Minimizing the Variance of a Weighted Average

It is common practice in science to take a weighted average of estimators of a single parameter. If the original estimators are unbiased, any weighted average will be an unbiased estimator as well. The best estimator among the weighted averages can be obtained by choosing weights that minimize the variance of the weighted average. If the variances of the individual estimators are given, the ideal weights have long been known to be the inverse of the variance. Nonetheless, I have not found a formal proof of this result in the literature. In this article, I provide three different proofs of the ideal weights.

On Weighted Possibilistic Mean,Variance and Correlation of Interval-valued Fuzzy Numbers

In this paper, the concept of weighted possibilistic mean of interval- valued fuzzy number is first introduced. Further, the notions of weighted possibilistic variance, covariance and correlation of interval-valued fuzzy numbers are presented. Meantime, some important properties of them and relationships between them are studied.

Developmental Genetic Analysis of Brown Rice Weight Under Different Environmental Conditions in indica Rice

Analysis of genetic main effects and genotype x environment (GE) interaction effects for brown rice weight (BRW) at four different filling stages in indica lice (Oryza sativa L.) was conducted for two-year experimental data by using developmental genetic models and corresponding statistical approaches for quantitative traits of seeds in cereal crops. It was indicated that the genetic main effects and their GE interaction effects of triploid endosperm, cytoplasmic and diploid maternal plant genes were important for BRW at different filling stages of rice, especially for endosperm or maternal additive main effects and their additive interaction effects. Because of the higher additive effects and additive interaction effects for BRW at different filling stages, the better improving effects for this trait could be expected by selection in rice breeding. The results of conditional genetic variance components showed that the new expression of quantitative genes in endosperm and maternal plant for BRW was mostly found at all different filling stages of rice. The gene expression, however, was most active at the early filling stages especially for the first (1-7 d) and the second filling stages (8-14 d after flowering). The phenomena that some genes were spasmodically expressible among filling stages of rice were detected for some genetic effects especially for net cytoplasmic main effects or its interaction effects and net dominance main effects. Predicted genetic effects at different filling stages of rice showed that some parents such as V20 and Zuo 5 were better than others for improving the BRW.

免疫细胞与膝骨关节炎之间因果作用:一项两样本双向孟德尔随机化分析

背景:膝骨关节炎是一种常见的关节软骨及周围组织损伤的慢性炎症性疾病,而免疫细胞在膝骨关节炎免疫炎症反应中起到重要作用,但其中的具体机制仍有待深入研究。目的:采用孟德尔随机化方法来评估731种免疫细胞表型与膝骨关节炎风险之间的潜在因果关系。方法:使用全基因组关联分析(GWAS)目录中公开获取731种免疫细胞表型的全基因组关联分析统计数据(从GCST0001391到GCST0002121)和IEUGWAS数据库中膝骨关节炎的全基因组关联分析数据(ebi-a-GCST007090)。采用逆方差加权法、MR-Egger回归法、加权中位数法、加权模型法和简单模型法来研究免疫细胞与膝骨关节炎之间的因果关系。敏感性分析用于检验孟德尔随机化分析结果是否可靠,然后以同样方法进行反向孟德尔随机化分析。结果与结论:①正向分析结果表明,共有4种免疫细胞表型与膝骨关节炎有显著的因果关系(FDR<0.20),其中B细胞中的CD27 on CD24+CD27+(OR=1.026,P=0.00026,Pfdr=0.18)、髓系细胞中的CD33 on CD33dim HLA DR-(OR=1.014,P=0.00050,Pfdr=0.18)以及Treg细胞中的CD45RA+CD28-CD8br%CD8br(OR=1.001,P=0.00078,Pfdr=0.18)与膝骨关节炎风险呈直接的正向因果关联;单核细胞中PDL-1 on monocyte(OR=0.952,P=0.00098,Pfdr=0.18)与膝骨关节炎风险呈直接的负向因果关联。②反向分析结果表明,当膝骨关节炎作为暴露数据时,与731种免疫细胞表型均不具有显著因果关系(FDR<0.20)。③敏感性分析结果显示:双向孟德尔随机化的Cochran’s Q检验和MR-Egger回归法结果P值均大于0.05,表明免疫细胞表型与膝骨关节炎之间的因果效应分析不存在显著的异质性和多效性。④上述结果证实,CD27 on CD24+CD27+,CD33 on CD33dim HLA DR-,CD45RA+CD28-CD8br%CD8br以及PDL-1 on monocyte免疫细胞表型与膝骨关节炎之间可能具有较为显著的潜在因果关系,这为研究膝骨关节炎的生物学机制及探索膝骨关节炎的早期防治提供有价值的线索,也为干预性药物的开发提供了新的方向。

