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Kernelized Gram matrix $W$ constructed from data points $\{x_i\}_{i=1}^N$ as $W_{ij}= k_0( \frac{ \| x_i - x_j \|^2} {\sigma ^2} ) $ is widely used in graph-based geometric data analysis and unsupervised learning. An important question is how to choose the kernel bandwidth $\sigma $, and a common practice called self-tuned kernel adaptively sets a $\sigma _i$ at each point $x_i$ by the $k$-nearest neighbor (kNN) distance. When $x_i$s are sampled from a $d$-dimensional manifold embedded in a possibly high-dimensional space, unlike with fixed-bandwidth kernels, theoretical results of graph Laplacian convergence with self-tuned kernels have been incomplete. This paper proves the convergence of graph Laplacian operator $L_N$ to manifold (weighted-)Laplacian for a new family of kNN self-tuned kernels $W^{(\alpha )}_{ij} = k_0( \frac{ \| x_i - x_j \|^2}{ \epsilon \hat{\rho }(x_i) \hat{\rho }(x_j)})/\hat{\rho }(x_i)^\alpha \hat{\rho }(x_j)^\alpha $, where $\hat{\rho }$ is the estimated bandwidth function by kNN and the limiting operator is also parametrized by $\alpha $. When $\alpha = 1$, the limiting operator is the weighted manifold Laplacian $\varDelta _p$. Specifically, we prove the point-wise convergence of $L_N f $ and convergence of the graph Dirichlet form with rates. Our analysis is based on first establishing a $C^0$ consistency for $\hat{\rho }$ which bounds the relative estimation error $|\hat{\rho } - \bar{\rho }|/\bar{\rho }$ uniformly with high probability, where $\bar{\rho } = p^{-1/d}$ and $p$ is the data density function. Our theoretical results reveal the advantage of the self-tuned kernel over the fixed-bandwidth kernel via smaller variance error in low-density regions. In the algorithm, no prior knowledge of $d$ or data density is needed. The theoretical results are supported by numerical experiments on simulated data and hand-written digit image data.

In this paper, we extend the diffusion maps algorithm on a family of heat kernels that are either local (having exponential decay) or nonlocal (having polynomial decay), arising in various applications. For example, these kernels have been used as a regularizer in various supervised learning tasks for denoising images. Importantly, these heat kernels give rise to operators that include (but are not restricted to) the generators of the classical Laplacian associated to Brownian processes as well as the fractional Laplacian associated with β-stable Levy processes. For local kernels, while the method is a version of the diffusion maps algorithm, we show that the applications with non-Gaussian local heat kernels approximate temporally rescaled Laplace-Beltrami operators. For the non-local heat kernels, we modify the diffusion maps algorithm to estimate fractional Laplacian operators. Here, the graph distance is used to approximate the geodesic distance with appropriate error bounds. While this approximation becomes numerically expensive as the number of data points increases, it produces an accurate operator estimation that is robust to the choice of the kernel bandwidth parameter value. In contrast, the local kernels are numerically more efficient but more sensitive to the choice of kernel bandwidth parameter value. In an application to estimate non-smooth regression functions, we find that using the nonlocal kernel as a regularizer produces a more robust and accurate estimate than using local kernels. For manifolds with boundary, we find that the proposed fractional diffusion maps framework implemented with non-local kernels approximates the regional fractional Laplacian.

Traditional interacting multiple model Kalman filter (IMM-KF) was derived for linear discrete time system with Markovian coefficients and it works well under Gaussian noise. In practice, when the system is affected by the impulsive noise, the performance of the traditional IMM-KF model becomes poor. Then, maximum correntropy Kalman filter (MCKF) which has advantages in dealing with non-Gaussian noise was proposed. However, the performance of traditional MCKF becomes poor when the motion state is not single. In this brief, we propose a new algorithm called interacting multiple model based on maximum correntropy Kalman filter (IMM-MCKF) by combining interacting multiple model (IMM) with MCKF to deal with the impulsive noise. Also, the traditional MCKF works poorly when the kernel bandwidth is small, we made a further improvement by changing the kernel. Simulation examples are given to demonstrate the effectiveness of the new algorithms.

