function. In fact, points in different positions have different effects on the regressor. Inspired by it, we propose an asymmetric ν-twin support vector regression based on pinball loss function (Asy- ν-TSVR). The new algorithm can effectively control the fitting error by tuning the parameters ν and p. Therefore, it enhances the generalization ability. Moreover, we study the distribution of samples and give the upper bounds for the samples locating in different positions. Numerical experiments on one artificial dataset, eleven benchmark datasets and a real wheat dataset demonstrate the validity of our proposed algorithm.

Statistical consistency of coefficient-based conditional quantile regression

This study focuses on the coefficient-based conditional quantile regression associated with l q -regularization term, where 1 ≤ q ≤ 2 . Error analysis is investigated based on the capacity of the hypothesis space. The linear piecewise nature of the pinball loss function for quantile regression and the l q -penalty of the learning algorithm lead to some difficulties in the theoretical analysis. In order to overcome the difficulties, we introduce a novel error decomposition formula. The prolix iteration is then circumvented in the error analysis. Some satisfactory learning rates are achieved in a general setting under mild conditions.

A maximum margin and minimum volume hyper-spheres machine with pinball loss for imbalanced data classification

The twin hyper-sphere support vector machine (THSVM) classifies two classes of samples via two hyper-spheres instead of a pair of nonparallel hyper-planes as in the conversional twin support vector machine (TSVM). Moreover THSVM avoids the matrix inverse operation when solving two dual quadratic programming problems (QPPs). However it cannot yield a desirable result when dealing with the imbalanced data classification. To improve the generalization performance, we propose a maximum margin and minimum volume hyper-spheres machine with pinball loss (Pin-M3HM) for the imbalanced data classification in this paper. The basic idea is to construct two hyper-spheres with different centers and radiuses in a sequential order. The first one contains as many examples in majority class as possible, and the second one covers minority class of examples as possible. Moreover the margin between two hyper-spheres is as large as possible. Besides, the pinball loss function is introduced into it to avoid the noise disturbance. Experimental results on 24 imbalanced datasets from the repositories of UCI and KEEL, and a real spectral dataset of Chinese grape wines indicate that our proposed Pin-M3HM yields a good generalization performance for the imbalanced data classification.

Probabilistic Load Forecasting via Quantile Regression Averaging on Sister Forecasts

The majority of the load forecasting literature has been on point forecasting, which provides the expected value for each step throughout the forecast horizon. In the smart grid era, the electricity demand is more active and less predictable than ever before. As a result, probabilistic load forecasting, which provides additional information on the variability and uncertainty of future load values, is becoming of great importance to power systems planning and operations. This paper proposes a practical methodology to generate probabilistic load forecasts by performing quantile regression averaging on a set of sister point forecasts. There are two major benefits of the proposed approach. It can leverage the development in the point load forecasting literature over the past several decades and it does not rely so much on high-quality expert forecasts, which are rarely achievable in load forecasting practice. To demonstrate the effectiveness of the proposed approach and make the results reproducible to the load forecasting community, we construct a case study using the publicly available data from the Global Energy Forecasting Competition 2014. Compared with several benchmark methods, the proposed approach leads to dominantly better performance as measured by the pinball loss function and the Winkler score.

Consistency of support vector machines using additive kernels for additive models

Support vector machines (SVMs) are special kernel based methods and have been among the most successful learning methods for more than a decade. SVMs can informally be described as kinds of regularized M-estimators for functions and have demonstrated their usefulness in many complicated real-life problems. During the last few years a great part of the statistical research on SVMs has concentrated on the question of how to design SVMs such that they are universally consistent and statistically robust for nonparametric classification or nonparametric regression purposes. In many applications, some qualitative prior knowledge of the distribution P or of the unknown function f to be estimated is present or a prediction function with good interpretability is desired, such that a semiparametric model or an additive model is of interest. The question of how to design SVMs by choosing the reproducing kernel Hilbert space (RKHS) or its corresponding kernel to obtain consistent and statistically robust estimators in additive models is addressed. An explicit construction of such RKHSs and their kernels, which will be called additive kernels, is given. SVMs based on additive kernels will be called additive support vector machines. The use of such additive kernels leads, in combination with a Lipschitz continuous loss function, to SVMs with the desired properties for additive models. Examples include quantile regression based on the pinball loss function, regression based on the ϵ-insensitive loss function, and classification based on the hinge loss function.

Sequential Quantile Prediction of Time Series

Motivated by a broad range of potential applications, we address the quantile prediction problem of real-valued time series. We present a sequential quantile forecasting model based on the combination of a set of elementary nearest neighbor-type predictors called “experts” and show its consistency under a minimum of conditions. Our approach builds on the methodology developed in recent years for prediction of individual sequences and exploits the quantile structure as a minimizer of the so-called pinball loss function. We perform an in-depth analysis of real-world data sets and show that this nonparametric strategy generally outperforms standard quantile prediction methods.