Linear Mixed Effects (LME) models have been widely applied in clustered data analysis in many areas including marketing research, clinical trials, and biomedical studies. Inference can be conducted using maximum likelihood approach if assuming Normal distributions on the random effects. However, in many applications of economy, business and medicine, it is often essential to impose constraints on the regression parameters after taking their real-world interpretations into account. Therefore, in this paper we extend the classical (unconstrained) LME models to allow for sign constraints on its overall coefficients. We propose to assume a symmetric doubly truncated Normal (SDTN) distribution on the random effects instead of the unconstrained Normal distribution which is often found in classical literature. With the aforementioned change, difficulty has dramatically increased as the exact distribution of the dependent variable becomes analytically intractable. We then develop likelihood-based approaches to estimate the unknown model parameters utilizing the approximation of its exact distribution. Simulation studies have shown that the proposed constrained model not only improves real-world interpretations of results, but also achieves satisfactory performance on model fits as compared to the existing model.
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This paper presents a convex-analytic framework to learn sparse graphs from data. While our problem formulation is inspired by an extension of the graphical lasso using the so-called combinatorial graph Laplacian framework, a key difference is the use of a nonconvex alternative to the $\ell_1$ norm to attain graphs with better interpretability. Specifically, we use the weakly-convex minimax concave penalty (the difference between the $\ell_1$ norm and the Huber function) which is known to yield sparse solutions with lower estimation bias than $\ell_1$ for regression problems. In our framework, the graph Laplacian is replaced in the optimization by a linear transform of the vector corresponding to its upper triangular part. Via a reformulation relying on Moreau's decomposition, we show that overall convexity is guaranteed by introducing a quadratic function to our cost function. The problem can be solved efficiently by the primal-dual splitting method, of which the admissible conditions for provable convergence are presented. Numerical examples show that the proposed method significantly outperforms the existing graph learning methods with reasonable CPU time.
Measurement error is a pervasive issue which renders the results of an analysis unreliable. The measurement error literature contains numerous correction techniques, which can be broadly divided into those which aim to produce exactly consistent estimators, and those which are only approximately consistent. While consistency is a desirable property, it is typically attained only under specific model assumptions. Two techniques, regression calibration and simulation extrapolation, are used frequently in a wide variety of parametric and semiparametric settings. However, in many settings these methods are only approximately consistent. We generalize these corrections, relaxing assumptions placed on replicate measurements. Under regularity conditions, the estimators are shown to be asymptotically normal, with a sandwich estimator for the asymptotic variance. Through simulation, we demonstrate the improved performance of the modified estimators, over the standard techniques, when these assumptions are violated. We motivate these corrections using the Framingham Heart Study, and apply the generalized techniques to an analysis of these data.
We derive consistency and asymptotic normality results for quasi-maximum likelihood methods for drift parameters of ergodic stochastic processes observed in discrete time in an underlying continuous-time setting. The special feature of our analysis is that the stochastic integral part is unobserved and non-parametric. Additionally, the drift may depend on the (unknown and unobserved) stochastic integrand. Our results hold for ergodic semi-parametric diffusions and backward SDEs. Simulation studies confirm that the methods proposed yield good convergence results.
Principal Component Analysis (PCA) is the workhorse tool for dimensionality reduction in this era of big data. While often overlooked, the purpose of PCA is not only to reduce data dimensionality, but also to yield features that are uncorrelated. Furthermore, the ever-increasing volume of data in the modern world often requires storage of data samples across multiple machines, which precludes the use of centralized PCA algorithms. This paper focuses on the dual objective of PCA, namely, dimensionality reduction and decorrelation of features, but in a distributed setting. This requires estimating the eigenvectors of the data covariance matrix, as opposed to only estimating the subspace spanned by the eigenvectors, when data is distributed across a network of machines. Although a few distributed solutions to the PCA problem have been proposed recently, convergence guarantees and/or communications overhead of these solutions remain a concern. With an eye towards communications efficiency, this paper introduces a feedforward neural network-based one time-scale distributed PCA algorithm termed Distributed Sanger's Algorithm (DSA) that estimates the eigenvectors of the data covariance matrix when data is distributed across an undirected and arbitrarily connected network of machines. Furthermore, the proposed algorithm is shown to converge linearly to a neighborhood of the true solution. Numerical results are also provided to demonstrate the efficacy of the proposed solution.
