Covariance Function Research Articles

Which parameterization of the Matérn covariance function?

The Matérn family of covariance functions is currently the most popularly used model in spatial statistics, geostatistics, and machine learning to specify the correlation between two geographical locations based on spatial distance. Compared to existing covariance functions, the Matérn family has more flexibility in data fitting because it allows the control of the field smoothness through a dedicated parameter. Moreover, it generalizes other popular covariance functions. However, fitting the smoothness parameter is computationally challenging since it complicates the optimization process. As a result, some practitioners set the smoothness parameter at an arbitrary value to reduce the optimization convergence time. In the literature, studies have used various parameterizations of the Matérn covariance function, assuming they are equivalent. This work aims at studying the effectiveness of different parameterizations under various settings. We demonstrate the feasibility of inferring all parameters simultaneously and quantifying their uncertainties on large-scale data using the ExaGeoStat parallel software. We also highlight the importance of the smoothness parameter by analyzing the Fisher information of the statistical parameters. We show that the various parameterizations have different properties and differ from several perspectives. In particular, we study the three most popular parameterizations in terms of parameter estimation accuracy, modeling accuracy and efficiency, prediction efficiency, uncertainty quantification, and asymptotic properties. We further demonstrate their differing performances under nugget effects and approximated covariance. Lastly, we give recommendations for parameterization selection based on our experimental results.

Locally Anisotropic Nonstationary Covariance Functions on the Sphere

Locally Anisotropic Nonstationary Covariance Functions on the Sphere

Continuity in law for solutions of SPDES with space-time homogeneous Gaussian noise

In this paper, we study the continuity in law of the solutions of two linear multiplicative SPDEs (the parabolic Anderson model and the hyperbolic Anderson model) with respect to the spatial parameter of the noise. The solution is interpreted in the Skorohod sense, using Malliavin calculus. We consider two cases: (i) the regular noise, whose spatial covariance is given by the Riesz kernel of order [Formula: see text], in spatial dimension [Formula: see text]; (ii) the rough noise, which is fractional in space with Hurst index [Formula: see text], in spatial dimension [Formula: see text]. We assume that the noise is colored in time.

THU500 Geospatial Analysis To Quantify Spatial Heterogeneity Of Tumor Microenvironment

Abstract Disclosure: J. Yoo: None. C. Mitra: None. Y. Yue: None. A. Soliman: None. Z. Madak Erdogan: None. Metabolic heterogeneity, a concept describing distinct differences in nutrient metabolism within the cells of a healthy or diseased tissue, is a common challenge in development and cancer research. Current methods, such as histology and Magnetic Resonance Imaging (MRI), cannot visualize nutrient uptake into single cells, making it difficult to understand the heterogeneity that affects cellular growth and metabolism. Visium Spatial Transcriptomics, a method that shows regional gene expression, is emerging as a powerful spatial analysis tool, and we aim to integrate this transcriptomic method with other biologically relevant datasets to highlight the importance of cell-microenvironment interactions. We propose to leverage geostatistical tools to model the multivariate covariance structure of genes expression profiles to allow: 1) understanding of their spatial characteristics and covariance changes under certain conditions (e.g. drug treatment or damage), 2) using these covariance structures to simulate synthetic samples at a scale sufficient to train the state of art neural network architectures, and 3) incorporating spatial awareness directly in the machine learning/deep learning models by enriching the spatial transcriptomics data using distance proximity functions. We will quantify the tumor microenvironment spatial heterogeneity using different approaches. First, we will analyze the degree of spatial auto-correlation to estimate the ranges of spatial heterogeneity within a tumor. We also plan to quantify the degree of anisotropy to identify if the tumor heterogeneity is happening along specific axis. Different local clustering indicators will be also calculated, such as the Getis-Ord (Gi*) statistic and Local Moran's I to identify the prevalence of hot and cold spots in the tumor. These hot and cold spots will be used to segment the tumor area into sub-regions and further quantify the local heterogeneity with a single tumor. We hypothesize that transcriptional heterogeneity reflects phenotypic differences in different regions of healthy or diseased tissue, rendering cells in certain regions to display a differential response to external stimuli (e.g. diet or drugs). Integration of multiple information (e.g. retrospective patient data) with different spatial heterogeneity indicators identified from the geospatial analyses could aid in identification of patients who are resistant to cancer therapies and further advance personalized therapy. Presentation: Thursday, June 15, 2023

