arXiv subjects

Anatoly Yambartsev

Publications and source records attributed to Anatoly Yambartsev.

12 recordsLinked to original sources

arXiv2021

Lack of phase transitions in staggered magnetic systems. A comparison of uniqueness criteria

We study a ferromagnetic Ising model with a staggered cell-board magnetic field previously proposed for image processing [Maruani et al., Markov Processes Relat. Fields 1 (1995) \cite{MPS}]. We complement previous results on the existence of phase transitions at low temperature [González-Navarrete et al., J. Stat. Phys. 162 (2016)] by determining bounds to the region of uniqueness of Gibbs measures. We establish sufficient rigorous uniqueness conditions derived from three different criteria: (1) Dobrushin criterion [Dobrushin, Theory Probab. Appl. 13 (1968)], (2) Disagreement percolation [van den Berg and Maes, Ann. Probab. 22 (1994)] and (3) Dobrushin-Shlosman criteria [Dobrushin and Shlosman, in Statistical Physics and Dynamical Systems. Rigorous Results. (1985)]. These conditions are subsequently solved numerically and the resulting uniqueness regions compared.

math-ph↗

arXiv2021

The C-SHIFT algorithm for normalizing covariances

Omics technologies are powerful tools for analyzing patterns in gene expression data for thousands of genes. Due to a number of systematic variations in experiments, the raw gene expression data is often obfuscated by undesirable technical noises. Various normalization techniques were designed in an attempt to remove these non-biological errors prior to any statistical analysis. One of the reasons for normalizing data is the need for recovering the covariance matrix used in gene network analysis. In this paper, we introduce a novel normalization technique, called the covariance shift (C-SHIFT) method. This normalization algorithm uses optimization techniques together with the blessing of dimensionality philosophy and energy minimization hypothesis for covariance matrix recovery under additive noise (in biology, known as the bias). Thus, it is perfectly suited for the analysis of logarithmic gene expression data. Numerical experiments on synthetic data demonstrate the method's advantage over the classical normalization techniques. Namely, the comparison is made with Rank, Quantile, cyclic LOESS (locally estimated scatterplot smoothing), and MAD (median absolute deviation) normalization methods. We also evaluate the performance of C-SHIFT algorithm on real biological data.

q-bio.GN↗

arXiv2021

Order book dynamics with liquidity fluctuations: limit theorems and large deviations

We propose a class of stochastic models for a dynamics of limit order book with different type of liquidities. Within this class of models we study the one where a spread decreases uniformly, belonging to the class of processes known as a population processes with uniform catastrophes. The law of large numbers (LLN), central limit theorem (CLT) and large deviations (LD) are proved for our model with uniform catastrophes. Our results allow us to satisfactorily explain the volatility and local trends in the prices, relevant empirical characteristics that are observed in this type of markets. Furthermore, it shows us how these local trends and volatility are determined by the typical values of the bid-ask spread. In addition, we use our model to show how large deviations occur in the spread and prices, such as those observed in flash crashes.

q-fin.TR↗

arXiv2019

Cross-Multiplicative Coalescent Processes and Applications

We introduce and analyze a novel type of coalescent processes called cross-multiplicative coalescent that models a system with two types of particles, $A$ and $B$. The bonds are formed only between the pairs of particles of opposite types with the same rate for each bond, producing connected components made of particles of both types. We analyze and solve the Smoluchowski coagulation system of equations obtained as a hydrodynamic limit of the corresponding Marcus-Lushnikov process. We establish that the cross-multiplicative kernel is a gelling kernel, and find the gelation time. As an application, we derive the limiting mean length of a minimal spanning tree on a complete bipartite graph $K_{α[n], β[n]}$ with partitions of sizes $α[n]=αn +o(\sqrt{n})$ and $β[n]=βn +o(\sqrt{n})$ and independent edge weights, distributed uniformly over $[0, 1]$.

math.PR↗

arXiv2019

Network reconstruction with local partial correlation: a comparative evaluation

Over the past decade, various methods have been proposed for the reconstruction of networks modeled as Gaussian Graphical Models. In this work, we analyzed three different approaches: the Graphical Lasso (GLasso), the Graphical Ridge (GGMridge), and the Local Partial Correlation (LPC). For the evaluation of the methods, we used high dimensional data generated from simulated random graphs (Erdös-Rényi, Barabási-Albert, Watts-Strogatz). The performance was assessed through the Receiver Operating Characteristic (ROC) curve. In addition, the methods were used to reconstruct the co-expression network for differentially expressed genes in human cervical cancer data. The LPC method outperformed the GLasso in most simulated cases. The GGMridge produced better ROC curves then both the other methods. Finally, LPC and GGMridge obtained similar outcomes in real data studies.

