STANDARD ERRORS IN THE EIGENVALUES OF A CROSS-PRODUCT MATRIX:THEORY AND APPLICATIONS

New expressions are derived for the standard errors in the eigenvalues of a cross-product matrix by themethod of error propagation.Cross-product matrices frequently arise in multivariate data analysis,especially in principal component analysis (PCA).The derived standard errors account for the variabilityin the data as a result of measurement noise and are therefore essentially different from the standarderrors developed in multivariate statistics.Those standard errors were derived in order to account for thefinite number of observations on a fixed number of variables,the so-called sampling error.They can beused for making inferences about the population eigenvalues.Making inferences about the populationeigenvalues is often not the purposes of PCA in physical sciences,This is particularly true if themeasurements are performed on an analytical instrument that produces two-dimensional arrays for onechemical sample:the rows and columns of such a data matrix cannot be identified with observations onvariables at all.However,PCA can still be used as a general data reduction technique,but now the effectof measurement noise on the standard errors in the eigenvalues has to be considered.The consequencesfor significance testing of the eigenvalues as well as the usefulness for error estimates for scores andloadings of PCA,multiple linear regression (MLR) and the generalized rank annihilation method(GRAM) are discussed.The adequacy of the derived expressions is tested by Monte Carlo simulations.     

RESIDUAL BILINEARIZATION.PART 1:THEORY AND ALGORITHMS

When using hyphenated methods in analytical chemistry,the data obtained for each sample are given asa matrix.When a regression equation is set up between an unknown sample (a matrix) and a calibrationset (a stack of matrices),the residual is a matrix R.The regression equation is usually solved by minimizing the sum of squares of R.If the sample containssome constituent not calibrated for,this approach is not valid.In this paper an algorithm is presentedwhich partitions R into one matrix of low rank corresponding to the unknown constituents,and onerandom noise matrix to which the least squares restrictions are applied.Properties and possibleapplications of the algorithm are also discussed.In Part 2 of this work an example from HPLC with diode array detection is presented and the resultsare compared with generalized rank annihilation factor analysis (GRAFA).     

NONPARAMETRIC FACTOR ANALYSIS IN URBAN GEOGRAPHY: METHOD AND VALIDATION

The present paper demonstrates that Kelly's (1955) method of hand factor analyzing the data matrices derived from repertory grids can be employed as a general method of multivariate analysis in geography. This brings the advantages of a non-parametric and noncomputer dependent approach to areas such as factorial ecology, classifications of towns and cities, and urban behavioral analyses, where multivariate techniques have customarily been employed. The method is initially explained by recourse to a simple hypothetical urban retailing data set. Subsequently, more complex real world examples involving multivariate analyses of housing data for Barbados, West Indies, and urban consumers' cognitions of a single store are presented. It is shown that the nonparametric method gives results that are virtually identical to those obtained from traditional computer-based factor analyses. Throughout the paper, the pedagogic and practical virtues of the nonparametric method are considered.     

Practical Incorporation of Multivariate Parameter Uncertainty in Geostatistical Resource Modeling

There is a need to bridge theory and practice for incorporating parameter uncertainty in geostatistical simulation modeling workflows. Simulation workflows are a standard practice in natural resource and recovery modeling, but the incorporation of multivariate parameter uncertainty into those workflows is challenging. However, the objectives can be met without considerable extra effort and programming. The sampling distributions of statistics comprise the core theoretical notion with the addition of the spatial degrees of freedom to account for the redundancy in the spatially correlated data. Prior parameter uncertainty is estimated from multivariate spatial resampling. Simulation-based transfer of prior parameter uncertainty results in posterior distributions which are updated by data conditioning and the model domain extents and configuration. The results are theoretically tractable and practical to achieve, providing realistic assessments of uncertainty by accounting for large-scale parameter uncertainty, which is often the most important component impacting a project. A simulation-based multivariate workflow demonstrates joint modeling of intrinsic shale properties and uncertainty in estimated ultimate recovery in a shale gas project. The multivariate workflow accounts for joint prior parameter uncertainty given the current well locations and results in posterior estimates on global distributions of all modeled properties. This is achieved by transferring the joint prior parameter uncertainty through conditional simulations.     

