Confidence Regions Research Articles

Application of output constraints to a physics-informed hybrid model for the prediction of the threshold of deep-penetration laser welding

Physics-informed hybrid models, the combination of physics and machine learning, have already shown considerable benefits for quantitative predictions of process constraints, such as the threshold of deep-penetration laser welding. However, despite the improved prediction accuracy and extrapolation capability of such models, there can still be cases where the predictions of the model, including the confidence region, result in values that are not consistent with physical boundary conditions. Therefore, this paper presents the application of additional output constraints to a physics-informed hybrid model to further improve the compliance of the model with physics. Gaussian processes are used for the machine learning model and output warping is used to incorporate the output constraints directly into the model. The approach is demonstrated at the example of a hybrid model for the prediction of the threshold of deep-penetration laser welding.

Limit theorems for local polynomial estimation of regression for functional dependent data

<p>Local polynomial fitting exhibits numerous compelling statistical properties, particularly within the intricate realm of multivariate analysis. However, as functional data analysis gains prominence as a dynamic and pertinent field in data science, the exigency arises for the formulation of a specialized theory tailored to local polynomial fitting. We explored the intricate task of estimating the regression function operator and its partial derivatives for stationary mixing random processes, denoted as $ (Y_i, X_i) $, using local higher-order polynomial fitting. Our key contributions include establishing the joint asymptotic normality of the estimates for both the regression function and its partial derivatives, specifically in the context of strongly mixing processes. Additionally, we provide explicit expressions for the bias and the variance-covariance matrix of the asymptotic distribution. Demonstrating uniform strong consistency over compact subsets, along with delineating the rates of convergence, we substantiated these results for both the regression function and its partial derivatives. Importantly, these findings rooted in reasonably broad conditions that underpinned the underlying models. To demonstrate practical applicability, we leveraged our results to compute pointwise confidence regions. Finally, we extended our ideas to the nonparametric conditional distribution, and obtained its limiting distribution.</p>

An integrated approach for parameter estimation of a propyl propionate synthesis model

This paper introduces an innovative approach to the ProPro synthesis process, aiming to improve the modeling and parameter estimation by integrating and optimizing existing methodologies. It combines well-established models available in the literature with a hybrid solution approach, emphasizing computational efficiency. Additionally, it assesses and enhance the modeling results by calculating confidence regions and uncertainties in both kinetic and thermodynamic modeling. To this end, a batch reactor with Amberlyst 46 catalyst is modeled. The parameter estimation approach combines Particles Swarm Optimization and Gradient Method to estimate both thermodynamic and kinetic parameters, sequentially and simultaneously. Confidence regions and uncertainties are determined for the sequential approach, guiding the search space for the simultaneous estimation. By utilizing PSO, the parameters are estimated, and their confidence regions are determined, providing a comprehensive understanding of their uncertainty. The uncertainty is propagated through the model predictions of the batch reactor, allowing for a comprehensive analysis of the system's behavior. The results demonstrate a good agreement between the model predictions and experimental data, with the expanded uncertainty adequately reflecting the statistical variability of the experimental observations. Furthermore, the heterogeneous Langmuir-Hinshelwood model yields improved representation of the experimental data in both transient and steady states compared to the pseudo-homogeneous model. The Root Mean Square Deviation (RMSD) for parameter estimation shows significant improvement compared to literature values. The simultaneous parameter estimation approach exhibits the best fit performance for the model predictions, confirming its effectiveness in accurately representing the system.

Cosmological Parameter Estimation with Genetic Algorithms

Genetic algorithms are a powerful tool in optimization for single and multimodal functions. This paper provides an overview of their fundamentals with some analytical examples. In addition, we explore how they can be used as a parameter estimation tool in cosmological models to maximize the likelihood function, complementing the analysis with the traditional Markov chain Monte Carlo methods. We analyze that genetic algorithms provide fast estimates by focusing on maximizing the likelihood function, although they cannot provide confidence regions with the same statistical meaning as Bayesian approaches. Moreover, we show that implementing sharing and niching techniques ensures an effective exploration of the parameter space, even in the presence of local optima, always helping to find the global optima. This approach is invaluable in the cosmological context, where an exhaustive space exploration of parameters is essential. We use dark energy models to exemplify the use of genetic algorithms in cosmological parameter estimation, including a multimodal problem, and we also show how to use the output of a genetic algorithm to obtain derived cosmological functions. This paper concludes that genetic algorithms are a handy tool within cosmological data analysis, without replacing the traditional Bayesian methods but providing different advantages.

