Consider the nonparametric regression model Y=go（T）+u,where Y is real-valued, u is a random error,T ranges over a nondegenerate compact interval,say[0,1],and go（·）is an unknown regression function,which is m...Consider the nonparametric regression model Y=go（T）+u,where Y is real-valued, u is a random error,T ranges over a nondegenerate compact interval,say[0,1],and go（·）is an unknown regression function,which is m（m≥0）times continuously differentiable and its ruth derivative,g<sub>0</sub><sup>（m）</sup>,satisfies a H■lder condition of order γ（m +γ】1/2）.A piecewise polynomial L<sub>1</sub>- norm estimator of go is proposed.Under some regularity conditions including that the random errors are independent but not necessarily have a common distribution,it is proved that the rates of convergence of the piecewise polynomial L<sub>1</sub>-norm estimator are o(n<sup>-2（m+γ）+1/m+γ-1/δ</sup>almost surely and o(n<sup>-2（m+γ）+1/m+γ-δ</sup>)in probability,which can arbitrarily approach the optimal rates of convergence for nonparametric regression,where δ is any number in （0, min（（m+γ-1/2）/3,γ））.展开更多
Probabilistic load forecasting(PLF)is able to present the uncertainty information of the future loads.It is the basis of stochastic power system planning and operation.Recent works on PLF mainly focus on how to develo...Probabilistic load forecasting(PLF)is able to present the uncertainty information of the future loads.It is the basis of stochastic power system planning and operation.Recent works on PLF mainly focus on how to develop and combine forecasting models,while the feature selection issue has not been thoroughly investigated for PLF.This paper fills the gap by proposing a feature selection method for PLF via sparse L1-norm penalized quantile regression.It can be viewed as an extension from point forecasting-based feature selection to probabilistic forecasting-based feature selection.Since both the number of training samples and the number of features to be selected are very large,the feature selection process is casted as a large-scale convex optimization problem.The alternating direction method of multipliers is applied to solve the problem in an efficient manner.We conduct case studies on the open datasets of ten areas.Numerical results show that the proposed feature selection method can improve the performance of the probabilistic forecasting and outperforms traditional least absolute shrinkage and selection operator method.展开更多
On one hand,to find the sparsest solution to the system of linear equations has been a major focus since it has a large number of applications in many areas;and on the other hand,the system of absolute value equations...On one hand,to find the sparsest solution to the system of linear equations has been a major focus since it has a large number of applications in many areas;and on the other hand,the system of absolute value equations(AVEs)has attracted a lot of attention since many practical problems can be equivalently transformed as a system of AVEs.Motivated by the development of these two aspects,we consider the problem to find the sparsest solution to the system of AVEs in this paper.We first propose the model of the concerned problem,i.e.,to find the solution to the system of AVEs with the minimum l0-norm.Since l0-norm is difficult to handle,we relax the problem into a convex optimization problem and discuss the necessary and sufficient conditions to guarantee the existence of the unique solution to the convex relaxation problem.Then,we prove that under such conditions the unique solution to the convex relaxation is exactly the sparsest solution to the system of AVEs.When the concerned system of AVEs reduces to the system of linear equations,the obtained results reduce to those given in the literature.The theoretical results obtained in this paper provide an important basis for designing numerical method to find the sparsest solution to the system of AVEs.展开更多
基金Supported by the National Natural Science Foundation of China.
文摘Consider the nonparametric regression model Y=go（T）+u,where Y is real-valued, u is a random error,T ranges over a nondegenerate compact interval,say[0,1],and go（·）is an unknown regression function,which is m（m≥0）times continuously differentiable and its ruth derivative,g<sub>0</sub><sup>（m）</sup>,satisfies a H■lder condition of order γ（m +γ】1/2）.A piecewise polynomial L<sub>1</sub>- norm estimator of go is proposed.Under some regularity conditions including that the random errors are independent but not necessarily have a common distribution,it is proved that the rates of convergence of the piecewise polynomial L<sub>1</sub>-norm estimator are o(n<sup>-2（m+γ）+1/m+γ-1/δ</sup>almost surely and o(n<sup>-2（m+γ）+1/m+γ-δ</sup>)in probability,which can arbitrarily approach the optimal rates of convergence for nonparametric regression,where δ is any number in （0, min（（m+γ-1/2）/3,γ））.
基金supported by National Key R&D Program of China(No.2016YFB0900100).
文摘Probabilistic load forecasting(PLF)is able to present the uncertainty information of the future loads.It is the basis of stochastic power system planning and operation.Recent works on PLF mainly focus on how to develop and combine forecasting models,while the feature selection issue has not been thoroughly investigated for PLF.This paper fills the gap by proposing a feature selection method for PLF via sparse L1-norm penalized quantile regression.It can be viewed as an extension from point forecasting-based feature selection to probabilistic forecasting-based feature selection.Since both the number of training samples and the number of features to be selected are very large,the feature selection process is casted as a large-scale convex optimization problem.The alternating direction method of multipliers is applied to solve the problem in an efficient manner.We conduct case studies on the open datasets of ten areas.Numerical results show that the proposed feature selection method can improve the performance of the probabilistic forecasting and outperforms traditional least absolute shrinkage and selection operator method.
基金This work was supported in part by the National Natural Science Foundation of China(Nos.11171252,11201332 and 11431002).
文摘On one hand,to find the sparsest solution to the system of linear equations has been a major focus since it has a large number of applications in many areas;and on the other hand,the system of absolute value equations(AVEs)has attracted a lot of attention since many practical problems can be equivalently transformed as a system of AVEs.Motivated by the development of these two aspects,we consider the problem to find the sparsest solution to the system of AVEs in this paper.We first propose the model of the concerned problem,i.e.,to find the solution to the system of AVEs with the minimum l0-norm.Since l0-norm is difficult to handle,we relax the problem into a convex optimization problem and discuss the necessary and sufficient conditions to guarantee the existence of the unique solution to the convex relaxation problem.Then,we prove that under such conditions the unique solution to the convex relaxation is exactly the sparsest solution to the system of AVEs.When the concerned system of AVEs reduces to the system of linear equations,the obtained results reduce to those given in the literature.The theoretical results obtained in this paper provide an important basis for designing numerical method to find the sparsest solution to the system of AVEs.
