Local Markov Research Articles

Geodesics of Random Riemannian Metrics

We analyze the disordered Riemannian geometry resulting from random perturbations of the Euclidean metric. We focus on geodesics, the paths traced out by a particle traveling in this quenched random environment. By taking the point of the view of the particle, we show that the law of its observed environment is absolutely continuous with respect to the law of the random metric, and we provide an explicit form for its Radon-Nikodym derivative. We use this result to prove a "local Markov property" along an unbounded geodesic, demonstrating that it eventually encounters any type of geometric phenomenon. We also develop in this paper some general results on conditional Gaussian measures. Our Main Theorem states that a geodesic chosen with random initial conditions (chosen independently of the metric) is almost surely not minimizing. To demonstrate this, we show that a minimizing geodesic is guaranteed to eventually pass over a certain "bump surface," which locally has constant positive curvature. By using Jacobi fields, we show that this is sufficient to destabilize the minimizing property.

Depth video-based gait recognition for smart home using local directional pattern features and hidden Markov model

Gait recognition at smart home is considered as a primary function of the smart system nowadays. The significance of gait recognition is high especially for the elderly as gait is one of the basic activities to promote and preserve their health. In this work, a novel method was proposed for human gait recognition by processing depth videos from a depth camera. The gait recognition method utilizes local directional patterns (LDPs) for local feature extraction from depth silhouettes and hidden Markov models (HMMs) for recognition. The LDP features were first extracted from the depth silhouettes of a human body from each frame of a video containing human gait. The dimension of the LDP features was reduced by principal component analysis. Then, each HMM was trained using the LDP features. Finally, the recognition was done with a maximum likelihood calculation of the trained HMMs of different gaits. We focused on training and recognizing two kinds of gaits here, namely, normal and abnormal. The proposed approach shows superior recognition performance over other traditional methods of gait recognition.

Application of the local Markov approximation method for the analysis of information processes processing algorithms with unknown discontinuous parameters under violation of the consistency property of their estimates

Application of the local Markov approximation method for the analysis of information processes processing algorithms with unknown discontinuous parameters under violation of the consistency property of their estimates

Application of the local Markov approximation method for the analysis of information processes processing algorithms with unknown discontinuous parameters

We considered the local Markov approximation method for the definition of the precision characteristics of statistical analysis algorithms of information processes with unknown discontinuous parameters in the presence of Gaussian distortions 4470 O. V. Chernoyarov et al. and illustrated the use of the stated approach in practical applications for the analysis of operating efficiency of detectors and measurers of the quasi-determined and random signals.

A Local Poisson Graphical Model for Inferring Networks From Sequencing Data

Gaussian graphical models, a class of undirected graphs or Markov Networks, are often used to infer gene networks based on microarray expression data. Many scientists, however, have begun using high-throughput sequencing technologies such as RNA-sequencing or next generation sequencing to measure gene expression. As the resulting data consists of counts of sequencing reads for each gene, Gaussian graphical models are not optimal for this discrete data. In this paper, we propose a novel method for inferring gene networks from sequencing data: the Local Poisson Graphical Model. Our model assumes a Local Markov property where each variable conditional on all other variables is Poisson distributed. We develop a neighborhood selection algorithm to fit our model locally by performing a series of l1 penalized Poisson, or log-linear, regressions. This yields a fast parallel algorithm for estimating networks from next generation sequencing data. In simulations, we illustrate the effectiveness of our methods for recovering network structure from count data. A case study on breast cancer microRNAs (miRNAs), a novel application of graphical models, finds known regulators of breast cancer genes and discovers novel miRNA clusters and hubs that are targets for future research.

