Space Variables Research Articles

Cross-ValidatedFunctional Generalized Partially Linear Single-Functional Index Model

In this paper, we have introduced a functional approach for approximating nonparametric functions and coefficients in the presence of multivariate and functional predictors. By utilizing the Fisher scoring algorithm and the cross-validation technique, we derived the necessary components that allow us to explain scalar responses, including the functional index, the nonlinear regression operator, the single-index component, and the systematic component. This approach effectively addresses the curse of dimensionality and can be applied to the analysis of multivariate and functional random variables in a separable Hilbert space. We employed an iterative Fisher scoring procedure with normalized B-splines to estimate the parameters, and both the theoretical and practical evaluations demonstrated its favorable performance. The results indicate that the nonparametric functions, the coefficients, and the regression operators can be estimated accurately, and our method exhibits strong predictive capabilities when applied to real or simulated data.

Probabilistic Performance-Pattern Decomposition (PPPD): Analysis framework and applications to stochastic mechanical systems

Numerous research efforts have been devoted to developing quantitative solutions to stochastic mechanical systems. In general, the problem is perceived as “solved” when a complete or partial probabilistic description on quantities of interest (QoIs) is determined. However, in the presence of complex system behavior, there is a critical need to go beyond computing probabilities. In fact, to gain a better understanding of the system, it is crucial to extract physical characterizations from the probabilistic structure of the QoIs, especially when the QoIs are computed in a data-driven fashion. Motivated by this perspective, the paper proposes a framework to obtain structuralized characterizations on behaviors of stochastic systems. The framework is named Probabilistic Performance-Pattern Decomposition (PPPD). PPPD analysis aims to decompose complex response behaviors, conditional to a prescribed performance state, into meaningful patterns in the space of system responses, and to investigate how the patterns are triggered in the space of basic random variables. To illustrate the application of PPPD, the paper studies three numerical examples: (1) an illustrative example with hypothetical stochastic processes input and output; (2) a stochastic Lorenz system with periodic as well as chaotic behaviors; and (3) a simplified shear-building model subjected to a stochastic ground motion excitation.

A strong maximum principle for quasilinear elliptic differential equations in a variable exponent Sobolev space

A strong maximum principle is an important property for elliptic and parabolic equations. In this paper, we consider a strong maximum principle for a class of quasilinear elliptic equations containing p(⋅)-Laplacian and mean curvature operator.

Customizable Colorimetric Sensor Array via a High-Throughput Robot for Mitigation of Humidity Interference in Gas Sensing.

One challenge for gas sensors is humidity interference, as dynamic humidity conditions can cause unpredictable fluctuations in the response signal to analytes, increasing quantitative detection errors. Here, we introduce a concept: Select humidity sensors from a pool to compensate for the humidity signal for each gas sensor. In contrast to traditional methods that extremely suppress the humidity response, the sensor pool allows for more accurate gas quantification across a broader range of application scenarios by supplying customized, high-dimensional humidity response data as extrinsic compensation. As a proof-of-concept, mitigation of humidity interference in colorimetric gas quantification was achieved in three steps. First, across a ten-dimensional variable space, an algorithm-driven high-throughput experimental robot discovered multiple local optimum regions where colorimetric humidity sensing formulations exhibited high evaluations on sensitivity, reversibility, response time, and color change extent for 10-90% relative humidity (RH) in room temperature (25 °C). Second, from the local optimum regions, 91 sensing formulations with diverse variables were selected to construct a parent colorimetric humidity sensor array as the sensor pool for humidity signal compensation. Third, the quasi-optimal sensor subarrays were identified as customized humidity signal compensation solutions for different gas sensing scenarios across an approximately full dynamic range of humidity (10-90% RH) using an ingenious combination optimization strategy, and two accurate quantitative detections were attained: one with a mean absolute percentage error (MAPE) reduction from 4.4 to 0.75% and the other from 5.48 to 1.37%. Moreover, the parent sensor array's excellent humidity selectivity was validated against 10 gases. This work demonstrates the feasibility and superiority of robot-assisted construction of a customizable parent colorimetric sensor array to mitigate humidity interference in gas quantification.

