Dimensionality Reduction Techniques Research Articles

Learning Accurate Label-Specific Features From Partially Multilabeled Data.

Feature selection is an effective dimensionality reduction technique, which can speed up an algorithm and improve model performance such as predictive accuracy and result comprehensibility. The study of selecting label-specific features for each class label has attracted considerable attention since each class label might be determined by some inherent characteristics, where precise label information is required to guide label-specific feature selection. However, obtaining noise-free labels is quite difficult and impractical. In reality, each instance is often annotated by a candidate label set that comprises multiple ground-truth labels and other false-positive labels, termed partial multilabel (PML) learning scenario. Here, false-positive labels concealed in a candidate label set might induce the selection of false label-specific features while masking the intrinsic label correlations, which misleads the selection of relevant features and compromises the selection performance. To address this issue, a novel two-stage partial multilabel feature selection (PMLFS) approach is proposed, which elicits credible labels to guide accurate label-specific feature selection. First, the label confidence matrix is learned to help elicit ground-truth labels from the candidate label set via the label structure reconstruction strategy, each element of which indicates how likely a class label is ground truth. After that, based on distilled credible labels, a joint selection model, including label-specific feature learner and common feature learner, is designed to learn accurate label-specific features to each class label and common features for all class labels. Besides, label correlations are fused into the features selection process to facilitate the generation of an optimal feature subset. Extensive experimental results clearly validate the superiority of the proposed approach.

A Grid-Based Method for Removing Overlaps of Dimensionality Reduction Scatterplot Layouts.

Dimensionality Reduction (DR) scatterplot layouts have become a ubiquitous visualization tool for analyzing multidimensional datasets. Despite their popularity, such scatterplots suffer from occlusion, especially when informative glyphs are used to represent data instances, potentially obfuscating critical information for the analysis under execution. Different strategies have been devised to address this issue, either producing overlap-free layouts that lack the powerful capabilities of contemporary DR techniques in uncovering interesting data patterns or eliminating overlaps as a post-processing strategy. Despite the good results of post-processing techniques, most of the best methods typically expand or distort the scatterplot area, thus reducing glyphs' size (sometimes) to unreadable dimensions, defeating the purpose of removing overlaps. This article presents Distance Grid (DGrid), a novel post-processing strategy to remove overlaps from DR layouts that faithfully preserves the original layout's characteristics and bounds the minimum glyph sizes. We show that DGrid surpasses the state-of-the-art in overlap removal (through an extensive comparative evaluation considering multiple different metrics) while also being one of the fastest techniques, especially for large datasets. A user study with 51 participants also shows that DGrid is consistently ranked among the top techniques for preserving the original scatterplots' visual characteristics and the aesthetics of the final results.

Comparative Analysis of Manifold Learning-Based Dimension Reduction Methods: A Mathematical Perspective

Manifold learning-based approaches have emerged as prominent techniques for dimensionality reduction. Among these methods, t-Distributed Stochastic Neighbor Embedding (t-SNE) and Uniform Manifold Approximation and Projection (UMAP) stand out as two of the most widely used and effective approaches. While both methods share similar underlying procedures, empirical observations indicate two distinctive properties: global data structure preservation and computational efficiency. However, the underlying mathematical principles behind these distinctions remain elusive. To address this gap, this study presents a comparative analysis of the subprocesses involved in these methods, aiming to elucidate the mathematical mechanisms underlying the observed distinctions. By meticulously examining the equation formulations, the mathematical mechanisms contributing to global data structure preservation and computational efficiency are elucidated. To validate the theoretical analysis, data are collected through a laboratory experiment, and an open-source dataset is utilized for validation across different datasets. The consistent alignment of results obtained from both balanced and unbalanced datasets robustly confirms the study’s findings. The insights gained from this study provide a deeper understanding of the mathematical underpinnings of t-SNE and UMAP, enabling more informed and effective use of these dimensionality reduction techniques in various applications, such as anomaly detection, natural language processing, and bioinformatics.