Canonical correlation analysis in geomorphology with particular reference to drainage basin characteristics

Problems in the analysis and interpretation of the interaction patterns of drainage basin characteristics are outlined. The canonical correlation technique is suggested as capable of simplifying and producing a high level of generalization of the complex interaction patterns. The technique is described and employed in structuring the relationships between the morphometric and topologic attributes of a sample of 36 monolithologic third-order drainage basins in the Udi-Awgu cuesta, southeastern Nigeria, and the factors controlling their development. The canonical solution identifies three significant patterns of association between basin planimetric properties and their causative and deterministic factors. The texture of dissection is strongly associated with soil and vegetation characteristics while the stream network size and the stream bifurcation ratio are closely related to the stage of basin development and basin relief respectively. Some general problems of the canonical correlation analysis are outlined.     

REVIEW ARTICLE MULTIVARIATE EXPLORATORY DATA ANALYSIS AND GRAPHICS:A TUTORIAL

Exploratory data analysis(EDA)is a toolbox of data manipulation methods for looking at data to seewhat they seem to say,i.e.one tries to let the data speak for themselves.In this way there is hope thatthe data will lead to indications about'models'of relationships not expected a priori.In this respect EDAis a pre-step to confirmatory data analysis which delivers measures of how adequate a model is.In thistutorial the focus is on multivariate exploratory data analysis for quantitative data using linear methodsfor dimension reduction and prediction.Purely graphical multivariate tools such as 3D rotation andscatterplot matrices are discussed after having introduced the univariate and bivariate tools on which theyare based.The main tasks of multivariate exploratory data analysis are identified as'search for structure'by dimension reduction and'model selection'by comparing predictive power.Resampling is used tosupport validity,and variables selection to improve interpretability.     

LATENT FEATURES WITH MAXIMAL CORRELATION BETWEEN TWO EQUAL SETS OF VARIABLES

If in two data tables X and Y objects are characterized by the same variables(measured at differentoccasions),then looking for common latent features should be more appropriate than choosing latentfeatures in X and Y separately(e.g.as in canonical correlation or PLS).The procedure to be proposedhere is a slight modification of the method of linear characteristics(according to De Groot and Li)bydisclaiming the assumption of equal inner-block covariance matrices.In order to find weights defining alatent feature with maximal correlation between X and Y,a system of non-linear equations has to besolved.The procedure is applied to the investigation of heavy metal concentrations in different humantissues.     

Processing seismic ambient noise data to obtain reliable broad-band surface wave dispersion measurements

Ambient noise tomography is a rapidly emerging field of seismological research. This paper presents the current status of ambient noise data processing as it has developed over the past several years and is intended to explain and justify this development through salient examples. The ambient noise data processing procedure divides into four principal phases: (1) single station data preparation, (2) cross-correlation and temporal stacking, (3) measurement of dispersion curves (performed with frequency–time analysis for both group and phase speeds) and (4) quality control, including error analysis and selection of the acceptable measurements. The procedures that are described herein have been designed not only to deliver reliable measurements, but to be flexible, applicable to a wide variety of observational settings, as well as being fully automated. For an automated data processing procedure, data quality control measures are particularly important to identify and reject bad measurements and compute quality assurance statistics for the accepted measurements. The principal metric on which to base a judgment of quality is stability, the robustness of the measurement to perturbations in the conditions under which it is obtained. Temporal repeatability, in particular, is a significant indicator of reliability and is elevated to a high position in our assessment, as we equate seasonal repeatability with measurement uncertainty. Proxy curves relating observed signal-to-noise ratios to average measurement uncertainties show promise to provide useful expected measurement error estimates in the absence of the long time-series needed for temporal subsetting.     

ADVANCES IN SECOND-ORDER CALIBRATION

Several multivariate methods are now available for the calibration of second-order or hyphenatedinstruments(e.g.GC/MS).When applied to bilinear data,it has been shown that calibration can beperformed in the presence of unknown interferences-a significant advantage over first-order calibration.In this paper,non-bilinear rank annihilation(NBRA),a method which has the potential of handling,second-order non-bi-linear data,is studied through theoretical analysis and computer simulation.It isfound that the second-order advantage can be carried over to non-bilinear data if a property defined asnet analyte rank(NAR)holds for the analyte of interest.The net analyte signal(NAS)is definedaccordingly for second-order calibration and the analogy to and difference from lower-order calibrationare discussed.With NAS,some analytical figures of merit such as signal-to noise ratio,selectivity,sensitivity and limit of determination can be calculated for second order calibration.An application toMS/MS data is also given.     