Exploratory Space–time Analysis of Inbound Tourism Flows to China Cities

ABSTRACTThe integration of space and time generates much closer interactions between geography and other social sciences in general while providing new perspectives for the role of geography in tourism flow studies. This paper aims to apply exploratory space–time analysis to provide insights into inbound tourism flow patterns in China's cities over the period of 2000–2009 using city‐level Geographical Information System datasets. First, this paper ascertains that there exists geographic neighborhood effect at city level by testifying significantly positive spatial autocorrelation. The hot spots and temporal stability are identified. The hot spots are locally concentrated in metropolitan areas. Local Markov matrix has also shown significant transitions with negative or positive influence of geographic neighbors upon a city. Specifically, three‐dimensional surface is created to visualize tourism flows. The spatial inequalities have the pattern of ‘the east more dense than the west, the south more dense than the north’. There simultaneously exist spatial agglomeration and spatial dispersion. The intensity of spatial agglomeration has become stronger. At the same time, smaller peaks of tourism flow centers developed around the main cities and spread toward surrounding cities. These potential tourism flow centers have gradually emerged and grown larger. Furthermore, the space–time covariance matrix uncovers correlation, which shows strong regional integration over years. Eight hundred and ninety links are identified and visualized. Two hundred and sixty nine links forming 10 spatial clusters represent strong positive correlation. One hundred and thirty‐four links are negative correlation, which is about 15.06 per cent of all links. In conclusion, our comprehensive evidences offer deeper insights and have important policy implications. These enable the tourism‐oriented governmental agencies, as well as the tourism industry professionals, to better understand the changes of inbound tourism flows in China's cities and relevant tourism partners/competitors for cities. Copyright © 2013 John Wiley & Sons, Ltd.

Characteristics of quazi-likelihood estimation of the image area in the presence of spatial noise

Asymptotic expressions for the characteristics of quasi-likelihood estimation of image area have been obtained by using the local Markov approximation method. It was shown that the accuracy of area estimation is determined by the magnitude of intensity jump over the boundary limiting the area occupied by image.

Quantum-correlation breaking channels, broadcasting scenarios, and finite Markov chains

One of the classical results concerning quantum channels is the characterization of entanglement-breaking channels [M. Horodecki, P. W. Shor, and M. B. Ruskai, Rev. Math. Phys. 15, 629 (2003)]. We address the question whether there exists a similar characterization on the level of quantum correlations which may go beyond entanglement. The answer is fully affirmative in the case of breaking quantum correlations down to the, so-called, QC (quantum-classical) type, while it is no longer true in the CC (classical-classical) case. The corresponding channels turn out to be measurement maps. Our study also reveals an unexpected link between quantum state and local correlation broadcasting and finite Markov chains. We present a possibility of broadcasting via non von Neumann measurements, which relies on the Perron-Frobenius theorem. Surprisingly, this is not the typical generalized controlled-not (c-not) gate scenario appearing naturally in this context.

Superparsing

This paper presents a simple and effective nonparametric approach to the problem of image parsing, or labeling image regions (in our case, superpixels produced by bottom-up segmentation) with their categories. This approach requires no training, and it can easily scale to datasets with tens of thousands of images and hundreds of labels. It works by scene-level matching with global image descriptors, followed by superpixel-level matching with local features and efficient Markov random field (MRF) optimization for incorporating neighborhood context. Our MRF setup can also compute a simultaneous labeling of image regions into semantic classes (e.g., tree, building, car) and geometric classes (sky, vertical, ground). Our system outperforms the state-of-the-art non-parametric method based on SIFT Flow on a dataset of 2,688 images and 33 labels. In addition, we report per-pixel rates on a larger dataset of 15,150 images and 170 labels. To our knowledge, this is the first complete evaluation of image parsing on a dataset of this size, and it establishes a new benchmark for the problem.

SAR-Based Terrain Classification Using Weakly Supervised Hierarchical Markov Aspect Models

We introduce the hierarchical Markov aspect model (HMAM), a computationally efficient graphical model for densely labeling large remote sensing images with their underlying terrain classes. HMAM resolves local ambiguities efficiently by combining the benefits of quadtree representations and aspect models-the former incorporate multiscale visual features and hierarchical smoothing to provide improved local label consistency, while the latter sharpen the labelings by focusing them on the classes that are most relevant for the broader local image context. The full HMAM model takes a grid of local hierarchical Markov quadtrees over image patches and augments it by incorporating a probabilistic latent semantic analysis aspect model over a larger local image tile at each level of the quadtree forest. Bag-of-word visual features are extracted for each level and patch, and given these, the parent-child transition probabilities from the quadtree and the label probabilities from the tile-level aspect models, an efficient forwards-backwards inference pass allows local posteriors for the class labels to be obtained for each patch. Variational expectation-maximization is then used to train the complete model from either pixel-level or tile-keyword-level labelings. Experiments on a complete TerraSAR-X synthetic aperture radar terrain map with pixel-level ground truth show that HMAM is both accurate and efficient, providing significantly better results than comparable single-scale aspect models with only a modest increase in training and test complexity. Keyword-level training greatly reduces the cost of providing training data with little loss of accuracy relative to pixel-level training.