Enhancing IoT (Internet of Things) feature selection: A two-stage approach via an improved whale optimization algorithm

Feature selection is a critical task for optimizing system performance and reducing computational overhead in the context of Internet of Things (IoT) applications. This paper presents a two-stage feature selection approach specifically designed for IoT scenarios. In the first stage, a variety of feature dimensionality reduction techniques are employed to significantly reduce the dimensionality of the original feature set by more than 50%. This process results in the creation of a highly effective feature subset, which serves as a solid foundation for subsequent feature selection in the second stage. In the second stage, feature selection is performed on the feature subset by an evolutionary algorithm to obtain high accuracy. Notably, we propose an improved whale optimization algorithm (WOA-HA), which incorporates several improvement factors such as a chaotic Hénon map mechanism (HMM), adaptive coefficient vector (ACV), and a binary operator. To assess the effectiveness of our approach, we compare the performance of WOA-HA with other evolutionary algorithms in terms of feature selection outcomes. Through extensive experiments, our proposed approach achieves an average accuracy of up to 95.5% on Aalto IoT dataset and 98.8% on RT-IoT 2022 dataset, respectively. Meanwhile, the average number of selected features reduced by about 82.5% on Aalto IoT dataset and about 62.3% in the RT-IoT 2022 dataset, respectively. Our proposed approach consistently outperforms other methods and achieves the best performance on most datasets with higher accuracy and fewer features.

Prediction of Sea Level Using Double Data Decomposition and Hybrid Deep Learning Model for Northern Territory, Australia

Sea level rise (SLR) attributed to the melting of ice caps and thermal expansion of seawater is of great global significance to vast populations of people residing along the world’s coastlines. The extent of SLR’s impact on physical coastal areas is determined by multiple factors such as geographical location, coastal structure, wetland vegetation and related oceanic changes. For coastal communities at risk of inundation and coastal erosion due to SLR, the modelling and projection of future sea levels can provide the information necessary to prepare and adapt to gradual sea level rise over several years. In the following study, a new model for predicting future sea levels is presented, which focusses on two tide gauge locations (Darwin and Milner Bay) in the Northern Territory (NT), Australia. Historical data from the Australian Bureau of Meteorology (BOM) from 1990 to 2022 are used for data training and prediction using artificial intelligence models and computation of mean sea level (MSL) linear projection. The study employs a new double data decomposition approach using Multivariate Variational Mode Decomposition (MVMD) and Successive Variational Mode Decomposition (SVMD) with dimensionality reduction techniques of Principal Component Analysis (PCA) for data modelling using four artificial intelligence models (Support Vector Regression (SVR), Adaptive Boosting Regressor (AdaBoost), Multilayer Perceptron (MLP), and Convolutional Neural Network–Bidirectional Gated Recurrent Unit (CNN-BiGRU). It proposes a deep learning hybrid CNN-BiGRU model for sea level prediction, which is benchmarked by SVR, AdaBoost, and MLP. MVMD-SVMD-CNN-BiGRU hybrid models achieved the highest performance values of 0.9979 (d), 0.996 (NS), 0.9409 (L); and 0.998 (d), 0.9959 (NS), 0.9413 (L) for Milner Bay and Darwin, respectively. It also attained the lowest error values of 0.1016 (RMSE), 0.0782 (MABE), 2.3699 (RRMSE), and 2.4123 (MAPE) for Darwin and 0.0248 (RMSE), 0.0189 (MABE), 1.9901 (RRMSE), and 1.7486 (MAPE) for Milner Bay. The mean sea level (MSL) trend analysis showed a rise of 6.1 ± 1.1 mm and 5.6 ± 1.5 mm for Darwin and Milner Bay, respectively, from 1990 to 2022.

Risk factor aggregation and stress testing

Stress testing refers to the application of adverse financial or macroeconomic scenarios to a portfolio. For this purpose, financial or macroeconomic risk factors are linked with asset returns, typically via a factor model. We expand the range of risk factors by adapting dimension-reduction techniques from unsupervised learning, namely PCA and autoencoders. This results in aggregated risk factors, encompassing a global factor, factors representing broad geographical regions, and factors specific to cyclical and defensive industries. As the adapted PCA and autoencoders provide an interpretation of the latent factors, this methodology is also valuable in other areas where dimension-reduction and explainability are critical.

