Kernel estimation of regression function gradient

Abstract

This paper is focused on kernel estimation of the gradient of a multivariate regression function. Despite the importance of this topic, the progress in this area is rather slow. Our aim is to construct a gradient estimator using the idea of local linear estimator for a regression function. The quality of this estimator is expressed in terms of the Mean Integrated Square Error. We focus on a choice of bandwidth matrix. Further, we present some data-driven methods for its choice and develop a new approach. The performance of presented methods is illustrated using a simulation study and real data example.     

Invexity: an updated survey and new results for nonsmooth functions

The purpose of this paper is to introduce two recent developments in optimization theory which have the potential for significant impact when applied to the many optimization problems in mathematical statistics. The needs of optimization theory have served as the catalyst for the introduction and recent development of two important concepts: invexity and generalized differentiability. The former is a significant generalization of the simple yet powerful concept of convexity, while the latter extends the differentiabe calculus to functions that are not differentiabe in any traditional two-sided sense. For the most part these concepts have been developed independently in the literature. In this paper we provide an updated review of many of the results concerning smooth invex functions and merge the two concepts by introducing invexity for nonsmooth functions. For both real-valued and vector-valued nonsmooth functions we present recent results on optimality conditions, duality, and converse duality. No previous knowledge of invexity or nonsmooth analysis is assumed; a self-contained section on generalized differentiability for Lipschitz functions is included.     

Comparison of various machine learning algorithms for estimating generalized propensity score

In this paper, we conducted a simulation study to evaluate the performance of four algorithms: multinomial logistic regression (MLR), bagging (BAG), random forest (RF), and gradient boosting (GB), for estimating generalized propensity score (GPS). Similar to the propensity score (PS), the ultimate goal of using GPS is to estimate unbiased average treatment effects (ATEs) in observational studies. We used the GPS estimates computed from these four algorithms with the generalized doubly robust (GDR) estimator to estimate ATEs in observational studies. We evaluated these ATE estimates in terms of bias and mean squared error (MSE). Simulation results show that overall, the GB algorithm produced the best ATE estimates based on these evaluation criteria. Thus, we recommend using the GB algorithm for estimating GPS in practice.     

修正共轭梯度恒模阵列算法研究

提出一种基于修正共轭梯度算法的恒模(MCG-CMA)盲干扰抑制算法,该算法将修正共轭梯度方法引入到恒模算法中,克服了传统恒模算法收敛缓慢、LS-CMA运算量大的缺点,保留了较好的计算复杂度和数值稳定性,理论推导了算法失调量的显式表达式。仿真结果表明该算法不需要波达方向估计,与传统的LS-CMA算法、SCG-CMA算法相比,具有较好的收敛性能和输出信干噪比(SINR)。     

The population dynamics of the voleClethrionomys rufocanus in Hokkaido, Japan

Population dynamics of the gray sided-vole,Clethrionomys rufocanus, in Hokkaido, Japan were described on the basis of 225 time series (being from 12 to 31 years long); 194 of the time series have a length of 23 years or longer. The time series were classified into 11 groups according to geographic proximity and topographic characteristics of the island of Hokkaido. Mean abundance varied among populations from 1.07 to 21.07 individuals per 150 trap-nights. The index of variability for population fluctuation (s-index) ranged from 0.204 to 0.629. Another index for population variability (amplitude on log-10 scale) ranged from 0.811 to 2.743. Mean abundance and variability of populations were higher in the more northern and eastern regions of the island. Most populations, except for the southernmost populations, exhibited significant direct density-dependence in population growth. Detection rate for delayed density-dependence varied among groups from 0% to 22.6%. Both direct and delayed density-dependence tended to be stronger in the more northern and eastern populations. The proportion of cyclic populations was higher in the northern-eastern areas than that in the southern-western areas. There was a clear gradient from the asynchronous populations in southwest, to the highly synchronized populations in the northeast.     

