Theoretical Analysis of Principal Components in an Umbrella Model of Intraspecific Evolution

Principal component analysis (PCA) is one of the most frequently-used approach to describe population structure from multilocus genotype data. Regarding geographic range expansions of modern humans, interpretations of PCA have, however, been questioned, as there is uncertainty about the wave-like patterns that have been observed in principal components. It has indeed been argued that wave-like patterns are mathematical artifacts that arise generally when PCA is applied to data in which genetic differentiation increases with geographic distance. Here, we present an alternative theory for the observation of wave-like patterns in PCA. We study a coalescent model -- the umbrella model -- for the diffusion of genetic variants. The model is based on a hierarchy of splits from an ancestral population without any particular geographical structure. In the umbrella model, splits occur almost continuously in time, giving birth to small daughter populations at a regular pace. Our results provide detailed mathematical descriptions of eigenvalues and eigenvectors for the PCA of sampled genomic sequences under the model. Removing variants uniquely represented in the sample, the PCA eigenvectors are defined as cosine functions of increasing periodicity, reproducing wave-like patterns observed in equilibrium isolation-by-distance models. Including rare variants in the analysis, the eigenvectors corresponding to the largest eigenvalues exhibit complex wave shapes. The accuracy of our predictions is further investigated with coalescent simulations. Our analysis supports the hypothesis that highly structured wave-like patterns could arise from genetic drift only, and may not always be artificial outcomes of spatially structured data. Genomic data related to the peopling of the Americas are reanalyzed in the light of our new theory.

Recurrence eigenvalues of movements from brain signals

The ability to characterize muscle activities or skilled movements controlled by signals from neurons in the motor cortex of the brain has many useful implications, ranging from biomedical perspectives to brain–computer interfaces. This paper presents the method of recurrence eigenvalues for differentiating moving patterns in non-mammalian and human models. The non-mammalian models of Caenorhabditis elegans have been studied for gaining insights into behavioral genetics and discovery of human disease genes. Systematic probing of the movement of these worms is known to be useful for these purposes. Study of dynamics of normal and mutant worms is important in behavioral genetic and neuroscience. However, methods for quantifying complexity of worm movement using time series are still not well explored. Neurodegenerative diseases adversely affect gait and mobility. There is a need to accurately quantify gait dynamics of these diseases and differentiate them from the healthy control to better understand their pathophysiology that may lead to more effective therapeutic interventions. This paper attempts to explore the potential application of the method for determining the largest eigenvalues of convolutional fuzzy recurrence plots of time series for measuring the complexity of moving patterns of Caenorhabditis elegans and neurodegenerative disease subjects. Results obtained from analyses demonstrate that the largest recurrence eigenvalues can differentiate phenotypes of behavioral dynamics between wild type and mutant strains of Caenorhabditis elegans; and walking patterns among healthy control subjects and patients with Parkinson’s disease, Huntington’s disease, or amyotrophic lateral sclerosis.

34 Dimensionality of Genomic Information and Its Impact on GWA and Variant Selection: A Simulation Study

The ability to identify true-positive variants increases as more genotyped animals are available. Although thousands of animals can be genotyped, the dimensionality of the genomic information is limited. Therefore, there is a certain number of animals that represent all chromosome segments (Me) segregating in the population. The number of Me can be approximated from the eigenvalue decomposition of the genomic relationship matrix (G). Thus, the limited dimensionality may help to identify the number of animals to be used in genome-wide association (GWA). The first objective of this study was to examine different discovery set sizes for GWA, with set sizes based on the number of largest eigenvalues explaining a certain proportion of variance in G. Additionally, we investigated the impact of incorporating variants selected from different set sizes to regular SNP chip used for genomic prediction. Sequence data were simulated that contained 500k SNP and 2k QTL, where the genetic variance was fully explained by QTL. The GWA was conducted using the number of genotyped animals equal to the number of largest eigenvalues of G (EIG) explaining 50, 60, 70, 80, 90, 95, 98, and 99 percent of the variance in G. Significant SNP had a p-value lower than 0.05 with Bonferroni correction. Further, SNP with the largest effect size (top10, 100, 500, 1k, 2k, and 4k) were also selected to be incorporated into the 50k regular chip. Genomic predictions using the 50k combined with selected SNP were conducted using single-step GBLUP (ssGBLUP). Using the number of animals corresponding to at least EIG98 enabled the identification of the largest effect size QTL. The greatest accuracy of prediction was obtained when the top 2k SNP was combined to the 50k chip. The dimensionality of genomic information should be taken into account for variant selection in GWAS.

Evaluating structural edge importance in temporal networks

To monitor risk in temporal financial networks, we need to understand how individual behaviours affect the global evolution of networks. Here we define a structural importance metric—which we denote as $l_{e}$ l e —for the edges of a network. The metric is based on perturbing the adjacency matrix and observing the resultant change in its largest eigenvalues. We then propose a model of network evolution where this metric controls the probabilities of subsequent edge changes. We show using synthetic data how the parameters of the model are related to the capability of predicting whether an edge will change from its value of $l_{e}$ l e . We then estimate the model parameters associated with five real financial and social networks, and we study their predictability. These methods have applications in financial regulation whereby it is important to understand how individual changes to financial networks will impact their global behaviour. It also provides fundamental insights into spectral predictability in networks, and it demonstrates how spectral perturbations can be a useful tool in understanding the interplay between micro and macro features of networks.