Online learning methods are designed to establish timely predictive models for machine learning problems. The methods for online learning of nonlinear systems are usually developed in the reproducing kernel Hilbert space (RKHS) associated with Gaussian kernel in which the kernel bandwidth is manually selected and remains steady during the entire modeling process in most cases. This setting may make the learning model rigid and inappropriate for complex data streams. Since the bandwidth appears in a nonlinear term of the kernel model, it raises substantial challenges in the development of learning methods with an adaptive bandwidth. In this article, we propose a novel approach to address this important open issue. By a carefully casted linearization scheme, the nonlinear learning problem is reasonably transformed into a state feedback control problem for a series of controllable systems. Then, by employing optimal control techniques, an effective algorithm is developed, and the parameters in the learning model including kernel bandwidth can be efficiently updated in a real-time manner. By taking advantage of the particular structure of the Gaussian kernel model, a theoretical analysis on the convergence and rationality of the proposed method is also provided. Compared with the kernel algorithms with a fixed bandwidth, our novel learning framework can not only achieve adaptive learning results with a better prediction accuracy but also show performance that is more robust with a faster convergence speed. Encouraging numerical results are provided to demonstrate the advantages of our new method.

The choice of a bandwidth can affect dramatically the accuracy of classification methods relying on it. Recently, a large number of methods to choose the bandwidth have been developed. This article...

Large-scale agent systems have foreseeable applications in the near future. Estimating their macroscopic density is critical for many density-based optimization and control tasks, such as sensor deployment and city traffic scheduling. In this paper, we study the problem of estimating their dynamically varying probability density, given the agents' individual dynamics (which can be nonlinear and time-varying) and their states observed in real-time. The density evolution is shown to satisfy a linear partial differential equation uniquely determined by the agents' dynamics. We present a density filter which takes advantage of the system dynamics to gradually improve its estimation and is scalable to the agents' population. Specifically, we use kernel density estimators (KDE) to construct a noisy measurement and show that, when the agents' population is large, the measurement noise is approximately ``Gaussian''. With this important property, infinite-dimensional Kalman filters are used to design density filters. It turns out that the covariance of measurement noise depends on the true density. This state-dependence makes it necessary to approximate the covariance in the associated operator Riccati equation, rendering the density filter suboptimal. The notion of input-to-state stability is used to prove that the performance of the suboptimal density filter remains close to the optimal one. Simulation results suggest that the proposed density filter is able to quickly recognize the underlying modes of the unknown density and automatically ignore outliers, and is robust to different choices of kernel bandwidth of KDE.

In this paper, an improved maximum correntropy cubature kalman filter(IMCCKF) is proposed to address the measurement outliers in cooperative localization(CL) of autonomous underwater vehicles (AUVs). The estimated performance of the maximum correntropy cubature kalman filter(MCCKF) algorithm is affected by the kernel bandwidth(KB). The selection value of the KB cannot be determined only by experience in practical CL of AUVs, which will greatly reduce the practical application value of the MCCKF algorithm. The adaptive factor is constructed by comparing the trace size of innovation matrix and the trace size of quantity prediction error matrix, and the KB in the MCCKF is adjusted online by the adaptive factor. Finally, the validity of the proposed IMCCKF method is verified by the lake test data. The experimental results show that the proposed method has the ability to adjust the KB in real time and quickly obtain the optimal value of the KB, and the IMCCKF algorithm can effectively improve the positioning performance of CL system with measurement outliers.

This paper develops a novel approach to track power system state evolution based on the maximum correntropy criterion, due to its robustness against non-Gaussian errors. It includes the temporal aspects on the estimation process within a maximum-correntropy-based extended Kalman filter (MCEKF), which is able to deal with both nonlinear supervisory control and data acquisition (SCADA) and phasor measurement unit (PMU) measurement models. By representing the behavior of the state variables with a nonparametric model within the kernel density estimation, it is possible to include abrupt state transitions as part of the process noise with non-Gaussian characteristics. Also, a novel strategy to update the size of Parzen windows in the kernel estimation is proposed to suppress the effects of suspect samples. By properly adjusting the kernel bandwidth, the proposed MCEKF keeps its accuracy during sudden load changes and contingencies, or in the presence of bad data. Simulations with IEEE test systems and the Brazilian interconnected system are carried out. The results show that the method deals with non-Gaussian noises in both the process and measurement, and provides accurate estimates of the system state under normal and abnormal conditions.