An efficient hybrid approach for numerical study of two-dimensional time-fractional Cattaneo model with Riesz distributed-order space-fractional operator along with stability analysis

In this article, we study and analyze the two-dimensional time-fractional Cattaneo model with Riesz space distributed-order. To obtain approximate solutions of this type of fractional model the combined and effective numerical approach based on the ADI Galerkin method and the Legendre spectral method used the ADI Galerkin numerical method uses the finite difference approach. The ADI Galerkin numerical method is used to approximate the proposed model in terms of the time variable, and the Legendre spectral method is applied to discretize the fractional model with respect to the space variable. Also, the convergence analysis and stability of the proposed method are discussed and reviewed in this manuscript. In the end, some numerical examples are tested for the effectiveness and accuracy of the proposed method. As well as, in the numerical examples section, the presented numerical approach is compared with two numerical methods and the results are reported in a table.

Analytical solution to the two-dimensional inverse heat conduction problem in an axisymmetric cylindrical coordinate system

The inverse heat conduction problem involves estimating an unknown cooling or heating action based on internal temperature histories in a body. Such a problem is ill-posed because the estimated results are highly sensitive to input data noise. An analytical solution is developed for the two-dimensional inverse heat conduction problem in an axisymmetric cylindrical coordinate system using the Laplace transform technique. On the basis of the known temperatures at multiple measuring points distributed on two cylindrical measuring surfaces along the axial direction inside the solid, the expressions for the surface temperature and heat flux are explicitly obtained in the form of a power series of time and eigenfunctions of the space variable. A comprehensive estimation error assessment is conducted using the internal temperature distribution that is obtained using a direct heat conduction analysis under the prescribed stationary and moving boundary conditions. The constant thermal condition is well reproduced using the present inverse solution, except in the vicinity of the discontinuity point. In addition, the effective heat flux and effective heat transfer coefficient are introduced to obtain the best estimation of the transient heat transfer in the major cooling area of the rotating cylinder.

Output related fault detection and diagnosis based on multi block modified orthogonal Broyden-Fletcher-Goldfarb-Shanno algorithm

In recent years, some progress has been made in fault diagnosis of multi-block regression model, but multi-block method uses the relationship between variables in Hilbert space to block, while the relationship between variables in complex industrial systems often exists in Banach space. In addition, Bayesian fusion method is used in fault monitoring, which can not accurately process the sub-block information and judge whether the fault information is related to the output. In this paper, a new multi-block method is proposed. Firstly, this method uses Ball Covariance (BCov) suitable for Banach space to describe the correlation between variables, and combines K-means clustering method to block variables to ensure that the blocking method is more reasonable. Secondly, an improved orthogonal decomposition Broyden-Fletcher -Goldfarb-Shanno (BFGS) algorithm is proposed. The MOBFGS algorithm performs orthogonal projection on potential structures, removing information orthogonal to the output from the input block. Furthermore, through complete orthogonal decomposition between the input block and the output block, information related to and unrelated to the output is completely separated, and used to detect output related and unrelated faults, respectively. Thirdly, at the block level, MOBFGS algorithm is used for fault detection and contribution graph method for fault diagnosis. Finally, numerical examples and Tennessee Eastman (TE) process simulation verify the effectiveness of this method.

Mining Alzheimer's disease clinical data: reducing effects of natural aging for predicting progression and identifying subtypes.

Because Alzheimer's disease (AD) has significant heterogeneity in encephalatrophy and clinical manifestations, AD research faces two critical challenges: eliminating the impact of natural aging and extracting valuable clinical data for patients with AD. This study attempted to address these challenges by developing a novel machine-learning model called tensorized contrastive principal component analysis (T-cPCA). The objectives of this study were to predict AD progression and identify clinical subtypes while minimizing the influence of natural aging. We leveraged a clinical variable space of 872 features, including almost all AD clinical examinations, which is the most comprehensive AD feature description in current research. T-cPCA yielded the highest accuracy in predicting AD progression by effectively minimizing the confounding effects of natural aging. The representative features and pathogenic circuits of the four primary AD clinical subtypes were discovered. Confirmed by clinical doctors in Tangdu Hospital, the plaques (18F-AV45) distribution of typical patients in the four clinical subtypes are consistent with representative brain regions found in four AD subtypes, which further offers novel insights into the underlying mechanisms of AD pathogenesis.

