Global mapping analysis: stochastic approximation for multidimensional scaling.

In this paper, we propose global mapping analysis (GMA) as a new method to solve multidimensional scaling (MDS). By GMA, MDS is done by an online learning rule based on stochastic approximation. GMA need not directly calculate the disparity matrix for carrying out MDS, as Oja's PCA network do not calculate the correlation matrix. So, GMA is expected to be useful for multivariate data analysis on a large scale. Actually, it was verified by numerical experiments based on artificial data that GMA can work well even if the number of the attribute N is quite large (N=10,000.)     

Biplots for multifactorial analysis of distance

Many data sets in practice fit a multivariate analysis of variance (MANOVA) structure, but do not accord with MANOVA assumptions for their analysis. One way forward is to calculate the matrix of dissimilarities or distances between every pair of individuals, and then to conduct an analysis of distance on the resulting data. Various metric scaling plots can be used to interpret the results of the analysis. However, developments to date of this approach have focused mainly on the individuals in the sample, and little attention has been paid to the assessment of influence of the original variables on the results. The present article attempts to rectify this omission. We discuss the inclusion of biplots on all forms of metric scaling representations in the analysis of distance. Exact biplots will often be nonlinear so we propose a simple linear approximation, and contrast it with other simple linear possibilities. An example from ecology illustrates the methodology.     

Multidimensional scaling for large genomic data sets

Background  

Multi-dimensional scaling (MDS) is aimed to represent high dimensional data in a low dimensional space with preservation of the similarities between data points. This reduction in dimensionality is crucial for analyzing and revealing the genuine structure hidden in the data. For noisy data, dimension reduction can effectively reduce the effect of noise on the embedded structure. For large data set, dimension reduction can effectively reduce information retrieval complexity. Thus, MDS techniques are used in many applications of data mining and gene network research. However, although there have been a number of studies that applied MDS techniques to genomics research, the number of analyzed data points was restricted by the high computational complexity of MDS. In general, a non-metric MDS method is faster than a metric MDS, but it does not preserve the true relationships. The computational complexity of most metric MDS methods is over O(N ² ), so that it is difficult to process a data set of a large number of genes N, such as in the case of whole genome microarray data.     

生态学中的尺度问题——尺度上推

尺度推绎是生态学理论和应用的核心。如何在一个异质景观中进行尺度推绎仍然是一个悬而未决的科学难题,是对当今生态学家在全球变化背景下研究环境问题的重大挑战。就目前的研究,一般可分为四大类尺度推绎途径:空间分析法(如分维分析法和小波分析法)、基于相似性的尺度上推方法、基于局域动态模型的尺度上推方法、随机(模型)法。基于相似性的尺度上推方法来源于生物学上的异量关联,可将其思想延伸至空间上,研究物种丰富度、自然河网、地形特征、生态学格局或过程变量和景观指数等。基于局域动态模型的尺度上推方法需要首先确定是否进行跨尺度推绎,以及是否考虑空间单元之间的水平相互作用和反馈,然后再应用具体的方法或途径,如简单聚合法、有效值外推法、直接外推法、期望值外推、显式积分法和空间相互作用模拟法等。随机(模型)法以其它尺度上推方法为基础,根据研究的是单个景观,还是多个景观,采用不同的途径。理解、定量和降低尺度推绎结果的不确定性已经变得越来越重要,但相关研究仍然极少。以上所有有关尺度推绎的方法、途径和结果分析共同构成了尺度推绎的概念框架。     

DISSIMILARITIES IN COGNITION OF FLAVOUR TERMS RELATED TO VARIOUS SENSORY LABORATORIES IN A MULTIVARIATE STUDY

A study of dissimilarities in cognitive perception of 20 common flavor terms was carried out by the Sensory Group of Norway. An average cognitive pattern of the flavor terms was revealed from multidimensional scaling (MDS) and cluster analysis (CLU). In general, small but interesting deviations between different sensory laboratories working with various food products were found by a multivariate pattern recognition technique based on principal component analysis (PCA). Suggestions for finding general reference standards for flavor terms are discussed.     

