Eigenvector mapping techniques are widely used by ecologists and evolutionary biologists to describe and control for spatial and/or phylogenetic patterns in their data. The selection of an ...appropriate subset of eigenvectors is a critical step (misspecification can lead to highly biased results and interpretations), and there is no consensus yet on how to proceed. We conducted a ten‐year review of the practices of eigenvector selection and highlighted three main procedures: selecting the subset of descriptors minimising the Akaike information criterion (AIC), using a forward selection with double stopping criterion after testing the global model significance (FWD), and selecting the subset minimising the autocorrelation in the model residuals (MIR). We compared the type I error rates, statistical power, and R² estimation accuracy of these methods using simulated data. Finally, a real dataset was analysed using variation partitioning analysis to illustrate to what extent the different selection approaches affected the ecological interpretation of the results. We show that, while the FWD and MIR approaches presented a correct type I error rate and were accurate, the AIC approach displayed extreme type I error rates (100%), and strongly overestimated the R². Moreover, the AIC approach resulted in wrong ecological interpretations, as it overestimated the pure spatial fraction (and the joint spatial‐environmental fraction to a lesser extent) of the variation partitioning. Both the FWD and MIR methods performed well at broad and medium scales but had a very low power to detect fine‐scale patterns. The FWD approach selected more eigenvectors than the MIR approach but also returned more accurate R² estimates. Hence, we discourage any future use of the AIC approach, and advocate choosing between the MIR and FWD approaches depending on the objective of the study: controlling for spatial or phylogenetic autocorrelation (MIR) or describing the patterns as accurately as possible (FWD).
Forward selection of explanatory variables Blanchet, F. Guillaume; Legendre, Pierre; Borcard, Daniel
Ecology (Durham),
September 2008, Letnik:
89, Številka:
9
Journal Article
Recenzirano
This paper proposes a new way of using forward selection of explanatory variables in regression or canonical redundancy analysis. The classical forward selection method presents two problems: a ...highly inflated Type I error and an overestimation of the amount of explained variance. Correcting these problems will greatly improve the performance of this very useful method in ecological modeling. To prevent the first problem, we propose a two-step procedure. First, a global test using all explanatory variables is carried out. If, and only if, the global test is significant, one can proceed with forward selection. To prevent overestimation of the explained variance, the forward selection has to be carried out with two stopping criteria: (1) the usual alpha significance level and (2) the adjusted coefficient of multiple determination ($R_{a}^{2}$) calculated using all explanatory variables. When forward selection identifies a variable that brings one or the other criterion over the fixed threshold, that variable is rejected, and the procedure is stopped. This improved method is validated by simulations involving univariate and multivariate response data. An ecological example is presented using data from the Bryce Canyon National Park, Utah, USA.
Spatial weights matrices used in quantitative geography furnish maps with their individual latent eigenvectors, whose geographic distributions portray distinct spatial autocorrelation (SA) ...components. These polygon patterns on maps have specific meaning, partially in terms of geographic scale, which this article describes. The goal of this description is to enable spatial analysts to better understand and interpret these maps individually, as well as mixtures of them, when accounting for SA in a spatial analysis. Linear combinations of Moran eigenvector maps supply a powerful and relatively simple tool that can explain SA in regression residuals, with an ability to render reasonably accurate reproductions of empirical geographic distributions with or without the aid of substantive covariates. The focus of this article is positive SA, the most commonly encountered nature of autocorrelation in georeferenced data. The principal innovative contribution of this article is establishing a better clarification of what the synthetic SA variates extracted from spatial weights matrices epitomize with regard to global, regional, and local clusters of similar values on a map. This article shows that the Getis-Ord G
i
* statistic provides a useful tool for classifying Moran eigenvector maps into these three qualitative categories, illustrating findings with a range of specimen geographic landscapes.
We develop the necessary methodology to conduct principal component analysis at high frequency. We construct estimators of realized eigenvalues, eigenvectors, and principal components, and provide ...the asymptotic distribution of these estimators. Empirically, we study the high-frequency covariance structure of the constituents of the S&P 100 Index using as little as one week of high-frequency data at a time, and examines whether it is compatible with the evidence accumulated over decades of lower frequency returns. We find a surprising consistency between the low- and high-frequency structures. During the recent financial crisis, the first principal component becomes increasingly dominant, explaining up to 60% of the variation on its own, while the second principal component drives the common variation of financial sector stocks. Supplementary materials for this article are available online.