Multifidelity Methodology for Reduced-Order Models with High-Dimensional Inputs

In the early stages of aerospace design, reduced-order models (ROMs) are crucial for minimizing computational costs associated with using physics-rich field information in many-query scenarios requiring multiple evaluations. The intricacy of aerospace design demands the use of high-dimensional design spaces to capture detailed features and design variability accurately. However, these spaces introduce significant challenges, including the curse of dimensionality, which stems from both high-dimensional inputs and outputs necessitating substantial training data and computational effort. To address these complexities, this study introduces a novel multifidelity, parametric, and nonintrusive ROM framework designed for high-dimensional contexts. It integrates machine learning techniques for manifold alignment and dimension reduction—employing proper orthogonal decomposition and model-based active subspace—with multifidelity regression for ROM construction. Our approach is validated through two test cases: the 2D RAE 2822 airfoil and the 3D NASA CRM wing, assessing various fidelity levels, training data ratios, and sample sizes. Compared to the single-fidelity principal component–active subspace (PCAS) method, our multifidelity solution offers improved cost-accuracy benefits and achieves better predictive accuracy with reduced computational demands. Moreover, our methodology outperforms the manifold-aligned ROM method by 50% in handling scenarios with large input dimensions, underscoring its efficacy in addressing the complex challenges of aerospace design.

Caritas’s Work for the Goals of Agenda 2030: A Study on the Services Provided in Campania

The United Nations’ Agenda 2030 has established a series of Sustainable Development Goals to address global challenges, including poverty, food insecurity, access to education, and social inequality. In this context, charitable organizations such as Caritas play a crucial role in mitigating the negative effects of these challenges and promoting fair and sustainable development. This study aims to analyze prevalent needs among individuals seeking assistance from Caritas in Campania and examine how the organization contributes to achieving the Agenda 2030 Goals in the region. The statistical investigation techniques considered include tandem analysis a dimension-reduction technique, such as multiple factor analysis, and then a cluster analysis to identify similar groups of individuals. These exploratory data analysis methods have allowed for the identification of common needs, including food assistance, support for education, employment, and housing assistance. Subsequently, Caritas programs and initiatives aimed at meeting these needs and promoting sustainable development are explored. The results indicate that Caritas plays a significant role in addressing the urgent needs of the vulnerable population in Campania and contributes to the goals of Agenda 2030, particularly those related to poverty alleviation, immigration, health promotion, education, employment, and the reduction of social inequalities. This study provides an important perspective on the relevance and effectiveness of Caritas’s work in the context of Agenda 2030.

The role of morphometric characteristics in predicting 20-meter sprint performance through machine learning

The aim of this study was to test the morphometric features affecting 20-m sprint performance in children at the first level of primary education using machine learning (ML) algorithms. In this study, 130 male and 152 female volunteers aged between 6 and 11 years were included. After obtaining demographic information of the participants, skinfold thickness, diameter and circumference measurements, and 20-m sprint performance were determined. The study conducted three distinct experiments to determine the optimal ML technique for predicting outcomes. Initially, the entire feature space was utilized for training the ML models to establish a baseline performance. In the second experiment, only significant features identified through correlation analysis were used for training and testing the models, enhancing the focus on relevant predictors. Lastly, Principal Component Analysis (PCA) was employed to reduce the feature space, aiming to streamline model complexity while retaining data variance. These experiments collectively aimed to evaluate different feature selection and dimensionality reduction techniques, providing insights into the most effective strategies for optimizing predictive performance in the given context. The correlation-based selected features (Age, Height, waist circumference, hip circumference, leg length, thigh length, foot length) has produced a minimum Mean Squared Error (MSE) value of 0.012 for predicting the sprint performance in children. The effective utilization of correlation analysis in the selection of relevant features for our regression model suggests that the features selected exhibit robust linear associations with the target variable and can be relied upon as predictors.

Organelle phenotyping and multi-dimensional microscopy identify C1q as a novel regulator of microglial function.

Microglia, the immune cells of the central nervous system, are dynamic and heterogenous cells. While single cell RNA sequencing has become the conventional methodology for evaluating microglial state, transcriptomics do not provide insight into functional changes, identifying a critical gap in the field. Here, we propose a novel organelle phenotyping approach in which we treat live human induced pluripotent stem cell-derived microglia (iMGL) with organelle dyes staining mitochondria, lipids, lysosomes and acquire data by live-cell spectral microscopy. Dimensionality reduction techniques and unbiased cluster identification allow for recognition of microglial subpopulations with single-cell resolution based on organelle function. We validated this methodology using lipopolysaccharide and IL-10 treatment to polarize iMGL to an "inflammatory" and "anti-inflammatory" state, respectively, and then applied it to identify a novel regulator of iMGL function, complement protein C1q. While C1q is traditionally known as the initiator of the complement cascade, here we use organelle phenotyping to identify a role for C1q in regulating iMGL polarization via fatty acid storage and mitochondria membrane potential. Follow up evaluation of microglia using traditional read outs of activation state confirm that C1q drives an increase in microglia pro-inflammatory gene production and migration, while suppressing microglial proliferation. These data together validate the use of a novel organelle phenotyping approach and enable better mechanistic investigation of molecular regulators of microglial state.