Implementation of the Iterative Proportion Fitting Algorithm for Geostatistical Facies Modeling

In geostatistics, most stochastic algorithm for simulation of categorical variables such as facies or rock types require a conditional probability distribution. The multivariate probability distribution of all the grouped locations including the unsampled location permits calculation of the conditional probability directly based on its definition. In this article, the iterative proportion fitting (IPF) algorithm is implemented to infer this multivariate probability. Using the IPF algorithm, the multivariate probability is obtained by iterative modification to an initial estimated multivariate probability using lower order bivariate probabilities as constraints. The imposed bivariate marginal probabilities are inferred from profiles along drill holes or wells. In the IPF process, a sparse matrix is used to calculate the marginal probabilities from the multivariate probability, which makes the iterative fitting more tractable and practical. This algorithm can be extended to higher order marginal probability constraints as used in multiple point statistics. The theoretical framework is developed and illustrated with estimation and simulation example.     

基于Moran统计量的空间自相关理论发展和方法改进

本文旨在发展基于Moran指数的空间自相关分析理论和方法。首先,利用线性代数知识对基于Moran统计量的空间自相关过程的数学表示进行规范化整理;其次,基于变换中的不变性思想给出Moran指数的理论解释;第三,对空间权重矩阵的数理性质、建设方法和应用范围提出新的见解。总结并发展了Moran指数的三种计算方法——三步求值法、矩阵标度法和回归分析法,将空间权重矩阵划分为四种基本类型——局域关联型、准局域关联型、准长程关联型和长程关联型。以河南省鹤壁市乡镇体系为实证对象,以本文改进的理论和方法为依据,提供了一个空间自相关分析的简明案例。     

DISCRIMINANT PRINCIPAL COMPONENTS ANALYSIS

When the number of variables exceeds the number of samples, one method of multivariate discriminationis to use principal components analysis to reduce the dimensionality and then to perform canonicalvariates analysis (PC-CVA). This paper proposes an alternative approach in which discriminant analysisis carried out by a weighted principal component analysis of the group means (DPCA). This method doesnot require prior data reduction and produces discriminant factors that are orthogonal in the original dataspace. The theory and performance of the two methods are compared. Although the individual factors ofDPCA are found to be less discriminating than PC-CVA, the overall discrimination, calculated bymultivariate analysis of variance, and the predictive value, estimated by the leaving-one-out error rate,are broadly comparable.     

雅砻江上游径流及影响因素关系研究

为研究雅砻江上游径流的变化及其积雪、气温和降水对径流的影响,首先根据相关性将年径流周期分为枯水期、融雪影响期和汛期,其次,结合MODIS 8天积雪产品、研究区气温和降水数据,采用相关分析和归因分析法分析了径流与影响因素的相关性以及各影响因素对径流变化的影响程度,最后用逐步多元回归分析法得出枯水期和融雪影响期径流的预测方程。结果表明：2000-2014年间雅砻江上游径流整体呈上升趋势,冬季积雪面积的减少导致径流减少了24.89%,汛期降水增加导致径流增加了79.38%,采用相关分析和逐步多元回归方法可有效分析径流与影响因素的关系及影响程度。     

SHORT COMMUNICATION PRINCIPAL COMPONENT ANALYSIS AND POLLUTION BY TRACE ELEMENTS IN A MUSSEL SURVEY

The total content of nine trace elements(Mn,Fe,Co,Ni,Cu,Zn,Cd,Hg,Pb)in the soft part of mussels(Mytilus galloprovincialis Lamarck)sampled in two sites was considered.Wild,polluted molluscs weresampled in Muggia Bay(Gulf of Trieste,Northern Adriatic Sea)in the proximity of an important sewerof the city.Edible,unpolluted mussels were simultaneously sampled in a hatchery just off the Bay.Principal component analysis has been applied to correlation matrices obtained from data matrices fromthe literature.The nine variables were reduced to three or two principal components,which explained70-80% of the total variance.The unrotated and orthogonally rotated matrix of the correlations ofvariables with principal components showed that the clusters of elements are positively associated to thefirst two eigenvectors.The origin of some toxic elements in the soft part of mussels from Muggia Bay isdiscussed.The projection on the first two eigenvectors of all data as component scores allow a nearlycomplete separation of polluted from unpolluted molluscs.     

