A Generalized Majorization Method for Least Squares Multidimensional Scaling of Pseudodistances that may be Negative

Human land cover can degrade estuaries directly through habitat loss and fragmentation or indirectly through nutrient inputs that reduce water quality. Strong precipitation events are occurring more frequently, causing greater hydrological connectivity between watersheds and estuaries. Nutrient enrichment and dissolved oxygen depletion that occur following these events are known to limit populations of benthic macroinvertebrates and commercially harvested species, but the consequences for top consumers such as birds remain largely unknown. We used non-metric multidimensional scaling (MDS) and structural equation modeling (SEM) to understand how land cover and annual variation in rainfall interact to shape waterbird community composition in Chesapeake Bay, USA. The MDS ordination indicated that urban subestuaries shifted from a mixed generalist-specialist community in 2002, a year of severe drought, to generalist-dominated community in 2003, of year of high rainfall. The SEM revealed that this change was concurrent with a sixfold increase in nitrate-N concentration in subestuaries. In the drought year of 2002, waterbird community composition depended only on the direct effect of urban development in watersheds. In the wet year of 2003, community composition depended both on this direct effect and on indirect effects associated with high nitrate-N inputs to northern parts of the Bay, particularly in urban subestuaries. Our findings suggest that increased runoff during periods of high rainfall can depress water quality enough to alter the composition of estuarine waterbird communities, and that this effect is compounded in subestuaries dominated by urban development. Estuarine restoration programs often chart progress by monitoring stressors and indicators, but rarely assess multivariate relationships among them. Estuarine management planning could be improved by tracking the structure of relationships among land cover, water quality, and waterbirds. Unraveling these complex relationships may help managers identify and mitigate ecological thresholds that occur with increasing human land cover.

A new procedure is discussed which fits either the weighted or simple Euclidian model to data that may (a) be defined at either the nominal, ordinal, interval or ratio levels of measurement; (b) have missing observations; (c) be symmetric or asymmetric; (d) be conditional or unconditional; (e) be replicated or unreplicated; and (f) be continuous or discrete. Various special cases of the procedure include the most commonly used individual differences multidimensional scaling models, the familiar nonmetric multidimensional scaling model, and several other previously undiscussed variants.The procedure optimizes the fit of the model directly to the data (not to scalar products determined from the data) by an alternating least squares procedure which is convergent, very quick, and relatively free from local minimum problems.The procedure is evaluated via both Monte Carlo and empirical data. It is found to be robust in the face of measurement error, capable of recovering the true underlying configuration in the Monte Carlo situation, and capable of obtaining structures equivalent to those obtained by other less general procedures in the empirical situation.

Research shows that college students are deficient in statistical skills. There is a need today for the development of learning tools that can help students overcome their anxiety of statistical applications. This study seeks to respond to that need by describing a student assignment involving multivariate statistics. The assignment integrates aspects of perceptual mapping via multidimensional scaling and multi-attribute models. A feedback questionnaire revealed that the assignment was well-received by students and successful in increasing their knowledge of statistics. Students felt the assignment did increase their knowledge of perceptual maps, multi-attribute models, and multidimensional scaling. Students showed an interest in learning more about statistics and marketing research. Since the assignment increased the student's interest in statistics and willingness to learn and use statistics, findings suggest that the students anxiety regarding statistics was reduced.

This paper discusses how Multidimensional Scaling (MDS) can provide an exploratory technique for identifying major growth profiles, which may be indicative of growth associated with subgroups. It briefly overviews the conventional growth models and growth mixture models, examines the assumptions related to these models, and indicates some limitations associated with these models. It then proposes an exploratory growth profile analysis using the MDS model as a complement to more specification-oriented techniques. It describes the Profile Analysis via Multidimensional Scaling model (PAMS) and extends the model for longitudinal data. The MDS profile model can solve for the growth parameters such that each MDS dimension corresponds to a major growth profile. It is argued that the MDS model provides an exploratory tool for identifying growth trends and studying individual differences with respect to those growth trends. Since MDS has not traditionally been used for longitudinal studies, the MDS growth analysis can serve as the basis for studies of the kind discussed in the paper.

