Original Variables Research Articles

Hydrodynamic Shape Optimization of a Naval Destroyer by Machine Learning Methods

This paper explores the integration of advanced machine learning (ML) techniques within simulation-based design optimization (SBDO) processes for naval applications, focusing on the hydrodynamic shape optimization of the DTMB 5415 destroyer model. The use of unsupervised learning for design-space dimensionality reduction, combined with supervised learning through active learning-based multi-fidelity surrogate modeling, allows for significant improvements in computational efficiency while addressing complex, high-dimensional design spaces. By applying these ML techniques to both single- and multi-objective optimizations, aimed at minimizing resistance and enhancing seakeeping performance, the proposed framework demonstrates its practical value in hydrodynamic design. This approach provides a scalable and efficient solution, reducing the reliance on high-fidelity simulations while accelerating the optimization process, without substantial modifications to existing toolchains. A design-space dimensionality reduction of approximately 70% is achieved, reducing the design variables from 22 to 7 while retaining 95% of the original geometric variance. Additionally, computational cost reductions of 65% to 98% are observed, compared to using the full design space and high-fidelity simulations only.

Dynamically regularized Lagrange multiplier schemes with energy dissipation for the incompressible Navier-Stokes equations

In this paper, we present efficient numerical schemes based on the Lagrange multiplier approach for the Navier-Stokes equations. By introducing a dynamic equation (involving the kinetic energy, the Lagrange multiplier, and a regularization parameter), we form a new system which incorporates the energy evolution process but is still equivalent to the original equations. Such nonlinear system is then discretized in time based on the backward differentiation formulas, resulting in a dynamically regularized Lagrange multiplier (DRLM) method. First- and second-order DRLM schemes are derived and shown to be unconditionally energy stable with respect to the original variables. The proposed schemes require only the solutions of two linear Stokes systems and a scalar quadratic equation at each time step. Moreover, with the introduction of the regularization parameter, the Lagrange multiplier can be uniquely determined from the quadratic equation, even with large time step sizes, without affecting accuracy and stability of the numerical solutions. Fully discrete energy stability is also proved with the Marker-and-Cell (MAC) discretization in space. Various numerical experiments in two and three dimensions verify the convergence and energy dissipation as well as demonstrate the accuracy and robustness of the proposed DRLM schemes.

Validation of digital cardiovascular risk score (DiCAVA) in the United States All of Us dataset

Abstract Introduction The cardiovascular disease (CVD) risk score DiCAVA was developed in 2021 using UK Biobank data with the aim to be used for population health screening by deriving its predictions only from variables readily available to every individual, omitting difficult-to-obtain measurements such as blood pressure and invasive blood tests for metrics like cholesterol or HbA1c, which are commonly used in clinical CVD risk assessment tools. The original set of variables was reduced to 30 in the DiCAVA Lite score to further improve usability in a digital solution accessible to the general population, while maintaining its discrimination ability. Purpose Like many countries, the US has a large burden of CVD and would benefit from easily accessible screening tools. However, to ensure generalisability of the UK-developed score, it is necessary to evaluate its performance in a population where the score is intended to be used. Methods Data from a large real-world modern observational study in the US, All of Us, was used to evaluate discrimination and calibration metrics of the risk score. Participants with pre-existing CVD and without available electronic health records were excluded from the study. Variables from the All of Us dataset were matched to the original variables and any missing values were imputed using multiple imputation. Results 216,985 participants with a median follow-up time of 3.44 years were included in the study. The median age was 51 years and 63% of the population were female. Discrimination ability in the All of Us dataset (C-index 0.76) slightly surpassed the discrimination achieved at training with UK Biobank (C-index 0.73), and that of Framingham risk score calculated for the All of Us population (C-index 0.72), demonstrating the score’s usability in the US. The score showed very good discrimination in both males (C-index 0.74) and females (C-index 0.77), all ethnic groups and age groups, with the exception of the population over 65 years of age, where the C-index dropped to 0.65 as expected (similar to that observed in the UK Biobank). Calibration of the original model was assessed at 3 years and showed under-estimation of the predicted risk (calibration-in-the-large of 2.37%). Following re-calibration, the algorithm demonstrated excellent results with the average predicted 3-year risk showing 0.59% difference from the observed CVD rate (calibration plot in Figure). Conclusions Overall, the DiCAVA Lite score has shown very good discrimination in the US, showing its generalisability in different demographic populations. Its re-calibrated form is now validated for use in the US population and because of its ability to be utilised outside clinical settings, it can be deployed widely in the general population as an initial screening tool and potentially boost awareness of cardiovascular health in an effort to drive preventative care of CVD.Re-calibration of the DiCAVA Lite score