Design and Usability of an Enhanced Geographic Information System for Exploration of Multivariate Health Statistics*

The multidimensional nature of many types of data in modern geography calls for creative and innovative approaches to their analysis. Statisticians have recently developed methods for exploring and visualizing large, multivariate datasets, but cartographers and geographers in general have only recently begun to integrate these methods for use with spatial and spatiotemporal datasets that are multivariate in character. This article will present an example of such an integration—an environment for visualization of health statistics—as a case study to demonstrate the philosophical and practical advantages of geovisualization systems for the exploration of complex spatiotemporal information. Emphasis is placed on the encouragement of creative thinking about geographic phenomena through the use of such data‐rich graphical tools.     

重庆地区MODIS/NDVI时间序列数据重建研究

基于时间分辨率为逐月、空间分辨率为1 km的MODIS/NDVI数据,利用WS、S-G、A-G和D-L这4种重建方法对重庆市2010~2014年间历年逐月的NDVI时间序列数据进行重建,并采用视觉对比、分地类像元对比、相关系数（R）、均方根误差（RMSE）、赤池信息准则（AIC）和贝叶斯信息准则（BIC）对4种重建结果进行了评价。结果表明：S-G和WS重建后噪声少,S-G的曲线最为平滑,A-G与WS保真性较好。其中,A-G的R>0.8和RMSE<0.05的分布面积最大,分别占总面积的89.41%和66.40 %;WS次之,占72.76%和59.37 %。此外,在模型效果分析中A-G的AIC和BIC评价结果最佳,WS在其他3种方法BIC评价结果较差的渝西地区也有较好的评价结果。     

基于路网相关性的分布式增量交通流大数据预测方法

针对城市道路拥堵问题的日益加剧的问题,智能化城市交通管理平台是缓解拥堵问题的有效方法,利用交通流大数据预测结果进行交通诱导,能够指导用户调整出行方案,有效缓解交通压力。研究了交通流大数据的分布式增量聚合方法,对海量交通流数据进行清洗统计,为交通流预测提供数据基础,基于交通流在路网中上下游路段的相关性分析,利用路口转弯率多阶分配将该相关性量化,构建基于路网相关性的空间权重矩阵,完成对于STARIMA模型的改进。通过应用试验证明,该方法能更准确的进行交通流预测,为交通诱导信息发布提供依据。     

基于交通网络空间权重的区域经济空间相关分析——以甘肃省为例

在区域经济中,交通是联系地理空间和区域经济活动的纽带,交通的发达程度决定了各地理单元空间相互作用的广度与深度.针对不同交通模式赋以不同的权重,基于最短加权交通网络,提出新的空间权重矩阵构建方法,构建了交通网络空间权重,与各种传统空间权重一起,对比研究甘肃省各县域单元之间的区域经济的空间相关性.研究结果表明,利用交通网络空间权重生成的空间权重矩阵,能更真实地反映区域间实际的空间过程;甘肃省的区域经济具有空间相关性但不显著,核心城市经济外溢现象不明显.     

USE OF REPLICATION AND SIGNAL-TO-NOISE RATIOS IN THE IDENTIFICATION AND ESTIMATION OF THE COMPOSITION OF LUBRICANT BASESTOCK MIXTURES USING ~(13)C NUCLEAR MAGNETIC RESONANCE SPECTROSCOPY AND PROJECTION INTO PRINCIPAL COMPONENT/CANONICAL VARIATES SPACE

The theory of experimental error in analysis of mixture experiments by abstract factor analysis or targettransformation factor analysis is considered. The theoretical implications of using signal-to-noise ratios(as weights) or canonical variates analysis to reduce the level of imbedded error in the factor model areexamined. The approach is illustrated by application to ~(13)C NMR spectra of lubricant basestockmixtures.