Although geographically weighted Poisson regression (GWPR) is a popular regression for spatially indexed count data, its development is relatively limited compared to that found for linear geographically weighted regression (GWR), where many extensions (e.g., multiscale GWR, scalable GWR) have been proposed. The weak development of GWPR can be attributed to the computational cost and identification problem in the underpinning Poisson regression model. This study proposes linearized GWPR (L-GWPR) by introducing a log-linear approximation into the GWPR model to overcome these bottlenecks. Because the L-GWPR model is identical to the Gaussian GWR model, it is free from the identification problem, easily implemented, computationally efficient, and offers similar potential for extension. Specifically, L-GWPR does not require a double-loop algorithm, which makes GWPR slow for large samples. Furthermore, we extended L-GWPR by introducing ridge regularization to enhance its stability (regularized L-GWPR). The results of the Monte Carlo experiments confirmed that regularized L-GWPR estimates local coefficients accurately and computationally efficiently. Finally, we compared GWPR and regularized L-GWPR through a crime analysis in Tokyo.
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The Gaussian graphical model (GGM) incorporates an undirected graph to represent the conditional dependence between variables, with the precision matrix encoding partial correlation between pair of variables given the others. To achieve flexible and accurate estimation and inference of GGM, we propose the novel method FLAG, which utilizes the random effects model for pairwise conditional regression to estimate the precision matrix and applies statistical tests to recover the graph. Compared with existing methods, FLAG has several unique advantages: (i) it provides accurate estimation without sparsity assumptions on the precision matrix, (ii) it allows for element-wise inference of the precision matrix, (iii) it achieves computational efficiency by developing an efficient PX-EM algorithm and a MM algorithm accelerated with low-rank updates, and (iv) it enables joint estimation of multiple graphs using FLAG-Meta or FLAG-CA. The proposed methods are evaluated using various simulation settings and real data applications, including gene expression in the human brain, term association in university websites, and stock prices in the U.S. financial market. The results demonstrate that FLAG and its extensions provide accurate precision estimation and graph recovery.
In Japan, the Housing and Land Survey (HLS) provides municipality-level grouped data on household incomes. Although these data can be used for effective local policymaking, their analyses are hindered by several challenges, such as limited information attributed to grouping, the presence of non-sampled areas, and the very low frequency of implementing surveys. To address these challenges, we propose a novel grouped-data-based spatio-temporal finite mixture model to model the income distributions of multiple spatial units at multiple time points. A unique feature of the proposed method is that all the areas share common latent distributions and that the mixing proportions that include the spatial and temporal effects capture the potential area-wise heterogeneity. Thus, incorporating these effects can smooth out the quantities of interest over time and space, impute missing values, and predict future values. By treating the HLS data with the proposed method, we obtain complete maps of the income and poverty measures at an arbitrary time point, which can be used to facilitate rapid and efficient policymaking with fine granularity.
In the problem of aggregation, the aim is to combine a given class of base predictors to achieve predictions nearly as accurate as the best one. In this flexible framework, no assumption is made on the structure of the class or the nature of the target. Aggregation has been studied in both sequential and statistical contexts. Despite some important differences between the two problems, the classical results in both cases feature the same global complexity measure. In this paper, we revisit and tighten classical results in the theory of aggregation in the statistical setting by replacing the global complexity with a smaller, local one. Some of our proofs build on the PAC-Bayes localization technique introduced by Catoni. Among other results, we prove localized versions of the classical bound for the exponential weights estimator due to Leung and Barron and deviation-optimal bounds for the Q-aggregation estimator. These bounds improve over the results of Dai, Rigollet and Zhang for fixed design regression and the results of Lecu\'e and Rigollet for random design regression.
Consider the family of power divergence statistics based on $n$ trials, each leading to one of $r$ possible outcomes. This includes the log-likelihood ratio and Pearson's statistic as important special cases. It is known that in certain regimes (e.g., when $r$ is of order $n^2$ and the allocation is asymptotically uniform as $n\to\infty$) the power divergence statistic converges in distribution to a linear transformation of a Poisson random variable. We establish explicit error bounds in the Kolmogorov (or uniform) metric to complement this convergence result, which may be applied for any values of $n$, $r$ and the index parameter $\lambda$ for which such a finite-sample bound is meaningful. We further use this Poisson approximation result to derive error bounds in Gaussian approximation of the power divergence statistics.
Models with intractable normalizing functions have numerous applications. Because the normalizing constants are functions of the parameters of interest, standard Markov chain Monte Carlo cannot be used for Bayesian inference for these models. A number of algorithms have been developed for such models. Some have the posterior distribution as their asymptotic distribution. Other ``asymptotically inexact'' algorithms do not possess this property. There is limited guidance for evaluating approximations based on these algorithms. Hence it is very hard to tune them. We propose two new diagnostics that address these problems for intractable normalizing function models. Our first diagnostic, inspired by the second Bartlett identity, is in principle broadly applicable to Monte Carlo approximations beyond the normalizing function problem. We develop an approximate version of this diagnostic that is applicable to intractable normalizing function problems. Our second diagnostic is a Monte Carlo approximation to a kernel Stein discrepancy-based diagnostic introduced by Gorham and Mackey (2017). We provide theoretical justification for our methods and apply them to several algorithms in challenging simulated and real data examples including an Ising model, an exponential random graph model, and a Conway--Maxwell--Poisson regression model, obtaining interesting insights about the algorithms in these contexts.
