Global Kernels Research Articles

A Fractional-Order Investigation of Vaccinated SARS-CoV-2 Epidemic Model with Caputo Fractional Derivative

In this paper, we consider a fractional-order mathematical system comprising four different compartments for the recent pandemic of SARS-CoV-2 with regard to global and singular kernels of Caputo fractional operator. The SARS-CoV-2 fractional mathematical model is analyzed for series-type solution by Laplace–Adomian decomposition techniques (LADM) and homotopy perturbation method (HPM). The whole quantity of each compartment is divided into small parts, and then the sum of these all parts is written as a series solution for each agent of the system, while the nonlinear part is decomposed using the Adomian polynomial. The model is also checked for approximate solution by HPM through a comparison of the parameter power, p , for each equation. The numerical simulation for both methods is provided in different fractional orders along with comparison with each other as well as with natural order 1.

An Efficient PCA-GA-HKSVM-Based Disease Diagnostic Assistant.

Disease diagnosis faces challenges such as misdiagnosis, lack of diagnosis, and slow diagnosis. There are several machine learning techniques that have been applied to address these challenges, where a set of symptoms is applied to a classification model that predicts the presence or absence of a disease. To improve on the performance of these techniques, this paper presents a technique which involves feature selection using principal component analysis (PCA), a hybrid kernel-based support vector machine (HKSVM) classification model and hyperparameter optimization using genetic algorithm (GA). The HKSVM in this paper introduces a new way of combining three kernels: Radial basis function (RBF), linear, and polynomial. Combining local (RBF) and global (linear and polynomial) kernels has the effect of improved model performance. This is because the local kernels are better able to distinguish points closer to each other while the global kernels are more suited to distinguish points that are far away from each other. The PCA-GA-HKSVM is used on 7 different medical datasets, with two datasets being multiclass datasets and 5 datasets being binary. Performance evaluation metrics used were accuracy, precision, and recall. It was observed that the PCA-GA-HKSVM offered better performance than the single kernel support vector machines (SVMs).

Spectral-Similarity-Based Kernel of SVM for Hyperspectral Image Classification

Spectral similarity measures can be regarded as potential metrics for kernel functions, and can be used to generate spectral-similarity-based kernels. However, spectral-similarity-based kernels have not received significant attention from researchers. In this paper, we propose two novel spectral-similarity-based kernels based on spectral angle mapper (SAM) and spectral information divergence (SID) combined with the radial basis function (RBF) kernel: Power spectral angle mapper RBF (Power-SAM-RBF) and normalized spectral information divergence-based RBF (Normalized-SID-RBF) kernels. First, we prove these spectral-similarity-based kernels to be Mercer’s kernels. Second, we analyze their efficiency in terms of local and global kernels. Finally, we consider three hyperspectral datasets to analyze the effectiveness of the proposed spectral-similarity-based kernels. Experimental results demonstrate that the Power-SAM-RBF and SAM-RBF kernels can obtain an impressive performance, particularly the Power-SAM-RBF kernel. For example, when the ratio of the training set is 20 % , the kappa coefficient of Power-SAM-RBF kernel (0.8561) is 1.61 % , 1.32 % , and 1.23 % higher than that of the RBF kernel on the Indian Pines, University of Pavia, and Salinas Valley datasets, respectively. We present three conclusions. First, the superiority of the Power-SAM-RBF kernel compared to other kernels is evident. Second, the Power-SAM-RBF kernel can provide an outstanding performance when the similarity between spectral signatures in the same hyperspectral dataset is either extremely high or extremely low. Third, the Power-SAM-RBF kernel provides even greater benefits compared to other commonly used kernels when the sizes of the training sets increase. In future work, multiple kernels combining with the spectral-similarity-based kernel are expected to be provide better hyperspectral classification.

Extrinsic Meshless Collocation Methods for PDEs on Manifolds

We proposed ways to implement meshless collocation methods extrinsically for solving elliptic PDEs on smooth, closed, connected, and complete Riemannian manifolds with arbitrary codimensions. Our methods are based on strong-form collocations with oversampling and least-squares minimizations, which can be implemented either analytically or approximately. By restricting global kernels to the manifold, our methods resemble their easy-to-implement domain-type analogies, i.e., Kansa methods. Our main theoretical contribution is the robust convergence analysis under some standard smoothness assumptions for high-order convergence. Numerical demonstrations are provided to verify the proven convergence rates, and we simulate reaction-diffusion equations for generating Turing patterns on manifolds.