Predictive Deep Learning for High‐Dimensional Inverse Modeling of Hydraulic Tomography in Gaussian and Non‐Gaussian Fields

AbstractInverse modeling of hydraulic tomography (HT) is computationally expensive for estimating high‐dimensional hydrogeologic parameter fields. In this work, we develop a novel method called HT‐INV‐NN, which combines dimensionality reduction techniques with a predictive deep learning (DL) model to estimate high‐dimensional Gaussian and non‐Gaussian channel fields. The HT‐INV‐NN model consists of a predictor that directly learns the inverse process from hydraulic head measurements to latent variables of random fields, and a decoder that generates high‐dimensional parameter fields from predicted latent variables. For Gaussian spatially correlated fields, the decoder utilizes principal components derived from spatial covariance, and for non‐Gaussian channel fields, a generative adversarial network (GAN) is trained using generated realizations based on a training image (TI). The predictor is a deep neural network calibrated using the reference data obtained from HT forward simulations, which can be implemented in parallel. HT‐INV‐NN is successfully tested in multiple numerical experiments including steady‐state and transient HT for estimating Gaussian fields in 2D and 3D, as well as binary discontinuous or continuous non‐Gaussian channel fields. The training process is efficient, and the model structure demonstrates robustness for input data with perturbations. The model performance on multiple validation data sets are satisfactory when compared with other numerical and deep learning methods.

DEVELOPMENT OF SEASONAL AUTOREGRESSIVE INTEGRATED MOVING AVERAGE MODEL WITH ERROR PROCESS

This research work investigated with error process. The auto covariance function maximum likelihood methods, iterative method and chi-squares test statistics is used to develop SARIMA model corrupted with error process that is used to estimate the true parameter of the SARIMA model. The forecast performance measurement and properties of error with different value are also investigated. Test of seasonal unit roots are also carried out in the work. The simulation, and real data of Zamfara state monthly Rain fall from 1998 to 2022 is used to validate the results with R - Statistical software version 4.1.1 and mini tab statistical software version 14. The result showed a significance p- value of 0.000, the proposed model provides a generalization and more flexible specification than the existing models of AR (1) error and ARMA (1,1) error in fitting time series processes in the presence of errors . Hence, the studies showed that, the finding is closely to the true parameter of the process and would be useful to researchers in the prediction and handling of natural calamities that disturb the otherwise stability of a system.

Dosimetric Predictors for Acute Esophagitis during Radiation Therapy for Lung Cancer: An Update of a Large Statewide Observational Study

Acute esophagitis remains a clinical challenge during the treatment of locally advanced non-small cell lung cancer (NSCLC). Here, we analyze the dosimetric and patient-level characteristics associated with acute grade 2+ and 3+ esophagitis in patients undergoing radiation therapy for NSCLC across a statewide consortium. Demographic, dosimetric, and acute toxicity data were prospectively collected for patients undergoing definitive radiation therapy +/- chemotherapy for stage II-III NSCLC from 2012-2022 across the Michigan Radiation Oncology Quality Consortium (MROQC). Logistic regression models were used to characterize the risk of grade 2+ and 3+ esophagitis as a function of dosimetric and clinical covariates. Multivariate regression models were fitted to predict the 50% risk of grade 2 esophagitis or 3% risk of grade 3 esophagitis at each dose value. Of 1760 patients evaluated, 84.2% had stage III disease and 85.3% received concurrent chemotherapy. 79.2% of patients had an ECOG performance status ≤1. Rates of acute grade 2+ and 3+ esophagitis were 48.4% and 2.2%, respectively. On multivariate analyses, performance status, mean esophageal dose and esophageal D2cc were significantly associated with grade 2+ and 3+ esophagitis. Concurrent chemotherapy was associated with grade 2+ but not grade 3+ esophagitis. Dose-response relationships were modeled for grade 2+ and 3+ esophagitis by mean esophageal dose and esophageal D2cc, stratified by performance status and/or receipt of concurrent chemotherapy. For all patients, mean esophageal dose of 29 Gy and esophageal D2cc of 61 Gy corresponded to a 3% risk of acute grade 3+ esophagitis. For patients receiving chemotherapy, mean esophageal dose of 22 Gy and esophageal D2cc of 49.5 Gy corresponded to a 50% risk of acute grade 2+ esophagitis. Performance status, concurrent chemotherapy, mean esophageal dose and esophageal D2cc are associated with acute esophagitis during definitive treatment of NSCLC. Models that quantitatively account for these factors can be useful in individualizing radiation plans. Mean esophageal dose of 29 Gy and esophageal D2cc of 61 Gy corresponded to a 3% risk of acute grade 3+ esophagitis and merit consideration as contemporary treatment planning constraints.