A Quantitative Comparison of the Similarity between Genes and Geography in Worldwide Human Populations

Multivariate statistical techniques such as principal components analysis (PCA) and multidimensional scaling (MDS) have been widely used to summarize the structure of human genetic variation, often in easily visualized two-dimensional maps. Many recent studies have reported similarity between geographic maps of population locations and MDS or PCA maps of genetic variation inferred from single-nucleotide polymorphisms (SNPs). However, this similarity has been evident primarily in a qualitative sense; and, because different multivariate techniques and marker sets have been used in different studies, it has not been possible to formally compare genetic variation datasets in terms of their levels of similarity with geography. In this study, using genome-wide SNP data from 128 populations worldwide, we perform a systematic analysis to quantitatively evaluate the similarity of genes and geography in different geographic regions. For each of a series of regions, we apply a Procrustes analysis approach to find an optimal transformation that maximizes the similarity between PCA maps of genetic variation and geographic maps of population locations. We consider examples in Europe, Sub-Saharan Africa, Asia, East Asia, and Central/South Asia, as well as in a worldwide sample, finding that significant similarity between genes and geography exists in general at different geographic levels. The similarity is highest in our examples for Asia and, once highly distinctive populations have been removed, Sub-Saharan Africa. Our results provide a quantitative assessment of the geographic structure of human genetic variation worldwide, supporting the view that geography plays a strong role in giving rise to human population structure.     

Simplification and shift in cognition of political difference: applying the geometric modeling to the analysis of semantic similarity judgment

Perceiving differences by means of spatial analogies is intrinsic to human cognition. Multi-dimensional scaling (MDS) analysis based on Minkowski geometry has been used primarily on data on sensory similarity judgments, leaving judgments on abstractive differences unanalyzed. Indeed, analysts have failed to find appropriate experimental or real-life data in this regard. Our MDS analysis used survey data on political scientists' judgments of the similarities and differences between political positions expressed in terms of distance. Both distance smoothing and majorization techniques were applied to a three-way dataset of similarity judgments provided by at least seven experts on at least five parties' positions on at least seven policies (i.e., originally yielding 245 dimensions) to substantially reduce the risk of local minima. The analysis found two dimensions, which were sufficient for mapping differences, and fit the city-block dimensions better than the Euclidean metric in all datasets obtained from 13 countries. Most city-block dimensions were highly correlated with the simplified criterion (i.e., the left-right ideology) for differences that are actually used in real politics. The isometry of the city-block and dominance metrics in two-dimensional space carries further implications. More specifically, individuals may pay attention to two dimensions (if represented in the city-block metric) or focus on a single dimension (if represented in the dominance metric) when judging differences between the same objects. Switching between metrics may be expected to occur during cognitive processing as frequently as the apparent discontinuities and shifts in human attention that may underlie changing judgments in real situations occur. Consequently, the result has extended strong support for the validity of the geometric models to represent an important social cognition, i.e., the one of political differences, which is deeply rooted in human nature.     

An interdisciplinary evaluation of the status and health of African lake fisheries using a rapid appraisal technique

Data from biological, economic, sociological, and technological attribute lists for 32 African lake fisheries were analysed with multivariate statistics. Multidimensional scaling (MDS) was used to create two-dimensional graphic ordinations of the fisheries for each of these four attributes lists. An overall MDS ordination was also generated, based on the fisheries' scores in the four ordinations. Groupings in each MDS ordination were achieved through cluster analysis. Multiple correlations was used to help determine the attributes which were most important in creating the MDS ordinations. This work showed that relatively fast and simple assessments of the status of African lake and other fisheries can be achieved. The technique also provided insight to help resolve the confounding information often associated with fisheries assessment. Diagnostic information may also be a product of such investigations as this research identified African lake fisheries at risk of declining standards. Indeed, many of the fisheries studied demonstrated similar interdisciplinary symptoms of fisheries decline as other small-scale tropical fisheries.     