VW-UM模型在采场稳定性评价中的应用

为避免地下采场稳定性常权分析时容易出现"状态失衡"的问题,将变权重理论(VW)引入其中,结合未确知测度理论(UM)建立采场稳定性VW-UM评价模型。基于对采场稳定性影响最大的14项指标,计算其未确知测度(UM)函数,运用层次分析法(AHP)确定各指标常权重,再通过VW理论细化各指标权重的取值,最后根据置信度识别准则预测采场风险等级。用VW-UM模型分析新桥矿和焦家矿5个采场的稳定性,得出各采场的风险等级。实例分析表明,指标权重在常权重的基础上需根据指标值确定,这样使得分析结果的准确度比传统评价模型要高。

肠道菌群与骨质疏松症的遗传关系:来自英国数据库211个肠道微生物群分析

背景:骨质疏松症被定义为一种慢性代谢性骨病,大量的证据表明肠道菌群与骨质疏松症有关。然而,肠道菌群对骨质疏松症的因果关系尚不清楚。目的:采用双样本孟德尔随机化方法来评估肠道菌群与骨质疏松症之间的潜在因果关系。方法:使用MiBioGen联盟肠道菌群的全基因组关联分析汇总统计数据和英国生物样本数据库中骨质疏松症的全基因组关联分析数据。采用逆方差加权法(Inverse variance weighting,IVW)、MR-Egger回归法、加权中位数法、加权模型法和简单模型法来研究肠道菌群与骨质疏松症之间的因果关系。敏感性分析用于检验孟德尔随机化分析结果是否可靠。结果与结论:逆方差加权法结果表明,肠道菌群与骨质疏松症之间存在因果关系。克里斯滕森菌科R7属(MR Egger:β=-0.007;IVW:β=-0.004,P=0.028)、粪球菌3属(MR Egger:β=-0.008;IVW:β=-0.003,P=0.046)和毛螺菌属(MR Egger:β=-0.009;IVW:β=-0.004,P=0.003),可能是骨质疏松的保护性因素,而霍氏菌属(MREgger:β=0.006;IVW:β=0.002,P=0.033)和氧化真杆菌属(MREgger:β=0.001;IVW:β=0.003,P=0.046),可能为骨质疏松症潜在危险性因素。数据分析结果显示,氧化真杆菌属和霍氏菌属可增加骨质疏松风险,克里斯滕森菌科R7属、粪球菌3属和毛螺菌属可降低骨质疏松风险。此结论是否也适用于非欧洲人群,未来需要不同群体的大样本临床试验来验证。