Groundwater LNAPL Contamination Source Identification Based on Stacking Ensemble Surrogate Model

Groundwater LNAPL (Light Non-Aqueous Phase Liquid) contamination source identification (GLCSI) is essential for effective remediation and risk assessment. Addressing the GLCSI problem often involves numerous repetitive forward simulations, which are computationally expensive and time-consuming. Establishing a surrogate model for the simulation model is an effective way to overcome this challenge. However, how to obtain high-quality samples for training the surrogate model and which method should be used to develop the surrogate model with higher accuracy remain important questions to explore. To this end, this paper innovatively adopted the quasi-Monte Carlo (QMC) method to sample from the prior space of unknown variables. Then, this paper established a variety of individual machine learning surrogate models, respectively, and screened three with higher training accuracy among them as the base-learning models (BLMs). The Stacking ensemble framework was utilized to integrate the three BLMs to establish the ensemble surrogate model for the groundwater LNAPL multiphase flow numerical simulation model. Finally, a hypothetical case of groundwater LNAPL contamination was designed. After evaluating the accuracy of the Stacking ensemble surrogate model, the differential evolution Markov chain (DE-MC) algorithm was applied to jointly identify information on groundwater LNAPL contamination source and key hydrogeological parameters. The results of this study demonstrated the following: (1) Employing the QMC method to sample from the prior space resulted in more uniformly distributed and representative samples, which improved the quality of the training data. (2) The developed Stacking ensemble surrogate model had a higher accuracy than any individual surrogate model, with an average R2 of 0.995, and reduced the computational burden by 99.56% compared to the inversion process based on the simulation model. (3) The application of the DE-MC algorithm effectively solved the GLCSI problem, and the mean relative error of the identification results of unknown variables was less than 5%.

Compact Extended Formulations for Low-Rank Functions with Indicator Variables

We study the mixed-integer epigraph of a special class of convex functions with nonconvex indicator constraints, which are often used to impose logical constraints on the support of the solutions. The class of functions we consider are defined as compositions of low-dimensional nonlinear functions with affine functions. Extended formulations describing the convex hull of such sets can easily be constructed via disjunctive programming although a direct application of this method often yields prohibitively large formulations, whose size is exponential in the number of variables. In this paper, we propose a new disjunctive representation of the sets under study, which leads to compact formulations with size exponential in the dimension of the nonlinear function but polynomial in the number of variables. Moreover, we show how to project out the additional variables for the case of dimension one, recovering or generalizing known results for the convex hulls of such sets (in the original space of variables). Our computational results indicate that the proposed approach can significantly improve the performance of solvers in structured problems. Funding: This work was supported by the National Science Foundation Division of Computing and Communication Foundations [Grant 2006762].

A fast numerical algorithm for finding all real solutions to a system of N nonlinear equations in a finite domain

AbstractA highly recurrent traditional bottleneck in applied mathematics, for which the most popular codes (Mathematica, Matlab, and Python as examples) do not offer a solution, is to find all the real solutions of a system of n nonlinear equations in a certain finite domain of the n-dimensional space of variables. We present two similar algorithms of minimum length and computational weight to solve this problem, in which one resembles a graphical tool of edge detection in an image extended to n dimensions. To do this, we discretize the n-dimensional space sector in which the solutions are sought. Once the discretized hypersurfaces (edges) defined by each nonlinear equation of the n-dimensional system have been identified in a single, simultaneous step, the coincidence of the hypersurfaces in each n-dimensional tile or cell containing at least one solution marks the approximate locations of all the hyperpoints that constitute the solutions. This makes the final Newton-Raphson step rapidly convergent to all the existent solutions in the predefined space sector with the desired degree of accuracy.

