多指标综合评价的非参数方法和缺失数据的聚类方法研究
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摘要
多指标综合评价概指对以多属性体系结构描述的对象系统做出全局性、整体性的评价,是利用数学及统计方法,将反映评价对象不同属性的多个统计指标的信息转化成无量纲的相对评价值,并综合这些评价值以求得评价对象的优劣等级的一种评价方法。综合评价方法的研究一直是评价研究领域中的热点问题。论文对多指标综合评价方法进行了概述,重点介绍了属于运筹学和其它数学方法范畴的几种常用的综合评价方法,包括层次分析法(Analytic Hierarchy Process, AHP)、模糊综合评判法(Fuzzy Comprehensive Evaluation,FCE)、数据包络分析法(Data Envelopment Analysis,DEA)、灰色综合评价法(Grey Comprehensive Evaluation)、 TOPSIS评价法(Technique for order preference by similarity to ideal solution)等,从综合评价方法的概念和原理、模型和步骤、优缺点评析等方面作了较为详细的阐述。最后讨论了评价方法的集成、存在的问题及研究趋势。
聚类分析是研究分类问题的一种多元统计分析方法,是一种重要的数据分析手段。它把一个没有类别标记的数据集按照某种相似性准则划分成若干个子集(类)。聚类的主要依据是类内对象的相似性尽可能大,而类间对象的相似性尽可能小。通过聚类分析,能有效地发现隐含在数据集中的数据分布特性和典型模式,从而为进一步充分、有效地利用数据奠定良好的基础。聚类分析现已成为数据挖掘中的一项重要技术和主要方法之一。多年来,众多学者对聚类算法进行了广泛而深入的研究。论文对基于划分的(partitioning method)、基于层次的(hierarchical method)、基于密度的(density-based method)、基于网格的(grid-based method)和基于模型的(model-based method)五大类聚类算法进行了综述,并在每一类中,介绍了一些经典的聚类算法。
在此基础上,论文针对多指标综合评价结果的差异显著性测验以及缺失数据聚类的统计分析两个问题开展了初步研究,主要结果如下：
(1)发展了一种多性状综合评价的统计假设测验方法(非参数的秩和与秩和差测验)
目前国内外关于多性状综合评价的方法很多,但它们都只是提供了各不相同的优劣判别方法,其评价结论表现为一定的综合评价值以及相应的优劣排序,而无法提供各评价对象与其平均水平的差异显著性。论文给出了一种多性状综合评价的统计假设测验方法(非参数的秩和测验),在“H0：各评价对象在各性状上的秩次随机分布”假设下,导出多性状秩和的理论分布,并据之提出获得秩和测验显著性临界值的一般化方法及计算程序。通过定义任意长度整数C++运算律,解决了当评价对象和性状数较多时,因常用软件内置数据类型有效位数不足所导致的计算误差问题。最后,以糯玉米12个品种5个淀粉粘度性状为例演示了分析程序。
以上多性状综合评价的秩和测验方法,虽可测验各评价对象与其平均水平的差异显著性,但无法实现各评价对象两两之间的差异显著性测验。论文以秩和理论分布为基础,利用组合数学方法,进一步导出了多性状秩和差的理论分布,并据之给出了多性状综合评价秩和差测验的显著性临界值。通过秩和差测验,确定评价对象两两之间的差异显著性,从而实现评价对象间的多重比较。
(2)给出了一种带有缺失数据的基于模型的动态聚类方法
聚类分析是把数据集中的对象按某种相似性准则聚集成多个类的多元统计分析方法。通常情况下,聚类过程需要基于完全数据集,但在许多实际问题的研究中,其数据是不完全的,这给聚类分析带来一定的困难。论文研究了带有缺失数据的基于模型的动态聚类方法,利用相关变量的辅助信息,对缺失数据进行推估,确定其合理的替代值,从而构造出一个“完全”数据集。在此基础上以EM算法循环迭代,参数的估计值和缺失数据的替代值都将逐渐收敛,以相应的贝叶斯后验概率判别个体的归类,进而实现动态聚类。模拟研究表明,缺值替代法具有较好的收敛性,对有缺失的数据基本都可正确地聚类。
Multi-index comprehensive evaluation refers to making an overall and entirely assessments on observation system described by multiple attributes structure. It is an evaluation method which transfers a number of statistical index information reflecting the different attributes of target observation into dimensionless relative evaluation value through the mathematical and statistical methods and achieves the merit ranks of target observation by using them. The study on comprehensive evaluation method has always been a hot issue in the field of evaluation research. The dissertation gives a brief review of the multi-index comprehensive evaluation method. The focus is on several commonly used comprehensive evaluation methods belonging to Operations Research and other mathematical method categories, which includes analytic hierarchy process, fuzzy comprehensive evaluation, data envelopment analysis, grey comprehensive evaluation method, TOPSIS evaluation method and so on. It also makes a detailed interpretation on comprehensive evaluation method from the aspects of its definition and principle, pattern and procedure, advantages and disadvantages analysis etc. Finally, it discusses the integration of evaluation methods, existing problems and research trends.