Propagation of material uncertainty in modal parameters and its influence in damage quantification of shear buildings

Structural health monitoring (SHM) plays a crucial role in post-disaster mitigation in investigating the serviceability of an existing structure. The physical properties, namely mass density, cross-sectional area, Poisson’s ratio, modulous of elasticity, etc., vary spatially throughout a real system. These uncertainties propagate in the modal parameters involved in SHM techniques. In this study, the elastic modulous has been considered the primary source of uncertainty. A detailed literature survey has been performed to identify the suitable probability distribution recommended for defining material uncertainty. Although the Gaussian distribution cannot replicate realistic material behaviour, its explicit implementation makes it prevalent. However, based on the principle of maximum entropy, the gamma distribution is the most appropriate recommendation among all non-Gaussian continuous distributions ranging [0, ∞). The Karhunen–Loève (KL) expansion has been adopted to discretize the stochastic field to determine the spectral mass and stiffness of a multi-damaged axially vibrating prismatic cantilever bar-type structure. Closed-form orthogonal pairs of eigensolutions of the KL expansion considering (Adhikari and Friswell, 2010) exponential covariance function have been derived. The expressions for spectral mass and stiffness for a multi-damaged cantilever bar considering material uncertainty have been developed. The uncertainty propagation in the modal properties has been estimated. The complete procedure has been followed to identify uncertainty propagation in damage severity of a common class of civil structures like shear buildings. The influence of material uncertainty in determining the damage severity of a structure with numerable degrees of freedom using a non-destructive approach has been explored. Extensive simulations have been performed to evaluate the effectiveness of the present study in a multi-damaged bar as well as shear building. Statistical parameters, namely mean and standard deviation of natural frequencies, modal displacement and damage severity have been determined. The uncertainty propagation has been reflected in the mean value for varying input stochasticity. However, the standard deviation remained consistent for undamaged as well as damaged structure irrespective of material uncertainty type. Considering gamma distribution in material stochasticity, the spread of uncertainty propagation has been more than the Gaussian distribution. However, experimental studies need to be performed further to investigate the stochastic behaviour in modal parameters of a real structure.

A time variant uncertainty propagation method for high-dimensional dynamic structural system via K-L expansion and Bayesian deep neural network.

In this paper, a time variant uncertainty propagation (TUP) method for dynamic structural system with high-dimensional input variables is proposed. Firstly, an arbitrary stochastic process simulation (ASPS) method based on Karhunen-Loève (K-L) expansion and numerical integration is developed, expressing the stochastic process as the combination of its marginal distributions and eigen functions at several discrete time points. Secondly, the iterative sorting method is implemented to the statistic samples of marginal distributions for matching the constraints of covariance function. Since marginal distributions are directly used to express the stochastic process, the proposed ASPS is suitable for stationary or non-stationary stochastic processes with arbitrary marginal distributions. Thirdly, the high-dimensional TUP problem is converted into several high-dimensional static uncertainty propagation (UP) problems after implementing ASPS. Then, the Bayesian deep neural network based UP method is used to compute the marginal distributions as well as the eigen functions of dynamic system response, the high-dimensional TUP problem can thus be solved. Finally, several numerical examples are used to validate the effectiveness of the proposed method. This article is part of the theme issue 'Physics-informed machine learning and its structural integrity applications (Part 1)'.