A new adaptive elastic loss for robust unsupervised feature selection

The sparse-based unsupervised dimension reduction technique has been demonstrated to be an effective tool for dealing with high-dimensional data. Among them, the sparse regularization technique is used to introduce the prior knowledge of the model and helps unsupervised dimension reduction algorithms to complete the feature selection. However, the sparse regularization only considers the sparsity of representation. It ignores the group effect of variables among which the pairwise correlations are very high. To solve these issues, we present a novel regularization approach based on adaptive elastic loss in this study. Then, utilizing the regularization method, an efficient unsupervised feature selection methodology is given. Additionally, we devise an efficient alternative optimization algorithm to resolve the complex optimization issue of our proposed method and conduct a theoretical analysis of its computational complexity and convergence. Eventually, we conduct comprehensive experiments to test the validity of our proposed method, and real-world data and a simulation study demonstrate that our proposed method outperforms many state-of-the-art unsupervised feature selection methods in terms of clustering and classification.

A Comprehensive Framework for Pathology Classification Bridging Precision and Interpretability

Pathology classification is an indispensable component of medical diagnostics, facilitating accurate disease identification, prognosis determination, and treatment planning. However, the increasing complexity and heterogeneity of pathological manifestations pose significant challenges to traditional classification methodologies. This abstract presents a novel framework that integrates advanced machine learning techniques with domain-specific expertise to enhance the precision and interpretability of pathology classification. Our framework adopts a multi-modal approach, leveraging diverse data sources including histopathological images, clinical records, genomic profiles, and molecular biomarkers. Through feature fusion and dimensionality reduction techniques, we effectively capture intricate patterns and latent relationships embedded within the data, enabling robust classification across diverse pathological conditions. Furthermore, interpretability is prioritized through the incorporation of explainable AI methodologies, facilitating the identification of salient features and decision rationales underlying classification outcomes. This ensures transparency and trustworthiness in the diagnostic process, empowering clinicians to make informed decisions and refine treatment strategies. Validation of our framework across various pathological contexts demonstrates superior performance compared to conventional approaches, exhibiting high accuracy, sensitivity, and specificity. Moreover, its modular architecture facilitates customization and scalability, accommodating evolving diagnostic needs and emerging technological advancements. In conclusion, our proposed framework represents a significant advancement in pathology classification, offering a synergistic blend of computational sophistication and clinical relevance. By seamlessly integrating cutting-edge technologies with domain knowledge, it holds promise for revolutionizing diagnostic practices and improving patient outcomes in the realm of precision medicine.

Explaining Conformational Diversity in Protein Families through Molecular Motions

Proteins play a central role in biological processes, and understanding their conformational variability is crucial for unraveling their functional mechanisms. Recent advancements in high-throughput technologies have enhanced our knowledge of protein structures, yet predicting their multiple conformational states and motions remains challenging. This study introduces Dimensionality Analysis for protein Conformational Exploration (DANCE) for a systematic and comprehensive description of protein families conformational variability. DANCE accommodates both experimental and predicted structures. It is suitable for analysing anything from single proteins to superfamilies. Employing it, we clustered all experimentally resolved protein structures available in the Protein Data Bank into conformational collections and characterized them as sets of linear motions. The resource facilitates access and exploitation of the multiple states adopted by a protein and its homologs. Beyond descriptive analysis, we assessed classical dimensionality reduction techniques for sampling unseen states on a representative benchmark. This work improves our understanding of how proteins deform to perform their functions and opens ways to a standardised evaluation of methods designed to sample and generate protein conformations.