Privacy‐Preserving Optimization Algorithm for Distributed Energy Management Over Time‐Varying Graphs: A State Decomposition Method

ABSTRACTThe rapid advancements of intelligent technologies have brought about the potential vulnerability of confidential gradient information linked to cost functions when solving distributed optimization and its related problems. Within the context of distributed energy management, safeguarding such private information has risen to paramount importance. This article investigates a distributed energy management problem (DEMP) to minimize cost while simultaneously satisfying multiple local constraints and protecting the private gradient information of the cost function. To this end, a new privacy‐preserving distributed optimization algorithm under the framework of gradient tracking over time‐varying graphs is proposed for solving the DEMP. Specifically, the auxiliary variables are designed for each node in the algorithm to update the gradient while the original state variables are responsible for the interaction with original neighbors and auxiliary variables. Consequently, the devised algorithm can protect the confidentiality of private cost gradient information, even in the presence of eavesdroppers within the network. In contrast to the homomorphic encryption, signal masking method, and some other algorithms without utilizing the state decomposition, the designed algorithm does not require extra information or computing resources. Moreover, it is proven that the designed algorithm could theoretically converge to the exact optimum of the DEMP at a rate of under some mild assumptions.

Type 2 diabetes and susceptibility to COVID-19: a machine learning analysis

BackgroundType 2 diabetes mellitus (T2DM) was one of the most prevalent comorbidities among patients with coronavirus disease 2019 (COVID-19). Interactions between different metabolic parameters contribute to the susceptibility to the virus; thereby, this study aimed to rank the importance of clinical and laboratory variables as risk factors for COVID-19 or as protective factors against it by applying machine learning methods.MethodThis study is a retrospective cohort conducted at a single center, focusing on a population with T2DM. The patients attended the Yazd Diabetes Research Center in Yazd, Iran, from February 20, 2020, to October 21, 2020. Clinical and laboratory data were collected within three months before the onset of the COVID-19 pandemic in Iran. 59 patients were infected with COVID-19, while 59 were not. The dataset was split into 70% training and 30% test sets. Principal Component Analysis (PCA) was applied to the data. The most important components were selected using a ‘sequential feature selector’ and scored by a Linear Discriminant Analysis model. PCA loadings were then multiplied by the PCs’ scores to determine the importance of the original variables in contracting COVID-19.ResultsHDL-C, followed by eGFR, showed a strong negative correlation with the risk of contracting the virus. Higher levels of HDL-C and eGFR offer protection against COVID-19 in the T2DM population. But, the ratio of BUN to creatinine did not show any correlation. Conversely, the AIP, TyG index and TG showed the most positive correlation with susceptibility to COVID-19 in such a way that higher levels of these factors increase the risk of contracting the virus. The positive correlation of diastolic BP, TyG-BMI index, MAP, BMI, weight, TC, FPG, HbA1C, Cr, systolic BP, BUN, and LDL-C with the risk of COVID-19 decreased, respectively.ConclusionThe atherogenic index of plasma, triglyceride glucose index, and triglyceride levels are the most significant risk factors for COVID-19 contracting in individuals with T2DM. Meanwhile, high-density lipoprotein cholesterol is the most protective factor.