Parameter selection method for support vector machine based on adaptive fusion of multiple kernel functions and its application in fault diagnosis

A new model parameter selection method for support vector machine based on adaptive fusion of multiple kernel functions is proposed in this paper. Characteristics of local kernels, global kernels, mixtures of kernels and multiple kernels were analyzed. Fusion coefficients of the multiple kernel function, kernel function parameters and regression parameters are combined to form the parameters of the state vector. Thus, the model selection problem is transformed into a nonlinear system state estimation problem. Then, we use a fifth-degree cubature Kalman filter to estimate the parameters. In this way, we realize adaptive selection of the multiple kernel function weighted coefficient, the kernel parameters and the regression parameters. A simulation experiment was performed to interpret the PE process for fault diagnosis.

Blowup of Volterra Integro-Differential Equations and Applications to Semi-Linear Volterra Diffusion Equations

AbstractIn this paper, we discuss the blowup of Volterra integro-differential equations (VIDEs) with a dissipative linear term. To overcome the fluctuation of solutions, we establish a Razumikhin-type theorem to verify the unboundedness of solutions. We also introduce leaving-times and arriving-times for the estimation of the spending-times of solutions to ∞. Based on these two typical techniques, the blowup and global existence of solutions to VIDEs with local and global integrable kernels are presented. As applications, the critical exponents of semi-linear Volterra diffusion equations (SLVDEs) on bounded domains with constant kernel are generalized to SLVDEs on bounded domains and ℝN with some local integrable kernels. Moreover, the critical exponents of SLVDEs on both bounded domains and the unbounded domain ℝN are investigated for global integrable kernels.

Wheat Monitoring by Using Kernel Based Possibilistic c-Means Classifier: Bi-sensor Temporal Multi-spectral Data

The existence of mixed pixels in the satellite images has always been an area of concern. Adding to the challenge is an occurrence of non-linearity between the classes, which is generally overlooked. The study makes an attempt to solve the two frequently occurring problems by kernel based fuzzy approach. This research work deals with Possibilistic c-Means (PCM) classifier with local, global, spectral angle and hyper tangent kernels for wheat crop (Triticum aestivum) identification in Haridwar, Uttarakhand, India. The multi-temporal vegetation index data of Formosat-2 have been used which covers the whole phenology of wheat crop. The additional sensor Landsat-8 OLI imagery has been filled the crucial gap of Formosat-2 temporal datasets. Nine types of proposed kernels based PCM classifier have been applied on three temporal datasets (four, five and six date combinations) to classify two classes early sown and late sown wheat crop. These test results have been concluded that at optimized weighted constant KMOD and polynomial kernel was found effective to separate wheat crop. The five and six date combination were sufficient to discriminate early sown and late sown wheat crop.

Transient analysis of non-Markovian models using stochastic state classes

The method of stochastic state classes approaches the analysis of Generalised Semi Markov Processes (GSMPs) through the symbolic derivation of probability density functions over supports described by Difference Bounds Matrix (DBM) zones. This makes steady state analysis viable, provided that at least one regeneration point is visited by every cyclic behaviour of the model.We extend the approach providing a way to derive transient probabilities. To this end, stochastic state classes are extended with a supplementary timer that enables the symbolic derivation of the distribution of time at which a class can be entered. The approach is amenable to efficient implementation when model timings are given by expolynomial distributions, and it can be applied to perform transient analysis of GSMPs within any given time bound. In the special case of models underlying a Markov Regenerative Process (MRGP), the method can also be applied to the symbolic derivation of local and global kernels, which in turn provide transient probabilities through numerical integration of generalised renewal equations. Since much of the complexity of this analysis is due to the local kernel, we propose a selective derivation of its entries depending on the specific transient measure targeted by the analysis.