Introduction.- Data management and software.- Advice for teachers.- Exploration.- Linear regression.- Generalised linear modelling.- Additive and generalised additive modelling.- Introduction to mixed modelling.- Univariate tree models.- Measures of association.- Ordination--first encounter.- Principal component analysis and redundancy analysis.- Correspondence analysis and canonical correspondence analysis.- Introduction to discriminant analysis.- Principal coordinate analysis and non-metric multidimensional scaling.- Time series analysis--Introduction.- Common trends and sudden changes.- Analysis and modelling lattice data.- Spatially continuous data analysis and modelling.- Univariate methods to analyse abundance of decapod larvae.- Analysing presence and absence data for flatfish distribution in the Tagus estuary, Portugual.- Crop pollination by honeybees in an Argentinean pampas system using additive mixed modelling.- Investigating the effects of rice farming on aquatic birds with mixed modelling.- Classification trees and radar detection of birds for North Sea wind farms.- Fish stock identification through neural network analysis of parasite fauna.- Monitoring for change: using generalised least squares, nonmetric multidimensional scaling, and the Mantel test on western Montana grasslands.- Univariate and multivariate analysis applied on a Dutch sandy beach community.- Multivariate analyses of South-American zoobenthic species--spoilt for choice.- Principal component analysis applied to harbour porpoise fatty acid data.- Multivariate analysis of morphometric turtle data--size and shape.- Redundancy analysis and additive modelling applied on savanna tree data.- Canonical correspondence analysis of lowland pasture vegetation in the humid tropics of Mexico.- Estimating common trends in Portuguese fisheries landings.- Common trends in demersal communities on the Newfoundland-Labrador Shelf.- Sea level change and salt marshes in the Wadden Sea: a time series analysis.- Time series analysis of Hawaiian waterbirds.- Spatial modelling of forest community features in the Volzhsko-Kamsky reserve.

Big Bayes is the computationally intensive co-application of big data and large, expressive Bayesian models for the analysis of complex phenomena in scientific inference and statistical learning. Standing as an example, Bayesian multidimensional scaling (MDS) can help scientists learn viral trajectories through space-time, but its computational burden prevents its wider use. Crucial MDS model calculations scale quadratically in the number of observations. We partially mitigate this limitation through massive parallelization using multi-core central processing units, instruction-level vectorization and graphics processing units (GPUs). Fitting the MDS model using Hamiltonian Monte Carlo, GPUs can deliver more than 100-fold speedups over serial calculations and thus extend Bayesian MDS to a big data setting. To illustrate, we employ Bayesian MDS to infer the rate at which different seasonal influenza virus subtypes use worldwide air traffic to spread around the globe. We examine 5392 viral sequences and their associated 14 million pairwise distances arising from the number of commercial airline seats per year between viral sampling locations. To adjust for shared evolutionary history of the viruses, we implement a phylogenetic extension to the MDS model and learn that subtype H3N2 spreads most effectively, consistent with its epidemic success relative to other seasonal influenza subtypes. Finally, we provide MassiveMDS, an open-source, stand-alone C++ library and rudimentary R package, and discuss program design and high-level implementation with an emphasis on important aspects of computing architecture that become relevant at scale.

Detecting significant change in stream benthic macroinvertebrate communities in wilderness areas

This review covers metric and nonmetric multidimensional scaling(MDS)studies. Topics reviewed include metric vs. nonmetric methods, similarities, distances and the Minkowski power metrics, distance vs. content methods, the additive constant problem in the2-way and the3-way MDS, computational problems(gradient vs. rank image methods, the alternating least squares method, the N-way Eckart-Young decomposition, maximum likelihood, convex function analysis, continuous transformation, lincar programming), computer programs, individual difference scaling of similarity and preference data(the3-way scaling and factor analysis, the unfolding vs. vector models), Monte Carlo studies, incmplete designs, interactive scaling, integrated approaches, books on MDS, and a quantitative look at main contributors to the MDS literature. Because of the limited space, applications of MDS are not reviewed, and an extensive bibliography on MDS had to be deleted from(2).

Dimension reduction has been used to visualise the distribution of multidimensional microbiome data, but the composite variables calculated by the dimension reduction methods have not been widely used to investigate the relationship of the human gut microbiome with lifestyle and disease. In the present study, we applied several dimension reduction methods, including principal component analysis, principal coordinate analysis (PCoA), non-metric multidimensional scaling (NMDS), and non-negative matrix factorization, to a microbiome dataset from 186 subjects with symptoms of allergic rhinitis (AR) and 106 controls. All the dimension reduction methods supported that the distribution of microbial data points appeared to be continuous rather than discrete. Comparison of the composite variables calculated from the different dimension reduction methods showed that the characteristics of the composite variables differed depending on the distance matrices and the dimension reduction methods. The first composite variables calculated from PCoA and NMDS with the UniFrac distance were strongly associated with AR (FDR adjusted P = 2.4 × 10–4 for PCoA and P = 2.8 × 10–4 for NMDS), and also with the relative abundance of Bifidobacterium and Prevotella. The abundance of Bifidobacterium was also linked to intake of several nutrients, including carbohydrate, saturated fat, and alcohol via composite variables. Notably, the association between the composite variables and AR was much stronger than the association between the relative abundance of individual genera and AR. Our results highlight the usefulness of the dimension reduction methods for investigating the association of microbial composition with lifestyle and disease in clinical research.

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 releves 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.

