Bayesian parameter estimation for latent Markov random fields and social networks

Undirected graphical models are widely used in statistics, physics and machine vision. However Bayesian parameter estimation for undirected models is extremely challenging, since evaluation of the posterior typically involves the calculation of an intractable normalising constant. This problem has received much attention, but very little of this has focussed on the important practical case where the data consists of noisy or incomplete observations of the underlying hidden structure. This paper specifically addresses this problem, comparing two alternative methodologies. In the first of these approaches particle Markov chain Monte Carlo (Andrieu et al., 2010) is used to efficiently explore the parameter space, combined with the exchange algorithm (Murray et al., 2006) for avoiding the calculation of the intractable normalising constant (a proof showing that this combination targets the correct distribution in found in a supplementary appendix online). This approach is compared with approximate Bayesian computation (Pritchard et al., 1999). Applications to estimating the parameters of Ising models and exponential random graphs from noisy data are presented. Each algorithm used in the paper targets an approximation to the true posterior due to the use of MCMC to simulate from the latent graphical model, in lieu of being able to do this exactly in general. The supplementary appendix also describes the nature of the resulting approximation.

Random prism: an alternative to random forests

Ensemble learning techniques generate multiple classifiers, so called base classifiers, whose combined classification results are used in order to increase the overall classification accuracy. In most ensemble classifiers the base classifiers are based on the Top Down Induction of Decision Trees (TDIDT) approach. However, an alternative approach for the induction of rule based classifiers is the Prism family of algorithms. Prism algorithms produce modular classification rules that do not necessarily fit into a decision tree structure. Prism classification rulesets achieve a comparable and sometimes higher classification accuracy compared with decision tree classifiers, if the data is noisy and large. Yet Prism still suffers from overfitting on noisy and large datasets. In practice ensemble techniques tend to reduce the overfitting, however there exists no ensemble learner for modular classification rule inducers such as the Prism family of algorithms. This article describes the first development of an ensemble learner based on the Prism family of algorithms in order to enhance Prism’s classification accuracy by reducing overfitting.

Parallel random prism: a computationally efficient ensemble learner for classification

Generally classifiers tend to overfit if there is noise in the training data or there are missing values. Ensemble learning methods are often used to improve a classifier's classification accuracy. Most ensemble learning approaches aim to improve the classification accuracy of decision trees. However, alternative classifiers to decision trees exist. The recently developed Random Prism ensemble learner for classification aims to improve an alternative classification rule induction approach, the Prism family of algorithms, which addresses some of the limitations of decision trees. However, Random Prism suffers like any ensemble learner from a high computational overhead due to replication of the data and the induction of multiple base classifiers. Hence even modest sized datasets may impose a computational challenge to ensemble learners such as Random Prism. Parallelism is often used to scale up algorithms to deal with large datasets. This paper investigates parallelisation for Random Prism, implements a prototype and evaluates it empirically using a Hadoop computing cluster.

Probability of survival in a random exchange economy with dependent agents

In this paper I analyze the general equilibrium in a random Walrasian economy. Dependence among agents is introduced in the form of dependency neighborhoods. Under the uncertainty, an agent may fail to survive due to a meager endowment in a particular state (direct effect), as well as due to unfavorable equilibrium price system at which the value of the endowment falls short of the minimum needed for survival (indirect terms-of-trade effect). To illustrate the main result I compute the stochastic limit of equilibrium price and probability of survival of an agent in a large Cobb-Douglas economy.

