In this article, we introduce mixture representations for likelihood ratio ordered distributions. Essentially, the ratio of two probability densities, or mass functions, is monotone if and only if one can be expressed as a mixture of one-sided truncations of the other. To illustrate the practical value of the mixture representations, we address the problem of density estimation for likelihood ratio ordered distributions. In particular, we propose a nonparametric Bayesian solution which takes advantage of the mixture representations. The prior distribution is constructed from Dirichlet process mixtures and has large support on the space of pairs of densities satisfying the monotone ratio constraint. Posterior consistency holds under reasonable conditions on the prior specification and the true unknown densities. To our knowledge, this is the first posterior consistency result in the literature on order constrained inference. With a simple modification to the prior distribution, we can test the equality of two distributions against the alternative of likelihood ratio ordering. We develop a Markov chain Monte Carlo algorithm for posterior inference and demonstrate the method in a biomedical application.
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Dynamical systems across the sciences, from electrical circuits to ecological networks, undergo qualitative and often catastrophic changes in behavior, called bifurcations, when their underlying parameters cross a threshold. Existing methods predict oncoming catastrophes in individual systems but are primarily time-series-based and struggle both to categorize qualitative dynamical regimes across diverse systems and to generalize to real data. To address this challenge, we propose a data-driven, physically-informed deep-learning framework for classifying dynamical regimes and characterizing bifurcation boundaries based on the extraction of topologically invariant features. We focus on the paradigmatic case of the supercritical Hopf bifurcation, which is used to model periodic dynamics across a wide range of applications. Our convolutional attention method is trained with data augmentations that encourage the learning of topological invariants which can be used to detect bifurcation boundaries in unseen systems and to design models of biological systems like oscillatory gene regulatory networks. We further demonstrate our method's use in analyzing real data by recovering distinct proliferation and differentiation dynamics along pancreatic endocrinogenesis trajectory in gene expression space based on single-cell data. Our method provides valuable insights into the qualitative, long-term behavior of a wide range of dynamical systems, and can detect bifurcations or catastrophic transitions in large-scale physical and biological systems.
We investigate the effect of the well-known Mycielski construction on the Shannon capacity of graphs and on one of its most prominent upper bounds, the (complementary) Lov\'asz theta number. We prove that if the Shannon capacity of a graph, the distinguishability graph of a noisy channel, is attained by some finite power, then its Mycielskian has strictly larger Shannon capacity than the graph itself. For the complementary Lov\'asz theta function we show that its value on the Mycielskian of a graph is completely determined by its value on the original graph, a phenomenon similar to the one discovered for the fractional chromatic number by Larsen, Propp and Ullman. We also consider the possibility of generalizing our results on the Sperner capacity of directed graphs and on the generalized Mycielsky construction. Possible connections with what Zuiddam calls the asymptotic spectrum of graphs are discussed as well.
Discovering underlying structures of causal relations from observational studies poses a great challenge in scientific research where randomized trials or intervention-based studies are infeasible. This challenge pertains to the lack of knowledge on pre-specified roles of cause and effect in observations studies. Leveraging Shannon's seminal work on information theory, we propose a new conceptual framework of asymmetry where any causal link between putative cause and effect is captured by unequal information flows from one variable to another. We present an entropy-based asymmetry coefficient that not only enables us to assess for whether one variable is a stronger predictor of the other, but also detects an imprint of the underlying causal relation in observational studies. Our causal discovery analytics can accommodate low-dimensional confounders naturally. The proposed methodology relies on scalable non-parametric density estimation using fast Fourier transformation, making the resulting estimation method manyfold faster than the classical bandwidth-based density estimation while maintaining comparable mean integrated squared error rates. We investigate key asymptotic properties of our methodology and utilize a data-splitting and cross-fitting technique to facilitate inference for the direction of causal relations. We illustrate the performance of our methodology through simulation studies and real data examples.
In this paper we present a low-rank method for conforming multipatch discretizations of compressible linear elasticity problems using Isogeometric Analysis. The proposed technique is a non-trivial extension of [M. Montardini, G. Sangalli, and M. Tani. A low-rank isogeometric solver based on Tucker tensors. Comput. Methods Appl. Mech. Engrg., page 116472, 2023.] to multipatch geometries. We tackle the model problem using an overlapping Schwarz method, where the subdomains can be defined as unions of neighbouring patches. Then on each subdomain we approximate the blocks of the linear system matrix and of the right-hand side vector using Tucker matrices and Tucker vectors, respectively. We use the Truncated Preconditioned Conjugate Gradient as a linear solver, coupled with a suited preconditioner. The numerical experiments show the advantages of this approach in terms of memory storage. Moreover, the number of iterations is robust with respect to the relevant parameters.
