Space Variables Research Articles

Estimating Latent-Variable Panel Data Models Using Parameter-Expanded SEM Methods

This article presents new estimation algorithms for three types of dynamic panel data models with latent variables: factor models, discrete choice models, and persistent-transitory quantile processes. The new methods combine the parameter expansion (PX) ideas of Liu, Rubin, and Wu with the stochastic expectation-maximization (SEM) algorithm in likelihood and moment-based contexts. The goal is to facilitate convergence in models with a large space of latent variables by improving algorithmic efficiency. This is achieved by specifying expanded models within the M step. Effectively, we are proposing new estimators for the pseudo-data within iterations that take into account the fact that the model of interest is misspecified for draws based on parameter values far from the truth. We establish the asymptotic equivalence of the likelihood-based PX-SEM to an alternative SEM algorithm with a smaller expected fraction of missing information compared to the standard SEM based on the original model, implying a faster global convergence rate. Finally, in simulations we show that the new algorithms significantly improve the convergence speed relative to standard SEM algorithms, sometimes dramatically so, by reducing the total computing time from hours to a few minutes.

A stochastic approach for parameter optimization of feature detection algorithms for non-target screening in mass spectrometry.

Feature detection plays a crucial role in non-target screening (NTS), requiring careful selection of algorithm parameters to minimize false positive (FP) features. In this study, a stochastic approach was employed to optimize the parameter settings of feature detection algorithms used in processing high-resolution mass spectrometry data. This approach was demonstrated using four open-source algorithms (OpenMS, SAFD, XCMS, and KPIC2) within the patRoon software platform for processing extracts from drinking water samples spiked with 46 per- and polyfluoroalkyl substances (PFAS). The designed method is based on a stochastic strategy involving random sampling from variable space and the use of Pearson correlation to assess the impact of each parameter on the number of detected suspect analytes. Using our approach, the optimized parameters led to improvement in the algorithm performance by increasing suspect hits in case of SAFD and XCMS, and reducing the total number of detected features (i.e., minimizing FP) for OpenMS. These improvements were further validated on three different drinking water samples as test dataset. The optimized parameters resulted in a lower false discovery rate (FDR%) compared to the default parameters, effectively increasing the detection of true positive features. This work also highlights the necessity of algorithm parameter optimization prior to starting the NTS to reduce the complexity of such datasets.

A uniformly convergent method for two-parameter singularly perturbed parabolic partial differential equations with a large shift

A class of singularly perturbed two-parameter parabolic partial differential equations with a large shift is studied. These equations include time-dependent variables and exhibit sensitivity to small perturbations in the system parameters. Moreover, the shift captures the influence of neighbouring states or events on the current evolution of the system. It adds memory-like behaviour to the problem, making their analysis and numerical solution more intricate. The solution of these equations exhibits a multiscale character. The interior and boundary layers exist across which the solution has a steep gradient. A higher-order accurate numerical method is proposed on a non-uniform mesh to solve the problem. The method combines an upwind difference scheme in space and a Crank-Nicolson scheme in time. The method is unconditionally stable, and parameters uniformly convergent with second-order accuracy in time and first-order accuracy in the space variable. Besides, the paper presents numerical results and illustrations to support theoretical estimates.

Particles with constant speed and random velocity in 3+1 space-time: separation of the space variables

We consider a particle in 3+1 space-time dimensions which constantly moves at speed of light c, randomly changing its velocity which can be represented by a point on the surface of a sphere of radius c. The velocity performs an isotropic Wiener process on this surface so that the velocity trajectories are almost everywhere continuous although not differentiable, while the position trajectories are continuous and differentiable. Together with the process that describes the particle in the ‘rest frame’, where the diffusion of velocity on the surface of the sphere is isotropic, the entire family of anisotropic processes which result from Lorentz boosts is also described. The focus of this article is on stochastic evolution in space. We identify a reduced set of variables whose stochastic evolution is autonomous from the remaining variables, but, nevertheless, carry all the relevant information concerning the spatial properties of the process. The associated stochastic equations as well the Forward Kolmogorov equation are considerably simplified compared to those of the complete set of position and velocity variables.