Modelling multivariate spatio-temporal data with identifiable variational autoencoders

Modelling multivariate spatio-temporal data with complex dependency structures is a challenging task but can be simplified by assuming that the original variables are generated from independent latent components. If these components are found, they can be modelled univariately. Blind source separation aims to recover the latent components by estimating the unknown linear or nonlinear unmixing transformation based on the observed data only. In this paper, we extend recently introduced identifiable variational autoencoder to the nonlinear nonstationary spatio-temporal blind source separation setting and demonstrate its performance using comprehensive simulation studies. Additionally, we introduce two alternative methods for the latent dimension estimation, which is a crucial task in order to obtain the correct latent representation. Finally, we illustrate the proposed methods using a meteorological application, where we estimate the latent dimension and the latent components, interpret the components, and show how nonstationarity can be accounted and prediction accuracy can be improved by using the proposed nonlinear blind source separation method as a preprocessing method.

Physical aspects of quantile residual lifetime sequence

Abstract The modeling of count data is found in many fields, such as statistical physics, public health, medicine, epidemiology, applied science, sociology, and agriculture. In many physical situations, it has been observed that many times in the real world, the original variables may be continuous in nature, but discrete by observation. In this study, the α {\rm{\alpha }} -quantile residual life function for discrete lifetime models is defined and some attributes are investigated. The relation between this measure and the hazard rate function is studied. We discuss how this measure could be useful for finding the burn-in time of a lifetime dataset. Then, a new stochastic order based on the α {\rm{\alpha }} -quantile residual life is proposed and studied.

Key performance indicators and reference values for turn performance in elite youth, junior and adult swimmers

ABSTRACT This study aimed to determine kinematic and kinetic key performance indicators (KPI) of swimming turn performance using principal component analysis (PCA) and multiple linear regression analysis and provide reference values using percentiles. Touch and tumble turn performances of male (n = 68) and female (n = 48) Swiss national team members from three age categories—adult (20.2 ± 2.7 yrs, 790 ± 57 points), junior (16.2 ± 0.8 yrs, 729 ± 53 points) and youth swimmers (14.4 ± 1.0 years of age, 667 ± 53 World Aquatics swimming points, respectively)—were assessed with a motion analysis system equipped with a force plate on the pool wall, one over- and four underwater cameras sampling forces at 500 Hz and footages at 100 Hz. The PCA reduced the 27 original variables by up to 15% depending on turn type and age category using Varimax component loading of >0.6 and explained up to 91% of the total variance. The highest Varimax component loadings for each principal component were used to determine KPI for each turn type and age category using multiple-regression analysis with total turn time as dependent variable. These KPI should be used to interpret turn performances and identify individual swimmers’ strengths, weaknesses and future potentials with the help of the percentiles as reference values.

Multivariate process capability analysis with decision-maker preferences

A process capability analysis is an invaluable technique for assessing the efficacy of a system by determining its alignment with customer expectations. In the context of a manufacturing process with multiple correlated variables, the use of multivariate methods is essential for the analysis of quality characteristics. In the literature, there have been various proposals for MPCI (multivariate process capability indices) based on different approaches, including the proportion of non-conforming items or PCA (principal component analysis). In this context, a novel approach to capability analysis with multiple responses of varying degrees of importance has been proposed, termed WAMP (Weighted Arithmetic Mean with Prioritization). The findings indicated that the WAMP method demonstrated greater robustness than the SAM (simple arithmetic mean) and SGM (simple geometric mean) approaches. However, the WAM (weighted arithmetic mean) and WGM (weighted geometric mean) methods also exhibited satisfactory performance in accordance with the established acceptance criteria. The method was evaluated using both simulated and experimental data, demonstrating its efficacy in scenarios with varying degrees of correlation. When a greater weight is assigned to a single original variable, the MPCI is observed to be similar in value to that of the original variable with the highest weight when considered univariately. When equal weights are assigned to two original variables, the MPCI is situated within the capability indices of the original variables with the highest weight. Furthermore, it was observed that the performance of the MPCI improves with an increase in correlation. In conclusion, WAMP has been demonstrated to be a valuable and effective tool for capability analysis in systems with multiple correlated responses of varying degrees of importance.

