The accuracy of Earth system models is compromised by unknown and/or unresolved dynamics, making the quantification of systematic model errors essential. While a model parameter estimation, which allows parameters to change spatio-temporally, shows promise in quantifying and mitigating systematic model errors, the estimation of the spatio-temporally distributed model parameters has been practically challenging. Here we present an efficient and practical method to estimate time-varying parameters in high-dimensional spaces. In our proposed method, Hybrid Offline and Online Parameter Estimation with ensemble Kalman filtering (HOOPE-EnKF), model parameters estimated by EnKF are constrained by results of offline batch optimization, in which the posterior distribution of model parameters is obtained by comparing simulated and observed climatological variables. HOOPE-EnKF outperforms the original EnKF in synthetic experiments using a two-scale Lorenz96 model and a simple global general circulation model. One advantage of HOOPE-EnKF over traditional EnKFs is that its performance is not greatly affected by inflation factors for model parameters, thus eliminating the need for extensive tuning of inflation factors. We thoroughly discuss the potential of HOOPE-EnKF as a practical method for improving parameterizations of process-based models and prediction in real-world applications such as numerical weather prediction.
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How do score-based generative models (SBMs) learn the data distribution supported on a low-dimensional manifold? We investigate the score model of a trained SBM through its linear approximations and subspaces spanned by local feature vectors. During diffusion as the noise decreases, the local dimensionality increases and becomes more varied between different sample sequences. Importantly, we find that the learned vector field mixes samples by a non-conservative field within the manifold, although it denoises with normal projections as if there is an energy function in off-manifold directions. At each noise level, the subspace spanned by the local features overlap with an effective density function. These observations suggest that SBMs can flexibly mix samples with the learned score field while carefully maintaining a manifold-like structure of the data distribution.
We introduce general tools for designing efficient private estimation algorithms, in the high-dimensional settings, whose statistical guarantees almost match those of the best known non-private algorithms. To illustrate our techniques, we consider two problems: recovery of stochastic block models and learning mixtures of spherical Gaussians. For the former, we present the first efficient $(\epsilon, \delta)$-differentially private algorithm for both weak recovery and exact recovery. Previously known algorithms achieving comparable guarantees required quasi-polynomial time. For the latter, we design an $(\epsilon, \delta)$-differentially private algorithm that recovers the centers of the $k$-mixture when the minimum separation is at least $ O(k^{1/t}\sqrt{t})$. For all choices of $t$, this algorithm requires sample complexity $n\geq k^{O(1)}d^{O(t)}$ and time complexity $(nd)^{O(t)}$. Prior work required minimum separation at least $O(\sqrt{k})$ as well as an explicit upper bound on the Euclidean norm of the centers.
Bayesian modeling provides a principled approach to quantifying uncertainty in model parameters and model structure and has seen a surge of applications in recent years. Within the context of a Bayesian workflow, we are concerned with model selection for the purpose of finding models that best explain the data, that is, help us understand the underlying data generating process. Since we rarely have access to the true process, all we are left with during real-world analyses is incomplete causal knowledge from sources outside of the current data and model predictions of said data. This leads to the important question of when the use of prediction as a proxy for explanation for the purpose of model selection is valid. We approach this question by means of large-scale simulations of Bayesian generalized linear models where we investigate various causal and statistical misspecifications. Our results indicate that the use of prediction as proxy for explanation is valid and safe only when the models under consideration are sufficiently consistent with the underlying causal structure of the true data generating process.
Generalized linear models (GLMs) are popular for data-analysis in almost all quantitative sciences, but the choice of likelihood family and link function is often difficult. This motivates the search for likelihoods and links that minimize the impact of potential misspecification. We perform a large-scale simulation study on double-bounded and lower-bounded response data where we systematically vary both true and assumed likelihoods and links. In contrast to previous studies, we also study posterior calibration and uncertainty metrics in addition to point-estimate accuracy. Our results indicate that certain likelihoods and links can be remarkably robust to misspecification, performing almost on par with their respective true counterparts. Additionally, normal likelihood models with identity link (i.e., linear regression) often achieve calibration comparable to the more structurally faithful alternatives, at least in the studied scenarios. On the basis of our findings, we provide practical suggestions for robust likelihood and link choices in GLMs.
This work presents an abstract framework for the design, implementation, and analysis of the multiscale spectral generalized finite element method (MS-GFEM), a particular numerical multiscale method originally proposed in [I. Babuska and R. Lipton, Multiscale Model.\;\,Simul., 9 (2011), pp.~373--406]. MS-GFEM is a partition of unity method employing optimal local approximation spaces constructed from local spectral problems. We establish a general local approximation theory demonstrating exponential convergence with respect to local degrees of freedom under certain assumptions, with explicit dependence on key problem parameters. Our framework applies to a broad class of multiscale PDEs with $L^{\infty}$-coefficients in both continuous and discrete, finite element settings, including highly indefinite problems (convection-dominated diffusion, as well as the high-frequency Helmholtz, Maxwell and elastic wave equations with impedance boundary conditions), and higher-order problems. Notably, we prove a local convergence rate of $O(e^{-cn^{1/d}})$ for MS-GFEM for all these problems, improving upon the $O(e^{-cn^{1/(d+1)}})$ rate shown by Babuska and Lipton. Moreover, based on the abstract local approximation theory for MS-GFEM, we establish a unified framework for showing low-rank approximations to multiscale PDEs. This framework applies to the aforementioned problems, proving that the associated Green's functions admit an $O(|\log\epsilon|^{d})$-term separable approximation on well-separated domains with error $\epsilon>0$. Our analysis improves and generalizes the result in [M. Bebendorf and W. Hackbusch, Numerische Mathematik, 95 (2003), pp.~1-28] where an $O(|\log\epsilon|^{d+1})$-term separable approximation was proved for Poisson-type problems.
