Principal differential analysis with a continuous covariate: low-dimensional approximations for functional data

Given a collection of n curves that are independent realizations of a functional variable, we are interested in finding patterns in the curve data by exploring low-dimensional approximations to the curves. It is assumed that the data curves are noisy samples from the vector space span <texlscub>f ₁, …, f _m </texlscub>, where f ₁, …, f _m are unknown functions on the real interval (0, T) with square-integrable derivatives of all orders m or less, and m<n. Ramsay [Principal differential analysis: Data reduction by differential operators, J. R. Statist. Soc. Ser. B 58 (1996), pp. 495–508] first proposed the method of regularized principal differential analysis (PDA) as an alternative to principal component analysis for finding low-dimensional approximations to curves. PDA is based on the following theorem: there exists an annihilating linear differential operator (LDO) ? of order m such that ?f _i =0, i=1, …, m [E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955, Theorem 6.2]. PDA specifies m, then uses the data to estimate an annihilating LDO. Smooth estimates of the coefficients of the LDO are obtained by minimizing a penalized sum of the squared norm of the residuals. In this context, the residual is that part of the data curve that is not annihilated by the LDO. PDA obtains the smooth low dimensional approximation to the data curves by projecting onto the null space of the estimated annihilating LDO; PDA is thus useful for obtaining low-dimensional approximations to the data curves whether or not the interpretation of the annihilating LDO is intuitive or obvious from the context of the data. This paper extends PDA to allow for the coefficients in the LDO to smoothly depend upon a single continuous covariate. The estimating equations for the coefficients allowing for a continuous covariate are derived; the penalty of Eilers and Marx [Flexible smoothing with B-splines and penalties, Statist. Sci. 11(2) (1996), pp. 89–121] is used to impose smoothness. The results of a small computer simulation study investigating the bias and variance properties of the estimator are reported.     

Compare diagnostic tests using transformation-invariant smoothed ROC curves()

Receiver operating characteristic (ROC) curve, plotting true positive rates against false positive rates as threshold varies, is an important tool for evaluating biomarkers in diagnostic medicine studies. By definition, ROC curve is monotone increasing from 0 to 1 and is invariant to any monotone transformation of test results. And it is often a curve with certain level of smoothness when test results from the diseased and non-diseased subjects follow continuous distributions. Most existing ROC curve estimation methods do not guarantee all of these properties. One of the exceptions is Du and Tang (2009) which applies certain monotone spline regression procedure to empirical ROC estimates. However, their method does not consider the inherent correlations between empirical ROC estimates. This makes the derivation of the asymptotic properties very difficult. In this paper we propose a penalized weighted least square estimation method, which incorporates the covariance between empirical ROC estimates as a weight matrix. The resulting estimator satisfies all the aforementioned properties, and we show that it is also consistent. Then a resampling approach is used to extend our method for comparisons of two or more diagnostic tests. Our simulations show a significantly improved performance over the existing method, especially for steep ROC curves. We then apply the proposed method to a cancer diagnostic study that compares several newly developed diagnostic biomarkers to a traditional one.     

Small area estimation using reduced rank regression models

Abstract

Small area estimation techniques have got a lot of attention during the last decades due to their important applications in survey studies. Mixed linear models and reduced rank regression analysis are jointly used when considering small area estimation. Estimates of parameters are presented as well as prediction of random effects and unobserved area measurements.     

测井地质结合定量评价低渗透储层成岩相

高邮凹陷南断阶低渗透储层同一沉积微相产能差异变化大。利用铸体薄片、阴极发光、扫描电镜等资料，对高邮凹陷南断阶阜宁组储层成岩作用类型及特征进行了综合研究，在此基础上结合实际生产应用，将研究区阜一段储层划分为弱压实成岩相、中等压实胶结成岩相、中等压实溶解成岩相、强压实胶结成岩相和强压实溶解成岩相5 种成岩相类型，建立了不同类型成岩相测井曲线特征，确定了不同成岩相的定量判别标准，利用判别标准定量识别研究区阜宁组阜一段储层成岩相，结合产能评价有利成岩相。     

Gerber抛物线在疲劳可靠性设计的应用研究

介绍了国际上五种简化疲劳极限应力(σm-σa)图,分析了Good man图、简化疲劳极限折线图、Soderberg直线图、Биргер抛物线图和Gerber抛物线图应用特点。针对Gerber抛物线在可靠性强度计算中疲劳极限的均值和标准差的计算,与实验数据比较符合,并对比折线方程的优点加以分析说明,为可靠性设计研究奠定了理论基础。     

Hierarchical bayesian curve fitting and smoothing

Estimation of a smooth function is considered when observations on this function added with Gaussian errors are observed. The problem is formulated as a general linear model, and a hierarchical Bayesian approach is then used to study it. Credible bands are also developed for the function. Sensitivity analysis is conducted to determine the influence of the choice of priors on hyperparameters. Finally, the methodology is illustrated using real and simulated examples where it is compared with classical cubic splines. It is also shown that our approach provides a Bayesian solution to some problems in discrete time series.     

理论频率曲线及其在工程问题上的应用

本文从频率曲线的定义出发，推导出理论频率曲线。然后对理论频率曲线的实际应用进行详细的探讨。     

Modelling and analyzing density-dependent population processes: Comparison between wild and laboratory strains of the bean weevil,Callosobruchus chinensis (L.)

Three models were constructed for analyzing the population characteristics ofC. chinensis on stored beans; model A describing the whole reproductive process with a single equation, model B describing the three age-specific processes (oviposition, egg survival and larval survival) with separate equations, and model C which describes all these processes not for the whole habitat but for the individual beans comprizing it. The logit equation was employed here as a common basis to describe the density-response relationship involved. All three models showed very good fit to the experimental data obtained for both laboratory and wild strains of the weevil. The parameter values characterizing the population dynamics were, however, widely different between the two strains; the laboratory one which had been reared for some 500 generations showed significantly higher reproductive capacity, less sensitive and gentler response to crowding in both adult and egg stages, and more uniform egg distribution among individual beans, as compared with the wild strain newly introduced. Sensitivity analyses using these models suggested that these changes in population characteristics have been attained by the process of domestication or adaptation to stable laboratory conditions through a long period of time. This process seemed in effect to have optimized the population's performances in the laboratory environment. Evolutionary significance of such optimization was discussed with reference to the selection pressure which may have acted upon individuals.     

沪深股指差额非对称的CARCH效应分析

根据非对称的CARCH理论，通过在模型分析中引入冲击因素和波动系数，对沪深股指差额波动率分析后发现：回归方程残差呈现显著的“聚集”现象，说明存在着ARCH效应；拟合CARCH模型冲击因素的显著性表明沪深股指差额波动率存在着明显的杠杆效应，负面的信息冲击比正面的信息冲击带来的波动更大；CARCH模型中的长期波动率表明，波动率收敛于稳态的速度更快。     

参数曲线在水泵叶轮设计中的应用

针对水泵设计所需的曲线及其特点，采用B样条和参数样条曲线对设计型线进行建模，根据设计准则和实践经验进行调整、修改和优化。基于曲线的参数化建模，从理论上推导出轴面流线和流线展开线的数学模型。改变了传统手工设计方法计算机化的常规做法，为在真正意义上实现水泵计算机辅助设计提供了理论基础和实现方法。