Use of NIR in COVID-19 Screening: Proof of Principles for Future Application.

The COVID-19 pandemic that affected the world between 2019 and 2022 showed the need for new tools to be tested and developed to be applied in global emergencies. Although standard diagnostic tools exist, such as the reverse-transcription polymerase chain reaction (RT-PCR), these tools have shown severe limitations when mass application is required. Consequently, a pressing need remains to develop a rapid and efficient screening test to deliver reliable results. In this context, near-infrared spectroscopy (NIRS) is a fast and noninvasive vibrational technique capable of identifying the chemical composition of biofluids. This study aimed to develop a rapid NIRS testing methodology to identify individuals with COVID-19 through the spectral analysis of swabs collected from the oral cavity. Swab samples from 67 hospitalized individuals were analyzed using NIR equipment. The spectra were preprocessed, outliers were removed, and classification models were constructed using partial least-squares for discriminant analysis (PLS-DA). Two models were developed: one with all the original variables and another with a limited number of variables selected using ordered predictors selection (OPS-DA). The OPS-DA model effectively reduced the number of redundant variables, thereby improving the diagnostic metrics. The model achieved a sensitivity of 92%, a specificity of 100%, an accuracy of 95%, and an AUROC of 94% for positive samples. These preliminary results suggest that NIRS could be a potential tool for future clinical application. A fast methodology for COVID-19 detection would facilitate medical diagnoses and laboratory routines, helping to ensure appropriate treatment.

Investigation of a new similarity solution for the (2+1)-dimensional Schwarzian Korteweg–de Vries equation

We study the (2+1)-dimensional Schwarzian Korteweg–de Vries equation (SKdV). The explored solutions describe new Lump soliton colliding with visible soliton, an interaction between multi-soliton waves with one soliton, multi peaks of waves moving in a curved path, two hyperbolic waves moving together without interaction and some of periodic waves. We examine the commutative product between multi unknown Lie infinitesimals for the (2+1)-dimensional (SKdV) equation, and this study result in some new Lie vectors. The commutative product generates a system of nonlinear ODEs which had been solved manually. Through two stages of Lie symmetry reduction, SKdV equation is reduced to non-solvable nonlinear ODEs using various combinations of optimal Lie vectors. Using the Riccati–Bernoulli sub-ODE and Integration methods, we investigate new analytical solutions for these ODEs. Back substituting for the original variables generates new solutions for SKdV. Some selected solutions are illustrated through three-dimensional plots.

Digital exclusion as a barrier to accessing healthcare: a summary composite indicator and online tool to explore and quantify local differences in levels of exclusion

AbstractExisting disparities in digital access were exacerbated with the accelerated shift to online provision of services during the COVID-19 pandemic, particularly for already disadvantaged groups. Metrics to quantify relative local differences in levels of digital exclusion are a necessary pre-requisite for the targeting of interventions to address these disparities. In this study we developed a composite indicator and an interactive dashboard ‘The Lincolnshire Digital Health Toolkit’ to explore digital exclusion in Lincolnshire, UK. To develop the indicator, individual variables were normalised and aggregated, intra-variable correlations explored, and factor analysis used to determine variable weightings. Three underlying factors were identified that explained a significant proportion of the variance in the original variables, the first two predominantly related to socioeconomic deprivation and lack of activity. In general, coastal areas in the east of the county had higher levels of digital exclusion, with significant local variation particularly within urban areas. Long travel times to reach medical facilities are an additional barrier in some communities. The toolkit has been used to support the evidence-based geographic targeting of interventions to address barriers to accessing digitally based health information and services. The impact of digital exclusion must be addressed to reduce marginalisation and isolation. The Lincolnshire Digital Health Toolkit provides a novel composite metric tailored to the conditions of this largely rural county and an interactive dashboard to support decisions on resource allocation.

