Positive data are very common in many scientific fields and applications; for these data, it is known that estimation and inference based on relative error criterion are superior to that of absolute error criterion. In prediction problems, conformal prediction provides a useful framework to construct flexible prediction intervals based on hypothesis testing, which has been actively studied in the past decade. In view of the advantages of the relative error criterion for regression problems with positive responses, in this paper, we combine the relative error criterion (REC) with conformal prediction to develop a novel REC-based predictive inference method to construct prediction intervals for the positive response. The proposed method satisfies the finite sample global coverage guarantee and to some extent achieves the local validity. We conduct extensive simulation studies and two real data analysis to demonstrate the competitiveness of the new proposed method.
In this paper, the structure of finite groups in which maximal subgroups of some Sylow subgroups have a
Assessing the influence of individual observations of the functional linear models is important and challenging, especially when the observations are subject to missingness. In this paper, we introduce three case-deletion diagnostic measures to identify influential observations in functional linear models when the covariate is functional and observations on the scalar response are subject to nonignorable missingness. The nonignorable missing data mechanism is modeled via an exponential tilting semiparametric functional model. A semiparametric imputation procedure is developed to mitigate the effects of missing data. Valid estimations of the functional coefficients are based on functional principal components analysis using the imputed dataset. A smoothed bootstrap sampling method is introduced to estimate the diagnostic probability for each proposed diagnostic measure, which is helpful to unveil which observations have the larger influence on estimation and prediction. Simulation studies and a real data example are conducted to illustrate the finite performance of the proposed methods.
The method of moving surfaces is an effective tool to implicitize rational parametric surfaces, and it has been extensively studied in the past two decades. An essential step in surface implicitization using the method of moving surfaces is to compute a
This paper proposes a novel method to compute the diffeomorphic registration of 3D surfaces with point and curve feature landmarks. First the surfaces are mapped to the canonical domain by a curve constrained harmonic map, where the landmark curves are straightened to line segments and their positions and inclining angles are determined intrinsically by the surface geometry and its curve landmarks. Then, the canonical domains are registered by aligning the corresponding point and straight line segments using the dynamic quasiconformal map (DQCM), which introduces the combinatorial diagonal switches to the quasiconformal optimization such that the resultant map is diffeomorphic. The end points of the source curve landmarks are mapped to their corresponding points on the target surface, while the interior points of the source curves can slide on the corresponding target curves, which provides more freedom for the surface registration than the point-based registration methods. Experiments on the real surfaces with point and curve landmarks demonstrate the efficiency, efficacy and robustness of the proposed method.
In this work, we consider numerical approximations of the phase-field model of diblock copolymer melt confined in Hele–Shaw cell, which is a very complicated coupled nonlinear system consisting of the Darcy equations and the Cahn–Hilliard type equations with the Ohta–Kawaski potential. Through the combination of a novel explicit-Invariant Energy Quadratization approach and the projection method, we develop the first full decoupling, energy stable, and second-order time-accurate numerical scheme. The introduction of two auxiliary variables and the design of two auxiliary ODEs play a vital role in obtaining the full decoupling structure while maintaining energy stability. The scheme is also linear and unconditional energy stable, and the practical implementation efficiency is also very high because it only needs to solve a few elliptic equations with constant coefficients at each time step. We strictly prove that the scheme satisfies the unconditional energy stability and give a detailed implementation process. Numerical experiments further verify the convergence rate, energy stability, and effectiveness of the developed algorithm.
In this paper, we define signed zero-divisor graphs over commutative rings and investigate the interplay between the algebraic properties of the rings and the combinatorial properties of their corresponding signed zero-divisor graphs. We investigate the structure of signed zero-divisor graphs, the relation between ideals and signed zero-divisor graphs, and the adjacency matrices and the spectra of signed zero-divisor graphs.