Data driven latent variable adaptive control for nonlinear multivariable processes

This article aims at a new partial least squares (PLS) control design and analysis in a data-driven framework for nonlinear multivariable processes whose mechanistic models are completely unknown. First, a general nonlinear autoregressive moving average with the exogenous input (NARMAX) model is used as the dynamic nonlinear PLS model. Then, by introducing a dynamic linearisation approach in each latent variable (LV) space, the unknown NARMAX-based PLS model is transformed to a linear dynamic PLS data model (dPLSDM), which can be improved in real time by estimating its unknown parameter using the latent input and output (I/O) data. Next, a data-driven latent variable adaptive control (DDLVAC) is proposed in each LV loop. By virtue of the dPLSDM, the multivariable nonlinear process is decoupled into multiple single-loop systems and the high dimensions of the process data are reduced such that the corresponding DDLVAC is simplified. Further, the DDLVAC only depends on the I/O data without requiring any model information of the original process. Theoretical analysis confirms the validity of the DDLVAC. The simulation study demonstrates the advantages of the DDLVAC such as less storage space, smaller computation burden, less control cost, as well as more robustness against uncertainties.

Hybrid reliability analysis method for systems with random and non-parameterized p-boxes based on weight coefficients

This paper establishes a hybrid variable system failure probability optimization model based on sampling methods and weighting coefficients. By introducing auxiliary input variables, important sampling functions, and p-box, failure samples are mapped from the random variable space to the p-box variable space. The new weight coefficients are constructed, including important sampling weights and interval weights. Combining discretization methods and Monte Carlo simulation (MCS), the interval weights are transformed into variables, and constraints conforming to the p-box variable distribution are constructed. After calculating the weighting coefficients for all failure samples, the new failure probability optimization model is built. This model is independent of the performance functions and does not involve cyclic optimization, with computational complexity only related to the dimensions. Six cases are used for method comparison, validating that the new method exhibits higher efficiency and accuracy.

Determination of soluble solids content in loquat using near-infrared spectroscopy coupled with broad learning system and hybrid wavelength selection strategy

Soluble solids content (SSC) serves as a crucial metric in assessing loquat quality. To achieve fast, non-destructive, and accurate detection of SSC, the combination of near-infrared (NIR) spectroscopy and broad learning system (BLS) model was developed in this study. Meanwhile, the influence of six spectral preprocessing methods and six wavelength selection algorithms on SSC prediction was explored, and the performance of the BLS model was compared with the widely used partial least squares regression (PLSR) and least squares support vector machine (LS-SVM) models. The results showed that Savitzky-Golay smoothing combined with standard normal transformation (SG-SNV) offered the optimal pretreatment, and the hybrid method, namely interval variable iterative space shrinking analysis and successive projections algorithm (iVISSA-SPA), emerged as the most effective wavelength selection method. The BLS model outperformed both PLSR and LS-SVM models. Specifically, the BLS model based on the iVISSA-SPA method achieved the optimal SSC prediction results, with RP2 = 0.8646, RMSEP = 0.5104, and RPD = 2.7481. Therefore, NIR spectroscopy coupled with the BLS model and hybrid wavelength selection strategy could rapidly, non-destructively and accurately detect the SSC of loquat, providing a viable alternative for SSC prediction in fruit.

A Sensitive SERS Sensor Combined with Intelligent Variable Selection Models for Detecting Chlorpyrifos Residue in Tea.

Chlorpyrifos is one of the most widely used broad-spectrum insecticides in agriculture. Given its potential toxicity and residue in food (e.g., tea), establishing a rapid and reliable method for the determination of chlorpyrifos residue is crucial. In this study, a strategy combining surface-enhanced Raman spectroscopy (SERS) and intelligent variable selection models for detecting chlorpyrifos residue in tea was established. First, gold nanostars were fabricated as a SERS sensor for measuring the SERS spectra. Second, the raw SERS spectra were preprocessed to facilitate the quantitative analysis. Third, a partial least squares model and four outstanding intelligent variable selection models, Monte Carlo-based uninformative variable elimination, competitive adaptive reweighted sampling, iteratively retaining informative variables, and variable iterative space shrinkage approach, were developed for detecting chlorpyrifos residue in a comparative study. The repeatability and reproducibility tests demonstrated the excellent stability of the proposed strategy. Furthermore, the sensitivity of the proposed strategy was assessed by estimating limit of detection values of the various models. Finally, two-tailed paired t-tests confirmed that the accuracy of the proposed strategy was equivalent to that of gas chromatography-mass spectrometry. Hence, the proposed method provides a promising strategy for detecting chlorpyrifos residue in tea.