HCS-hierarchical algorithm for simulation of omics datasets.

Analysis of the omics data with the help of machine learning (ML) methods is limited by small sample sizes and a large number of variables. One possible approach to deal with such data is using algorithms for feature selection and reducing the dataset to include only those variables that are related to the studied phenomena. Existing simulators of the omics data were mostly developed with the goal of improving the methods for generations of high-quality data, that correspond with the highest possible fidelity to the real level of molecular markers in the biological materials. The current study aims to simulate the data on a higher level of generalization. Such datasets can then be used to perform tests of the feature selection and ML algorithms on systems that have structures mimicking those of real data, but where the ground truth may be implanted by design. They can also be used to generate contrast variables with the desired correlation structure for the feature selection. We proposed the algorithm for the reconstruction of the omic dataset that, with high fidelity, preserves the correlation structure of the original data with a reduced number of parameters. It is based on the hierarchical clustering of variables and uses principal components of the clusters. It reproduces well topological descriptors of the correlation structure. The correlation structure of the principal components of the clusters then is used to obtain datasets with correlation structures similar to the original data but not correlated with the original variables. The code and data is available at: https://github.com/p100mma/hcrs_omics.

Socioeconomic determinants of small and medium-sized dairy farms in the Ecuador-Colombia border area

The socioeconomic factors influencing small-scale dairy producers in the border area between Ecuador and Colombia were meticulously identified. Employing a non-experimental design, the study leveraged multivariate statistical analysis to discern key determinants. Data processing was executed using the statistical software SPSS v27, facilitating comprehensive analysis. A random survey was administered to 532 small and medium-scale dairy producers in the Carchi province of Ecuador, employing a structured questionnaire supplemented with a Likert scale for nuanced insights. Based on 35 original variables, seven determining factors were identified in dairy farms: political representation, adequate housing, equipment, innovation, empathy, profitability, social welfare, which combined explain 60.95% of the system’s variability. Such factors affect production, the level of household income, as well as their effect on the standard of living of households. Three groups were formed, the first with a low perception of economic development (Traditionalists 33.3%); the second with a better expectation of economic development (Modernizers 27.6%); and the third, identified with greater economic development (Innovators 10.3%). Each group presents cases with a low to high standard of living perspective. The groups have peculiarities in terms of their performance that can be applied to the entire population. A significant relation was established between socioeconomic factors and standard of living.

Rational Polynomial Coefficient Estimation via Adaptive Sparse PCA-Based Method

The Rational Function Model (RFM) is composed of numerous highly correlated Rational Polynomial Coefficients (RPCs), establishing a mathematical relationship between two-dimensional images and three-dimensional spatial coordinates. Due to the existence of ill-posedness and overparameterization, the estimated RPCs are sensitive to any slight perturbations in the observation data, particularly when handling a limited number of Ground Control Points (GCPs). Recently, Principal Component Analysis (PCA) has demonstrated significant performance improvements in the RFM optimization problem. In the PCA-based RFM, each Principal Component (PC) is a linear combination of all variables in the design matrix. However, some original variables are noise related and have very small or almost zero contributions to the construction of PCs, which leads to the overparameterization problem and makes the RPC estimation process ill posed. To address this problem, in this paper, we propose an Adaptive Sparse Principal Component Analysis-based RFM method (ASPCA-RFM) for RPC estimation. In this method, the Elastic Net sparsity constraint is introduced to ensure that each PC contains only a small number of original variables, which automatically eliminates unnecessary variables during PC computation. Since the optimal regularization parameters of the Elastic Net vary significantly in different scenarios, an adaptive regularization parameter approach is proposed to dynamically adjust the regularization parameters according to the explained variance of PCs and degrees of freedom. By adopting the proposed method, the noise and error in the design matrix can be reduced, and the ill-posedness and overparameterization of the RPC estimation can be significantly mitigated. Additionally, we conduct extensive experiments to validate the effectiveness of our method. Compared to existing state-of-the-art methods, the proposed method yields markedly improved or competitive performance.