Validating the reported random errors of ACE‐FTS measurements

In order to validate the reported precision of space‐based atmospheric composition measurements, validation studies often focus on measurements in the tropical stratosphere, where natural variability is weak. The scatter in tropical measurements can then be used as an upper limit on single‐profile measurement precision. Here we introduce a method of quantifying the scatter of tropical measurements which aims to minimize the effects of short‐term atmospheric variability while maintaining large enough sample sizes that the results can be taken as representative of the full data set. We apply this technique to measurements of O3, HNO3, CO, H2O, NO, NO2, N2O, CH4, CCl2F2, and CCl3F produced by the Atmospheric Chemistry Experiment–Fourier Transform Spectrometer (ACE‐FTS). Tropical scatter in the ACE‐FTS retrievals is found to be consistent with the reported random errors (RREs) for H2O and CO at altitudes above 20 km, validating the RREs for these measurements. Tropical scatter in measurements of NO, NO2, CCl2F2, and CCl3F is roughly consistent with the RREs as long as the effect of outliers in the data set is reduced through the use of robust statistics. The scatter in measurements of O3, HNO3, CH4, and N2O in the stratosphere, while larger than the RREs, is shown to be consistent with the variability simulated in the Canadian Middle Atmosphere Model. This result implies that, for these species, stratospheric measurement scatter is dominated by natural variability, not random error, which provides added confidence in the scientific value of single‐profile measurements.

Random Prism: a noise-tolerant alternative to Random Forests

Ensemble learning can be used to increase the overall classification accuracy of a classifier by generating multiple base classifiers and combining their classification results. A frequently used family of base classifiers for ensemble learning are decision trees. However, alternative approaches can potentially be used, such as the Prism family of algorithms that also induces classification rules. Compared with decision trees, Prism algorithms generate modular classification rules that cannot necessarily be represented in the form of a decision tree. Prism algorithms produce a similar classification accuracy compared with decision trees. However, in some cases, for example, if there is noise in the training and test data, Prism algorithms can outperform decision trees by achieving a higher classification accuracy. However, Prism still tends to overfit on noisy data; hence, ensemble learners have been adopted in this work to reduce the overfitting. This paper describes the development of an ensemble learner using a member of the Prism family as the base classifier to reduce the overfitting of Prism algorithms on noisy datasets. The developed ensemble classifier is compared with a stand-alone Prism classifier in terms of classification accuracy and resistance to noise.

Sparse polynomial approximation in positive order Sobolev spaces with bounded mixed derivatives and applications to elliptic problems with random loading

In the present paper we study the approximation of functions with bounded mixed derivatives by sparse tensor product polynomials in positive order tensor product Sobolev spaces. We introduce a new sparse polynomial approximation operator which exhibits optimal convergence properties in L2 and tensorized View the MathML source simultaneously on a standard k-dimensional cube. In the special case k=2 the suggested approximation operator is also optimal in L2 and tensorized H1 (without essential boundary conditions). This allows to construct an optimal sparse p-version FEM with sparse piecewise continuous polynomial splines, reducing the number of unknowns from O(p2), needed for the full tensor product computation, to View the MathML source, required for the suggested sparse technique, preserving the same optimal convergence rate in terms of p. We apply this result to an elliptic differential equation and an elliptic integral equation with random loading and compute the covariances of the solutions with View the MathML source unknowns. Several numerical examples support the theoretical estimates.

On the spectra and pseudospectra of a class of non-self-adjoint random matrices and operators

In this paper we develop and apply methods for the spectral analysis of non-selfadjoint tridiagonal infinite and finite random matrices, and for the spectral analysis of analogous deterministic matrices which are pseudo-ergodic in the sense of E. B. Davies (Commun. Math. Phys. 216 (2001), 687–704). As a major application to illustrate our methods we focus on the “hopping sign model” introduced by J. Feinberg and A. Zee (Phys. Rev. E 59 (1999), 6433–6443), in which the main objects of study are random tridiagonal matrices which have zeros on the main diagonal and random ±1’s as the other entries. We explore the relationship between spectral sets in the finite and infinite matrix cases, and between the semi-infinite and bi-infinite matrix cases, for example showing that the numerical range and p-norm ε - pseudospectra (ε > 0, p ∈ [1,∞] ) of the random finite matrices converge almost surely to their infinite matrix counterparts, and that the finite matrix spectra are contained in the infinite matrix spectrum Σ. We also propose a sequence of inclusion sets for Σ which we show is convergent to Σ, with the nth element of the sequence computable by calculating smallest singular values of (large numbers of) n×n matrices. We propose similar convergent approximations for the 2-norm ε -pseudospectra of the infinite random matrices, these approximations sandwiching the infinite matrix pseudospectra from above and below.