Discrimination of tea seed oil adulteration based on near-infrared spectroscopy and combined preprocessing method

Near-infrared spectroscopy and chemometrics was used to qualitatively distinguish the types of adulterated oils in binary adulteration of tea seed oil in this study. To address the limitations of a single preprocessing method, nine preprocessing methods from four categories were combined, and the impact of preprocessing method order on model accuracy was assessed. Additionally, variable iterative space shrinkage approach (VISSA), interval combinatorial optimization (ICO), and uninformative variables elimination (UVE) were used to screen characteristic wavelengths. Subsequently, a discriminative model for tea seed oil adulteration was constructed using two strategies. The results indicate that the order of preprocessing methods significantly influences model accuracy, and combining preprocessing methods can effectively enhance model accuracy. All three characteristic wavelength selection methods effectively screened characteristic variables. Both two strategies demonstrate good discriminant capabilities for binary adulteration in tea seed oil. In strategy 1, identification accuracies for the calibration, prediction and external datasets are 98.67 %, 100 % and 94.44 %, respectively. In strategy 2, identification accuracies for the calibration, prediction and external datasets are 100 %, 98 % and 94.44 %, respectively. Therefore, integrating NIRS with combined preprocessing and variable screening can effectively discern the types of adulterated oils in tea seed oil, serving as a potent detection tool.

Robot-assisted optimized array design for accurate multi-component gas quantification

Designing sensor arrays is a common strategy for detecting mixtures. However, a sensor array designed based on human experience leads to inaccuracies due to cross-sensitivity and information overlap. This difficulty led to the realization that arrays should be optimized as a whole. The conventional array optimizing method involves selecting the best subarray from a limited sensor pool. However, this method did not consider the vast continuous multi-dimensional variable space of each sensing element’s recipe, thus only generating a simpler array with limited detection capacity. To address this problem, we developed a Robot-assisted Optimized Array Design (ROAD) method. This innovative method holistically optimizes the sensor array across a continuous and vast variable space, overcoming the bottlenecks of high dimensionality and high-quality data collection. We applied ROAD to an optoelectronic nose, tasked with detecting a mixture of CO2, NH3, and water vapor. The method explored an immense space, estimated at the order of 2^(10^7). The implementation of ROAD led to a revolutionary quantitative accuracy, achieving an average relative standard deviation of 1.90%. This advancement has propelled the application of optoelectronic nose from qualitative to quantitative. The successful ROAD of system-wide array optimization has been demonstrated in the optoelectronic nose, validating the potential of the holistically optimized sensor array for accurately quantifying each component in a gas mixture.

Joint Probability Densities on Riemannian Manifolds are Symmetric Tensor Densities

This paper presents the tensor properties of joint probability densities on a Riemannian manifold. Initially, we develop a binary data matrix to record the values of a large number of particles confining in a closed system at a certain time in order to retrieve the joint probability densities of related variables. By introducing the particle-oriented coordinate and the generalized inner product as a multi-linear operation on the basis of this coordinate, we extract the set of joint probabilities and prove them to meet covariant tensor properties on a general Riemannian space of variables. Based on the Taylor expansion of scalar fields in Riemannian manifolds, it has been shown that the symmetrized iterative covariant derivatives of the cumulative probability function defined on Riemannian manifolds also give the set of related joint probability densities equivalent to the aforementioned multi-linear method. We show these covariant tensors reduce to classical ordinary partial derivatives in ordinary Euclidean space with Cartesian coordinates and give the formal definition of joint probabilities by partial derivatives of the cumulative distribution function. The equivalence between the symmetrized covariant derivative and the generalized inner product has been concluded. Some examples of well-known physical tensors clarify that many deterministic physical variables are presented as tensor densities with an interpretation similar to probability densities.