脂肪酸种类及含量与膝骨关节炎的发病风险

背景:近年来,流行病学研究显示肥胖是膝骨关节炎的危险因素,而脂肪酸的摄入、代谢和生物合成与肥胖的发生发展密切相关,但脂肪酸与膝骨关节炎之间是否存在因果关联仍未知。目的:应用孟德尔随机化分析方法探讨5种脂肪酸表型与膝骨关节炎的因果关联。方法:使用来自英国生物银行(met-D)的脂肪酸比例的全基因组关联研究数据和来自EBI-A数据库的膝骨关节炎的全基因组关联研究数据进行汇总,将单核苷酸多态性(single nucleotide polymorphism,SNP)作为工具变量并选择敏感的SNPs进行分析,通过双样本孟德尔随机化分析,以评估脂肪酸与膝骨关节炎结局风险的因果关系。采用逆方差加权法、MR-Egger回归法、加权中位数法、加权模型法和简单模型法来研究脂肪酸与膝骨关节炎之间的因果关系。进一步以同样方法进行反向孟德尔随机化分析,确保结果的有效性。结果与结论:①正向分析逆方差加权法结果表明,共有3种脂肪酸表型与膝骨关节炎有因果关系,其中饱和脂肪酸占总脂肪酸的比例(OR=1.825,95%CI:1.230-2.706,P=0.003)与膝骨关节炎的发病风险呈正相关、Ω3多不饱和脂肪酸占总脂肪酸的比例(OR=0.822,95%CI:0.689-0.981,P=0.03)与膝骨关节炎的发病风险呈负相关以及Ω6多不饱和脂肪酸占总脂肪酸的比例(OR=1.268,95%CI:1.079-1.491,P=0.004)与膝骨关节炎的发病风险呈正相关;有2种脂肪酸表型与膝骨关节炎不具有因果关系,为总脂肪酸(OR=0.925,95%CI:0.804-1.066,P=0.283)、单不饱和脂肪酸占总脂肪酸的比例(OR=0.877,95%CI:0.756-1.018,P=0.084)。②反向分析结果表明,当膝骨关节炎作为暴露数据时,与脂肪酸的表型均不具有显著因果关系。③敏感性分析结果显示:双向孟德尔随机化的Cochran’s Q检验和MR-Egger回归法的结果P值均大于0.05,表明脂肪酸表型与膝骨关节炎之间的因果效应分析不存在显著的异质性和多效性。④上述结果证实,降低饱和脂肪酸、Ω6多不饱和脂肪酸的含量和提升Ω3多不饱和脂肪酸的含量能降低膝骨关节炎发病的风险,这为研究膝骨关节炎的生物学机制以及探索膝骨关节炎的早期防治提供有价值的线索,也为干预性药物的开发提供了新的方向。

Retrieval Single-Doppler Radar Wind with Variational Assimilation Method-Part I: Objective Selection of Functional Weighting Factors

In variational problem, the selection of functional weighting factors (FWF) is one of the key points for discussing many relevant studies. To overcome arbitrariness and subjectivity of the empirical selecting methods used widely at present, this paper tries to put forward an optimal objective selecting method of FWF. The focus of the study is on the weighting factors optimal selection in the variation retrieval single-Doppler radar wind field with the simple adjoint models. Weighting factors in the meaning of minimal variance are calculated out with the matrix theory and the finite difference method of partial differential equation. Experiments show that the result is more objective comparing with the factors obtained with the empirical method.

Self-tuning weighted measurement fusion Kalman filter and its convergence

For multisensor systems,when the model parameters and the noise variances are unknown,the consistent fused estimators of the model parameters and noise variances are obtained,based on the system identification algorithm,correlation method and least squares fusion criterion.Substituting these consistent estimators into the optimal weighted measurement fusion Kalman filter,a self-tuning weighted measurement fusion Kalman filter is presented.Using the dynamic error system analysis （DESA） method,the convergence of the self-tuning weighted measurement fusion Kalman filter is proved,i.e.,the self-tuning Kalman filter converges to the corresponding optimal Kalman filter in a realization.Therefore,the self-tuning weighted measurement fusion Kalman filter has asymptotic global optimality.One simulation example for a 4-sensor target tracking system verifies its effectiveness.

基于VW-PCA的盾构刀盘驱动液压系统故障诊断

为提高盾构刀盘液压系统故障诊断的快速性和准确性,引入权值向量来改进传统主元分析法(PCA),并推导变量加权主元分析法(VW-PCA)公式。通过AMESim仿真软件建立6 450 mm土压平衡盾构刀盘驱动液压系统模型,选取12个测试变量,模拟仿真6种故障状态;运用仿真故障数据得到每一类故障的加权向量,建立变量加权主元分析模型(VW-PCA),并使用该模型对测试故障样本进行研究。结果表明:变量加权(VW)能够增强各系统变量对不同故障类型的贡献程度,变量加权主元模型(VWPCA)能够显著提高系统诊断的正确率,最大可提高50%以上。

Minimum MSE Weights of Adjusted Summary Estimator of Risk Difference in Multi-Center Studies