A scalable expressive ensemble learning using Random Prism: a MapReduce approach

The induction of classification rules from previously unseen examples is one of the most important data mining tasks in science as well as commercial applications. In order to reduce the influence of noise in the data, ensemble learners are often applied. However, most ensemble learners are based on decision tree classifiers which are affected by noise. The Random Prism classifier has recently been proposed as an alternative to the popular Random Forests classifier, which is based on decision trees. Random Prism is based on the Prism family of algorithms, which is more robust to noise. However, like most ensemble classification approaches, Random Prism also does not scale well on large training data. This paper presents a thorough discussion of Random Prism and a recently proposed parallel version of it called Parallel Random Prism. Parallel Random Prism is based on the MapReduce programming paradigm. The paper provides, for the first time, novel theoretical analysis of the proposed technique and in-depth experimental study that show that Parallel Random Prism scales well on a large number of training examples, a large number of data features and a large number of processors. Expressiveness of decision rules that our technique produces makes it a natural choice for Big Data applications where informed decision making increases the user’s trust in the system.

A moment problem for random discrete measures

Let X be a locally compact Polish space. A random measure on X is a probability measure on the space of all (nonnegative) Radon measures on X. Denote by K(X) the cone of all Radon measures η on X which are of the form η =

Recurrence and higher ergodic properties for quenched random Lorentz tubes in dimension bigger than two

We consider the billiard dynamics in a non-compact set of ℝ d that is constructed as a bi-infinite chain of translated copies of the same d-dimensional polytope. A random configuration of semi-dispersing scatterers is placed in each copy. The ensemble of dynamical systems thus defined, one for each global realization of the scatterers, is called quenched random Lorentz tube. Under some fairly general conditions, we prove that every system in the ensemble is hyperbolic and almost every system is recurrent, ergodic, and enjoys some higher chaotic properties.

Recurrence for quenched random Lorentz tubes

We consider the billiard dynamics in a striplike set that is tessellated by countably many translated copies of the same polygon. A random configuration of semidispersing scatterers is placed in each copy. The ensemble of dynamical systems thus defined, one for each global choice of scatterers, is called quenched random Lorentz tube. We prove that under general conditions, almost every system in the ensemble is recurrent.

Systematic and random errors between collocated satellite ice water path observations

There remains large disagreement between ice-water path (IWP) in observational data sets, largely because the sensors observe different parts of the ice particle size distribution. A detailed comparison of retrieved IWP from satellite observations in the Tropics (!30 " latitude) in 2007 was made using collocated measurements. The radio detection and ranging(radar)/light detection and ranging (lidar) (DARDAR) IWP data set, based on combined radar/lidar measurements, is used as a reference because it provides arguably the best estimate of the total column IWP. For each data set, usable IWP dynamic ranges are inferred from this comparison. IWP retrievals based on solar reflectance measurements, in the moderate resolution imaging spectroradiometer (MODIS), advanced very high resolution radiometer–based Climate Monitoring Satellite Applications Facility (CMSAF), and Pathfinder Atmospheres-Extended (PATMOS-x) datasets, were found to be correlated with DARDAR over a large IWP range (~20–7000 g m -2 ). The random errors of the collocated data sets have a close to lognormal distribution, and the combined random error of MODIS and DARDAR is less than a factor of 2, which also sets the upper limit for MODIS alone. In the same way, the upper limit for the random error of all considered data sets is determined. Data sets based on passive microwave measurements, microwave surface and precipitation products system (MSPPS), microwave integrated retrieval system (MiRS), and collocated microwave only (CMO), are largely correlated with DARDAR for IWP values larger than approximately 700 g m -2 . The combined uncertainty between these data sets and DARDAR in this range is slightly less MODIS-DARDAR, but the systematic bias is nearly an order of magnitude.