Nonparametric Estimation of Nonstationary Spatial Covariance Structure

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.

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.

BackgroundAlzheimer disease (AD) has a long preclinical phase with pathophysiological changes such as β‐amyloid (Aβ) accumulation in early stages followed by tau deposition. Graph theory may provide additional insights into the disease processes underlying patterns of Aβ and tau deposition (Pereira et al., 2018; Hoenig,2018). This project uses graph theory to study the spatial covariance structure of Aβ‐PET and tau‐PET changes across disease stages.MethodThis study includes 254 cognitively normal participants (Clinical Dementia Rating™ (CDR™) =0) and 41 impaired participants (CDR>0) from the Washington University Knight Alzheimer Disease Research Center (ADRC, Funding: P50AG005681; P01AG003991; P01AG026276), who underwent MRI, Aβ‐PET 18F‐AV45, and tau‐PET (18F‐AV1451). Regions of interest were defined using the Freesurfer Desikin atlas. Regional Aβ and tau standardized uptake value ratios (SUVRs) were adjusted by gender and age, and used to generate Pearson correlation matrix. Aβ and tau spatial covariance structure can be built as a collection of nodes representing the mean SUVR values from different brain regions, connected by edges representing correlated deposition quantity. BRAPH toolbox (Mijalkov, 2017) were used to calculate Binary Undirected Density (BUD) graph measures. To test for significant differences between these two groups, BRAPH performs non‐parametric permutation tests. Global graph theory measures (global and local efficiency, and modularity) are used to assess the spatial covariance structure of Aβ‐PET and tau‐PET changes.ResultNo significant difference between the CDR=0 and CDR>0 groups was observed for the global measures generated from the Aβ‐PET data (Table 1 and Figure 1A‐C). However, there were significant differences between the two groups for the global measures generated from the tau‐PET data (Table 1 and Figure 1D‐F). The global and local efficiency were lower in the CDR>0 group (p=0.003 and p=0.018, respectively) indicating a higher global and local path length in the covariance structure. Moreover, the modularity was lower (p=0.005) for CDR>0 group, indicating lower fragmentation between regions in this group.ConclusionChanges in spatial correlation associated with conversion to clinical impairment are happening in the tau pattern. These spatial covariance structure changes need further evaluation to assess their utility as a biomarker of disease progression in the preclinical phase of AD.

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.

Climate variability occurs over wide ranges of spatial and temporal scales. It exhibits a complex spatial covariance structure, which depends on geographic location (e.g., tropics vs extratropics) and also consists of a superposition of (i) components with gradually decaying positive correlation functions and (ii) teleconnections that often involve anticorrelations. In addition, there are indications that the spatial covariance structure depends on frequency. Thus, a comprehensive assessment of the spatiotemporal covariance structure of climate variability would require an extensive set of statistical diagnostics. Therefore, it is often desirable to characterize the covariance structure by a simple summarizing metric that is easy to compute from datasets. Such summarizing metrics are useful, for example, in the context of comparisons between climate models or between models and observations. Here we introduce a frequency-dependent version of a simple measure of the effective spatial degrees of freedom. The measure is based on the temporal variance of the global average of some climate variable, and its novel aspect consists in its frequency dependence. We also provide a clear geometric interpretation of the measure. Its easy applicability is demonstrated using near-surface temperature and precipitation fields obtained from a paleoclimate model simulation. This application reveals a distinct scaling behavior of the spatial degrees of freedom as a function of frequency, ranging from monthly to millennial scales.

The usual convergence proof of the SMACOF algorithm model for least squares multidimensional scaling critically depends on the assumption of nonnegativity of the quantities to be fitted, called the pseudodistances. When this assumption is violated, erratic convergence behavior is known to occur. Three types of circumstances in which some of the pseudodistances may become negative are outlined: nonmetric multidimensional scaling with normalization on the variance, metric multidimensional scaling including an additive constant, and multidimensional scaling under the city-block distance model. A generalization of the SMACOF method is proposed to resolve the difficulty that is based on the same rationale frequently involved in robust fitting with least absolute residuals.