On estimation of covariance function for functional data with detection limits

In many studies on disease progression, biomarkers are restricted by detection limits, hence informatively missing. Current approaches ignore the problem by just filling in the value of the detection limit for the missing observations for the estimation of the mean and covariance function, which yield inaccurate estimation. Inspired by our recent work [Liu and Houwing-Duistermaat (2022), ‘Fast Estimators for the Mean Function for Functional Data with Detection Limits’, Stat, e467.] in which novel estimators for mean function for data subject to detection limit are proposed, in this paper, we will propose a novel estimator for the covariance function for sparse and dense data subject to a detection limit. We will derive the asymptotic properties of the estimator. We will compare our method to the standard method, which ignores the detection limit, via simulations. We will illustrate the new approach by analysing biomarker data subject to a detection limit. In contrast to the standard method, our method appeared to provide more accurate estimates of the covariance. Moreover its computation time is small.

Simulation of non-stationary and non-Gaussian stochastic processes by the AFD-Type Sparse Representations

Simulation of a stochastic process to simultaneously meet a given marginal distribution condition and be compatible with a given covariance function is a well-known problem called the two-match problem in the sequel. We propose the adaptive Fourier decomposition (AFD) type methods as constructive steps to solve the two-match problem. AFD-Type methods used in the physical domain (e.g., time or frequency) are replacements of the Karhunen–Loève expansion, or spectral decomposition, in general, while, when being used in the probability space are replacements of the polynomial chaos or other types of chaos methods. The AFD-Type methods in both circumstances play the role of an approximation. Being compared with the Karhunen–Loève expansion, the proposed AFD-Type methods do not need to compute out the eigenvalues and eigenfunctions of the kernel integral operator defined by the target covariance function, while compared with the polynomial chaos or other chaos methods the AFD-Type methods offer a great deal of flexibility and efficiency. The proposed methods can be applied to stationary and non-stationary, weakly, and strongly non-Gaussian stochastic processes. We provide several examples to show effectiveness and efficiency of the AFD-Type methods.

A cross-session motor imagery classification method based on Riemannian geometry and deep domain adaptation

Recently, more and more studies have begun to use deep learning to decode and classify EEG signals. The use of deep learning has led to an increase in the classification accuracy of motor imagery (MI), but the problem of taking a long time to calibrate in brain–computer interface (BCI) applications has not been solved. To address this problem, we propose a novel Riemannian geometry and deep domain adaptation network (RGDDANet) for MI classification. Specifically, two one-dimensional convolutions are designed to extract temporal and spatial features from the EEG signals, and then the spatial covariance matrices are utilized to map the extracted features to Riemannian manifolds for processing. In order to align the source and target features’ distributions on the Riemannian manifold, we propose a Symmetric Positive Definite (SPD) matrix mean discrepancy loss (SMMDL) to minimize the distance between two domains. To analyze the feasibility of the method, we conducted extensive experiments on BCIC IV 2a and BCIC IV 2b datasets, respectively, and the results showed that the proposed method achieved better performance than some state-of-the-art methods.

Detection of Interaction Effects in a Nonparametric Concurrent Regression Model.

Many methods have been developed to study nonparametric function-on-function regression models. Nevertheless, there is a lack of model selection approach to the regression function as a functional function with functional covariate inputs. To study interaction effects among these functional covariates, in this article, we first construct a tensor product space of reproducing kernel Hilbert spaces and build an analysis of variance (ANOVA) decomposition of the tensor product space. We then use a model selection method with the L1 criterion to estimate the functional function with functional covariate inputs and detect interaction effects among the functional covariates. The proposed method is evaluated using simulations and stroke rehabilitation data.