Clustering analysis is a kind of multivariate statistical analysis method on classification issue, and an important mean of data analysis. Clustering analysis is to divide a non-tag labeled data set into several sub-sets (classes) in accordance with a certain similarity. The main basis of clustering is on the similarity of objects within the class as large as possible, while the similarity between classes objects as small as possible. Clustering analysis can effectively find the data distribution feature and typical pattern hidden in the data set, which lays a good foundation for the further use of data fully and effectively. Clustering analysis has become an important technique and main method to data mining. Over the years, many scholars have made a broad and deep research on the clustering algorithm. The dissertation made an overview of five clustering algorithms, including partitioning method, hierarchical method, density-based method, grid-based method and model-based method, and introduced some classical clustering algorithms in each category.
On this basis, the dissertation launched a preliminary study on two issues: difference significance test method on multi-index comprehensive evaluation results and statistical analysis of missing data clustering. The major research results include:
(1) Developed a statistical hypothesis test method of multiple traits comprehensive evaluation (non-parametric rank-sum and rank-sum-difference test)
At present, there are many multiple traits comprehensive evaluation methods at home and abroad, while they can only provide the different distinguishing methods, the evaluation findings showed a certain comprehensive evaluation value and the corresponding merit ordering, but they can not provide the difference significance from each evaluation object and its average level. The dissertation presents a statistical hypothesis testing method of multiple traits comprehensive evaluation (non-parametric rank-sum test). Under null hypothesis:the variety's ranking on each trait is random, the theoretical distribution of sum of ranks (SR) was firstly derived and further used to obtain the critical values for multi-trait comprehensive evaluation in rank-sum test. A new C++class and its basic arithmetic were defined to deal with the miscount caused by the precision limitation of built-in data type in common statistical software under large number of varieties and traits. Finally, an application of the theoretical results was demonstrated using five starch viscosity traits of12glutinous maize varieties.
The above rank-sum testing method for multi-trait comprehensive evaluation can test the significance of difference between evaluation objects and the average level, but it can not realize the significance of difference testing between two evaluation objects. Based on the theoretical distribution of rank-sum, the dissertation deduced the theoretical distribution of multi-trait rank-sum-difference by using combinatorial mathematics method, on which it presented the significant critical values for the rank-sum-difference testing in multi-trait comprehensive evaluation. Finally, it tested the difference significance between two evaluation objects through rank-sum-difference testing, and realized the multiple comparisons among various evaluation objects.