Thousands of permanent GNSS stations sample nowadays the 3D deformation of the Earth’s surface. The temporal covariance structure of the field of GNSS station displacements is well characterized and modelled. On the other hand, there lacks a general agreed-upon model of its spatial covariance structure, in part because the theory of random vector fields on the sphere remains hardly developed.In this contribution, we show how the well-established theory of random isotropic scalar fields on the sphere generalizes to the case of vector fields. We derive in particular a spectral representation of random isotropic vector fields on the sphere in the domain of vector spherical harmonics, from which several properties of their covariance functions follow. We then present several parametric families of covariance functions which could be used to describe and model the covariance structure of vector fields on the sphere, such as GNSS station displacements.Although this presentation focuses on theoretical aspects, it is given with future practical applications in mind. A realistic spatio-temporal covariance model of GNSS station displacements could indeed benefit different problems such as the estimation of long-term GNSS station velocities, the identification and mitigation of offsets in GNSS station position time series, the filtering of spatial “common modes” to isolate local deformation, or the spatial interpolation of GNSS station displacements into global maps. Applications may also be found in other domains involving vector quantities distributed on a sphere, e.g., winds, ocean currents, magnetic anomalies, etc.

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.]

A new cross-validation criterion for selecting covariance structures in multivariate analysis is introduced. The criterion relies on the use of the Frobenius matrix distance as discrepancy function between structure-based and sample covariance matrices. Its implementation only requires to specify i) the candidate covariance structures, and ii) an arbitrary structure-dependent estimator of the population covariance whose performance is assumed to be of primary interest. No further statistical specification is needed, which makes its use feasible in a wide range of practical situations. The cross- validatory choice through the minimization of this criterion has a simple statistical interpretation, as seeking the covariance structure for which the given estimator achieves minimum (generalized) mean squared error. Simulation results are presented to show the performance of the introduced method in a variety of factor covariance structures including settings with different eigenvalues, sample sizes and numbers of factors.

Acacia decurrens, originally from Australia, is a woody plant that has become a concern in the erupted areas of Merapi volcano in 2006. The purpose of this study was to quantitatively describe the ecology of Acacia decurrens, and its invasive potential. Vegetation analysis was carried out in four areas of Mount Merapi National Park (TNGM), namely Kalikuning, Kaliadem, Plawangan and Pranajiwa. Ordination using the Non Metric Multidimensional Scaling (NMDS) method and Canonical Correspondence (CCA) and Spearman's bivariate correlation were carried out in data analysis. The results of NMDS analysis (2D stress = 0.14) showed that open areas due to eruption in Kaliadem are now dominated by A. decurrens. The results of the analysis also showed a significant negative correlation (Spearman rho = 0.6) between the abundance of A. decurrens and the level of species diversity in the sampling location. From the results of CCA, A. decurrens in the seedling stage appeared to coexist with other types of ground cover such as Alangium javanicum and Araliaceae. But in the tree phase, this species tended to form pure stands and only occasionally appeared to live side by side with other types of Fabaceae such as Albizia lopantha. Tree-level A. decurrens seemed to prefer sites with lower pH levels while A. decurrens seedling levels were more commonly found in sites with higher pH. Acacia decurrens have the potential to become a weed in TNGM due to its dominant distribution and tendency to decrease the level of species diversity in TNGM.

Dissimilarity and the Classification of Female Singing Voices: A Preliminary Study