Adjusted feature screening for ultra-high dimensional missing response

ABSTRACT This paper presents a new method for the feature screening of ultra-high dimensional data with response missing at random. The distribution function of the missing response is completed by imputation technology, and then the distance correlation between the imputed distribution function of response and the distribution function of covariate is used as an index for feature screening. The proposed method has the following advantages. First, it is a nonparametric model-free method, and can detect the nonlinear relationship between variables. Second, it is robust to covariates with outliers and heavy-tailed distributions. Third, it can deal with multi-dimensional response variables directly. Under certain assumptions, this paper demonstrates the sure screening and ranking consistency properties. Simulation studies are conducted to examine the performance of the proposed procedure and to compare with existing methods. Finally, our method is applied to the data analysis of diffuse large B-cell lymphoma.

Revised and wider classes of isotropic space-time covariance functions

AbstractSeveral classes of space-time correlation models have been proposed by various authors in the last years. However, most of these families utilize non negative covariance functions to be adapted to different case studies: indeed, the traditional classes of covariances, such as the Whittle–Matern class and the several families constructed by applying the classical properties are not so flexible to describe covariance functions characterized by negative values. A recent analysis, regarding the difference between two isotropic covariance functions, has underlined that these new families of models are more flexible than the traditional ones because the same models are able to select covariance functions which are always positive in their domain, as well as covariance functions which could be negative in a subset of their field of definition. Moreover, within the same class of models, it is possible to select covariance models which present different behaviours in proximity of the origin. In this paper several families of isotropic space-time covariance functions, among the ones proposed in the literature, have been reviewed in order to enrich the same families including models characterized by negative values in a subset of their domain. Furthermore, the definition of separability has been revised in order to enlarge the classical definition. Apart from the theoretical importance related to the new aspects, these new classes of covariance models are characterized by an extremely simple formalism and can be easily adapted to several case studies.

Bayesian structural identification using Gaussian Process discrepancy models

Bayesian model updating based on Gaussian Process (GP) models has received attention in recent years, which incorporates kernel-based GPs to provide enhanced fidelity response predictions. Although most kernel functions provide high fitting accuracy in the training data set, their out-of-sample predictions can be highly inaccurate. This paper investigates this problem by reformulating the problem on a consistent probabilistic foundation, reviewing common choices of kernel covariance functions, and proposing a new Bayesian model selection for kernel function selection, aiming to create a balance between fitting accuracy, generalizability, and model parsimony. Computational aspects are addressed via Laplace approximation and sampling techniques, providing detailed algorithms and strategies. Numerical and experimental examples are included to demonstrate the accuracy and robustness of the proposed framework. As a result, an exponential-trigonometric covariance function is characterized and justified based on the Bayesian model selection approach and observations of the sample autocorrelation function of the response discrepancies.

Stationary nonseparable space-time covariance functions on networks

Abstract The advent of data science has provided an increasing number of challenges with high data complexity. This paper addresses the challenge of space-time data where the spatial domain is not a planar surface, a sphere, or a linear network, but a generalised network (termed a graph with Euclidean edges). Additionally, data are repeatedly measured over different temporal instants. We provide new classes of stationary nonseparable space-time covariance functions where space can be a generalised network, a Euclidean tree, or a linear network, and where time can be linear or circular (seasonal). Because the construction principles are technical, we focus on illustrations that guide the reader through the construction of statistically interpretable examples. A simulation study demonstrates that the correct model can be recovered when compared to misspecified models. In addition, our simulation studies show that we effectively recover simulation parameters. In our data analysis, we consider a traffic accident dataset that shows improved model performance based on covariance specifications and network-based metrics.