Over the past decades, linear mixed models have attracted considerable attention in various fields of applied statistics. They are popular whenever clustered, hierarchical or longitudinal data are investigated. Nonetheless, statistical tools for valid simultaneous inference for mixed parameters are rare. This is surprising because one often faces inferential problems beyond the pointwise examination of fixed or mixed parameters. For example, there is an interest in a comparative analysis of cluster-level parameters or subject-specific estimates in studies with repeated measurements. We discuss methods for simultaneous inference assuming a linear mixed model. Specifically, we develop simultaneous prediction intervals as well as multiple testing procedures for mixed parameters. They are useful for joint considerations or comparisons of cluster-level parameters. We employ a consistent bootstrap approximation of the distribution of max-type statistic to construct our tools. The numerical performance of the developed methodology is studied in simulation experiments and illustrated in a data example on household incomes in small areas.
An efficient simulation-based methodology is proposed for the rolling window estimation of state space models, called particle rolling Markov chain Monte Carlo (MCMC) with double block sampling. In our method, which is based on Sequential Monte Carlo (SMC), particles are sequentially updated to approximate the posterior distribution for each window by learning new information and discarding old information from observations. Th particles are refreshed with an MCMC algorithm when the importance weights degenerate. To avoid degeneracy, which is crucial for reducing the computation time, we introduce a block sampling scheme and generate multiple candidates by the algorithm based on the conditional SMC. The theoretical discussion shows that the proposed methodology with a nested structure is expressed as SMC sampling for the augmented space to provide the justification. The computational performance is evaluated in illustrative examples, showing that the posterior distributions of the model parameters are accurately estimated. The proofs and additional discussions (algorithms and experimental results) are provided in the Supplementary Material.
The problem of Approximate Nearest Neighbor (ANN) search is fundamental in computer science and has benefited from significant progress in the past couple of decades. However, most work has been devoted to pointsets whereas complex shapes have not been sufficiently treated. Here, we focus on distance functions between discretized curves in Euclidean space: they appear in a wide range of applications, from road segments to time-series in general dimension. For $\ell_p$-products of Euclidean metrics, for any $p$, we design simple and efficient data structures for ANN, based on randomized projections, which are of independent interest. They serve to solve proximity problems under a notion of distance between discretized curves, which generalizes both discrete Fr\'echet and Dynamic Time Warping distances. These are the most popular and practical approaches to comparing such curves. We offer the first data structures and query algorithms for ANN with arbitrarily good approximation factor, at the expense of increasing space usage and preprocessing time over existing methods. Query time complexity is comparable or significantly improved by our algorithms, our algorithm is especially efficient when the length of the curves is bounded.
Implicit probabilistic models are models defined naturally in terms of a sampling procedure and often induces a likelihood function that cannot be expressed explicitly. We develop a simple method for estimating parameters in implicit models that does not require knowledge of the form of the likelihood function or any derived quantities, but can be shown to be equivalent to maximizing likelihood under some conditions. Our result holds in the non-asymptotic parametric setting, where both the capacity of the model and the number of data examples are finite. We also demonstrate encouraging experimental results.
Discrete random structures are important tools in Bayesian nonparametrics and the resulting models have proven effective in density estimation, clustering, topic modeling and prediction, among others. In this paper, we consider nested processes and study the dependence structures they induce. Dependence ranges between homogeneity, corresponding to full exchangeability, and maximum heterogeneity, corresponding to (unconditional) independence across samples. The popular nested Dirichlet process is shown to degenerate to the fully exchangeable case when there are ties across samples at the observed or latent level. To overcome this drawback, inherent to nesting general discrete random measures, we introduce a novel class of latent nested processes. These are obtained by adding common and group-specific completely random measures and, then, normalising to yield dependent random probability measures. We provide results on the partition distributions induced by latent nested processes, and develop an Markov Chain Monte Carlo sampler for Bayesian inferences. A test for distributional homogeneity across groups is obtained as a by product. The results and their inferential implications are showcased on synthetic and real data.
In this paper we introduce a covariance framework for the analysis of EEG and MEG data that takes into account observed temporal stationarity on small time scales and trial-to-trial variations. We formulate a model for the covariance matrix, which is a Kronecker product of three components that correspond to space, time and epochs/trials, and consider maximum likelihood estimation of the unknown parameter values. An iterative algorithm that finds approximations of the maximum likelihood estimates is proposed. We perform a simulation study to assess the performance of the estimator and investigate the influence of different assumptions about the covariance factors on the estimated covariance matrix and on its components. Apart from that, we illustrate our method on real EEG and MEG data sets. The proposed covariance model is applicable in a variety of cases where spontaneous EEG or MEG acts as source of noise and realistic noise covariance estimates are needed for accurate dipole localization, such as in evoked activity studies, or where the properties of spontaneous EEG or MEG are themselves the topic of interest, such as in combined EEG/fMRI experiments in which the correlation between EEG and fMRI signals is investigated.
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