stat.AP↗

arXiv2016

A note on scaling limits for truncated birth-and-death processes with interaction

In this note we consider a Markov chain formed by a finite system of interacting birth-and-death processes on a finite state space. We study an asymptotic behaviour of the Markov chain as its state space becomes large. In particular, we show that the appropriately scaled Markov chain converges to a diffusion process, and derive conditions for existence of the stationary distribution of the limit diffusion process in special cases.

math.PR↗

arXiv2016

Stochastic Ising model with plastic interactions

We propose a new model based on the Ising model with the aim to study synaptic plasticity phenomena in neural networks. It is today well established in biology that the synapses or connections between certain types of neurons are strengthened when the neurons are co-active, a form of the so called synaptic plasticity. Such mechanism is believed to mediate the formation and maintenance of memories. The proposed model describes some features from that phenomenon. Together with the spin-flip dynamics, in our model the coupling constants are also subject to stochastic dynamics, so that they interact with each other. The evolution of the system is described by a continuous-time Markov jump process. Keyword Markov chain, Stochastic Ising model, synaptic plasticity, neural networks, transience

cond-mat.dis-nn↗

arXiv2015

Phase transition in ferromagnetic Ising model with a cell-board external field

We show the presence of a first-order phase transition for a ferromagnetic Ising model on $\mathbb{Z}^2$ with a periodical external magnetic field. The external field takes two values $h$ and $-h$, where $h>0$. The sites associated with positive and negative values of external field form a cell-board configuration with rectangular cells of sides $L_1\times L_2$ sites, such that the total value of the external field is zero. The phase transition holds if $h<\frac{2J}{L_1}+ \frac{2J}{L_2}$, where $J$ is an interaction constant. We prove a first-order phase transition using the reflection positivity (RP) method. We apply a key inequality which is usually referred to as the chessboard estimate.

math-ph↗

arXiv2015

Unexpected links reflect the noise in networks

Gene covariation networks are commonly used to study biological processes. The inference of gene covariation networks from observational data can be challenging, especially considering the large number of players involved and the small number of biological replicates available for analysis. We propose a new statistical method for estimating the number of erroneous edges in reconstructed networks that strongly enhances commonly used inference approaches. This method is based on a special relationship between sign of correlation (positive/negative) and directionality (up/down) of gene regulation, and allows for the identification and removal of approximately half of all erroneous edges. Using the mathematical model of Bayesian networks and positive correlation inequalities we establish a mathematical foundation for our method. Analyzing existing biological datasets, we find a strong correlation between the results of our method and false discovery rate (FDR). Furthermore, simulation analysis demonstrates that our method provides a more accurate estimate of network error than FDR.

q-bio.MN↗

arXiv2014

Reverse enGENEering of regulatory networks from Big Data: a guide for a biologist

Omics technologies enable unbiased investigation of biological systems through massively parallel sequence acquisition or molecular measurements, bringing the life sciences into the era of Big Data. A central challenge posed by such omics datasets is how to transform this data into biological knowledge. For example, how to use this data to answer questions such as: which functional pathways are involved in cell differentiation? Which genes should we target to stop cancer? Network analysis is a powerful and general approach to solve this problem consisting of two fundamental stages, network reconstruction and network interrogation. Herein, we provide an overview of network analysis including a step by step guide on how to perform and use this approach to investigate a biological question. In this guide, we also include the software packages that we and others employ for each of the steps of a network analysis workflow.

q-bio.MN↗

arXiv2014

Building complex networks through classical and Bayesian statistics - a comparison

This research is about studying and comparing two different ways of building complex networks. The main goal of our study is to find an effective way to build networks, particularly when we have fewer observations than variables. We construct networks estimating the partial correlation coefficient on Classic Statistics (Inverse Method) and on Bayesian Statistics (Normal - Inverse Wishart conjugate prior). In this current work, in order to solve the problem of having less observations than variables, we propose a new methodology called local partial correlation, which consists of selecting, for each pair of variables, the other variables most correlated to the pair.We applied these methods on simulated data and compared them through ROC curves. The most attractive result is that, even though it has high computational costs, to use Bayesian inference on trees is better when we have less observations than variables. In other cases, both approaches present satisfactory results.

stat.ME↗

arXiv2008

Phase transition for the Ising model on the Critical Lorentzian triangulation

Ising model without external field on an infinite Lorentzian triangulation sampled from the uniform distribution is considered. We prove uniqueness of the Gibbs measure in the high temperature region and coexistence of at least two Gibbs measures at low temperature. The proofs are based on the disagreement percolation method and on a variant of Peierls method. The critical temperature is shown to be constant a.s.

math.PR↗