Mixing time for the solid-on-solid model

We analyze the mixing time of a natural local Markov chain (the Glauber dynamics) on configurations of the solid-on-solid model of statistical physics. This model has been proposed, among other things, as an idealization of the behavior of contours in the Ising model at low temperatures. Our main result is an upper bound on the mixing time of Õ(n3.5), which is tight within a factor of Õ(√n). The proof, which in addition gives some insight into the actual evolution of the contours, requires the introduction of a number of novel analytical techniques that we conjecture will have other applications.

Equivalence of the Local Markov Inequality and a Kolmogorov Type Inequality in the Complex Plane

We prove that a compact subset of the complex plane satisfies a local Markov inequality if and only if it satisfies a Kolmogorov type inequality. This result generalizes a theorem established by Bos and Milman in the real case. We also show that every set satisfying the local Markov inequality is a sum of Cantor type sets which are regular in the sense of the potential theory.

TEXTURE ANALYSIS USING GAUSSIAN GRAPHICAL MODELS

Texture classification is a challenging and important problem in image analysis. graphical models (GM) are promising tools for texture analysis. In this paper, we address the problem of learning the structure of Gaussian graphical models (GGM) for texture models. GGM can be considered as regression problems due to the connection between the local Markov properties and conditional regression of a Gaussian random variable. We utilize L1-penalty regularization technique for appropriate neighborhood selection and parameter estimation simultaneously. The proposed algorithms are applied in texture synthesis and classification. Experimental results on Brodatz textures demonstrate that the proposed algorithms have good performance and prospects.

On the Risk Neutralization of Transition Matrix

Risk-neutral transition matrix term structure is an essential component of rating-based credit derivative pricing models. However, generation of suitable risk-neutral transition matrix term structure remains a challenging problem. Many calibration models in the literature either are unstable or result in poor fit to the market implied default probability term structure.In this paper, we propose a new risk-neutral transition matrix term structure calibration method in which the conditional rating transition matrix is a mixture of Markov chains. Numerical results indicate that the local Markov mixture model is capable of accurately calibrating to a variety of market implied CDS/PD term structure.

Mixing time of exponential random graphs

A variety of random graph models has been developed in recent years to study a range of problems on networks, driven by the wide availability of data from many social, telecommunication, biochemical and other networks. A key model, extensively used in sociology literature, is the exponential random graph model. This model seeks to incorporate in random graphs the notion of reciprocity, that is, the larger than expected number of triangles and other small subgraphs. Sampling from these distributions is crucial for parameter estimation hypothesis testing and more generally for understanding basic features of the network model itself. In practice, sampling is typically carried out using Markov chain Monte Carlo, in particular, either the Glauber dynamics or the Metropolis–Hastings procedure. In this paper we characterize the high and low temperature regimes of the exponential random graph model. We establish that in the high temperature regime the mixing time of the Glauber dynamics is Θ(n2 log n), where n is the number of vertices in the graph; in contrast, we show that in the low temperature regime the mixing is exponentially slow for any local Markov chain. Our results, moreover, give a rigorous basis for criticisms made of such models. In the high temperature regime, where sampling with Markov chain Monte Carlo is possible, we show that any finite collection of edges is asymptotically independent; thus, the model does not possess the desired reciprocity property and is not appreciably different from the Erdős–Rényi random graph.

Tame linear extension operators for smooth Whitney functions

For a compact set K ⊆ R d we present a rather easy construction of a linear extension operator E : E ( K ) → C ∞ ( R d ) for the space of Whitney jets E ( K ) which satisfies linear tame continuity estimates sup { | ∂ α E ( f ) ( x ) | : | α | ⩽ m , x ∈ R d } ⩽ C m , ε ‖ f ‖ ( r + ε ) m , where ‖ ⋅ ‖ s denotes the s-th Whitney norm. The construction turns out to be possible if and only if the local Markov inequality LMI ( s ) introduced by Bos and Milman holds for every s > r on K. In particular, E ( K ) admits a tame linear extension operator if and only if the local Markov inequality LMI ( s ) holds on K for some s ⩾ 1 .