Influenced mixed moving average fields are a versatile modeling class for spatio-temporal data. However, their predictive distribution is not generally known. Under this modeling assumption, we define a novel spatio-temporal embedding and a theory-guided machine learning approach that employs a generalized Bayesian algorithm to make ensemble forecasts. We employ Lipschitz predictors and determine fixed-time and any-time PAC Bayesian bounds in the batch learning setting. Performing causal forecast is a highlight of our methodology as its potential application to data with spatial and temporal short and long-range dependence. We then test the performance of our learning methodology by using linear predictors and data sets simulated from a spatio-temporal Ornstein-Uhlenbeck process.
Undoubtedly, Location-based Social Networks (LBSNs) provide an interesting source of geo-located data that we have previously used to obtain patterns of the dynamics of crowds throughout urban areas. According to our previous results, activity in LBSNs reflects the real activity in the city. Therefore, unexpected behaviors in the social media activity are a trustful evidence of unexpected changes of the activity in the city. In this paper we introduce a hybrid solution to early detect these changes based on applying a combination of two approaches, the use of entropy analysis and clustering techniques, on the data gathered from LBSNs. In particular, we have performed our experiments over a data set collected from Instagram for seven months in New York City, obtaining promising results.
In this paper, we propose a generic approach to perform global sensitivity analysis (GSA) for compartmental models based on continuous-time Markov chains (CTMC). This approach enables a complete GSA for epidemic models, in which not only the effects of uncertain parameters such as epidemic parameters (transmission rate, mean sojourn duration in compartments) are quantified, but also those of intrinsic randomness and interactions between the two. The main step in our approach is to build a deterministic representation of the underlying continuous-time Markov chain by controlling the latent variables modeling intrinsic randomness. Then, model output can be written as a deterministic function of both uncertain parameters and controlled latent variables, so that it becomes possible to compute standard variance-based sensitivity indices, e.g. the so-called Sobol' indices. However, different simulation algorithms lead to different representations. We exhibit in this work three different representations for CTMC stochastic compartmental models and discuss the results obtained by implementing and comparing GSAs based on each of these representations on a SARS-CoV-2 epidemic model.
In this paper, we present a rigorous analysis of root-exponential convergence of Hermite approximations, including projection and interpolation methods, for functions that are analytic in an infinite strip containing the real axis and satisfy certain restrictions on the asymptotic behavior at infinity within this strip. Asymptotically sharp error bounds in the weighted and maximum norms are derived. The key ingredients of our analysis are some remarkable contour integral representations for the Hermite coefficients and the remainder of Hermite spectral interpolations. Further extensions to Gauss--Hermite quadrature, Hermite spectral differentiations, generalized Hermite spectral approximations and the scaling factor of Hermite approximation are also discussed. Numerical experiments confirm our theoretical results.
We focus on online second price auctions, where bids are made sequentially, and the winning bidder pays the maximum of the second-highest bid and a seller specified reserve price. For many such auctions, the seller does not see all the bids or the total number of bidders accessing the auction, and only observes the current selling prices throughout the course of the auction. We develop a novel non-parametric approach to estimate the underlying consumer valuation distribution based on this data. Previous non-parametric approaches in the literature only use the final selling price and assume knowledge of the total number of bidders. The resulting estimate, in particular, can be used by the seller to compute the optimal profit-maximizing price for the product. Our approach is free of tuning parameters, and we demonstrate its computational and statistical efficiency in a variety of simulation settings, and also on an Xbox 7-day auction dataset on eBay.
Forecast combination involves using multiple forecasts to create a single, more accurate prediction. Recently, feature-based forecasting has been employed to either select the most appropriate forecasting models or to optimize the weights of their combination. In this paper, we present a multi-task optimization paradigm that focuses on solving both problems simultaneously and enriches current operational research approaches to forecasting. In essence, it incorporates an additional learning and optimization task into the standard feature-based forecasting approach, focusing on the identification of an optimal set of forecasting methods. During the training phase, an optimization model with linear constraints and quadratic objective function is employed to identify accurate and diverse methods for each time series. Moreover, within the training phase, a neural network is used to learn the behavior of that optimization model. Once training is completed the candidate set of methods is identified using the network. The proposed approach elicits the essential role of diversity in feature-based forecasting and highlights the interplay between model combination and model selection when optimizing forecasting ensembles. Experimental results on a large set of series from the M4 competition dataset show that our proposal enhances point forecast accuracy compared to state-of-the-art methods.
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