Validation of a chemotherapy toxicity prediction model in older adults with cancer in Taiwan.

223 Background: The Cancer and Aging Research Group (CARG) model is established for predicting chemotherapy-related toxicities in older patients; however, its applicability has not been validated in Taiwanese or non-solid tumor populations. Due to biopsychosocial distinctions in the Taiwanese demographic and inherent disparities between lymphoma and solid tumors, a modified CARG (mCARG) was assessed for its toxicity prediction capabilities. Methods: In this study, 337 consecutive patients aged ≥65 years with solid tumors (N=258) or lymphoma (N=79), slated for first-line chemotherapy, were recruited from a single medical center in Taiwan between 2018 and 2021, with follow-up until December 31, 2021. Patients were categorized into low-, medium-, and high-risk groups based on mCARG, excluding the cancer-type parameter from the original CARG variables. Validation of mCARG involved receiver operating characteristic (ROC) curves, and individual mCARG variables were analyzed using univariate analysis for their impact on chemotherapy toxicities and overall survival. Results: This study included 337 patients (mean age, 70 years; 160 female [47%]; 177 male [53%]) classified as low-(N=112, 33.2%), medium-(N=106, 31.5%), and high-(N=119, 35.3%) risk. Toxicities of grades ≥3 were 41.1%, 50.0%, and 71.4% ( P<0.001), respectively. ROC was 0.665 (95% CI: 0.608–0.722), indicating satisfactory discrimination. However, the lymphoma subgroup exhibited no significant differences in toxicity risk levels (70.0%, 74.1%, and 81.0%; P=0.39). Among mCARG variables, age ≥72 years ( P=0.008), polychemotherapy ( P=0.009), decreased hemoglobin ( P<0.001), falls ( P=0.008), and inability to walk one block ( P=0.01) were associated with toxicities. One-year overall survival rates were 80.6%, 69.4%, and 57.2%, respectively, in ascending-risk groups, with high-risk groups showing decreased survival ( P=0.001). Conclusions: This prognostic study validated the mCARG model in a heterogeneous cancer cohort in Taiwan but failed to do so in the lymphoma subgroup. Although unable to predict all toxicities, mCARG could offer insights into patient survival among older individuals with cancer.

The Exact Density of the Eigenvalues of the Wishart and Matrix-Variate Gamma and Beta Random Variables

The determination of the distributions of the eigenvalues associated with matrix-variate gamma and beta random variables of either type proves to be a challenging problem. Several of the approaches utilized so far yield unwieldy representations that, for instance, are expressed in terms of multiple integrals, functions of skew symmetric matrices, ratios of determinants, solutions of differential equations, zonal polynomials, and products of incomplete gamma or beta functions. In the present paper, representations of the density functions of the smallest, largest and jth largest eigenvalues of matrix-variate gamma and each type of beta random variables are explicitly provided as finite sums when certain parameters are integers and, as explicit series, in the general situations. In each instance, both the real and complex cases are considered. The derivations initially involve an orthonormal or unitary transformation whereby the wedge products of the differential elements of the eigenvalues can be worked out from those of the original matrix-variate random variables. Some of these results also address the distribution of the eigenvalues of a central Wishart matrix as well as eigenvalue problems arising in connection with the analysis of variance procedure and certain tests of hypotheses in multivariate analysis. Additionally, three numerical examples are provided for illustration purposes.