Ice hockey shooting event modeling with mixture hidden Markov model

In this paper we present a new event analysis framework based on mixture hidden Markov model (HMM) for ice hockey videos. Hockey is a competitive sport and hockey videos are hard to analyze because of the homogeneity of its frame features. However, the temporal dynamics of hockey videos is highly structured. Using the mixture representation of local observations and Markov chain property of hockey event structure, we successfully model the hockey event as a mixture HMM. Based on the mixture HMM, the hockey event could be classified with high accuracy. Two types of mixture HMMs, Gaussian mixture and independent component analysis (ICA) mixture, are compared for the hockey video event classification. The results confirm our analysis that the mixture HMM is a suitable model to deal with videos with intensive activities. The new mixture HMM hockey event model could be a very useful tool for hockey game analysis.

Reconstruction for Colorings on Trees

Consider -colorings of the complete tree of depth and branching factor . If we fix the coloring of the leaves, for what range of is the root uniformly distributed over all colors (in the limit )? This corresponds to the threshold for uniqueness of the infinite-volume Gibbs measure. It is straightforward to show the existence of colorings of the leaves which “freeze” the entire tree when . For , Jonasson proved the root is “unbiased” for any fixed coloring of the leaves, and thus the Gibbs measure is unique. What happens for a typical coloring of the leaves? When the leaves have a nonvanishing influence on the root in expectation, over random colorings of the leaves, reconstruction is said to hold. Nonreconstruction is equivalent to extremality of the free-boundary Gibbs measure. When , it is straightforward to show that reconstruction is possible (and hence the measure is not extremal). We prove that for and , nonreconstruction holds; i.e., the Gibbs measure is extremal. We prove a strong form of extremality: With high probability over the colorings of the leaves the influence at the root decays exponentially fast with the depth of the tree. Closely related results were also proven recently by Sly. The above strong form of extremality implies that a local Markov chain that updates constant sized blocks has inverse linear entropy constant and hence mixing time where is the number of vertices of the tree. Extremality on trees and random graphs has received considerable attention recently since it may have connections to the efficiency of local algorithms.

FITBAR: a web tool for the robust prediction of prokaryotic regulons

BackgroundThe binding of regulatory proteins to their specific DNA targets determines the accurate expression of the neighboring genes. The in silico prediction of new binding sites in completely sequenced genomes is a key aspect in the deeper understanding of gene regulatory networks. Several algorithms have been described to discriminate against false-positives in the prediction of new binding targets; however none of them has been implemented so far to assist the detection of binding sites at the genomic scale.ResultsFITBAR (Fast Investigation Tool for Bacterial and Archaeal Regulons) is a web service designed to identify new protein binding sites on fully sequenced prokaryotic genomes. This tool consists in a workbench where the significance of the predictions can be compared using different statistical methods, a feature not found in existing resources. The Local Markov Model and the Compound Importance Sampling algorithms have been implemented to compute the P-value of newly discovered binding sites. In addition, FITBAR provides two optimized genomic scanning algorithms using either log-odds or entropy-weighted position-specific scoring matrices. Other significant features include the production of a detailed genomic context map for each detected binding site and the export of the search results in spreadsheet and portable document formats. FITBAR discovery of a high affinity Escherichia coli NagC binding site was validated experimentally in vitro as well as in vivo and published.ConclusionsFITBAR was developed in order to allow fast, accurate and statistically robust predictions of prokaryotic regulons. This feature constitutes the main advantage of this web tool over other matrix search programs and does not impair its performance. The web service is available at http://archaea.u-psud.fr/fitbar.

Local Causal and Markov Blanket Induction for Causal Discovery and Feature Selection for Classification Part II: Analysis and Extensions

In part I of this work we introduced and evaluated the Generalized Local Learning (GLL) framework for producing local causal and Markov blanket induction algorithms. In the present second part we analyze the behavior of GLL algorithms and provide extensions to the core methods. Specifically, we investigate the empirical convergence of GLL to the true local neighborhood as a function of sample size. Moreover, we study how predictivity improves with increasing sample size. Then we investigate how sensitive are the algorithms to multiple statistical testing, especially in the presence of many irrelevant features. Next we discuss the role of the algorithm parameters and also show that Markov blanket and causal graph concepts can be used to understand deviations from optimality of state-of-the-art non-causal algorithms. The present paper also introduces the following extensions to the core GLL framework: parallel and distributed versions of GLL algorithms, versions with false discovery rate control, strategies for constructing novel heuristics for specific domains, and divide-and-conquer local-to-global learning (LGL) strategies. We test the generality of the LGL approach by deriving a novel LGL-based algorithm that compares favorably to the state-of-the-art global learning algorithms. In addition, we investigate the use of non-causal feature selection methods to facilitate global learning. Open problems and future research paths related to local and local-to-global causal learning are discussed.