In this paper, numerical methods based on Vieta-Lucas wavelets are proposed for solving a class of singular differential equations. The operational matrix of the derivative for Vieta-Lucas wavelets is derived. It is employed to reduce the differential equations into the system of algebraic equations by applying the ideas of the collocation scheme, Tau scheme, and Galerkin scheme respectively. Furthermore, the convergence analysis and error estimates for Vieta-Lucas wavelets are performed. In the numerical section, the comparative analysis is presented among the different versions of the proposed Vieta-Lucas wavelet methods, and the accuracy of the approaches is evaluated by computing the errors and comparing them to the existing findings.
Federated learning is an approach to collaboratively training machine learning models for multiple parties that prohibit data sharing. One of the challenges in federated learning is non-IID data between clients, as a single model can not fit the data distribution for all clients. Meta-learning, such as Per-FedAvg, is introduced to cope with the challenge. Meta-learning learns shared initial parameters for all clients. Each client employs gradient descent to adapt the initialization to local data distributions quickly to realize model personalization. However, due to non-convex loss function and randomness of sampling update, meta-learning approaches have unstable goals in local adaptation for the same client. This fluctuation in different adaptation directions hinders the convergence in meta-learning. To overcome this challenge, we use the historical local adapted model to restrict the direction of the inner loop and propose an elastic-constrained method. As a result, the current round inner loop keeps historical goals and adapts to better solutions. Experiments show our method boosts meta-learning convergence and improves personalization without additional calculation and communication. Our method achieved SOTA on all metrics in three public datasets.
Blockwise missing data occurs frequently when we integrate multisource or multimodality data where different sources or modalities contain complementary information. In this paper, we consider a high-dimensional linear regression model with blockwise missing covariates and a partially observed response variable. Under this framework, we propose a computationally efficient estimator for the regression coefficient vector based on carefully constructed unbiased estimating equations and a blockwise imputation procedure, and obtain its rate of convergence. Furthermore, building upon an innovative projected estimating equation technique that intrinsically achieves bias-correction of the initial estimator, we propose a nearly unbiased estimator for each individual regression coefficient, which is asymptotically normally distributed under mild conditions. Based on these debiased estimators, asymptotically valid confidence intervals and statistical tests about each regression coefficient are constructed. Numerical studies and application analysis of the Alzheimer's Disease Neuroimaging Initiative data show that the proposed method performs better and benefits more from unsupervised samples than existing methods.
We present a novel numerical method for solving the anisotropic diffusion equation in toroidally confined magnetic fields which is efficient, accurate and provably stable. The continuous problem is written in terms of a derivative operator for the perpendicular transport and a linear operator, obtained through field line tracing, for the parallel transport. We derive energy estimates of the solution of the continuous initial boundary value problem. A discrete formulation is presented using operator splitting in time with the summation by parts finite difference approximation of spatial derivatives for the perpendicular diffusion operator. Weak penalty procedures are derived for implementing both boundary conditions and parallel diffusion operator obtained by field line tracing. We prove that the fully-discrete approximation is unconditionally stable and asymptotic preserving. Discrete energy estimates are shown to match the continuous energy estimate given the correct choice of penalty parameters. Convergence tests are shown for the perpendicular operator by itself, and the ``NIMROD benchmark" problem is used as a manufactured solution to show the full scheme converges even in the case where the perpendicular diffusion is zero. Finally, we present a magnetic field with chaotic regions and islands and show the contours of the anisotropic diffusion equation reproduce key features in the field.
This paper presents the application of a new semi-analytical method of linear regression for Poisson count data to COVID-19 events. The regression is based on the Bonamente and Spence (2022) maximum-likelihood solution for the best-fit parameters, and this paper introduces a simple analytical solution for the covariance matrix that completes the problem of linear regression with Poisson data. The analytical nature for both parameter estimates and their covariance matrix is made possible by a convenient factorization of the linear model proposed by J. Scargle (2013). The method makes use of the asymptotic properties of the Fisher information matrix, whose inverse provides the covariance matrix. The combination of simple analytical methods to obtain both the maximum-likelihood estimates of the parameters, and their covariance matrix, constitute a new and convenient method for the linear regression of Poisson-distributed count data, which are of common occurrence across a variety of fields. A comparison between this new linear regression method and two alternative methods often used for the regression of count data -- the ordinary least-square regression and the $\chi^2$ regression -- is provided with the application of these methods to the analysis of recent COVID-19 count data. The paper also discusses the relative advantages and disadvantages among these methods for the linear regression of Poisson count data.
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