稳健随机均值模型在时空数据分析中的应用

项目名称： 稳健随机均值模型在时空数据分析中的应用

项目编号： No.11471283

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 张朋

作者单位： 浙江大学

项目金额： 70万元

中文摘要： 时空数据分析日趋重要，从空气污染数据到房地产发展预测都离不开时空数据统计分析。本项目将提出随机均值模型，致力于其在时空数据分析中应用的研究以及在这个模型基础上，对各种时空数据中出现的异常现象提出相应的稳健模型。时空数据特征决定了其时间和空间维度上的相关性非常复杂，随机均值模型中时间和空间的相关性将可以自由设定，可以解决混合效应模型协方差矩阵由随机效应决定这一缺陷。当时空数据中出现异常值使得正态分布假设不满足时，我们提出用多维t分布模型提供对异常观测的稳健分析；当随机效应分布不对称时，我们探讨用对数伽马分布随机均值模型提供稳健分析。在离散时空数据出现多零现象，并在因变量观测值有误差时，提出用截零泊松分布随机期望模型，结合测量误差模型技术探讨其估计偏差等性质，完善时空数据分析。

中文关键词： 生物统计；纵向数据；测量误差模型；混合模型；参数估计

英文摘要： There has been a tremendous growth in the statistical models and techniques to analyze spatio-temporal data such as air-pollution data in recent years. Spatio-temporal data arise in many other contexts, for example, disease mapping and economic monitoring of real estate prices. Often the primary interests in analyzing such data are to smooth and predict time evolution of some response variables over a certain spatial domain. Linear mixed effects models are main tools to estimate population effects and make statistical inference. However, the restriction that the covariance matrix is derived from the random effects included in the model not only limits their usage in practice but also reduces their efficiencies and even produces misleading conclusions in some cases. In this project we proposal a family of random mean models to overcome such problem.The sophisticated correlations can be set up in such models temporally and spatially. Spatio-temporal are prone to irregularities, including outliers,skewness,excess of zeros and measurement errors.We proposed to model the left skewed distribution of random effects in a linear mixed effects model with a log-gamma distribution in our previous studies. It effectively predicted the patient-specific disease progression rates. We propose to extend this framework in more sophisticated settings to facilitate modelling more complex data. When some of the distributions of random effects appear skewed shapes, the estimation of the correlations may be biased if the random effects are modelled by normal distributions, as usually do in routine mixed effects model fittings. Assuming a log-gamma distribution for the skewed random effects and using a upper triangle matrix for the correlations can correctly estimate not only the subject-specific process but also their correlations. Another issue in the spatio-temporal studies is the tail dependence. Multivariate normal distributions are known to be tail independent but multivariate t distributions are tail dependent and the dependence is determined by the correlation coefficient. We will explore this property of MT model in addition to the robustness against outliers shown in our previous studies. Similar issues may present in spatio-temporal data analysis where some of areas showing outstanding progressions than most of the others. Tail dependence find its own way in this setting as well. Random mean models, a generalization of random effects models that is capable of modelling autocorrelations and spatial correlations with multivariate t distributions, are proposed to deal with spatio-temporal data analysis. Excess zeros are often observed in spatio-temporal data. Measurements in response are prone to errors but it has not been discussed thoroghly in the literature. We propose to study a measurement error model with truncated-zero possion or binomial for the response variable in a spatio-temporal data analysis with discrete response variables.

英文关键词： Spatio-temporal data;longitudinal data;measurement error model;mixed effects model;robustness