An interpretable ensemble trees method with joint analysis of static and dynamic features for myocardial infarction detection

Objective. In recent years, artificial intelligence-based electrocardiogram (ECG) methods have been massively applied to myocardial infarction (MI). However, the joint analysis of static and dynamic features to achieve accurate and interpretable MI detection has not been comprehensively addressed. Approach. This paper proposes a simplified ensemble tree method with a joint analysis of static and dynamic features to solve this issue for MI detection. Initially, the dynamic features are extracted by modeling the intrinsic dynamics of ECG via dynamic learning in addition to extracting classical static features. Secondly, a two-stage feature selection strategy is designed to identify a few significant features, which substitute the original variables that are employed in constructing the ensemble tree. This approach enhances the discriminative ability by selecting significant static and dynamic features. Subsequently, this paper presents an interpretable classification method named StackTree by introducing a stacked ensemble scheme to modify the ensemble tree simplification algorithm. The representative rules of the raw ensemble trees are selected as the intermediate training data that is used to retrain a decision tree with performance close to that of the source ensemble model. Using this scheme, the significant precision and interpretability of MI detection are thus comprehensively addressed. Main results. The effectiveness of our method in detecting MI is evaluated using the Physikalisch-Technische Bundesanstalt (PTB) and clinical database. The findings suggest that our algorithm outperforms the traditional methods based on a single type of feature. Additionally, it is comparable to the conventional random forest, achieving 97.1% accuracy under the inter-patient framework on the PTB database. Furthermore, feature subsets trained on PTB are validated using the clinical database, resulting in an accuracy of 84.5%. The chosen important features demonstrate that both static and dynamic information have crucial roles in MI detection. Crucially, the proposed method provides clear internal workings in an easy-to-understand visual manner.

One-level and two-level operator splitting methods for the unsteady incompressible micropolar fluid equations with double diffusion convection

In this paper, a one-level operator splitting method (OOSM) and a two-level operator splitting method (TOSM) for the two-dimensional or three-dimensional (2D/3D) unsteady incompressible micropolar fluid equations with double diffusion convection (IMFDDC) are proposed and analyzed. Firstly, a OOSM is constructed, which consists of five steps. The projection strategy is adopted to get the values of linear velocity and pressure in the first two steps, then the three elliptic subproblems about the angular velocity, the temperature, and the concentration variables are solved in the last three steps. The original variables with the predefined boundary conditions need to be solved separately in each step. Next, in order to solve the problem efficiently, a TOSM is presented, which only solves the nonlinear equations in the coarse level subspaces and the Oseen type linearized equations in the fine level. The numerical analysis of stability and error estimates of the fully discrete methods show that TOSM can reach the same accuracy as the OOSM with H∼O(h2/3). Numerical examples are provided to confirm the effectiveness and reliability of the proposed methods.

Nonlinear Characteristics of Radio Variability in Quasar 3C 446

Quasars are characterized by violent and large amplitude variability in all the observation wavelengths, and the analysis of optical variability is useful for developing theoretical models. Long-term optical variability data of quasar 3C 446 were collected from the University of Michigan Radio Astronomy Observatory database in the 4.8, 8.0, and 14.5 GHz radio bands from 1976 to 2012. Due to the complexity of the variability data, it's hard to study by the linear time series analysis methods. For learning more about non-linear characteristics of the temporal evolution of quasar variability, the Ensemble Empirical Mode Decomposition and nonlinear analysis are used to analyse chaotic dynamics, fractal properties, and periodicity. This paper focuses on whether there is a significant difference between the periodic and non-linear properties of the quasar variability before and after the removal of the periodic or chaotic components. It turns out that the variability of quasar 3C 446 in the radio bands consists of periodic, trend, and chaotic components, and the periodic and trend components are dominant. The periods of the variability after removing the chaotic and trend components are exactly the same as the periods of the original variability data, but the chaotic and fractal characteristics of the two are significantly different. The saturated correlation dimension indicates that the reconstruction of the dynamical system requires more independent parameters than the original optical variables after the removal of the periodic and trend components. The Kolmogorov entropy indicates that the loss of information is greater for the former than the latter, and the system is more chaotic and more complex. The Hurst value indicates that the self-similarity and long-range correlation are slightly stronger for the latter than the former.