The maximum likelihood estimator (MLE) is pivotal in statistical inference, yet its application is often hindered by the absence of closed-form solutions for many models. This poses challenges in real-time computation scenarios, particularly within embedded systems technology, where numerical methods are impractical. This study introduces a generalized form of the MLE that yields closed-form estimators under certain conditions. We derive the asymptotic properties of the proposed estimator and demonstrate that our approach retains key properties such as invariance under one-to-one transformations, strong consistency, and an asymptotic normal distribution. The effectiveness of the generalized MLE is exemplified through its application to the Gamma, Nakagami, and Beta distributions, showcasing improvements over the traditional MLE. Additionally, we extend this methodology to a bivariate gamma distribution, successfully deriving closed-form estimators. This advancement presents significant implications for real-time statistical analysis across various applications.
Factor models have been widely used to summarize the variability of high-dimensional data through a set of factors with much lower dimensionality. Gaussian linear factor models have been particularly popular due to their interpretability and ease of computation. However, in practice, data often violate the multivariate Gaussian assumption. To characterize higher-order dependence and nonlinearity, models that include factors as predictors in flexible multivariate regression are popular, with GP-LVMs using Gaussian process (GP) priors for the regression function and VAEs using deep neural networks. Unfortunately, such approaches lack identifiability and interpretability and tend to produce brittle and non-reproducible results. To address these problems by simplifying the nonparametric factor model while maintaining flexibility, we propose the NIFTY framework, which parsimoniously transforms uniform latent variables using one-dimensional nonlinear mappings and then applies a linear generative model. The induced multivariate distribution falls into a flexible class while maintaining simple computation and interpretation. We prove that this model is identifiable and empirically study NIFTY using simulated data, observing good performance in density estimation and data visualization. We then apply NIFTY to bird song data in an environmental monitoring application.
Feedforward neural networks (FNNs) are typically viewed as pure prediction algorithms, and their strong predictive performance has led to their use in many machine-learning applications. However, their flexibility comes with an interpretability trade-off; thus, FNNs have been historically less popular among statisticians. Nevertheless, classical statistical theory, such as significance testing and uncertainty quantification, is still relevant. Supplementing FNNs with methods of statistical inference, and covariate-effect visualisations, can shift the focus away from black-box prediction and make FNNs more akin to traditional statistical models. This can allow for more inferential analysis, and, hence, make FNNs more accessible within the statistical-modelling context.
This study addresses the significant challenge of developing efficient decoding algorithms for classifying steady-state visual evoked potentials (SSVEPs) in scenarios characterized by extreme scarcity of calibration data, where only one calibration is available for each stimulus target. To tackle this problem, we introduce a novel cross-subject dual-domain fusion network (CSDuDoFN) incorporating task-related and task-discriminant component analysis (TRCA and TDCA) for one-shot SSVEP classification. The CSDuDoFN framework is designed to comprehensively transfer information from source subjects, while TRCA and TDCA are employed to exploit the single available calibration of the target subject. Specifically, we develop multi-reference least-squares transformation (MLST) to map data from both source subjects and the target subject into the domain of sine-cosine templates, thereby mitigating inter-individual variability and benefiting transfer learning. Subsequently, the transformed data in the sine-cosine templates domain and the original domain data are separately utilized to train a convolutional neural network (CNN) model, with the adequate fusion of their feature maps occurring at distinct network layers. To further capitalize on the calibration of the target subject, source aliasing matrix estimation (SAME) data augmentation is incorporated into the training process of the ensemble TRCA (eTRCA) and TDCA models. Ultimately, the outputs of the CSDuDoFN, eTRCA, and TDCA are combined for SSVEP classification. The effectiveness of our proposed approach is comprehensively evaluated on three publicly available SSVEP datasets, achieving the best performance on two datasets and competitive performance on one. This underscores the potential for integrating brain-computer interface (BCI) into daily life.
This paper presents a novel centralized, variational data assimilation approach for calibrating transient dynamic models in electrical power systems, focusing on load model parameters. With the increasing importance of inverter-based resources, assessing power systems' dynamic performance under disturbances has become challenging, necessitating robust model calibration methods. The proposed approach expands on previous Bayesian frameworks by establishing a posterior distribution of parameters using an approximation around the maximum a posteriori value. We illustrate the efficacy of our method by generating events of varying intensity, highlighting its ability to capture the systems' evolution accurately and with associated uncertainty estimates. This research improves the precision of dynamic performance assessments in modern power systems, with potential applications in managing uncertainties and optimizing system operations.
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