The modified Cholesky decomposition(MCD)is an efficient technique for estimating a covariance matrix.However,it is known that the MCD technique often requires a pre-specified variable ordering in the estimation proced...The modified Cholesky decomposition(MCD)is an efficient technique for estimating a covariance matrix.However,it is known that the MCD technique often requires a pre-specified variable ordering in the estimation procedure.In this work,we propose a weighted average ensemble covariance estimation for high-dimensional data based on the MCD technique.It canflexibly accommodatethehigh-dimensional case and ensure the positive definiteness property of the resultant estimate.Our key idea is to obtain different weights for different candidate estimates by minimizing an appropriate risk function with respect to the Frobenius norm.Different from the existing ensemble estimation based on the MCD,the proposed method provides a sparse weighting scheme such that one can distinguish which variable orderings employed in the MCD are useful for the ensemble matrix estimate.The asymptotically theoretical convergence rate of the proposed ensemble estimate is established under regularity conditions.The merits of the proposed method are examined by the simulation studies and a portfolio allocationexampleofrealstockdata.展开更多
Estimation of large covariance matrices is of great importance in multivariate analysis.The modified Cholesky decomposition is a commonly used technique in covariance matrix estimation given a specific order of variab...Estimation of large covariance matrices is of great importance in multivariate analysis.The modified Cholesky decomposition is a commonly used technique in covariance matrix estimation given a specific order of variables.However,information on the order of variables is often unknown,or cannot be reasonably assumed in practice.In this work,we propose a Choleskybased model averaging approach of covariance matrix estimation for high dimensional datawith proper regularisation imposed on the Cholesky factor matrix.The proposed method not only guarantees the positive definiteness of the covariance matrix estimate,but also is applicable in general situations without the order of variables being pre-specified.Numerical simulations are conducted to evaluate the performance of the proposed method in comparison with several other covariance matrix estimates.The advantage of our proposed method is further illustrated by a real case study of equity portfolio allocation.展开更多
A good visualisation method can greatly enhance human-machine collaboration in target contexts.To aid the optimal selection of visualisations for users,visualisation recommender systems have been developed to provide ...A good visualisation method can greatly enhance human-machine collaboration in target contexts.To aid the optimal selection of visualisations for users,visualisation recommender systems have been developed to provide the right visualisation method to the right person given specific contexts.A visualisation recommender system often relies on a user study to collect data and conduct analysis to provide personalised recommendations.However,a user study without employing an effective experimental design is typically expensive in terms of time and cost.In this work,we propose a prediction-oriented optimal design to determine the user-task allocation in the user study for the recommendation of visualisation methods.The proposed optimal design will not only encourage the learning of the similarity embedded in the recommendation responses(i.e.,users’preference),but also improve the modelling accuracy of the similarities captured by the covariates of contexts(i.e.,task attributes).A simulation study and a real-data case study are used to evaluate the proposed optimal design.展开更多
Smoothing spline is a popular method in non-parametric function estimation.For the analysis of data from real applications,specific shapes on the estimated function are often required to ensure the estimated function ...Smoothing spline is a popular method in non-parametric function estimation.For the analysis of data from real applications,specific shapes on the estimated function are often required to ensure the estimated function undeviating from the domain knowledge.In this work,we focus on constructing the exact shape constrained smoothing spline with efficient estimation.The‘exact’here is referred as to impose the shape constraint on an infinite set such as an interval in one-dimensional case.Thus the estimation becomes a so-called semi-infinite optimisation problem with an infinite number of constraints.The proposed method is able to establish a sufficient and necessary condition for transforming the exact shape constraints to a finite number of constraints,leading to efficient estimation of the shape constrained functions.The performance of the proposed methods is evaluated by both simulation and real case studies.展开更多
基金supported by National Natural Science Foundation of China[grant numbers 72232001 and 72371059]supported by National Natural Science Foundation of China[grant number 12001365].
文摘The modified Cholesky decomposition(MCD)is an efficient technique for estimating a covariance matrix.However,it is known that the MCD technique often requires a pre-specified variable ordering in the estimation procedure.In this work,we propose a weighted average ensemble covariance estimation for high-dimensional data based on the MCD technique.It canflexibly accommodatethehigh-dimensional case and ensure the positive definiteness property of the resultant estimate.Our key idea is to obtain different weights for different candidate estimates by minimizing an appropriate risk function with respect to the Frobenius norm.Different from the existing ensemble estimation based on the MCD,the proposed method provides a sparse weighting scheme such that one can distinguish which variable orderings employed in the MCD are useful for the ensemble matrix estimate.The asymptotically theoretical convergence rate of the proposed ensemble estimate is established under regularity conditions.The merits of the proposed method are examined by the simulation studies and a portfolio allocationexampleofrealstockdata.
基金National Science of Foundation of China[grant number NSFC-71531004]NNSF.
文摘Estimation of large covariance matrices is of great importance in multivariate analysis.The modified Cholesky decomposition is a commonly used technique in covariance matrix estimation given a specific order of variables.However,information on the order of variables is often unknown,or cannot be reasonably assumed in practice.In this work,we propose a Choleskybased model averaging approach of covariance matrix estimation for high dimensional datawith proper regularisation imposed on the Cholesky factor matrix.The proposed method not only guarantees the positive definiteness of the covariance matrix estimate,but also is applicable in general situations without the order of variables being pre-specified.Numerical simulations are conducted to evaluate the performance of the proposed method in comparison with several other covariance matrix estimates.The advantage of our proposed method is further illustrated by a real case study of equity portfolio allocation.
文摘A good visualisation method can greatly enhance human-machine collaboration in target contexts.To aid the optimal selection of visualisations for users,visualisation recommender systems have been developed to provide the right visualisation method to the right person given specific contexts.A visualisation recommender system often relies on a user study to collect data and conduct analysis to provide personalised recommendations.However,a user study without employing an effective experimental design is typically expensive in terms of time and cost.In this work,we propose a prediction-oriented optimal design to determine the user-task allocation in the user study for the recommendation of visualisation methods.The proposed optimal design will not only encourage the learning of the similarity embedded in the recommendation responses(i.e.,users’preference),but also improve the modelling accuracy of the similarities captured by the covariates of contexts(i.e.,task attributes).A simulation study and a real-data case study are used to evaluate the proposed optimal design.
基金supported by National Science Foundation[1634867].
文摘Smoothing spline is a popular method in non-parametric function estimation.For the analysis of data from real applications,specific shapes on the estimated function are often required to ensure the estimated function undeviating from the domain knowledge.In this work,we focus on constructing the exact shape constrained smoothing spline with efficient estimation.The‘exact’here is referred as to impose the shape constraint on an infinite set such as an interval in one-dimensional case.Thus the estimation becomes a so-called semi-infinite optimisation problem with an infinite number of constraints.The proposed method is able to establish a sufficient and necessary condition for transforming the exact shape constraints to a finite number of constraints,leading to efficient estimation of the shape constrained functions.The performance of the proposed methods is evaluated by both simulation and real case studies.