Faunal lists were compiled for the presence or absence of 94 genera from 16 regions. Four similarity coefficients (Simple Matching, Dice, Jaccard, and Otsuka) were computed for the data matrix in Q mode. The similarity matrices were clustered by both the unweighted pair group method and by single linkage. The dendrograms based on the Simple Matching coefficient are not readily interpretable: coefficients that consider negative matches are unsatisfactory for biogeographic studies. The remaining dendrograms have very similar major structure. Elimination of rare genera, those known from only one locality, reduced the number of genera to 64, but the resulting dendrograms differed little from those based on the full data matrix. With the present data set, the methods used are relatively robust. Ordination of the Dice similarity matrix by principal coordinates analysis and nonmetric multidimensional scaling was also employed. The latter technique produced the best model of the relations between the localities as assessed by correlation between the original distances in multidimensional space and in the reduced 3-dimensional space. In this respect, it is superior to principal coordinates and to both of the clustering techniques that were used. All the methods, however, revealed a broadly similar structure to the data. The nonmetric multidimensional scaling model can be readily interpreted. It confirms the previous division of faunas into these characteristic of the restricted shelf environment and these that inhabited the shelf margin—open-ocean regions. It allows recognition of a North American faunal province that may be divided into two subprovinces, one characteristic of the restricted shelf, the other typically found in the northeastern part of the continent and bearing a shelf margin—open-ocean fauna. We have no strong evidence of a homogeneous faunal belt peripheral to the entire cratonic restricted shelf subprovince. The available evidence shows that the faunas inhabiting this shelf-margin-open-ocean environment were, in contrast, relatively heterogeneous. Dependent on their geographic position, these faunas display, to varying extent, the effect of Asiatic and European influences.

Germination potential of baldcypress (Taxodium distichum) swamp soil seed bank along geographical gradients

This study concerned regularities in trait inference, the manner in which limited information about another person is used to form a consistent total impression. An individual differences model for multidimensional scaling was appraised as a method for faithfully strong both broad culturally- determined standards, and differently-structured points of view. Judgments of likelihood of joint occurrence of trait pairs and measures of cognition were obtained from 139 subjects. Scaling analyses resulted in two points of view, the largest of which yielded dimensions of interpersonal affectivity, harmfulness, and charitableness. S e w a t e analyses of randomly divided groups yielded indices of dimension stability ranging up to .99. The results support the view that the multidimensional scaling model can serve well to explain and order trait inference data.

Estimation of the covariance structure of spatial processes is a fundamental prerequisite for problems of spatial interpolation and the design of monitoring networks. We introduce a nonparametric approach to global estimation of the spatial covariance structure of a random function Z(x, t) observed repeatedly at times ti (i = 1, …, T) at a finite number of sampling stations xi (i = 1, 2, …, N) in the plane. Our analyses assume temporal stationarity but do not assume spatial stationarity (or isotropy). We analyze the spatial dispersions var(Z(xi, t) − Z(xj, t)) as a natural metric for the spatial covariance structure and model these as a general smooth function of the geographic coordinates of station pairs (xi, xj ). The model is constructed in two steps. First, using nonmetric multidimensional scaling (MDS) we compute a two-dimensional representation of the sampling stations for which a monotone function of interpoint distances δij approximates the spatial dispersions. MDS transforms the problem into one for which the covariance structure, expressed in terms of spatial dispersions, is stationary and isotropic. Second, we compute thin-plate splines to provide smooth mappings of the geographic representation of the sampling stations into their MDS representation. The composition of this mapping f and a monotone function g derived from MDS yields a nonparametric estimator of var(Z(xa, t) − Z(xb, t)) for any two geographic locations xa and xb (monitored or not) of the form g(|f(xa ) − f(xb )|). By restricting the monotone function g to a class of conditionally nonpositive definite variogram functions, we ensure that the resulting nonparametric model corresponds to a nonnegative definite covariance model. We use biorthogonal grids, introduced by Bookstein in the field of morphometrics, to depict the thin-plate spline mappings that embody the nature of the anisotropy and nonstationarity in the sample covariance matrix. An analysis of mesoscale variability in solar radiation monitored in southwestern British Columbia demonstrates this methodology.

Pimentel, R. A. (Department of Biological Sciences, California Polytechnic State Univ., San Luis Obispo, California) 1981. A comparative study of data and ordination techniques based on a hybrid swarm of sand verbenas (Abronia Juss.). Syst. Zool., 30:250-267.-The influence of kinds of variables, data errors, standardizations similarity coefficients, and ordination techniques are judged in reference to a model involving hybridization and introgression in three species of sand verbenas, Abronia (Juss.). Various criteria indicate that principal component analysis performs best, but inaccurately, with variable standardization and the correlation matrix, and that nonlinear mapping consistently performs poorly. Excellent results were obtained from principal coordinate analysis with Gower's general similarity coefficient based upon quantitative and multistate variables; a 'diagnostic' character measured with known error; and detailed color evaluation. This and other principal coordinate analysis results were further improved by nonmetric multidimensional scaling. Performance of methods is consistent with the nonlinearity of Abronia data. It is assumed that nonlinearity is common in taxonomic data. [Multivariate analysis; multivariate analysis assumptions; discriminant analysis; classification methods; principal component analysis; principal coQrdinate analysis; nonlinear mapping; nonmetric multidimensional scaling; ordination comparisons; data properties; nonlinearity; sand verbenas, Abronia.]