Preface. Acknowledgments. 1 Introduction. 1.1 The nature of paleontological data. 1.2 Advantages and pitfalls of paleontological data analysis. 1.3 Software. 2 Basic statistical methods. 2.1 Introduction. 2.2 Statistical distributions. 2.3 Shapiro-Wilk test for normal distribution. 2.4 F test for equality of variances. 2.5 Student's t test and Welch test for equality of means. 2.6 Mann-Whitney U test for equality of medians. 2.7 Kolmogorov-Smirnov test for equality of distributions. 2.8 Permutation and resampling. 2.9 One-way ANOVA. 2.10 Kruskal-Wallis test. 2.11 Linear correlation. 2.12 Non-parametric tests for correlation. 2.13 Linear regression. 2.14 Reduced major axis regression. 2.15 Nonlinear curve fitting. 2.16 Chi-square test. 3 Introduction to multivariate data analysis. 3.1 Approaches to multivariate data analysis. 3.2 Multivariate distributions. 3.3 Parametric multivariate tests. 3.4 Non-parametric multivariate tests. 3.5 Hierarchical cluster analysis. 3.5 K-means cluster analysis. 4 Morphometrics. 4.1 Introduction. 4.2 The allometric equation. 4.3 Principal components analysis (PCA). 4.4 Multivariate allometry. 4.5 Discriminant analysis for two groups. 4.6 Canonical variate analysis (CVA). 4.7 MANOVA. 4.8 Fourier shape analysis. 4.9 Elliptic Fourier analysis. 4.10 Eigenshape analysis. 4.11 Landmarks and size measures. 4.12 Procrustean fitting. 4.13 PCA of landmark data. 4.14 Thin-plate spline deformations. 4.15 Principal and partial warps. 4.16 Relative warps. 4.17 Regression of partial warp scores. 4.18 Disparity measures. 4.19 Point distribution statistics. 4.20 Directional statistics. Case study: The ontogeny of a Silurian trilobite. 5 Phylogenetic analysis. 5.1 Introduction. 5.2 Characters. 5.3 Parsimony analysis. 5.4 Character state reconstruction. 5.5 Evaluation of characters and tree topologies. 5.6 Consensus trees. 5.7 Consistency index. 5.8 Retention index. 5.9 Bootstrapping. 5.10 Bremer support. 5.11 Stratigraphical congruency indices. 5.12 Phylogenetic analysis with Maximum Likelihood. Case study: The systematics of heterosporous ferns. 6 Paleobiogeography and paleoecology. 6.1 Introduction. 6.2 Diversity indices. 6.3 Taxonomic distinctness. 6.4 Comparison of diversity indices. 6.5 Abundance models. 6.6 Rarefaction. 6.7 Diversity curves. 6.8 Size-frequency and survivorship curves. 6.9 Association similarity indices for presence/absence data. 6.10 Association similarity indices for abundance data. 6.11 ANOSIM and NPMANOVA. 6.12 Correspondence analysis. 6.13 Principal Coordinates analysis (PCO). 6.14 Non-metric Multidimensional Scaling (NMDS). 6.15 Seriation. Case study: Ashgill brachiopod paleocommunities from East China. 7 Time series analysis. 7.1 Introduction. 7.2 Spectral analysis. 7.3 Autocorrelation. 7.4 Cross-correlation. 7.5 Wavelet analysis. 7.6 Smoothing and filtering. 7.7 Runs test. Case study: Sepkoski's generic diversity curve for the Phanerozoic. 8 Quantitative biostratigraphy. 8.1 Introduction. 8.2 Parametric confidence intervals on stratigraphic ranges. 8.3 Non-parametric confidence intervals on stratigraphic ranges. 8.4 Graphic correlation. 8.5 Constrained optimisation. 8.6 Ranking and scaling. 8.7 Unitary Associations. 8.8 Biostratigraphy by ordination. 8.9 What is the best method for quantitative biostratigraphy?. Appendix A: Plotting techniques. Appendix B: Mathematical concepts and notation. References. Index

The data open sharing provides us opportunity to investigate mechanisms underlying autism spectrum disorder (ASD) by employing advanced techniques. In the current study, we employed structural covariance (SC) model to investigate the development of triple networks in ASD. Three hundred and seven ASD and 337 typical controls were collected and further classified into four distinct age cohorts. Night brain seeds belonging to default modal network, salience network, and central executive network were obtained. SC between those seeds, as well as its topological properties, was calculated for each group within each age cohort. Statistical analysis revealed that ASD had dynamic reconfigurations of SC of the triple networks, especially the right fronto-insular cortex. The results might indicate that ASD had specific mechanism within distinct age cohort. Additionally, the big data sharing, together with the SC modal, was able to facilitate understanding of the mechanism underlying ASD.