(2) Developed a model-based dynamic clustering method with missing data
Cluster analysis, as multivariate statistical method, is the assignment of a set of observations into subsets (called clusters) so that observations in the same cluster are similar in certain sense. Generally, the current clustering technique always depends on complete data. However, missing data set is often seen in practice, which brings difficulties in clustering analysis. The dissertation studied a pattern-based dynamic cluster method with missing data. It determines the reasonable alternative value and constructs a "complete" data set by using the auxiliary information of relevant variables and estimating out missing data. On this basis, with the EM algorithm iteration, the parameter estimates and alternative values of missing data will gradually converge, and judge the individual classification by corresponding Bayesian posterior probability so as to realize the dynamic clustering. Simulation studies show that missing value alternative method has good convergence, which can accurately cluster the missing data.
聚类分析是研究分类问题的一种多元统计分析方法,是一种重要的数据分析手段。它把一个没有类别标记的数据集按照某种相似性准则划分成若干个子集(类)。聚类的主要依据是类内对象的相似性尽可能大,而类间对象的相似性尽可能小。通过聚类分析,能有效地发现隐含在数据集中的数据分布特性和典型模式,从而为进一步充分、有效地利用数据奠定良好的基础。聚类分析现已成为数据挖掘中的一项重要技术和主要方法之一。多年来,众多学者对聚类算法进行了广泛而深入的研究。论文对基于划分的(partitioning method)、基于层次的(hierarchical method)、基于密度的(density-based method)、基于网格的(grid-based method)和基于模型的(model-based method)五大类聚类算法进行了综述,并在每一类中,介绍了一些经典的聚类算法。
在此基础上,论文针对多指标综合评价结果的差异显著性测验以及缺失数据聚类的统计分析两个问题开展了初步研究,主要结果如下：
(1)发展了一种多性状综合评价的统计假设测验方法(非参数的秩和与秩和差测验)
目前国内外关于多性状综合评价的方法很多,但它们都只是提供了各不相同的优劣判别方法,其评价结论表现为一定的综合评价值以及相应的优劣排序,而无法提供各评价对象与其平均水平的差异显著性。论文给出了一种多性状综合评价的统计假设测验方法(非参数的秩和测验),在“H0：各评价对象在各性状上的秩次随机分布”假设下,导出多性状秩和的理论分布,并据之提出获得秩和测验显著性临界值的一般化方法及计算程序。通过定义任意长度整数C++运算律,解决了当评价对象和性状数较多时,因常用软件内置数据类型有效位数不足所导致的计算误差问题。最后,以糯玉米12个品种5个淀粉粘度性状为例演示了分析程序。
以上多性状综合评价的秩和测验方法,虽可测验各评价对象与其平均水平的差异显著性,但无法实现各评价对象两两之间的差异显著性测验。论文以秩和理论分布为基础,利用组合数学方法,进一步导出了多性状秩和差的理论分布,并据之给出了多性状综合评价秩和差测验的显著性临界值。通过秩和差测验,确定评价对象两两之间的差异显著性,从而实现评价对象间的多重比较。
(2)给出了一种带有缺失数据的基于模型的动态聚类方法
聚类分析是把数据集中的对象按某种相似性准则聚集成多个类的多元统计分析方法。通常情况下,聚类过程需要基于完全数据集,但在许多实际问题的研究中,其数据是不完全的,这给聚类分析带来一定的困难。论文研究了带有缺失数据的基于模型的动态聚类方法,利用相关变量的辅助信息,对缺失数据进行推估,确定其合理的替代值,从而构造出一个“完全”数据集。在此基础上以EM算法循环迭代,参数的估计值和缺失数据的替代值都将逐渐收敛,以相应的贝叶斯后验概率判别个体的归类,进而实现动态聚类。模拟研究表明,缺值替代法具有较好的收敛性,对有缺失的数据基本都可正确地聚类。
Multi-index comprehensive evaluation refers to making an overall and entirely assessments on observation system described by multiple attributes structure. It is an evaluation method which transfers a number of statistical index information reflecting the different attributes of target observation into dimensionless relative evaluation value through the mathematical and statistical methods and achieves the merit ranks of target observation by using them. The study on comprehensive evaluation method has always been a hot issue in the field of evaluation research. The dissertation gives a brief review of the multi-index comprehensive evaluation method. The focus is on several commonly used comprehensive evaluation methods belonging to Operations Research and other mathematical method categories, which includes analytic hierarchy process, fuzzy comprehensive evaluation, data envelopment analysis, grey comprehensive evaluation method, TOPSIS evaluation method and so on. It also makes a detailed interpretation on comprehensive evaluation method from the aspects of its definition and principle, pattern and procedure, advantages and disadvantages analysis etc. Finally, it discusses the integration of evaluation methods, existing problems and research trends.