Open and Anisotropic Soft Regions in a Model Polymer Glass

The vibrational dynamics of a model polymer glass is studied by Molecular Dynamics simulations. The focus is on the “soft” monomers with high participation to the lower-frequency vibrational modes contributing to the thermodynamic anomalies of glasses. To better evidence their role, the threshold to qualify monomers as soft is made severe, allowing for the use of systems with limited size. A marked tendency of soft monomers to form quasi-local clusters involving up to 15 monomers is evidenced. Each chain contributes to a cluster up to about three monomers and a single cluster involves a monomer belonging to about 2–3 chains. Clusters with monomers belonging to a single chain are rare. The open and tenuous character of the clusters is revealed by their fractal dimension df<2. The inertia tensor of the soft clusters evidences their strong anisotropy in shape and remarkable linear correlation of the two largest eigenvalues. Owing to the limited size of the system, finite-size effects, as well as dependence of the results on the adopted polymer length, cannot be ruled out.

Eigenvalues and triangles in graphs

Bollobás and Nikiforov (J. Combin. Theory Ser. B.97 (2007) 859–865) conjectured the following. If G is a Kr+1-free graph on at least r+1 vertices and m edges, then ${\rm{\lambda }}_1^2(G) + {\rm{\lambda }}_2^2(G) \le (r - 1)/r \cdot 2m$ , where λ1 (G)and λ2 (G) are the largest and the second largest eigenvalues of the adjacency matrix A(G), respectively. In this paper we confirm the conjecture in the case r=2, by using tools from doubly stochastic matrix theory, and also characterize all families of extremal graphs. Motivated by classic theorems due to Erdős and Nosal respectively, we prove that every non-bipartite graph G of order n and size m contains a triangle if one of the following is true: (i) ${{\rm{\lambda }}_1}(G) \ge \sqrt {m - 1} $ and $G \ne {C_5} \cup (n - 5){K_1}$ , and (ii) ${{\rm{\lambda }}_1}(G) \ge {{\rm{\lambda }}_1}(S({K_{[(n - 1)/2],[(n - 1)/2]}}))$ and $G \ne S({K_{[(n - 1)/2],[(n - 1)/2]}})$ , where $S({K_{[(n - 1)/2],[(n - 1)/2]}})$ is obtained from ${K_{[(n - 1)/2],[(n - 1)/2]}}$ by subdividing an edge. Both conditions are best possible. We conclude this paper with some open problems.

Development for modification of Torgerson projection method using cumulative curve analysis in outlier detection problem for high-dimensional data

Рассмотрена задача выявления аномальных наблюдений в данных больших размерностей на основе метода многомерного шкалирования с учетом возможности построения качественной визуализации данных. Предложен алгоритм модифицированного метода главных проекций Торгерсона, основанный на построении подпространства проектирования исходных данных путем изменения способа факторизации матрицы скалярных произведений при помощи метода анализа кумулятивных кривых. Построено и проанализировано эмпирическое распределение F -меры для разных вариантов проектирования исходных данных Purpose. Purpose of the article. The paper aims at the development of methods for multidimensional data presentation for solving classification problems based on the cumulative curves analysis. The paper considers the outlier detection problem for high-dimensional data based on the multidimensional scaling, in order to construct high-quality data visualization. An abnormal observation (or outlier), according to D. Hawkins, is an observation that is so different from others that it may be assumed as appeared in the sample in a fundamentally different way. Methods. One of the conceptual approaches that allow providing the classification of sample observations is multidimensional scaling, representing by the classical Orlochi method, the Torgerson main projections and others. The Torgerson method assumes that when converting data to construct the most convenient classification, the origin must be placed at the gravity center of the analyzed data, after which the matrix of scalar products of vectors with the origin at the gravity center is calculated, the two largest eigenvalues and corresponding eigenvectors are chosen and projection matrix is evaluated. Moreover, the method assumes the linear partitioning of regular and anomalous observations, which arises rarely. Therefore, it is logical to choose among the possible axes for designing those that allow obtaining more effective results for solving the problem of detecting outlier observations. A procedure of modified CC-ABOD (Cumulative Curves for Angle Based Outlier Detection) to estimate the visualization quality has been applied. It is based on the estimation of the variances of angles assumed by particular observation and remaining observations in multidimensional space. Further the cumulative curves analysis is implemented, which allows partitioning out groups of closely localized observations (in accordance with the chosen metric) and form classes of regular, intermediate, and anomalous observations. Results. A proposed modification of the Torgerson method is developed. The F1-measure distribution is constructed and analyzed for different design options in the source data. An analysis of the empirical distribution showed that in a number of cases the best axes are corresponding to the second, third, or even fourth largest eigenvalues. Findings. The multidimensional scaling methods for constructing visualizations of multi-dimensional data and solving problems of outlier detection have been considered. It was found out that the determination of design is an ambiguous problem.

Variational Quantum Fidelity Estimation

Computing quantum state fidelity will be important to verify and characterize states prepared on a quantum computer. In this work, we propose novel lower and upper bounds for the fidelity F(ρ,σ) based on the ``truncated fidelity'' F(ρm,σ), which is evaluated for a state ρm obtained by projecting ρ onto its m-largest eigenvalues. Our bounds can be refined, i.e., they tighten monotonically with m. To compute our bounds, we introduce a hybrid quantum-classical algorithm, called Variational Quantum Fidelity Estimation, that involves three steps: (1) variationally diagonalize ρ, (2) compute matrix elements of σ in the eigenbasis of ρ, and (3) combine these matrix elements to compute our bounds. Our algorithm is aimed at the case where σ is arbitrary and ρ is low rank, which we call low-rank fidelity estimation, and we prove that no classical algorithm can efficiently solve this problem under reasonable assumptions. Finally, we demonstrate that our bounds can detect quantum phase transitions and are often tighter than previously known computable bounds for realistic situations.