This research concerns the estimation of latent linear or polychoric correlations from fuzzy frequency tables. Fuzzy counts are of particular interest to many disciplines including social and behavioral sciences and are especially relevant when observed data are classified using fuzzy categories—as for socioeconomic studies, clinical evaluations, content analysis, inter-rater reliability analysis—or when imprecise observations are classified into either precise or imprecise categories—as for the analysis of ratings data or fuzzy-coded variables. In these cases, the space of count matrices is no longer defined over naturals and, consequently, the polychoric estimator cannot be used to accurately estimate latent linear correlations. The aim of this contribution is twofold. First, we illustrate a computational procedure based on generalized natural numbers for computing fuzzy frequencies. Second, we reformulate the problem of estimating latent linear correlations from fuzzy counts in the context of expectation–maximization-based maximum likelihood estimation. A simulation study and two applications are used to investigate the characteristics of the proposed method. Overall, the results show that the fuzzy EM-based polychoric estimator is more efficient to deal with imprecise count data as opposed to standard polychoric estimators that may be used in this context.
In this paper, the functional central limit theorem is established for martingale like random vectors under the framework sub-linear expectations introduced by Shige Peng. As applications, the Lindeberg central limit theorem for independent random vectors is established, the sufficient and necessary conditions of the central limit theorem for independent and identically distributed random vectors are found, and a Lévy’s characterization of a multi-dimensional G-Brownian motion is obtained.
Katok’s entropy formula is an important formula in entropy theory. It plays significant roles in large deviation theories, multifractal analysis, quantitative recurrence and so on. This paper is devoted to establishing Katok’s entropy formula of unstable metric entropy which is the entropy caused by the unstable part of partially hyperbolic systems. We also construct a similar formula which can be used to study the quantitative recurrence in the unstable manifold for partially hyperbolic diffeomorphisms.
Although deep learning-based approximation algorithms have been applied very successfully to numerous problems, at the moment the reasons for their performance are not entirely understood from a mathematical point of view. Recently, estimates for the convergence of the overall error have been obtained in the situation of deep supervised learning, but with an extremely slow rate of convergence. In this note, we partially improve on these estimates. More specifically, we show that the depth of the neural network only needs to increase much slower in order to obtain the same rate of approximation. The results hold in the case of an arbitrary stochastic optimization algorithm with i.i.d. random initializations.
This paper studies variable selection using the penalized likelihood method for distributed sparse regression with large sample size n under a limited memory constraint. This is a much needed research problem to be solved in the big data era. A naive divide-and-conquer method solving this problem is to split the whole data into N parts and run each part on one of N machines, aggregate the results from all machines via averaging, and finally obtain the selected variables. However, it tends to select more noise variables, and the false discovery rate may not be well controlled. We improve it by a special designed weighted average in aggregation. Although the alternating direction method of multiplier can be used to deal with massive data in the literature, our proposed method reduces the computational burden a lot and performs better by mean square error in most cases. Theoretically, we establish asymptotic properties of the resulting estimators for the likelihood models with a diverging number of parameters. Under some regularity conditions, we establish oracle properties in the sense that our distributed estimator shares the same asymptotic efficiency as the estimator based on the full sample. Computationally, a distributed penalized likelihood algorithm is proposed to refine the results in the context of general likelihoods. Furthermore, the proposed method is evaluated by simulations and a real example.
The Harnack inequality for stochastic differential equation driven by G-Brownian motion with multiplicative noise is derived by means of the coupling by change of measure, which extends the corresponding results derived in Wang (Probab. Theory Related Fields 109:417–424) under the linear expectation. Moreover, we generalize the gradient estimate under nonlinear expectation appeared in Song (Sci. China Math. 64:1093–1108).
In this paper, we consider relativization of measure-theoretical- restricted sensitivity. For a given topological dynamical system, we define conditional measure-theoretical-restricted asymptotic rate with respect to sensitivity and obtain that it equals to the reciprocal of the Brin–Katok local entropy for almost every point under the conditional measure.
This paper considers the problem of numerically evaluating discrete barrier option prices when the underlying asset follows the jump-diffusion model with stochastic volatility and stochastic intensity. We derive the three-dimensional characteristic function of the log-asset price, the volatility and the jump intensity. We also provide the approximate formula of the discrete barrier option prices by the three-dimensional Fourier cosine series expansion (3D-COS) method. Numerical results show that the 3D-COS method is rather correct, fast and competent for pricing the discrete barrier options.