Multivariate oceanographic sampling

Increasing attention in recent years has been devoted to the application of statistical techniques in the analysis and interpretation of geologic and oceanographic data. Equally important, but less well explored, are methods for efficient experimental design. The theory of linear programming provides plans for optimal sampling of geologic and oceanographic phenomena. Of particular significance are solutions to problems of multivariate sampling. Often, a single field sample may be analyzed for a number of oxides, or a number of minerals, or a number of textural parameters. In general, these variables differ in the degree to which they are diagnostic of changes in the phenomenon of interest, and thus they must be known with different levels of precision if they are to be useful. Similarly, the variables differ in the ease with which they may be measured. If a sampling plan is to be most efficient, it must provide the requisite levels of precision for the minimum expenditure of time and effort. Sampling for a single variable may be optimized directly. Sampling for several variables simultaneously usually introduces special difficulties, but if the objective function can be generalized to hold for all variables, solutions can be determined even in this situation.     

Factor analysis: How it copes with complex geological problems

Factor analysis utilizing textural data from 81 bottom samples was used to analyze the surficial sediments covering a 40,000-sq km area, which is one input data point per 500 km. On the other hand, the surficial geology of the area studied is complex as some map units are only 1 km wide in places. Under these circumstances it is interesting to determine that factor analysis nonetheless aims toward a reasonable geological solution. If the premise is accepted that factor analysis provides a solution best-fitted to the data, the geologist has carried his research one step further and is left with the problem of interpreting the results of factor analysis correctly. In this experiment, the interpretation of the factors representing the gravel and the mud is relatively simple, although the two factors representing sands are more difficult to explain. The proper interpretation of factors leads naturally to an inquiry on the optimum number of factors to use, but this problem can be solved objectively by considering the factor loadings.     

A method for the mathematical analysis of the cross-sectional geometry of tidal inlet channels

Analysis of channel cross sections is hindered by lack of parameters to describe the shape of the cross section. In the situation of a sample of cross sections taken across tidal inlets, if the cross section is expressed as an observation vector, principal-component analysis can be used to derive eigenvectors for the data set. By neglecting eigenvectors that explain little variance, mathematical representation of the original data set is simplified by transformation to the eigenvector space. For 408 cross sections each represented by a 60-component vector, three eigenvectors explain 97.5 percent of the total variance in the data set. The three-dimensional representation simplifies the task of analyzing cross-sectional shape. The physical form of the first three eigenvectors have considerable resemblance to classical types of variation noted in inlet-channel studies. The method is applicable directly to analysis of other fluvial and estuarine channels.     

Trend-surface analysis of distribution of some heavy minerals in shelf sediments off Mangalore, India

The mineralogy of the 0.062–0.125 and 0.125–0.25 mm fractions of 58 selected samples from a set of 78 samples collected at a spacing of 1–2 km in the inner shelf of Mangalore were studied. Trend-surface analysis of distribution of the heavy minerals, hornblende, muscovite, garnet, and sillimanite, were made to determine the various factors controlling the mineral-distribution patterns in the area. Parts of the area showing contrasting environments were analyzed separately to determine the relationship of the regional and local trends and the various factors controlling the trends. The linear, quadratic and cubic trends for hornblende, garnet, and sillimanite were controlled by the source, westerly river flow and southerly currents; for mica, the response to processes has been the main factor. The mineral distribution in this area is mainly the result of dynamic interaction of process and response elements and a process-response model is suggested.Published with permission of the Director General, Geological Survey of India.     

Multivariate discriminant analysis of bioclastic turbidites

A piston core from the southern Blake Basin penetrated nine distinct bioclastic carbonate turbidites separated by pelagic terrigenous clay units. X-ray diffraction analysis of the carbonate mineralogy of the graded, laminated, and homogenous units of the individual turbidite units indicated a general loss of unstable carbonates in the homogenous unit beyond that contained in the lower two units. Univariate and multivariate statistical tests were employed to determine the significance of the changes and to determine if different units of a turbidite sequence could be consistently grouped by statistical methods. Principal components, discriminant function, and cluster analysis were used in the multivariate tests. The graded and laminated units were closely related in mineralogy with little significant differences. The mineralogy of the homogenous units was significantly different than that of the other two units except in one example. Cluster analysis generated five groupings of the samples with the end members consisting of graded and laminated units at one end and homogenous units at the other.     