Co-eigenvector graphs Van Mieghem, Piet; Jokić, Ivan
Linear algebra and its applications,
05/2024, Letnik:
689
Journal Article
Recenzirano
Odprti dostop
Except for the empty graph, we show that the orthogonal matrix X of the adjacency matrix A determines that adjacency matrix completely, but not always uniquely. The proof relies on interesting ...properties of the Hadamard product Ξ=X∘X. As a consequence of the theory, we show that irregular co-eigenvector graphs exist only if the number of nodes N≥6. Co-eigenvector graphs possess the same orthogonal eigenvector matrix X, but different eigenvalues of the adjacency matrix. Co-eigenvector graphs are the dual of co-spectral graphs, that share all eigenvalues of the adjacency matrix, but possess a different orthogonal eigenvector matrix. We deduce general properties of co-eigenvector graph and start to enumerate all co-eigenvector graphs on N=6 and N=7 nodes. Finally, we list many open problems.
Centrality is widely recognized as one of the most critical measures to provide insight into the structure and function of complex networks. While various centrality measures have been proposed for ...single-layer networks, a general framework for studying centrality in multilayer networks (i.e., multicentrality) is still lacking. In this study, a tensor-based framework is introduced to study eigenvector multicentrality, which enables the quantification of the impact of interlayer influence on multicentrality, providing a systematic way to describe how multicentrality propagates across different layers. This framework can leverage prior knowledge about the interplay among layers to better characterize multicentrality for varying scenarios. Two interesting cases are presented to illustrate how to model multilayer influence by choosing appropriate functions of interlayer influence and design algorithms to calculate eigenvector multicentrality. This framework is applied to analyze several empirical multilayer networks, and the results corroborate that it can quantify the influence among layers and multicentrality of nodes effectively.
Determining the principal energy-transfer pathways responsible for allosteric communication in biomolecules remains challenging, partially due to the intrinsic complexity of the systems and the lack ...of effective characterization methods. In this work, we introduce the eigenvector centrality metric based on mutual information to elucidate allosteric mechanisms that regulate enzymatic activity. Moreover, we propose a strategy to characterize the range of correlations that underlie the allosteric processes. We use the V-type allosteric enzyme imidazole glycerol phosphate synthase (IGPS) to test the proposed methodology. The eigenvector centrality method identifies key amino acid residues of IGPS with high susceptibility to effector binding. The findings are validated by solution NMR measurements yielding important biological insights, including direct experimental evidence for interdomain motion, the central role played by helix hα 1, and the short-range nature of correlations responsible for the allosteric mechanism. Beyond insights on IGPS allosteric pathways and the nature of residues that could be targeted by therapeutic drugs or site-directed mutagenesis, the reported findings demonstrate the eigenvector centrality analysis as a general cost-effective methodology to gain fundamental understanding of allosteric mechanisms at the molecular level.
In 1978, a fuzzy eigenequation with max−min product was introduced and studied by Sanchez. He provided us with its maximal solution. In this study, we examine the fuzzy eigenequation A∘x=λ⁎x, where ...⁎ is an increasing operation satisfying some additional assumptions. The assumptions for which the greatest solution exists in the eigenvector family are examined. The notion of max−⁎ subspace in 0,1 is introduced and the properties of this are studied. In particular, the relationships between families of eigenvectors for certain matrices and the results of operations on these matrices are studied.
Buckling-constrained structural design problems have conventionally prioritized optimizing the buckling load factor with less consideration given to the buckling mode shape. In this work, mode shape ...constraints are imposed within a topology optimization problem using an eigenvector aggregate constraint that is a weighted sum of homogeneous quadratic functions of the linearized buckling eigenvectors. A generalized formulation of the eigenvector aggregate is introduced, extending previous work. A new adjoint-based derivative evaluation technique is derived that is valid even in the presence of repeated eigenvalues. Numerical examples, including a clamped beam, a compressed column, and a square plate, demonstrate the effectiveness of the proposed approach. The results show the ability of the eigenvector aggregate to handle repeated eigenvalues, enable design space exploration, and capture mode shape switching.
•Eigenvector aggregates impose constraints on linearized buckling modal displacements.•Hyperbolic tangent generator function aggregates contributions from a range of modes.•Eigenvector aggregate derivatives are computed using a modified adjoint method.•Numerical examples demonstrate methods handle repeated eigenvalues and mode switching.•Designs optimized with the eigenvector aggregate may have better buckling performance.
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