The simple adjusted estimator of risk difference in each center is easy constructed by adding a value c on the number of successes and on the number of failures in each arm of the proportion estimator. Assessing a treatment effect in multi-center studies, we propose minimum MSE (mean square error) weights of an adjusted summary estimate of risk difference under the assumption of a constant of common risk difference over all centers. To evaluate the performance of the proposed weights, we compare not only in terms of estimation based on bias, variance, and MSE with two other conventional weights, such as the Cochran-Mantel-Haenszel weights and the inverse variance (weighted least square) weights, but also we compare the potential tests based on the type I error probability and the power of test in a variety of situations. The results illustrate that the proposed weights in terms of point estimation and hypothesis testing perform well and should be recommended to use as an alternative choice. Finally, two applications are illustrated for the practical use.

基于VW-FDA的盾构机液压系统故障诊断方法研究

盾构机规模庞大,施工过程中检测变量众多,液压系统故障易发。为了实现在线监测,提高液压系统故障诊断的准确性,探究与盾构机适应的快速准确有效的故障识别方法,提出使用加权因子与传统费歇尔判别法相结合的加权费歇尔判别法(Variable-Weighted-Fisher Discriminant Analysis,VW-FDA)。通过AMESim软件建立推进系统模型,仿真四类故障,运用仿真故障数据得到每一类故障的加权向量,建立VWFDA模型。结果表明:对于盾构机液压推进系统,VW-FDA比传统FDA具有更强的故障诊断能力,能够将故障信息进行有效的分类,正确地做出故障诊断。

Global variance reduction method for global Monte Carlo particle transport simulations of CFETR

It can be difficult to calculate some under-sampled regions in global Monte Carlo radiation transport calculations. The global variance reduction(GVR) method is a useful solution to the problem of variance reduction everywhere in a phase space. In this research, a GVR procedure was developed and applied to the Chinese Fusion Engineering Testing Reactor(CFETR). A cylindrical CFETR model was utilized for comparing various implementations of the GVR method to find the optimum.It was found that the flux-based GVR method could ensure more reliable statistical results, achieving an efficiency being 7.43 times that of the analog case. A mesh tally of the scalar neutron flux was chosen for the GVR method to simulate global neutron transport in the CFETR model.Particles distributed uniformly in the system were sampled adequately through ten iterations of GVR weight window.All voxels were scored, and the average relative error was 2.4% in the ultimate step of the GVR iteration.

OPTIMAL SELECTION FOR THE WEIGHTED COEFFICIENTS OF THE CONSTRAINED VARIATIONAL PROBLEMS

The aim is to put forward the optimal selecting of weights in variational problemin which the linear advection equation is used as constraint. The selection of the functionalweight coefficients ( FWC) is one of the key problems for the relevant research. It wasarbitrary and subjective to some extent presently. To overcome this difficulty, thereasonable assumptions were given for the observation field and analyzed field, variationalproblems with " weak constraints" and " strong constraints" were considered separately. Bysolving Euler' s equation with the matrix theory and the finite difference method of partialdifferential equation, the objective weight coefficients were obtained in the minimumvariance of the difference between the analyzed field and ideal field. Deduction results showthat theoretically the optimal selection indeed exists in the weighting factors of the costfunction in the means of the minimal variance between the analysis and ideal field in terms ofthe matrix theory and partial differential ( corresponding difference ) equation, if thereasonable assumption from the actual problem is valid and the differnece equation is stable.It may realize the coordination among the weight factors, numerical models and theobservational data. With its theoretical basis as well as its prospects of applications, thisobjective selecting method is probably a way towards the finding of the optimal weightingfactors in the variational problem.