The HulC: confidence regions from convex hulls

Abstract We develop and analyse the HulC, an intuitive and general method for constructing confidence sets using the convex hull of estimates constructed from subsets of the data. Unlike classical methods which are based on estimating the (limiting) distribution of an estimator, the HulC is often simpler to use and effectively bypasses this step. In comparison to the bootstrap, the HulC requires fewer regularity conditions and succeeds in many examples where the bootstrap provably fails. Unlike sub-sampling, the HulC does not require knowledge of the rate of convergence of the estimators on which it is based. The validity of the HulC requires knowledge of the (asymptotic) median bias of the estimators. We further analyse a variant of our basic method, called the Adaptive HulC, which is fully data-driven and estimates the median bias using sub-sampling. We discuss these methods in the context of several challenging inferential problems which arise in parametric, semi-parametric, and non-parametric inference. Although our focus is on validity under weak regularity conditions, we also provide some general results on the width of the HulC confidence sets, showing that in many cases the HulC confidence sets have near-optimal width.

Evaluating joint confidence region of hypervolume under ROC manifold and generalized Youden index.

In biomarker evaluation/diagnostic studies, the hypervolume under the receiver operating characteristic manifold ( ) and the generalized Youden index ( ) are the most popular measures for assessing classification accuracy under multiple classes. While is frequently used to evaluate the overall accuracy, provides direct measure of accuracy at the optimal cut-points. Simultaneous evaluation of and provides a comprehensive picture about the classification accuracy of the biomarker/diagnostic test under consideration. This article studies both parametric and non-parametric approaches for estimating the confidence region of and for a single biomarker. The performances of the proposed methods are investigated by an extensive simulation study and are applied to a real data set from the Alzheimer's Disease Neuroimaging Initiative.

Confidence in causal inference under structure uncertainty in linear causal models with equal variances

Abstract Inferring the effect of interventions within complex systems is a fundamental problem of statistics. A widely studied approach uses structural causal models that postulate noisy functional relations among a set of interacting variables. The underlying causal structure is then naturally represented by a directed graph whose edges indicate direct causal dependencies. In a recent line of work, additional assumptions on the causal models have been shown to render this causal graph identifiable from observational data alone. One example is the assumption of linear causal relations with equal error variances that we will take up in this work. When the graph structure is known, classical methods may be used for calculating estimates and confidence intervals for causal-effects. However, in many applications, expert knowledge that provides an a priori valid causal structure is not available. Lacking alternatives, a commonly used two-step approach first learns a graph and then treats the graph as known in inference. This, however, yields confidence intervals that are overly optimistic and fail to account for the data-driven model choice. We argue that to draw reliable conclusions, it is necessary to incorporate the remaining uncertainty about the underlying causal structure in confidence statements about causal-effects. To address this issue, we present a framework based on test inversion that allows us to give confidence regions for total causal-effects that capture both sources of uncertainty: causal structure and numerical size of non-zero effects.

Constrained evolution of Hamiltonian phase space distributions in the presence of natural, non-conservative forces

Confidence regions for spacecraft state can be constructed in phase space which encapsulate some region where there is a likelihood for the state to reside. These regions can be treated as phase space distributions or structures. Structures, such as surfaces or volumes, are constrained to preserve specific properties as they evolve in phase space under Hamiltonian dynamics. Thus, spacecraft uncertainty is then constrained by Hamiltonian flow which can provide insight into state determination. This work examines the modified constraints in the presence of non-conservative forces which relate to both probabilistic and geometric properties of the evolving uncertainty structure. The modified constraints are then derived for a Two-Body and drag environment and are shown to be valid after comparison with alternative methods. Applying the modified constraints, the constrained evolution of the confidence region is then tied to a simple physical explanation for the changing knowledge in our spacecraft state, in the atmospheric drag environment and Poynting–Robertson drag environment.

Statistical inference for dependence networks in topological data analysis.

Topological data analysis (TDA) provide tools that are becoming increasingly popular for analyzing multivariate time series data. One key aspect in analyzing multivariate time series is dependence between components. One application is on brain signal analysis. In particular, various dependence patterns in brain networks may be linked to specific tasks and cognitive processes. These dependence patterns may be altered by various neurological and cognitive impairments such as Alzheimer's and Parkinson's diseases, as well as attention deficit hyperactivity disorder (ADHD). Because there is no ground-truth with known dependence patterns in real brain signals, testing new TDA methods on multivariate time series is still a challenge. Our goal here is to develop novel statistical inference procedures via simulations. Simulations are useful for generating some null distributions of a test statistic (for hypothesis testing), forming confidence regions, and for evaluating the performance of proposed TDA methods. To the best of our knowledge, there are no methods that simulate multivariate time series data with potentially complex user-specified connectivity patterns. In this paper we present a novel approach to simulate multivariate time series with specific number of cycles/holes in its dependence network. Furthermore, we also provide a procedure for generating higher dimensional topological features.