Probing the quantum nature of gravity using a Bose-Einstein condensate

The effect of noise induced by gravitons has been investigated using a Bose-Einstein condensate. The general complex scalar field theory with a quadratic self-interaction term has been considered in the presence of a gravitational wave. The gravitational wave perturbation is then considered as a sum of discrete Fourier modes in the momentum space. Varying the action and making use of the principle of least action, one obtains two equations of motion corresponding to the gravitational perturbation and the time-dependent part of the pseudo-Goldstone boson. Coming to an operatorial representation and quantizing the phase space variables via appropriately introduced canonical commutation relations between the canonically conjugate variables corresponding to the graviton and bosonic part of the total system, one obtains a proper quantum gravity setup. Then we obtain the Bogoliubov coefficients from the solution of the time-dependent part of the pseudo-Goldstone boson and construct the covariance metric for the bosons initially being in a squeezed state. The entries of the covariance matrix now involves a stochastic contribution which results in an operatorial stochastic structure of the quantum Fisher information. Using the stochastic average of the Fisher information, we obtain a lower bound on the amplitude parameter of the gravitational wave. As the entire calculation is done at zero temperature, the bosonic system, by construction, will behave as a Bose-Einstein condensate. For a Bose-Einstein condensate with a single mode, we observe that the lower bound of the expectation value of the square of the uncertainty in the amplitude measurement does not become infinite when the total observational term approaches zero. It always has a finite value if the gravitons are initially in a squeezed state with high enough squeezing. In order to sum over all possible momentum modes, we next consider a noise term with a suitable Gaussian weight factor which decays over time. We then obtain the lower bound on the final expectation value of the square of the variance in the amplitude parameter. Because of the noise induced by the graviton, there is a minimum value of the measurement time below which it is impossible to detect any gravitational wave using a Bose-Einstein condensate. Finally, we consider interaction between the phonon modes of the Bose-Einstein condensate which results in a decoherence. We observe that the decoherence effect becomes significant for gravitons with minimal squeezing. Published by the American Physical Society 2024

Quantitative determination of oxides in cement raw meal based on near-infrared spectroscopy and hybrid feature selection strategy

Determining the content of components in cement raw meal is vital in Portland cement manufacturing. Near-infrared spectroscopy (NIRS)enables rapid, safe, and accurate quantitative analysis of the primary constituents of cement raw meal (CaO, SiO2, Al2O3, and Fe2O3), thereby assisting the cement industry in reducing the time delay in raw meal proportioning control and enhancing the quality of the raw meal. This study employed a hybrid feature selection strategy to identify effective feature points in the NIRS of raw meal samples. Effective feature points from the raw meal were integrated with partial least squares (PLS) to develop a regression model. Initially, NIRS data from raw meal samples were obtained using a Fourier NIR spectrometer. Subsequently, sample set partitioning based on joint X-Y distance (SPXY) was utilized to divide the samples into a calibration set (70 samples) and a validation set (22 samples). Next, A backward interval partial least squares (BiPLS) approach, along with various preprocessing algorithms, was then employed to initially screen the optimal waveband combinations of the NIRS. Within these combinations, effective feature points were further identified using competitive adaptive reweighted sampling (CARS), variable combined population analysis (VCPA) and weighted fusion variable space shrinkage approach (WFVSSA). Finally, we developed and compared PLS models based on the full spectrum and three hybrid feature selection methods. The results demonstrated that.WFVSSA offers a comprehensive assessment of wavelength importance, effectively preserving feature points while eliminating redundant information. BiPLS-WFVSSA-PLS outperformed BiPLS-CARS-PLS and BiPLS-VCPAPLS on the validation set, with R2, RMSEP, and RPD values of 0.8676, 0.1431, and 2.8129 for CaO; 0.9212, 0.1216, and 3.6461 for SiO2; 0.9198, 0.0607, and 3.5618 for Al2O3; and 0.8717, 0.0166, and 2.8614 for Fe2O3, respectively. These findings underscore the superior accuracy and stability of BiPLS-WFVSSA-PLS in determining cement raw meal component contents.