An algorithm andFortran-iv program for large-scaleQ-mode factor analysis and calculation of factor scores

An algorithm andFortran-iv computer program,cabfac, forQ-mode factor analysis is described. The program will accept up to 1500 items and 50 variables on a moderate-size computer.     

Study of factors influencing variation in size characteristics of fluvioglacial sediments

Factor analysis using promax oblique rotation was used in a study of fluvioglacial sediments of Late Weichsilian (Wurm) age. The use of oblique rotation allowed a more realistic interpretation of the factors and understanding of relationships between sedimentsize classes. At the second-order level, two factors are operatable, one producing variations in the coarse-size range, the other in the fine-size range. The factors are uncorrelated and mutally unrelated. At the first-order level are six factors, four representing aspects of the second-order factor of coarse size and two representing the fine size. The factor producing the variation in the coarse size is the most important one. At the lowest level in the hierarchy scale are the individual size variables. Results show that the number of factors required to account for the variation in a sediment-size data set is a function of the scale at which the problem is examined.     

Principal components analysis of three-way tables

A three-mode principal components method allows visualization of the structural or taxonomic relationships within three-way data tables. The fundamental model includes three sets of eigenvectors and a core matrix relating the principal components of each mode. Formal relationships between the method and the usual principal components formulation allow calculation of loadings and scores for each mode; taken with the core matrix, these provide a number of points of view in graphical analysis of three-mode data. The model compares favorably with alternative formulations in terms of simplicity of computation, generality, and symmetry of operation among the modes. An organic geochemical example illustrates the method.     

An oblique factor analysis solution for the analysis of mixtures

The problems involved in the factor analysis of data consisting of measurements of material quantity or proportion are discussed, and the inability of existing factor analysis solutions to properly model such data is pointed out. A new factor analysis solution using the linear programming technique is presented which properly analyzes mixture data. A numerical example is presented in which a body of suspended sediment grain-size distribution data is analyzed using the new factor analysis solution. FORTRAN-IV subroutines for computing this solution are included in an Appendix.     

Thoughts on use of principal component analysis in petrogenetic problems

The use of principal component analysis in studying chemical trends in volcanic rock suites is described. It is suggested that eigenvectors generated from a correlation matrix, rather than a covariance matrix, could be used in this context. In the latter situation many elements are swamped by silicon's numerical size and range. In the former situation the alkalies and titanium begin to show their true importance.     

Deciphering Groundwater Flow Systems in Oasis Valley, Nevada, Using Trace Element Chemistry, Multivariate Statistics, and Geographical Information System

The origin of groundwater discharging via evapotranspiration and from springs within Oasis Valley, Nevada, is of concern owing to the close proximity of the Nevada Test Site (NTS) and the possible contamination of groundwater as a result of underground nuclear testing. Principal components analysis, cluster analysis, and population partitioning, along with a Geographical Information System, were used to decipher groundwater flow patterns in Oasis Valley, Nevada. These multivariate statistical techniques were applied to the trace element chemistry of groundwater samples collected from 26 springs and wells within Oasis Valley, the NTS, and the Nellis Air Force Range. The results of all statistical analyses showed similar geographical trends in the trace element chemistry of the groundwaters included in this study. Differences are observed between the groundwaters from the NTS and those of Oasis Valley based on the concentrations of the elements Li, Ge, Mo, Rb, Ba, U, and Ru. A concentration gradient is observed from lower concentrations in the NTS to increasing concentrations toward Oasis Valley suggesting groundwater flow in an overall southwestward direction from the NTS. Also, a different trace element signature is observed for the waters collected in the northern and western region of Oasis Valley, suggesting another source of groundwater to this area.     

