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We study the properties of points in [0, 1]d generated by applying Hilbert's space filling curve to uniformly distributed points in [0, 1]. For deterministic sampling we obtain a discrepancy of O(n–1/d) for d ≥ 2. For random stratified sampling, and scrambled van der Corput points, we derive a mean-squared error of O(n–1–2/d) for integration of Lipschitz continuous integrands, when d ≥ 3. These rates are the same as those obtained by sampling on d-dimensional grids and they show a deterioration with increasing d. The rate for Lipschitz functions is, however, the best possible at that level of smoothness and is better than plain independent and identically distributed sampling. Unlike grids, space filling curve sampling provides points at any desired sample size, and the van der Corput version is extensible in n. We also introduce a class of piecewise Lipschitz functions whose discontinuities are in rectifiable sets described via Minkowski content. Although these functions may have infinite variation in the sense of Hardy and Krause, they can be integrated with a mean-squared error of O(n–1–1/d). It was previously known only that the rate was o(n–1). Other space filling curves, such as those due to Sierpinski and Peano, also attain these rates, whereas upper bounds for the Lebesgue curve are somewhat worse, as if the dimension were log2(3) times as high.

Markov chain Monte Carlo and sequential Monte Carlo methods have emerged as the two main tools to sample from high dimensional probability distributions. Although asymptotic convergence of Markov chain Monte Carlo algorithms is ensured under weak assumptions, the performance of these algorithms is unreliable when the proposal distributions that are used to explore the space are poorly chosen and/or if highly correlated variables are updated independently. We show here how it is possible to build efficient high dimensional proposal distributions by using sequential Monte Carlo methods. This allows us not only to improve over standard Markov chain Monte Carlo schemes but also to make Bayesian inference feasible for a large class of statistical models where this was not previously so. We demonstrate these algorithms on a non-linear state space model and a Lévy-driven stochastic volatility model.

We propose a methodology to sample sequentially from a sequence of probability distributions that are defined on a common space, each distribution being known up to a normalizing constant. These probability distributions are approximated by a cloud of weighted random samples which are propagated over time by using sequential Monte Carlo methods. This methodology allows us to derive simple algorithms to make parallel Markov chain Monte Carlo algorithms interact to perform global optimization and sequential Bayesian estimation and to compute ratios of normalizing constants. We illustrate these algorithms for various integration tasks arising in the context of Bayesian inference.

This paper introduces a class of Monte Carlo algorithms which are based on the simulation of a Markov process whose quasi-stationary distribution coincides with a distribution of interest. This differs fundamentally from, say, current Markov chain Monte Carlo methods which simulate a Markov chain whose stationary distribution is the target. We show how to approximate distributions of interest by carefully combining sequential Monte Carlo methods with methodology for the exact simulation of diffusions. The methodology introduced here is particularly promising in that it is applicable to the same class of problems as gradient-based Markov chain Monte Carlo algorithms but entirely circumvents the need to conduct Metropolis–Hastings type accept–reject steps while retaining exactness: the paper gives theoretical guarantees ensuring that the algorithm has the correct limiting target distribution. Furthermore, this methodology is highly amenable to ‘big data’ problems. By employing a modification to existing naive subsampling and control variate techniques it is possible to obtain an algorithm which is still exact but has sublinear iterative cost as a function of data size.

The ensemble Kalman filter is a sophisticated and powerful data assimilation method for filtering high dimensional problems arising in fluid mechanics and geophysical sciences. This Monte Carlo method can be interpreted as a mean-field McKean–Vlasov-type particle interpretation of the Kalman–Bucy diffusions. In contrast to more conventional particle filters and nonlinear Markov processes, these models are designed in terms of a diffusion process with a diffusion matrix that depends on particle covariance matrices. Besides some recent advances on the stability of nonlinear Langevin-type diffusions with drift interactions, the long-time behaviour of models with interacting diffusion matrices and conditional distribution interaction functions has never been discussed in the literature. One of the main contributions of the article is to initiate the study of this new class of models. The article presents a series of new functional inequalities to quantify the stability of these nonlinear diffusion processes. In the same vein, despite some recent contributions on the convergence of the ensemble Kalman filter when the number of sample tends to infinity very little is known on stability and the long-time behaviour of these mean-field interacting type particle filters. The second contribution of this article is to provide uniform propagation of chaos properties as well as 𝕃 n -mean error estimates w.r.t. to the time horizon. Our regularity condition is also shown to be sufficient and necessary for the uniform convergence of the ensemble Kalman filter. The stochastic analysis developed in this article is based on an original combination of functional inequalities and Foster–Lyapunov techniques with coupling, martingale techniques, random matrices and spectral analysis theory.