Molecular marker diversity of SCN-resistant sources in soybean.

Soybean cyst nematode (SCN) (Heterodera glycines Ichinohe; HG) is one of the most destructive pests of soybean (Glycine max (L.) Merr.) in the United States. Over 100 SCN-resistant accessions within the USDA Soybean Germplasm Collection have been identified, but little is known about the genetic diversity of this SCN-resistant germplasm. The objective of this research was to evaluate the genetic variation and determine the genetic relationships among SCN-resistant accessions. One hundred twenty-two genotypes were evaluated by 85 simple sequence repeat (SSR) markers from 20 linkage groups. Non-hierarchical (VARCLUS) and hierarchical (Ward's) clustering were combined with multidimensional scaling (MDS) to determine relationships among tested lines. The 85 SSR markers produced 566 allelic fragments with a mean polymorphic information content (PIC) value of 0.35. The 122 lines were grouped into 7 clusters by 2 different clustering methods and the MDS results consistently corresponded to the assigned clusters. Assigned clusters were dominated by genotypes that possess one or more unique SCN resistance genes and were associated with geographical origins. The results of analysis of molecular variance (AMOVA) showed that the variation differences among clusters and individual lines were significant, but the differences among individuals within clusters were not significant.     

Applying non-metric multidimensional scaling analysis for measuring the assemblage change of Chu-Lan Creek dike in Taiwan

Ensuring that water resources development in harmony with aquatic environment is the major water policy of Taiwan in the 21st century, Taiwan's water authority has adopted several methodologies, such as utilizing ecological engineering techniques, establishing integrated water resources management configuration, involving the public in decision-making processes, etc., and applying them in the field. Significant consequences in ecological engineering were obtained in several cases, such as for Ta-Chia Creek, where wire cages, tires, and boulders were installed to improve the stream habitat, and for Wu-Lao Creek, where natural water purification facilities were constructed to reduce river pollution. Although the sustainable methodologies have been widely accepted by hydraulic engineers in Taiwan, lack of engineer-friendly evaluation indices or methods hindered the further progress of river ecological engineering projects. This research applied a non-metric multidimensional–scaling (MDS) analysis to measure the assemblage change of river aquatic habitat. A dike construction project at Chu-Lan Creek was selected for verification in this study. The analyzed results showed that the dike construction project did affect the aquatic habitat in Chu-Lan Creek. The proposed MDS analysis successfully captured the effect of the construction. The MDS method could be used to evaluate the improvement or damage of aquatic habitat by a traditional hydraulic approach or a new ecological hydraulic developed technique in Taiwan.     

Improving the efficiency of multidimensional scaling in the analysis of high-dimensional data using singular value decomposition

MOTIVATION: Multidimensional scaling (MDS) is a well-known multivariate statistical analysis method used for dimensionality reduction and visualization of similarities and dissimilarities in multidimensional data. The advantage of MDS with respect to singular value decomposition (SVD) based methods such as principal component analysis is its superior fidelity in representing the distance between different instances specially for high-dimensional geometric objects. Here, we investigate the importance of the choice of initial conditions for MDS, and show that SVD is the best choice to initiate MDS. Furthermore, we demonstrate that the use of the first principal components of SVD to initiate the MDS algorithm is more efficient than an iteration through all the principal components. Adding stochasticity to the molecular dynamics simulations typically used for MDS of large datasets, contrary to previous suggestions, likewise does not increase accuracy. Finally, we introduce a k nearest neighbor method to analyze the local structure of the geometric objects and use it to control the quality of the dimensionality reduction. RESULTS: We demonstrate here the, to our knowledge, most efficient and accurate initialization strategy for MDS algorithms, reducing considerably computational load. SVD-based initialization renders MDS methodology much more useful in the analysis of high-dimensional data such as functional genomics datasets.     