Augmenting the power of time-delay cosmography in lens galaxy clusters by probing their member galaxies

We present a simple and promising new method to measure the expansion rate and the geometry of the universe that combines observations related to the time delays between the multiple images of time-varying sources, strongly lensed by galaxy clusters, and Type Ia supernovae, exploding in galaxies belonging to the same lens clusters. By means of two different statistical techniques that adopt realistic errors on the relevant quantities, we quantify the accuracy of the inferred cosmological parameter values. We show that the estimate of the Hubble constant is robust and competitive, and depends only mildly on the chosen cosmological model. Remarkably, the two probes separately produce confidence regions on the cosmological parameter planes that are oriented in complementary ways, thus providing in combination valuable information on the values of the other cosmological parameters. We conclude by illustrating the immediate observational feasibility of the proposed joint method in a well-studied lens galaxy cluster, with a relatively small investment of telescope time for monitoring from a 2 to 3 m class ground-based telescope.

Correcting for Antibody Waning in Cumulative Incidence Estimation From Sequential Serosurveys.

Serosurveys are a widely used tool to estimate the cumulative incidence-the fraction of a population that has been infected by a given pathogen. These surveys rely on serological assays that measure the level of pathogen-specific antibodies. Because antibody levels are waning, the fraction of previously infected individuals that have seroreverted increases with time past infection. To avoid underestimating the true cumulative incidence, it is therefore essential to correct for waning antibody levels. We present an empirically supported approach for seroreversion correction in cumulative incidence estimation when sequential serosurveys are conducted in the context of a newly emerging infectious disease. The correction is based on the observed dynamics of antibody titers in seropositive cases and validated using several in silico test scenarios. Furthermore, through this approach we revise a previous cumulative incidence estimate relying on the assumption of an exponentially declining probability of seroreversion over time, of severe acute respiratory syndrome coronavirus 2, of 76% in Manaus, Brazil, by October 2020 to 47.6% (95% confidence region: 43.5-53.5). This estimate has implications, for example, for the proximity to herd immunity in Manaus in late 2020.

Simultaneous confidence region of an embedded one-dimensional curve in multi-dimensional space

This paper focuses on the simultaneous confidence region of a one-dimensional curve embedded in multi-dimensional space. Local linear regression is applied component-wise to each variable in multi-dimensional data, which yields an estimator of the one-dimensional curve. A simultaneous confidence region of the curve is proposed based on this estimator and theoretical results for the estimator and the region are developed under some reasonable assumptions. Practically efficient algorithms to determine the thickness of the region are also addressed. The effectiveness of the region is investigated through simulation studies and applications to artificial and real datasets, which reveal that the proposed simultaneous confidence region works well.

Bootstrapping ARMA time series models after model selection

Inference after model selection is a very important problem. This article derives the asymptotic distribution of some model selection estimators for autoregressive moving average time series models. Under strong regularity conditions, the model selection estimators are asymptotically normal, but generally the asymptotic distribution is a non normal mixture distribution. Hence bootstrap confidence regions that can handle this complicated distribution were used for hypothesis testing. A bootstrap technique to eliminate selection bias is to fit the model selection estimator β ̂ M S ∗ to a bootstrap sample to find a submodel, then draw another bootstrap sample, and fit the same submodel to get the bootstrap estimator β ̂ M I X ∗ .

Statistical Inference for Partially Linear Varying Coefficient Spatial Autoregressive Panel Data Model

This paper studies the estimation and inference of a partially linear varying coefficient spatial autoregressive panel data model with fixed effects. By means of the basis function approximations and the instrumental variable methods, we propose a two-stage least squares estimation procedure to estimate the unknown parametric and nonparametric components, and meanwhile study the asymptotic properties of the proposed estimators. Together with an empirical log-likelihood ratio function for the regression parameters, which follows an asymptotic chi-square distribution under some regularity conditions, we can further construct accurate confidence regions for the unknown parameters. Simulation studies show that the finite sample performance of the proposed methods are satisfactory in a wide range of settings. Lastly, when applied to the public capital data, our proposed model can also better reflect the changing characteristics of the US economy compared to the parametric panel data models.

Robust estimation and bias-corrected empirical likelihood in generalized linear models with right censored data

In this paper, we study the robust estimation and empirical likelihood for the regression parameter in generalized linear models with right censored data. A robust estimating equation is proposed to estimate the regression parameter, and the resulting estimator has consistent and asymptotic normality. A bias-corrected empirical log-likelihood ratio statistic of the regression parameter is constructed, and it is shown that the statistic converges weakly to a standard χ 2 distribution. The result can be directly used to construct the confidence region of regression parameter. We use the bias correction method to directly calibrate the empirical log-likelihood ratio, which does not need to be multiplied by an adjustment factor. We also propose a method for selecting the tuning parameters in the loss function. Simulation studies show that the estimator of the regression parameter is robust and the bias-corrected empirical likelihood is better than the normal approximation method. An example of a real dataset from Alzheimer's disease studies shows that the proposed method can be applied in practical problems.