Calibration of Nondestructive Assay Instruments: An Application of Linear Regression and Propagation of Variance

Several nondestructive assay (NDA) methods to quantify special nuclear materials use calibration curves that are linear in the predictor, either directly or as an intermediate step. The linear response model is also often used to illustrate the fundamentals of calibration, and is the usual detector behavior assumed when evaluating detection limits. It is therefore important for the NDA community to have a common understanding of how to implement a linear calibration according to the common method of least squares and how to assess uncertainty in inferred nuclear quantities during the prediction stage following calibration. Therefore, this paper illustrates regression, residual diagnostics, effect of estimation errors in estimated variances used for weighted least squares, and variance propagation in a form suitable for implementation. Before the calibration can be used, a transformation of axes is required;this step, along with variance propagation is not currently explained in available NDA standard guidelines. The role of systematic and random uncertainty is illustrated and expands on that given previously for the chosen practical NDA example. A listing of open-source software is provided in the Appendix.

Estimation of Genetic Parameters for Live Weight in South African Holstein Cattle

Cow live weight is of economic importance in dairy production;however, it is not included in the selection objective for South African dairy cattle. Variance components estimates are a prerequisite to incorporating a trait in the breeding objective. Variance components were estimated for live weight of lactating Holstein cows on two South African dairy herds. Live weight records on 9843 lactating cows, collected over a period of three years, were used. An analysis of variance was carried out to determine fixed effects to include in the model by the least squares method, using the Generalised Linear Models procedure of the Statistical Analysis System. Variance components were estimated by the Restricted Maximum Likelihood procedure using the ASREML programme. A high heritability estimate of 0.74 ± 0.19 is found, which suggests that there is scope for significant response to selection on live weight in the South African Holstein cattle population. A repeatability estimate of 0.86 was obtained. These results form the basis for incorporating live weight in the breeding objective for South African Holstein cattle. Further work, however, needs to be done to develop estimates based on a representative sample of the whole population, as the current study is based on data from only two herds.

Modified Adaptive Weighted Averaging Filtering Algorithm for Noisy Image Sequences

In order to avoid the influence of noise variance on the filtering performances, a modified adaptive weighted averaging (MAWA) filtering algorithm is proposed for noisy image sequences. Based upon adaptive weighted averaging pixel values in consecutive frames, this algorithm achieves the filtering goal by assigning smaller weights to the pixels with inappropriate estimated motion trajectory for noise. It only utilizes the intensity of pixels to suppress noise and accordingly is independent of noise variance. To evaluate the performance of the proposed filtering algorithm, its mean square error and percentage of preserved edge points were compared with those of traditional adaptive weighted averaging and non-adaptive mean filtering algorithms under different noise variances. Relevant results show that the MAWA filtering algorithm can preserve image structures and edges under motion after attenuating noise, and thus may be used in image sequence filtering.

Comprehensive Study on the Estimation of the Variance Components of Traverse Nets

This paper advances a new simplified formula for estimating variance components ,sums up the basic law to calculate the weights of observed values and a circulation method using the increaments of weights when estimating the variance components of traverse nets,advances the charicteristic roots method to estimate the variance components of traveres nets and presents a practical method to make two real and symmetric matrices two diagonal ones.

Maximum Posterior Adjustment of Extended Network and Estimation Formulae of Its Variance Components

This paper derives the maximum posterior adjustment formulae of the extended network and the estimation formulaes of variance components of Helmert, Welsch and Frstner types when there are two types of uncorrelated observations in it, and perfects the theory of the maximum posterior adjustment.

基于IVW方法的MODIS AOD融合

气溶胶光学厚度(Aerosol Optical Depth,AOD)受高湿、多云等气候特征的影响而造成非随机缺失,为了最大限度地扩大AOD的覆盖范围,本文以华中三省为研究区,首先基于武汉大学自测的地面AOD数据,对华中地区C 6.1中等分辨率成像光谱仪(Moderate Resolution Imaging Spectroradiometer,MODIS)反演的2010—2017年10 km DT、10 km DB以及3 km DT AOD数据的精度进行验证,然后利用逆方差加权(Inverse Variance Weighting,IVW)方法,将10 km分辨率深蓝算法反演的AOD分别和10 km、3 km分辨率暗目标算法反演的AOD融合。结果表明,融合后10 km和3 km AOD与地面实测AOD的皮尔森相关系数分别为0.83和0.86;相对于MODIS标准10 km和3 km DT数据,IVW融合后的10 km和3 km AOD的平均覆盖率分别提高55.17%和48.00%。