EVALUATION AND APPLICATIONS OF ODOR PROFILING

An odor profiling procedure was developed based on the ASTM odor profiling method. This modified procedure involved using approximately twenty panelists. Panel sessions and data collection were controlled by computer. The results obtained by this panel compared favorably to results obtained by the ASTM panel for which 150 panelists evaluated each compound, indicating that a small panel can be used to produce replicable results. Statistical methods of finding similarities and dissimilarities among compounds using profile data are discussed and compared to results from a multidimensional scaling (MDS) study in which degrees of differences among compounds were judged directly. These results indicate that profile data can be used to define and map the degree of similarity/dissimilarity among compounds, as well as to define the sensory dimensions on which these compounds differ. The use of factor analysis to study the underlying sensory dimensions of the odor space is also discussed. It is hoped that this type of research will lead to a better understanding of the underlying dimensions used to describe odorants.     

Ordination of boreal heath-like vegetation with principal component analysis,correspondence analysis and multidimensional scaling

Four relatively homogeneous field data sets were analyzed, representing boreal, heath-like forest-floor and rock vegetation in Finland, corresponding to Finnish Calluna and Cladina site types. The methods used were principal component analysis (PCA) of covariance matrices, orthogonal correspondence analysis or reciprocal averaging (RA), detrended correspondence analysis (DCA), and linear and nonmetric multidimensional scaling (MDS). RA and DCA gave ordinations in which every species had nearly equal weight. MDS and PCA gave results determined mostly by a few dominant species. MDS and PCA ordinations were very similar to RA and DCA ones when the original data were standardized so that for each species the mean of positive occurrences was the same while quantitative differences within species were retained. RA and PCA were generally very good and reliable, providing that the impact of rare species and outlier relevés was removed in RA. DCA was slightly less reliable than RA. MDS was sensitive to uneven sampling patterns and was the least reliable method compared.     

早期生命形态演化研究中的定量方法

定量古生物学是现代古生物学的一个分支,提倡用定量的手段来研究地质历史时期生命的演化过程。我国从事定量古生物研究的群体较小,特别是对前寒武纪早期生命演化的定量研究还没有系统地展开。这篇文章将主要介绍如何利用定量手段来研究前寒武纪化石的形态演化。对于前寒武纪化石,由于大部分化石分类属性的不确定性,通常使用几何性状对化石的最基本形态结构进行分析,并用存在/缺失(1/0)这种离散变量对每个性状进行量化。非参数多维标量分析方法[Non-parametric multidimensional scaling analysis(MDS)]可以将高维度的离散数据投影到二维或者三维的形态空间上,进而探讨生物群在形态空间中所占有的范围;由离散变量计算得出的生物群的表形分异度(morphological disparity)可以用MDS方差或者平均差异参数[Mean dissimilarity coeffi-cient(MDC)]来计算。形态空间的范围(morphospace range)和表形分异度是相互联系的,如果形态空间范围是固定的,那么表形分异度实际上代表了生物群在形态空间中的分布密度。在解释数据之前,需要对可能存在的样本效应进行测试。常用的方法包括稀释法(rarefaction)、随机取样法(randomization)和自举法(bootstrapping)等。为了帮助读者进一步了解这些方法的使用,文中列举了三个实例:伊迪卡拉生物的形态演化,元古代宏观藻类的形态演化和元古代及寒武纪疑源类的演化。     

Detecting significant change in stream benthic macroinvertebrate communities in wilderness areas