Globally solvable time-periodic evolution equations in Gelfand–Shilov classes

AbstractIn this paper we consider a class of evolution operators with coefficients depending on time and space variables $$(t,x) \in {\mathbb {T}}\times {\mathbb {R}}^n$$ ( t , x ) ∈ T × R n , where $${\mathbb {T}}$$ T is the one-dimensional torus, and prove necessary and sufficient conditions for their global solvability in (time-periodic) Gelfand–Shilov spaces. The argument of the proof is based on a characterization of these spaces in terms of the eigenfunction expansions given by a fixed self-adjoint, globally elliptic differential operator on $${\mathbb {R}}^n$$ R n .

A note about almost uniform convergence on D-poset of intuitionistic fuzzy sets

The aim of this contribution is studying the almost uniform convergence on D-poset of intuitionistic fuzzy sets. We prove the connection between almost everywhere convergence of random variables in Kolmogorov probability space and almost uniform convergence of observables in the mentioned D-poset. We define a product operation on D-poset of intuitionistic fuzzy sets and prove the existence of a joint observable, too.

Active learning for adaptive surrogate model improvement in high-dimensional problems

This paper investigates a novel approach to efficiently construct and improve surrogate models in problems with high-dimensional input and output. In this approach, the principal components and corresponding features of the high-dimensional output are first identified. For each feature, the active subspace technique is used to identify a corresponding low-dimensional subspace of the input domain; then a surrogate model is built for each feature in its corresponding active subspace. A low-dimensional adaptive learning strategy is proposed to identify training samples to improve the surrogate model. In contrast to existing adaptive learning methods that focus on a scalar output or a small number of outputs, this paper addresses adaptive learning with high-dimensional input and output, with a novel learning function that balances exploration and exploitation, i.e., considering unexplored regions and high-error regions, respectively. The adaptive learning is in terms of the active variables in the low-dimensional space, and the newly added training samples can be easily mapped back to the original space for running the expensive physics model. The proposed method is demonstrated for the numerical simulation of an additive manufacturing part, with a high-dimensional field output quantity of interest (residual stress) in the component that has spatial variability due to the stochastic nature of multiple input variables (including process variables and material properties). Various factors in the adaptive learning process are investigated, including the number of training samples, range and distribution of the adaptive training samples, contributions of various errors, and the importance of exploration versus exploitation in the learning function.

Non-probabilistic reliability analysis with both multi-super-ellipsoidal input and fuzzy state

In real-world engineering scenarios, incomplete uncertainty information and ambiguous failure states persist and pose significant challenges for structural reliability analysis. This paper introduces a non-probabilistic fuzzy reliability analysis (NPFRA) model featuring fuzzy output states, where the input uncertainties are quantified by a multi-super-ellipsoidal model. Initially, we define both reliability and failure indices of NPFRA, and provide the corresponding Monte Carlo simulation (MCS) solution. Additionally, an extended variable space (EVS) method is established to transform the NPFRA problem into a conventional non-probabilistic reliability analysis (NPRA) one, and MCS based on EVS is derived accordingly. To address the efficiency issue of MCS, a novel method called active learning kriging with norm-constrained expected risk function (ALK-NERF) is developed explicitly for NPFRA. Four examples are adopted to verify the rationality and effectiveness of the proposed ALK-NERF for NPFRA.

A quest for precipitation attractors in weather radar archives

Abstract. Archives of composite weather radar images represent an invaluable resource to study the predictability of precipitation. In this paper, we compare two distinct approaches to construct empirical low-dimensional attractors from radar precipitation fields. In the first approach, the phase space variables of the attractor are defined using the domain-scale statistics of precipitation fields, such as the mean precipitation, fraction of rain, and spatial and temporal correlations. The second type of attractor considers the spatial distribution of precipitation and is built by principal component analysis (PCA). For both attractors, we investigate the density of trajectories in phase space, growth of errors from analogue states, and fractal properties. To represent different scales and climatic and orographic conditions, the analyses are done using multi-year radar archives over the continental United States (≈4000×4000 km2, 21 years) and the Swiss Alpine region (≈500×500 km2, 6 years).