We introduce a class of generative network models that insert edges by connecting the starting and terminal vertices of a random walk on the network graph. Within the taxonomy of statistical network models, this class is distinguished by permitting the location of a new edge to depend explicitly on the structure of the graph, but being nonetheless statistically and computationally tractable. In the limit of infinite walk length, the model converges to an extension of the preferential attachment model—in this sense, it can be motivated alternatively by asking what preferential attachment is an approximation to. Theoretical properties, including the limiting degree sequence, are studied analytically. If the entire history of the graph is observed, parameters can be estimated by maximum likelihood. If only the final graph is available, its history can be imputed by using Markov chain Monte Carlo methods. We develop a class of sequential Monte Carlo algorithms that are more generally applicable to sequential network models and may be of interest in their own right. The model parameters can be recovered from a single graph generated by the model. Applications to data clarify the role of the random-walk length as a length scale of interactions within the graph.

Sequential Monte Carlo (SMC) methods are a class of techniques to sample approximately from any sequence of probability distributions using a combination of importance sampling and resampling steps. This paper is concerned with the convergence analysis of a class of SMC methods where the times at which resampling occurs are computed online using criteria such as the effective sample size. This is a popular approach amongst practitioners but there are very few convergence results available for these methods. By combining semigroup techniques with an original coupling argument, we obtain functional central limit theorems and uniform exponential concentration estimates for these algorithms.

Many studies have quantified the effects of TV ad spendingor gross rating points on brand sales. Yet this effect is likelymoderated by the different types of brand-related messages or cues (e.g., logo, brand attributes) embedded in the ads and by the ways (e.g., explicitly or implicitly) these cues are conveyed to TV audiences. The authors thusmeasure 17 cues often usedwithin ads to build brand awareness (or salience) and brand image and investigate their influence on ad effectiveness. Technically, the study builds a dynamic model to quantify the effects of advertising on sales; builds a robust and interpretable (i.e., nonparametric and sparse) factor model that integrates correlated, left-censored branding cues; and thenmodels the effects of advertising as a function of the factors identified by these cues. An analysis of 177 campaigns aired by 62 brands finds that salience cues (e.g., logo) and benefit and attributemessagesmoderate ad effectiveness. It also finds that explicit cues aremore effective than implicitones; nonetheless, the primary drivers of ad effectiveness are visual salience cues: the duration and frequencywith which the logo and the duration with which the product are displayed. The study can thus suggest ways brand and ad agency managers can improve the effects of creative ad content on sales.

We model consumer journeys for user-created programs published in an online programming platform (OPP) and uncover factors that predict their occurrence. We build our model on a theoretical framework where consumer journeys involve three latent stages (Learn, Feel, Do), in which users gather information about, express fondness toward, and try the published items, respectively. Using a dataset from an OPP where users publish multimedia items and follow other users, we find that there is no one dominant consumer journey; instead, the sequences of stages in a journey (e.g., Learn → Feel → Do) vary across published items. Furthermore, we find that the social capital (i.e., social network) of a publisher influences the occurrence of spillover effects between latent stages (the phenomenon that one stage in a period triggers another stage in the next period) for the items posted by the publisher. We also find that a publisher’s social capital has only a transient impact on the consumer journeys for the publisher’s projects, underlining the importance of consistently making new network connections in order to promote the growth of user activities surrounding the publisher’s projects. We apply our findings to the publishers’ networking investment decisions to show that publishers’ networking investment would be severely suboptimal if journey heterogeneity is not considered.

Polymer based textile composites have gained much attention in recent years and gradually transformed the growth of industries especially automobiles, construction, aerospace and composites. The inclusion of natural polymeric fibres as reinforcement in carbon fibre reinforced composites manufacturing delineates an economic way, enhances their surface, structural and mechanical properties by providing better bonding conditions. Almost all textile-based products are associated with quality, price and consumer’s satisfaction. Therefore, classification of textiles products and fibre reinforced polymer composites is a challenging task. This paper focuses on the classification of various problems in textile processes and fibre reinforced polymer composites by artificial neural networks, genetic algorithm and fuzzy logic. Moreover, their limitations associated with state-of-the-art processes and some relatively new and sequential classification methods are also proposed and discussed in detail in this paper.