A major challenge in the biological monitoring of stream ecosystems in protected wilderness areas is discerning whether temporal changes in community structure are significantly outside of a reference condition that represents natural or acceptable annual variation in population cycles. Otherwise sites could erroneously be classified as impaired. Long-term datasets are essential for understanding these trends, to ascertain whether any changes in community structure significantly beyond the reference condition are permanent shifts or with time move back to within previous limits. To this end, we searched for long-term (>8 years) quantitative data sets of macroinvertebrate communities in wadeable rivers collected by similar methods and time of year in protected wilderness areas with minimal anthropogenic disturbance. Four geographic areas with datasets that met these criteria in the USA were identified, namely: McLaughlin Nature Reserve in California (1 stream), Great Smoky Mountains National Park in Tennesse-North Carolina (14 streams), Wind River Wilderness Areas in Wyoming (3 streams) and Denali National Park and Preserve in Alaska (6 streams).Two statistical approaches were applied: Taxonomic Distinctness (TD) to describe changes in diversity over time and non-metric multidimensional scaling (MDS) to describe changes over time in community persistence (Jaccards Index) and community stability (Bray–Curtis Index). Control charts were used to determine if years in MDS plots were significantly outside a reference condition. For Hunting Creek, TD showed three years outside natural variation which could be attributed to severe hydrological events but years outside the natural-variation funnel at sites in other geographical areas were inconsistent and could not be explained by environmental variables. TD identified simulated severe pollutant events which caused the removal of entire invertebrate assemblages but not simulated water temperature shifts.Within a region, both MDS analyses typically identified similar years as exceeding reference condition variation, illustrating the utility of the approach for identifying wider spatial scale effects that influence more than one stream. MDS responded to both simulated water temperature stress and a pollutant event, and generally outlying years on MDS plots could be explained by environmental variables, particularly higher precipitation. Multivariate control charts successfully identified whether shifts in community structure identified by MDS were significant and whether the shift represented a press disturbance (long-term change) or a pulse disturbance. We consider a combination of TD and MDS with control charts to be a potentially powerful tool for determining years significantly outside of a reference condition variation.     

Modeling fractal structure of city-size distributions using correlation functions

Zipf's law is one the most conspicuous empirical facts for cities, however, there is no convincing explanation for the scaling relation between rank and size and its scaling exponent. Using the idea from general fractals and scaling, I propose a dual competition hypothesis of city development to explain the value intervals and the special value, 1, of the power exponent. Zipf's law and Pareto's law can be mathematically transformed into one another, but represent different processes of urban evolution, respectively. Based on the Pareto distribution, a frequency correlation function can be constructed. By scaling analysis and multifractals spectrum, the parameter interval of Pareto exponent is derived as (0.5, 1]; Based on the Zipf distribution, a size correlation function can be built, and it is opposite to the first one. By the second correlation function and multifractals notion, the Pareto exponent interval is derived as [1, 2). Thus the process of urban evolution falls into two effects: one is the Pareto effect indicating city number increase (external complexity), and the other the Zipf effect indicating city size growth (internal complexity). Because of struggle of the two effects, the scaling exponent varies from 0.5 to 2; but if the two effects reach equilibrium with each other, the scaling exponent approaches 1. A series of mathematical experiments on hierarchical correlation are employed to verify the models and a conclusion can be drawn that if cities in a given region follow Zipf's law, the frequency and size correlations will follow the scaling law. This theory can be generalized to interpret the inverse power-law distributions in various fields of physical and social sciences.     

Pattern in the Co-occurrence of Fishes Inhabiting the Coral Reefs of Bonaire,Netherlands Antilles

Synopsis We conducted an analysis of species associations using fish diversity and abundance surveys conducted in Bonaire Marine Park by recreational divers. We used data from the REEF (Reef Environmental Education Foundation) Fish Survey Project to compute Bray–Curtis similarity coefficients for all species pairs for the 100 most abundant species. We quantified relationships between species using hierarchical agglomerative clustering and non-metric multidimensional scaling (MDS) of the matrix of Bray–Curtis similarity coefficients. We identified three clusters of species from the analysis. MDS results showed species clusters occupied distinct regions across a continuous gradient of species in two-dimensional space, rather than form distinct clusters. While differences in habitat requirements can explain some of the pattern in pairwise species interactions, these results suggest that there are significant direct and indirect behavioral interactions mediating the distribution and abundance of species. Studies conducted to elucidate patterns of species-habitat relationships have been central to conservation planning for marine protected areas (MPAs). However, the role of behavioral interactions between species driving the dynamics of species composition within MPA networks, designed for representation of biological diversity, should be considered when selecting sites in order to be effective.