Urban green spaces, self-rated air pollution and health: A sensitivity analysis of green space characteristics and proximity in four European cities

Exploring the influence of green space characteristics and proximity on health via air pollution mitigation, our study analysed data from 1,365 participants across Porto, Nantes, Sofia, and Høje-Taastrup. Utilizing OpenStreetMap and the AID-PRIGSHARE tool, we generated nine green space indicators around residential addresses at 15 distances, ranging from 100m to 1500m. We performed a mediation analysis for these 135 green space variables and revealed significant associations between self-rated air pollution and self-rated health for specific green space characteristics. In our study, indirect positive effects on health via air pollution were mainly associated with green corridors in intermediate Euclidean distances (800-1,000m) and the amount of accessible green spaces in larger network distances (1,400–1,500m). Our results suggest that the amount of connected green spaces measured in intermediate surroundings seems to be a prime green space characteristic that could drive the air pollution mitigation pathway to health.

A locking-free numerical method for the quasi-static linear thermo-poroelasticity model

In this paper, we propose a locking-free numerical method to solve the quasi-static linear thermo-poroelasticity model, which exhibits two types of locking phenomena: Poisson locking and nonphysical oscillations. By analyzing the regularity of solution of the original model, we find that the Poisson locking is caused by divu≈0 as λ→+∞. Moreover, when discretizing the diffusion equations using backward Euler method for time, one can see that divu1≈0 for some special parameters. If we directly use the continuous Galerkin mixed finite element method (FEM) to solve the original model, the pressure and temperature fields exhibit numerical oscillations at early times. To overcome these two locking phenomena, we introduce a new variable ξ=αp+βT−λdivu to reformulate the original problem into a new one. This new problem includes a built-in mechanism to maintain stability for the continuous Galerkin mixed FEM. We further prove the existence and uniqueness of the weak solution by using the standard Galerkin method in conjunction with regularity estimates. We also design a fully discrete time-stepping scheme that employs mixed FEM with P2−P1−P1−P1 element pairs for the space variables and the backward Euler method for the time variable. Optimal convergence is demonstrated in both space and time. Finally, numerical examples are provided to demonstrate the optimal convergence rates of the variables and the robustness of the proposed method with respect to ν, and to verify the absence of the locking phenomenon.

Projection-based reduced-order modelling of time-periodic problems, with application to flow past flapping hydrofoils

Simulating forced time-periodic flows in industrial applications presents significant computational challenges, partly due to the need to overcome costly transients before achieving time-periodicity. Reduced-order modelling emerges as a promising method to speed-up computations. We extend upon the work of Lotz et al. (2024) where a time-periodic space–time model is introduced. We present a time-periodic reduced-order model that directly finds the time-periodic solution without requiring extensive time integration. The reduced-order model gives a reduction in variables in both space and time. Our approach involves a POD-Galerkin reduced-order model based on a time-periodic full-order model that employs isogeometric analysis, residual-based variational multiscale turbulence modelling and weak boundary conditions. The projection-based reduced-order model inherits these features. We evaluate the reduced-order model with numerical experiments on moving hydrofoils. The motion is known a priori and we restrict ourselves to two spatial dimensions. In these experiments we vary the Strouhal and Reynolds numbers, and the motion profile respectively. Reduced-order model solutions agree well with those of the full-order model. The errors over the entire time period of thrust and lift forces are less than 0.2%. This includes complex scenarios such as the transition from drag to thrust production with increasing Strouhal number. Our time-periodic reduced-order model offers speed-ups ranging from O(102) to O(103) compared to the full-order model, depending upon the basis size. This makes it an appealing solution for prescribed time-periodic problems, with potential for additional speedup through nonlinear reduction techniques such as hyper-reduction.