Phasor-based analysis of a neuromorphic architecture for microwave sensing

This article presents a design procedure for implementing artificial neural networks (ANNs) using conventional microwave components at the hardware level with potential applications in radar and remote sensing. The main objective is to develop structured hardware design methods for implementing artificial neurons, utilizing microwave devices to create neuromorphic devices compatible with high-frequency electromagnetic waves. The research aims to address the challenge of encoding and modulating information in electromagnetic waves into a format suitable for the neuromorphic device by using frequency-modulated information instead of intensity-modulated information. It also proposes a method for integrating principal component analysis as a dimensionality reduction technique with the implementation of ANNs on a single hardware. As a dummy task, the process outlined here is used to implement an artificial neural network at the hardware level, with a specific emphasis on creating hardware that is capable of performing matrix multiplications in the form of dot products while also being able to extract the resulting data in an interpretable manner. The proposed implementation involves the use of directional couplers to implement weights and sample the resulting signal at specific intervals to obtain the multiplication result.

Contextualized dynamic meta embeddings based on Gated CNNs and self-attention for Arabic machine translation

PurposeThe paper aims to enhance Arabic machine translation (MT) by proposing novel approaches: (1) a dimensionality reduction technique for word embeddings tailored for Arabic text, optimizing efficiency while retaining semantic information; (2) a comprehensive comparison of meta-embedding techniques to improve translation quality; and (3) a method leveraging self-attention and Gated CNNs to capture token dependencies, including temporal and hierarchical features within sentences, and interactions between different embedding types. These approaches collectively aim to enhance translation quality by combining different embedding schemes and leveraging advanced modeling techniques.Design/methodology/approachRecent works on MT in general and Arabic MT in particular often pick one type of word embedding model. In this paper, we present a novel approach to enhance Arabic MT by addressing three key aspects. Firstly, we propose a new dimensionality reduction technique for word embeddings, specifically tailored for Arabic text. This technique optimizes the efficiency of embeddings while retaining their semantic information. Secondly, we conduct an extensive comparison of different meta-embedding techniques, exploring the combination of static and contextual embeddings. Through this analysis, we identify the most effective approach to improve translation quality. Lastly, we introduce a novel method that leverages self-attention and Gated convolutional neural networks (CNNs) to capture token dependencies, including temporal and hierarchical features within sentences, as well as interactions between different types of embeddings. Our experimental results demonstrate the effectiveness of our proposed approach in significantly enhancing Arabic MT performance. It outperforms baseline models with a BLEU score increase of 2 points and achieves superior results compared to state-of-the-art approaches, with an average improvement of 4.6 points across all evaluation metrics.FindingsThe proposed approaches significantly enhance Arabic MT performance. The dimensionality reduction technique improves the efficiency of word embeddings while preserving semantic information. Comprehensive comparison identifies effective meta-embedding techniques, with the contextualized dynamic meta-embeddings (CDME) model showcasing competitive results. Integration of Gated CNNs with the transformer model surpasses baseline performance, leveraging both architectures' strengths. Overall, these findings demonstrate substantial improvements in translation quality, with a BLEU score increase of 2 points and an average improvement of 4.6 points across all evaluation metrics, outperforming state-of-the-art approaches.Originality/valueThe paper’s originality lies in its departure from simply fine-tuning the transformer model for a specific task. Instead, it introduces modifications to the internal architecture of the transformer, integrating Gated CNNs to enhance translation performance. This departure from traditional fine-tuning approaches demonstrates a novel perspective on model enhancement, offering unique insights into improving translation quality without solely relying on pre-existing architectures. The originality in dimensionality reduction lies in the tailored approach for Arabic text. While dimensionality reduction techniques are not new, the paper introduces a specific method optimized for Arabic word embeddings. By employing independent component analysis (ICA) and a post-processing method, the paper effectively reduces the dimensionality of word embeddings while preserving semantic information which has not been investigated before especially for MT task.

Machine Learning Driven Feature Extraction and Dimensionality Reduction for Image Classification

Cancer is the leading cause of death globally, affecting various organs in the human body. Early diagnosis of gastric cancer is essential for improving survival rates. However, traditional diagnosis methods are time-consuming, require multiple tests, and rely on specialist availability. This motivates the development of automated techniques for diagnosing gastric cancer using image analysis. While existing computerized techniques have been proposed, challenges remain. These include difficulty distinguishing healthy from cancerous regions in images and extracting irrelevant features during analysis. This research addresses these challenges by proposing a novel deep learning-based method for gastric cancer classification. The method utilizes deep feature extraction, dimensionality reduction, and classification techniques applied to a gastric cancer image dataset. This approach achieves high accuracy (99.32%), sensitivity (99.13%), and specificity (99.64%) in classifying gastric cancer.