Clustering analysis is a kind of multivariate statistical analysis method on classification issue, and an important mean of data analysis. Clustering analysis is to divide a non-tag labeled data set into several sub-sets (classes) in accordance with a certain similarity. The main basis of clustering is on the similarity of objects within the class as large as possible, while the similarity between classes objects as small as possible. Clustering analysis can effectively find the data distribution feature and typical pattern hidden in the data set, which lays a good foundation for the further use of data fully and effectively. Clustering analysis has become an important technique and main method to data mining. Over the years, many scholars have made a broad and deep research on the clustering algorithm. The dissertation made an overview of five clustering algorithms, including partitioning method, hierarchical method, density-based method, grid-based method and model-based method, and introduced some classical clustering algorithms in each category.
On this basis, the dissertation launched a preliminary study on two issues: difference significance test method on multi-index comprehensive evaluation results and statistical analysis of missing data clustering. The major research results include:
(1) Developed a statistical hypothesis test method of multiple traits comprehensive evaluation (non-parametric rank-sum and rank-sum-difference test)
At present, there are many multiple traits comprehensive evaluation methods at home and abroad, while they can only provide the different distinguishing methods, the evaluation findings showed a certain comprehensive evaluation value and the corresponding merit ordering, but they can not provide the difference significance from each evaluation object and its average level. The dissertation presents a statistical hypothesis testing method of multiple traits comprehensive evaluation (non-parametric rank-sum test). Under null hypothesis:the variety's ranking on each trait is random, the theoretical distribution of sum of ranks (SR) was firstly derived and further used to obtain the critical values for multi-trait comprehensive evaluation in rank-sum test. A new C++class and its basic arithmetic were defined to deal with the miscount caused by the precision limitation of built-in data type in common statistical software under large number of varieties and traits. Finally, an application of the theoretical results was demonstrated using five starch viscosity traits of12glutinous maize varieties.
The above rank-sum testing method for multi-trait comprehensive evaluation can test the significance of difference between evaluation objects and the average level, but it can not realize the significance of difference testing between two evaluation objects. Based on the theoretical distribution of rank-sum, the dissertation deduced the theoretical distribution of multi-trait rank-sum-difference by using combinatorial mathematics method, on which it presented the significant critical values for the rank-sum-difference testing in multi-trait comprehensive evaluation. Finally, it tested the difference significance between two evaluation objects through rank-sum-difference testing, and realized the multiple comparisons among various evaluation objects.
(2) Developed a model-based dynamic clustering method with missing data
Cluster analysis, as multivariate statistical method, is the assignment of a set of observations into subsets (called clusters) so that observations in the same cluster are similar in certain sense. Generally, the current clustering technique always depends on complete data. However, missing data set is often seen in practice, which brings difficulties in clustering analysis. The dissertation studied a pattern-based dynamic cluster method with missing data. It determines the reasonable alternative value and constructs a "complete" data set by using the auxiliary information of relevant variables and estimating out missing data. On this basis, with the EM algorithm iteration, the parameter estimates and alternative values of missing data will gradually converge, and judge the individual classification by corresponding Bayesian posterior probability so as to realize the dynamic clustering. Simulation studies show that missing value alternative method has good convergence, which can accurately cluster the missing data.
引文
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