Innovative design of wood texture images for indoor furniture based on variable space

In the design of furniture wood texture images, image restoration is a key issue. This study proposes a Bregmanized operator splitting optimization algorithm based on variable space. This study combines variable spatial morphology to process texture images and effectively extract image features using different operators, thereby achieving image restoration. The results of comparing the proposed algorithm with other image processing algorithms showed that the research algorithm achieved a peak signal-to-noise ratio of 29.86 and a structural similarity index of 0.87 in image denoising, respectively, and had a good denoising effect. In terms of image deblurring, the research algorithm had the lowest root mean square error values on the France and Boat datasets, with values of 8.98 and 8.82, respectively, indicating that the image processed by the algorithm had a high similarity with the real image. In terms of image resolution reconstruction, the peak signal-to-noise ratio and root mean square error values of the research algorithm reached 29.74 and 12.67, respectively, indicating that the reconstructed image had the best fit with the original image and the smallest error. In summary, the proposed algorithm has shown good performance in image processing and can be effectively applied in fields such as image denoising, deblurring, and resolution reconstruction. It provides effective methods and technical support for innovative design of wood texture images in indoor furniture.

Bearing degradation prediction based on deep latent variable state space model with differential transformation

Rolling bearings are a critical component of mechanical transmission equipment. Predicting their degradation trend is crucial for ensuring safe and stable equipment operation. Most existing bearing degradation prediction methods based on state space models (SSMs) use either linear functions or limited nonlinear functions (e.g., exponential/power laws) to construct the state and measurement equations. As such, these models fail to adapt to the complex and varied nonlinear degradation processes that occur in real-world environments. To address this limitation, we developed a deep latent variable-driven state space degradation model and employed it for bearing degradation prediction. Owing to the powerful nonlinear modeling ability of deep learning models, the proposed method extends the applicability of state space equations. In addition, it inherits the advantages of SSMs and can model uncertainties in a structured manner. Furthermore, the model was integrated with differential pre-transformation to improve its long-term prediction performance. Finally, to validate the effectiveness of the proposed model in predicting bearing degradation, experiments were conducted using a bearing dataset from the PRONOSTIA platform and real wind turbine bearing data. Results showed that the proposed method yielded higher bearing degradation prediction accuracy than existing methods, thus demonstrating the superior performance of the proposed model in predicting bearing degradation.

An improved problem transformation algorithm for large-scale multi-objective optimization

Solving Large-Scale Multi-Objective Optimization Problems (LSMOPs) is the major challenge in evolution computation. Due to the large number of decision variables involved, it is not easy for existing multi-objective evolutionary algorithms to search the entire decision variable space with limited computation resources. To address the above problem, a large-scale multi-objective evolutionary algorithm based on problem transformation, called LSMOEA/PT, is presented in this paper. LSMOEA/PT uses an improved problem transformation strategy to transform LSMOPs into small-scale multi-objective problems to accelerate convergence speed and reduce computational resources. Specifically, The transformation strategy selects reference solutions to initialize a weight population by applying the opposite individual method. After optimizing the weight population, a dual line strategy is implemented to combine the weight vectors with the decision variables from the original population, thereby generating a new population. The new population is combined with the original population for environment selection. In LSMOEA/PT, a double learning swarm optimizer is introduced to speed up the convergence of the population. A uniform distribution strategy is conducted to update transformed solutions after the problem transformation process, increasing the diversity of the population. Experimental results show that LSMOEA/PT is competitive with eleven state-of-the-art algorithms on large-scale multi-objective optimization test problems with up to 2000 decision variables.

Global aerosol models considering their spatial heterogeneities based on AERONET measurements

The imprecision inherent in aerosol models poses a significant challenge in satellite- and surface- based aerosol retrieval. The correct classification of global aerosol types and understanding of their spatial heterogeneity are prerequisites for accurately deriving aerosol optical and microphysical properties and assessing their impacts on climate, air quality, and human health. In this study, a deep clustering algorithm is developed and employed to classify global aerosol types by using >160,000 records of AERONET measurements at 411 sites during 2000–2020. The proposed algorithm utilizes deep learning techniques to optimize the feature space of input variables, thereby improving the distinguishability among various aerosol types. This algorithm outperforms the commonly used K-means method with 77.4% of AERONET sites experiencing increased explained variance. Based on their unique properties (e.g., single scattering albedo, complex refractive index and dust ratio, etc.) and geographic distribution, global aerosols are classified into six types: mineral dust, smoke, two urban aerosols with different particle sizes, and two mixtures with different dominant components. The spatial heterogeneity of aerosol optical/microphysical properties is distinctly presented in our clustering results. Through the subdivision of the global domain into 11 sub-regions, a comprehensive elucidation has been undertaken to delineate the variances in volume size distribution, absorption, and scattering characteristics exhibited among disparate aerosol types, as well as within a same aerosol category. These new understandings of aerosol models and their spatial heterogeneities in this study will be highly helpful to update or optimize aerosol retrieval algorithms in the future.