A gradient Markov chain Monte Carlo algorithm for computing multivariate maximum likelihood estimates and posterior distributions: mixture dose-response assessment

Multivariate probability distributions, such as may be used for mixture dose‐response assessment, are typically highly parameterized and difficult to fit to available data. However, such distributions may be useful in analyzing the large electronic data sets becoming available, such as dose‐response biomarker and genetic information. In this article, a new two‐stage computational approach is introduced for estimating multivariate distributions and addressing parameter uncertainty. The proposed first stage comprises a gradient Markov chain Monte Carlo (GMCMC) technique to find Bayesian posterior mode estimates (PMEs) of parameters, equivalent to maximum likelihood estimates (MLEs) in the absence of subjective information. In the second stage, these estimates are used to initialize a Markov chain Monte Carlo (MCMC) simulation, replacing the conventional burn‐in period to allow convergent simulation of the full joint Bayesian posterior distribution and the corresponding unconditional multivariate distribution (not conditional on uncertain parameter values). When the distribution of parameter uncertainty is such a Bayesian posterior, the unconditional distribution is termed predictive. The method is demonstrated by finding conditional and unconditional versions of the recently proposed emergent dose‐response function (DRF). Results are shown for the five‐parameter common‐mode and seven‐parameter dissimilar‐mode models, based on published data for eight benzene–toluene dose pairs. The common mode conditional DRF is obtained with a 21‐fold reduction in data requirement versus MCMC. Example common‐mode unconditional DRFs are then found using synthetic data, showing a 71% reduction in required data. The approach is further demonstrated for a PCB 126‐PCB 153 mixture. Applicability is analyzed and discussed. Matlab^® computer programs are provided.     

A conceptual framework for the study of human ecosystems in urban areas

The need for integrated concepts, capable of satisfying natural and social scientists and supporting integrated research, motivates a conceptual framework for understanding the role of humans in ecosystems. The question is how to add humans to the ecological models used to understand urban ecosystems. The ecosystem concept can serve as the basis, but specific social attributes of humans and their institutions must be added. Learning and feedback between the human and natural components of urban ecosystems are key attributes of the integrated model. Parallels with familiar ecological approaches can help in understanding the ecology of urban ecosystems. These include the role of spatial heterogeneity and organizational hierarchies in both the social and natural components of urban ecosystems. Although urban watersheds are commonly highly altered, the watershed approach can serve as a spatial basis for organizing comparative studies of ecosystems exhibiting differing degrees of urbanization. The watershed concept can also spatially organize the hierarchically scaled linkages by which the integrated human ecosystem model can be applied. The study of urban ecosystems is a relatively new field, and the questions suggested by the integrated framework can be used to frame ecosystem research in and associated with urban and metropolitan areas.     

Partially projected gradient algorithms for computing nonparametric maximum likelihood estimates of mixing distributions

There exist primarily three different types of algorithms for computing nonparametric maximum likelihood estimates (NPMLEs) of mixing distributions in the literature, which are the EM-type algorithms, the vertex direction algorithms such as VDM and VEM, and the algorithms based on general constrained optimization techniques such as the projected gradient method. It is known that the projected gradient algorithm may run into stagnation during iterations. When a stagnation occurs, VDM steps need to be added. We argue that the abrupt switch to VDM steps can significantly reduce the efficiency of the projected gradient algorithm, and is usually unnecessary. In this paper, we define a group of partially projected directions, which can be regarded as hybrids of ordinary projected gradient directions and VDM directions. Based on these directions, four new algorithms are proposed for computing NPMLEs of mixing distributions. The properties of the algorithms are discussed and their convergence is proved. Extensive numerical simulations show that the new algorithms outperform the existing methods, especially when a NPMLE has a large number of support points or when high accuracy is required.     

Building adaptive estimating equations when inverse of covariance estimation is difficult

Summary. To construct an optimal estimating function by weighting a set of score functions, we must either know or estimate consistently the covariance matrix for the individual scores. In problems with high dimensional correlated data the estimated covariance matrix could be unreliable. The smallest eigenvalues of the covariance matrix will be the most important for weighting the estimating equations, but in high dimensions these will be poorly determined. Generalized estimating equations introduced the idea of a working correlation to minimize such problems. However, it can be difficult to specify the working correlation model correctly. We develop an adaptive estimating equation method which requires no working correlation assumptions. This methodology relies on finding a reliable approximation to the inverse of the variance matrix in the quasi-likelihood equations. We apply a multivariate generalization of the conjugate gradient method to find estimating equations that preserve the information well at fixed low dimensions. This approach is particularly useful when the estimator of the covariance matrix is singular or close to singular, or impossible to invert owing to its large size.     

一般约束极大极小问题的广义梯度投影算法

讨论了一类带等式、不等式约束的极大极小值问题，将其转化为带等式、不等式约束的非线性规划问题，利用辅助规划进行处理，给出了一个广义的梯度投影算法，解决了一般约束极大极小值问题。算法可在有限步达到最优点或产生一系列点列，其极限点则是最优点，并证明了该算法的全局收敛性。