Comparison of confidence regions for quantum state tomography

Abstract The quantum state associated to an unknown experimental preparation procedure can be determined by performing quantum state tomography. If the statistical uncertainty in the data dominates over other experimental errors, then a tomographic reconstruction procedure must express this uncertainty. A rigorous way to accomplish this is via statistical confidence regions (CRs) in state space. Naturally, the size of this region decreases when increasing the number of samples, but it also depends critically on the construction method of the region. We compare recent methods for constructing CRs as well as a reference method based on a Gaussian approximation. For the comparison, we propose an operational measure with the finding, that there is a significant difference between methods, but which method is preferable can depend on the details of the state preparation scenario.

Confidence region for distributed stochastic optimization problem via stochastic gradient tracking method

Since stochastic approximation (SA) based algorithms are easy to implement and need less memory, they are very popular in distributed stochastic optimization problems. Many works have focused on the consistency of the objective values and the iterates returned by the SA based algorithms. It is of fundamental interest to know how to quantify the uncertainty associated with SA solutions via the confidence regions of a prescribed level of significance for the true solution. In this paper, we discuss the framework of constructing the asymptotic confidence regions of the optimal solution to distributed stochastic optimization problem with a focus on the distributed stochastic gradient tracking method. To attain this goal, we first present the asymptotic normality of Polyak–Ruppert averaged distributed stochastic gradient tracking method. We then estimate the corresponding covariance matrix through online estimators. Finally, we provide a practical procedure to build the asymptotic confidence regions for the optimal solution. Numerical tests are also conducted to show the efficiency of the proposed methods.

Unifying Estimation and Inference for Linear Regression with Stationary and Integrated or Near-Integrated Variables

Abstract There is a discrepancy in the limiting distributions of least-squares estimators for stationary and integrated variables. For statistical inference, it must be decided which distribution should be used in advance. This motivates us to develop a unifying inference procedure based on weighted estimation. The asymptotic distributions of the proposed estimators are developed and a random weighting bootstrap method is proposed for constructing confidence regions. The proposed method outperforms existing methods (with time constant or time-varying error variance) in simulations. We further study the predictability of asset returns in a setting where some of our state variables are endogenous.

Constraints on cosmological model in Weyl-Cartan spacetime from astronomical measurements

In order to give some lights on the issue of Hubble tension, alternative to Riemannian spacetime, we investigate a cosmological model in Weyl-Cartan spacetime by astronomical measurements, where a new density parameter Ω ϕ is introduced to denote the non-Riemannian spacetime structure, which scales as ∼(1 + z)6. Based on Markov Chain Monte Carlo estimation and taking astronomical measurements of cosmic microwave background (Planck 2018), baryon acoustic oscillations and Type Ia supernova (Pantheon), model parameters of our considering case are constrained. Our results show that Hubble tension cannot be alleviated in Weyl-Cartan cosmology. However, constraints on the parameter Ω ϕ can not be distinguished from ΛCDM model in 1σ confidence region. Moreover, the space of the Universe is infinite and unbounded, and an upper limit of the model parameter Ω ϕ is also obtained. Additionally, it is worth noting that results obtained in this work are valid for models that the scaling behavior of density parameters is same to our considering case.

Sensitive method for total selenium determination in garlic and nuts by UV–photochemical vapor generation coupled to atomic absorption spectrometry using mild conditions

Photochemical vapor generation (PVG) coupled on–line to atomic absorption spectrometry using a flow injection mode was investigated for the determination of total Se in garlic and nut samples. In the presence of low molecular weight organic acid solutions, Se(IV) is converted by UV irradiation to volatile species which are transported to the heated quartz tube atomizer. Formic and acetic acids were tested as photochemical reagents and a mixture of 1.5 % (v/v) formic and 5 % (v/v) acetic acid was the best alternative. Central composite design was employed for parameters optimization. The effect of potential interfering elements such as hydride–forming elements, transition metals and nitric acid was investigated to ascertain possible effects of the matrix on Se(IV) signal. Under optimized conditions a detection limit (LOD) of 33 ng g–1 (3σ) was reached. Samples of garlic and nuts purchased at a local market were analyzed and Se concentrations spanned the range 0.36–248 µg g–1. These results were compared with ICP–MS measurements utilizing an elliptical joint confidence region (EJCR) with bilinear least squares analysis, and good results were obtained.