On the Cauchy problem for p-evolution equations with variable coefficients: A necessary condition for Gevrey well-posedness

In this paper we consider a class of p-evolution equations of arbitrary order with variable coefficients depending on time and space variables (t,x). We prove necessary conditions on the decay rates of the coefficients for the well-posedness of the related Cauchy problem in Gevrey spaces.

The Evaluation of Relationships between Milk Composition Traits and Breeds with Categorical Principal Component Analysis in Akkaraman and Awasi Sheep

Background: This study aims to determine the relationship between milk composition traits and breed in the Akkaraman and Awasi sheep as well as to provide ease of interpretation by showing the relationships structure between variables and between categories of variables in two-dimensional space with Categorical principal component analysis. Methods: Categorical principal component analysis determines relationships between continuous and categorical variables as well as ordinal variables. It aims to reduce system dimensionality through optimal scaling while maintaining variable measurement levels (nominal, multiple nominal, ordinal and interval). In this research, data obtained from Akkaraman and Awasi Breed Sheep Raised by Public Hands in Tuşba District of Van Province were used. In order to determine relationship with breed, the traits were divided into two categories, “low” and “high” and all variables (9 variables) were considered together and a Categorical principal components analysis was performed. Result: As a results, Dimension 1 accounted for 35.58% of the total variation while dimension 2 accounted for 15.21%. Two dimensions together accounted for 50.79% of the variation. Thus it can be noted that Categorical principal component analysis can be used in the analysis of data sets containing a large number of different types of variables with linear or non-linear relationships between them.

Velocity-Vorticity Geometric Constraints for the Energy Conservation of 3D Ideal Incompressible Fluids

In this paper we consider the 3D Euler equations and we first prove a criterion for energy conservation for weak solutions, where the velocity satisfies additional assumptions in fractional Sobolev spaces with respect to the space variables, balanced by proper integrability with respect to time. Next, we apply the criterion to study the energy conservation of solution of the Beltrami type, carefully applying properties of products in (fractional and possibly negative) Sobolev spaces and employing a suitable bootstrap argument.

Innovative coupling of s-stage one-step and spectral methods for non-smooth solutions of nonlinear problems

The behavior of nonlinear dynamical systems arising in mathematical physics through numerical tools is a challenging task for researchers. In this context, an efficient semi-spectral method is proposed and applied to observe the robust solutions for the mathematical physics problems. Firstly, the space variable is approximated by the Vieta-Lucas polynomials and then the s-stage one-step method is applied to discretize the temporal variable which transfers the problem in the form Cn+1=Cn+Δtϕ(x,t,Cn,F(un)). Novel operational matrices of integer order are developed to replace the spatial derivative terms presented in the discussed problem. Related theorems are included in the study to validate the approach mathematically. The proposed semi-spectral schemes convert the considered nonlinear problem to a system of linear algebraic equations which is easier to tackle. We also accomplish an investigation on the error bound and convergence to confirm the mathematical formulation of the computational algorithm. To show the accuracy and effectiveness of the suggested computational method numerous test problems, such as the advection-diffusion problem, generalized Burger-Huxley, sine-Gordon, and modified KdV–Burgers’ equations are considered. An inclusive comparative examination demonstrates the currently suggested computational method in terms of credibility, accuracy, and reliability. Moreover, the coupling of the spectral method with the fourth-order Runge-Kutta method seems outstanding to handle the nonlinear problem to examine the precise smooth and non-smooth solutions of physical problems. The computational order of convergence (COC) is computed numerically through numerous simulations of the proposed schemes. It is found that the proposed schemes are in exponential order of convergence in the spatial direction and the COC in the temporal direction validates the studies in the literature.