Consensus and discriminative non-negative matrix factorization for multi-view unsupervised feature selection

Multi-view unsupervised feature selection (MUFS) has been proven to be an efficient dimensionality reduction technique for multi-view data. Existing methods have two main challenges: (1) The consistency information from different views is not fully exploited. (2) The cluster structure of the original data is not well utilized. To settle these problems, we propose a novel consensus and discriminative non-negative matrix factorization (CDNMF) for multi-view unsupervised feature selection. Specifically, CDNMF obtains a robust low-dimensional latent space by NMF with an ℓ2,1-norm constraint. To select more discriminative features, we further impose a sparsity constraint on the learned latent features. Moreover, CDNMF performs k-means clustering on all views separately to obtain the pseudo-labels of each view, which are used to guide the learning of consensus information among views. To solve our model, we develop an efficient iterative optimization algorithm. Extensive experimental results on ten benchmark datasets demonstrate that the proposed significantly outperforms several existing feature selection methods in clustering tasks.

Association of M2 macrophages with EMT in glioma identified through combination of multi-omics and machine learning

BackgroundThe incidence of glioma, a prevalent brain malignancy, is increasing, particularly among the elderly population. This study aimed to elucidate the clinical importance of epithelial–mesenchymal transition (EMT) in gliomas and its association with malignancy and prognosis. BackgroundThe incidence of glioma, particularly among elderly individuals, is on the rise. The malignancy of glioma is determined not only by the oncogenic properties of tumor cells but also by the composition of the tumor microenvironment, which includes immune system macrophages. The prevalence of M2-type macrophages typically fosters tumor progression, yet the underlying mechanism remains elusive. Our study explored the clinical importance of epithelial–mesenchymal transition (EMT) in gliomas and its association with malignancy and prognosis. MethodsOur study used the gene set variation analysis (GSVA) algorithm to classify different levels of EMT activation based on the transcriptomic and multi-omics data. Machine learning (ML) and single-cell analysis were integrated into our model for comprehensive analysis. A predictive model was constructed and in vitro experiments were performed to validate our findings. ResultsOur study classified 1,641 samples into two clusters based on EMT activation: the EMT-hot group and the EMT-cold group. The EMT-hot group had elevated copy number loss, tumor mutational burden (TMB), and a poorer survival rate. Conversely, the EMT-cold group showed a better survival rate, likely attributed to lower stromal and immune cell scores, as well as decreased expression of human leukocyte antigen-related genes. Driving genes were identified through weighted gene coexpression network analysis (WGCNA) and dimensionality reduction techniques. These genes were then utilized in the construction of a prognostic model using ML and protein-protein interaction (PPI) network analysis. Furthermore, the impact of the core genes identified through single-cell analysis on glioma prognosis was examined. ConclusionOur research underscores the efficacy of our model in predicting glioma prognosis and elucidates the connection between the M2 macrophages and EMT. Additionally, core genes such as LY96, C1QB, LGALS1, CSPG5, S100A8, and CHGB were identified as pivotal for mediating the occurrence of EMT induced by M2 macrophages.

Analysis of Dynamic Behavior of Gravity Model Using the Techniques of Road Saturation and Hilbert Curve Dimensionality Reduction

Analysis of Dynamic Behavior of Gravity Model Using the Techniques of Road Saturation and Hilbert Curve Dimensionality Reduction

Exponential Stability of Impulsive Switched Nonlinear Systems With Partial Unmeasurable States on Time Scales

Abstract This paper investigates the stability of a class of partially unmeasurable nonlinear impulsive switched dynamical systems on time scales. It provides sufficient conditions for ensuring exponential stability of impulsive switched systems in hybrid time domains using Gronwall's inequality, dimensionality reduction technique, and time scale theory. These conditions apply to systems where nonlinearities affect both continuous and discrete subsystems. Stable strategies involving partial unmeasurable states are obtained. Furthermore, the proposed results allow the switched system to have incomplete measurable states. The effectiveness of these results is demonstrated by several examples, which are accompanied by simulations for comparison.