Online chaotic time series prediction using unbiased composite kernel machine via Cholesky factorization

The kernel method has proved to be an effective machine learning tool in many fields. Support vector machines with various kernel functions may have different performances, as the kernels belong to two different types, the local kernels and the global kernels. So the composite kernel, which can bring more stable results and good precision in classification and regression, is an inevitable choice. To reduce the computational complexity of the kernel machine's online modeling, an unbiased least squares support vector regression model with composite kernel is proposed. The bias item of LSSVR is eliminated by improving the form of structure risk in this model, and then the calculating method of the regression coefficients is greatly simplified. Simultaneously, through introducing the composite kernel to the LSSVM, the model can easily adapt to the irregular variation of the chaotic time series. Considering the real-time performance, an online learning algorithm based on Cholesky factorization is designed according to the characteristic of extended kernel function matrix. Experimental results indicate that the unbiased composite kernel LSSVR is effective and suitable for online time series with both the steep variations and the smooth variations, as it can well track the dynamic character of the series with good prediction precisions, better generalization and stability. The algorithm can also save much computation time comparing to those methods using matrix inversion, although there is a little more loss in time than that with the usage of single kernels.

Piecewise Global Volterra Nonlinear Modeling and Characterization for Aircraft Dynamics

Evaluating global aircraft dynamic behavior by Volterra theory as an efficient nonlinear approximation method is investigated. In previous research, Volterra theory was implemented to evaluate the system behavior in a selected local region, which restricts the ability to capture the aircraft behavior while moving from one flight region to another. The current research presents a global piecewise Volterra kernel approach, which uses submodel Volterra kernels to build global kernels. A piecewise interpolation is employed to switch between the submodels. An approximate nonlinear pitch-plunge model for a high-performance aircraft was used to assess the proposed approach. A set of experimental simulations is conducted to verify the capability of the proposed model. The approach is compared with a global linear approach while employing the nonlinear simulation as the benchmark.

View independent face detection based on horizontal rectangular features and accuracy improvement using combination kernel of various sizes

This paper proposes a view independent face detection method based on horizontal rectangular features, and accuracy improvement by combining kernels of various sizes. Since the view changes of faces induce large variation in appearance in the horizontal direction, local kernels are applied to horizontal rectangular regions to model such appearance changes. Local kernels are integrated by summation, and then used as a summation kernel for support vector machine (SVM). View independence is shown to be realized by the integration of local horizontal rectangular kernels. However, in general, local kernels (features) of various sizes have different similarity measures, such as detailed and rough similarity, and thus their error patterns are different. If the local and global kernels are combined well, the generalization ability is improved. This research demonstrates the comparative effectiveness of combining the global kernel and local kernels of various sizes as a summation kernel for SVM against use of only the global kernel, only the combination of local kernels and Adaboost with SVMs with a kind of local kernel.

Performance analysis of Markov regenerative reward models

Abstract The problem considered in the paper can not be solved by the traditional technique available for the analysis of Markov Regenerative Processes ( MRGP ). The widely used description of MRGPs , i.e. by the local and the global kernels, do not contain sufficient information on the process to evaluate the distribution of reward measures. A new analytical approach is proposed and studied to utilize better the Markov regenerative property for the analysis of Markov Regenerative Reward Models ( MRRM ). The distribution of the accumulated reward and the completion time of MRRMs is provided in transform domain, as well as considerations about the time domain numerical evaluation of these measures. As a simple application example the performance of a finite queueing system is analyzed.

Performance analysis of Markov regenerative reward models

The problem considered in the paper can not be solved by the traditional technique available for the analysis of Markov Regenerative Processes ( MRGP). The widely used description of MRGPs, i.e. by the local and the global kernels, do not contain sufficient information on the process to evaluate the distribution of reward measures. A new analytical approach is proposed and studied to utilize better the Markov regenerative property for the analysis of Markov Regenerative Reward Models ( MRRM). The distribution of the accumulated reward and the completion time of MRRMs is provided in transform domain, as well as considerations about the time domain numerical evaluation of these measures. As a simple application example the performance of a finite queueing system is analyzed.

Global residues and intersections on a complex manifold

This paper is the study of a class of forms η \eta on a complex manifold V which are smooth on V − W V - W and have poles of kernel type on a complex submanifold W of codimension d; such a form is one whose pull-back to the monoidal transform of V along W has a logarithmic pole. A global existence theorem is proved which asserts that any smooth form φ \varphi on W of filtration s (no (p, q) components with p > s p > s ) is the residue of a form η \eta of filtration s + d s + d such that d η d\eta is smooth on V. This result is used to construct global kernels for ∂ ¯ \bar \partial which establish similar global existence theorems for W with singularities. We then establish formulas connecting intersection and wedge product on the d-cohomology theory of Dolbeault which preserve the Hodge filtration. A number of results are also proved on the integrability of f ∗ η {f^\ast }\eta where f is a rather general holomorphic map.