Principal components analysis using the hypothetical closed array

The application of R-mode principal components analysis to a set of closed chemical data is described using previously published chemical analyses of rocks from Gough Island. Different measures of similarity have been used and the results compared by calculating the correlation coefficients between each of the elements of the extracted eigenvectors and each of the original variables. These correlations provide a convenient measure of the contribution of each variable to each of the principal components. The choice of similarity measure (variance-covariance or correlation coefficient)should reflect the nature of the data and the view of the investigator as to which is the proper weighting of the variables—according to their sample variance or equally. If the data are appropriate for principal components analysis, then the Chayes and Kruskal concept of the hypothetical open and closed arrays and the expected closure correlations would seem to be useful in defining the structure to be expected in the absence of significant departures from randomness. If the data are not multivariate normally distributed, then it is possible that the principal components will not be independent. This may result in significant nonzero covariances between various pairs of principal components.     

A generalization of extendedQ-mode factor analysis to data matrices with variable row sums

The extended method of Q-mode factor analysis developed by Miesch for data matrices with constant row sums is generalized to data matrices with variable row sums. With the algorithm provided it is possible to compute factor scores in the metric of the original data and compute goodness-of-fit statistics and model geological systems unconstrained by constancy of row sums of data points.     

应用主成分分析法研究渗透介质的渗透稳定问题

结合国内多个水电工程的大量渗透试验数据,在多维系统中选择了5项指标:干密度、孔隙比、不均匀系数、渗透系数和临界坡降,采用多元统计中的主成分分析法进行聚类, 运用回归分析建立了砂性土、砂卵石土、断层带物质三类渗透介质的渗透模型.将所建立的断层带物质渗透模型应用到单薄山梁断层带的渗透稳定分析预测上,得到了模型计算结果与渗透试验所得结果相一致的结论,说明主成分分析法在渗透稳定分析中具有一定的适用性.     

藏南硅质岩成岩信息多元统计分析

应用因子分析方法对藏南3条剖面硅质岩的常量、微量元素进行了分析,提取第一、第二、第三主因子,不同程度地反映了陆源组分、自生组分和热水来源组分的特征。结果显示,自彭错林、夏鲁到宗卓组剖面,陆源因子和自生因子所占方差贡献率逐渐升高,热水源因子的方差贡献率大幅下降,指示彭错林硅质岩以热水沉积为主,夏鲁硅质岩属正常沉积叠加热水沉积,宗卓组硅质岩为正常沉积。     

藏南硅质岩成岩信息多元统计分析

应用因子分析方法对藏南3条剖面硅质岩的常量、微量元素进行了分析,提取第一、第二、第三主因子,不同程度地反映了陆源组分、自生组分和热水来源组分的特征。结果显示,自彭错林、夏鲁到宗卓组剖面,陆源因子和自生因子所占方差贡献率逐渐升高,热水源因子的方差贡献率大幅下降,指示彭错林硅质岩以热水沉积为主,夏鲁硅质岩属正常沉积叠加热水沉积,宗卓组硅质岩为正常沉积。     

Multiple group principal component analysis

Common Principal Component Analysis is a generalization of standard principal components to several groups under the rigid mathematical assumption of equality of all latent vectors across groups (i.e., principal component directions), whereas the latent roots are allowed to vary between groups (differing inflations of dispersion ellipsoids). In practice, data that fulfill these strict requirements are relatively rare. Examples from palaeontology are used to illustrate the principles. Compositional data can be made to fit the Common Principal Component (CPC) model by the appropriate logratio covariance matrix.     

悬浮泥沙对藻类生长影响的实测与分析

通过对长江下游镇扬段的牛轭湖—内江的泥沙、水质、光强、藻类等进行现场监测和数据分析,探讨了内江悬浮泥沙对水下光强的衰减规律以及对藻类生长的影响。结果表明:内江水体中存在80%以上的粒径小于75μm的微细悬浮颗粒,其中不易沉降的粘土颗粒占到20%～50%左右,枯水期平均含量为40 mg/L,丰水期平均含量为70 mg/L,而藻细胞生物总量基本都在100万个/L以内,其中绿藻、硅藻为优势种。数据分析表明:泥沙对水下光强的衰减遵循指数函数衰减规律,光衰减系数与泥沙含量呈较好的线性相关性,内江的优势种绿藻、硅藻的生物量与泥沙含量呈显著的指数相关。认为,泥沙通过衰减光强间接地抑制藻类繁殖,这一抑制作用可以由泥沙含量和藻细胞密度之间的关系进行定量。