Applying machine learning to model radon using topsoil geochemistry

Radon is classified as a Class 1 carcinogen, being the leading cause of lung cancer in non-smokers. Understanding the prominent sources of radon helps to mitigate against the adverse effects of radon exposure. Considering soil-gas radon is the main contributor to indoor radon, it is possible that soil geochemistry can be used as a proxy for the soil radon emanation potential or geogenic radon classes for a particular location. This paper investigates the relationship between soil geochemistry and geogenic radon. A large area of 17,983 km2 from the West, Midlands and East of Ireland was selected to represent a range of geology types and radon categories. A rigorous assessment is presented to investigate the relationship of geogenic radon and topsoil geochemistry; using univariate processes (i.e. r2, Pearson r and heatmaps) and multivariate techniques (i.e. principle component analysis (PCA) and machine learning (ML) algorithms including Gaussian process regression, logistic regression and random forest). Here, PCA and ML techniques were used to test the utility of soil geochemistry to predict geogenic radon classes. Gaussian Process Regression yielded the highest accuracy (74%) and f1-score (0.74) of all models. The feature importance (i.e. highest ranking elements for predicting geogenic radon class) from the ML models outputs elements including [Y, Tl, Mn, Cr, Co, Be, Sc and Rb]. The PCA biplot demonstrates that these elements cluster in conjunction with higher geogenic radon categories. Multivariate data analysis reveals that certain elements important for predicting higher geogenic radon classes, also covary together within topsoil samples; here these are termed “radon-prone elements”. Spatial covariance of radon-prone elements permits soil geochemistry to be used as a tool for understanding the distribution of geogenic radon. The methodology presented in this paper provides a comprehensive geo-statistical approach to investigate the relation between topsoil geochemistry and geogenic radon. This approach could be applied as a diagnostic tool to assist radon mitigation measures, hence adding value to legacy soil geochemistry datasets.

Hybrid mechanical and data-driven probabilistic model for shear strength of RC beam-column joints

In order to overcome the limitations of traditional data-driven or mechanics-based methods, a novel hybrid mechanical and data-driven probabilistic model for the shear strength of reinforced concrete (RC) beam-column joints was proposed based on the modified Gaussian process (GP). A new mean function for the GP was developed first by considering the mechanical mechanisms of joint shear strength to improve the prediction accuracy and generalization ability. Then a hybrid mechanical and data-driven probabilistic model (HMDPM) for the joint shear strength was established based on the modified GP. Moreover, the hyper-parameters of the prior mean and covariance functions of the HMDPM were jointly optimized by maximum likelihood estimation (MLE). Finally, the HMDPM was compared with traditional GPs and mechanics-based methods by using 137 sets of test data. Comparisons show that the proposed model is less sensitive to training data sizes and has superior extrapolation ability for out-of-distribution predictions as compared to traditional GPs. In addition, the proposed model not only has satisfactory prediction performance, but can also reasonably describe the probabilistic characteristics of joint shear strength compared with traditional mechanics-based methods. Comparisons demonstrate that the proposed model has high prediction accuracy, robustness and generalizability, which provides an alternative approach to predicting joint shear strength.

Augmenting Stationary Covariance Functions with a Smoothness Hyperparameter and Improving Gaussian Process Regression Using a Structural Similarity Index

Gaussian process (GP) regression provides a probabilistic framework for modeling geochemistry in mineral resource estimation and environmental monitoring applications. An issue with this approach is that the kernel hyperparameters obtained by maximizing the log-marginal likelihood (LML) often produce GP posterior mean estimates that are overly smooth. This motivates the development of augmented kernels that are more capable of capturing the variability inherent in geological/geochemical processes than the existing stationary covariance functions like the Matérn kernels. This paper makes two contributions. First, it describes an extended class of stationary kernels that contain an extra smoothness hyperparameter (α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}) which can be learned from the input data. Valid intervals for α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} that lead to positive semi-definiteness are determined. Second, it uses a statistical measure called the structural similarity index (SSIM) to quantify smoothness and the spatial fidelity of GP solutions with respect to the input samples. This provides a new way for validating and optimizing GP models. Statistical and spectral analyses provide insights into the behavior of α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} in the augmented kernels which retain useful properties such as sparsity. Results on the Northern Great Basin geochemical dataset demonstrate that, all things being equal, (1) adjusting α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} increases spatial fidelity in GP regression; (2) SSIM is a more reliable spatial quality measure than LML; and (3) the optimal α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} value obtained is correlated with the Wiener entropy of the random process, which indicates the spectral flatness of the chemical signal in the Fourier domain. For GP regression, function smoothness may be defined using Sobolev space. The results show that α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} regulates over-smoothing by moderating the rate of decay in the power spectrum of the equivalent kernel.Graphic abstract