A novel estimator is proposed for estimating the energy distribution of photons incident upon a detector in X-ray spectroscopic systems. It is specifically designed for count-rate regimes where pulse pile-up is an issue. A key step in the derivation of the estimator is the novel reformulation of the problem as a decompounding problem of a compound Poisson process. A non-parametric decompounding algorithm is proposed for pile-up correction with finite-length data sets. Non-parametric estimation typically includes appropriately choosing a ‘kernel bandwidth’. Simulations demonstrate our data-driven bandwidth selection is close to optimal, and outperforms asymptotic-based selection in the typical regions of interest to spectroscopic applications. Non-parametric approaches are particularly useful when the shape of the detector response varies with each interaction. The method exhibits similar accuracy to other state-of-the-art non-parametric methods, while being much faster to compute.

The adaptive kernel filters are sequential learning algorithms that operate in a particular functional space called a reproducing kernel Hilbert space. However, their performance depends on the selection of two hyper-parameters, i.e., kernel bandwidth and learning-rate. Besides, as these algorithms train the model using a sequence of input vectors, their computation scales with the number of samples. In this work, we propose to address the previous challenges of these sequential learning algorithms. The proposed framework, unlike similar methods, maximizes the correntropy function to optimize the bandwidth and learning-rate parameters. Further, we introduce a sparsification strategy based on dimensionality reduction to remove redundant samples. The framework is tested on both synthetic and real-world data sets, showing convergence to relatively low values of mean-square-error.

We present a framework for Nesterov's accelerated gradient flows in probability space to design efficient mean-field Markov chain Monte Carlo (MCMC) algorithms for Bayesian inverse problems. Here four examples of information metrics are considered, including Fisher-Rao metric, Wasserstein-2 metric, Kalman-Wasserstein metric and Stein metric. For both Fisher-Rao and Wasserstein-2 metrics, we prove convergence properties of accelerated gradient flows. In implementations, we propose a sampling-efficient discrete-time algorithm for Wasserstein-2, Kalman-Wasserstein and Stein accelerated gradient flows with a restart technique. We also formulate a kernel bandwidth selection method, which learns the gradient of logarithm of density from Brownian-motion samples. Numerical experiments, including Bayesian logistic regression and Bayesian neural network, show the strength of the proposed methods compared with state-of-the-art algorithms.

This paper considers the estimation of the long memory parameter in nonparametric regression with strongly correlated errors. The key idea is to minimize a unified mean squared error of long memory parameter to select both kernel bandwidth and the number of frequencies used in exact local Whittle estimation. A unified mean squared error framework is more natural because it provides both goodness of fit and measure of strong dependence. The block bootstrap is applied to evaluate the mean squared error. Finite sample performance using Monte Carlo simulations shows the closest performance to the oracle. The proposed method outperforms existing methods especially when dependency and sample size increase. The proposed method is also illustreated to the volatility of exchange rate between Korean Won for US dollar.

The fifth-generation technology is emerging with various real-time applications that demand high data rate. The number of connected devices is expected to increase. However, spectral and energy resources are limited. Therefore, it is challenging to meet users satisfaction. Thus, the necessity for a structure that can sustain real-time performance of wireless communication systems is pressing. This paper presents an efficient radio resource allocation scheme using regularized particle filter with optimized kernel size and kernel bandwidth, and Monte Carlo Markov chain move step based on metropolis algorithm is used to estimate next state of channel. Channel state is calculated from the temporal channel correlation using maximum likelihood estimation. Throughput for the next user state is predicted using posterior probability with normalized weights. The main aim of this paper is to observe the user throughput. The problem is modeled as throughput maximization subject to channel state. The optimization problem is formulated as a non convex optimization and solved using branch and bound method. It has been shown in the simulation results that the average throughput is increased by ${\text{5.94}\%}{\text{--52.98}\%}$ compared to sequential importance sampling with resampling (SISR) method. Therefore, the RPF scheme has proved to be efficient by its remarkable increase in user throughput compared to the SISR particle filter algorithm.

This work considers the problem of output-only identification for time-varying structures in a recursive manner under non-Gaussian impulsive noise. Presently, the most existing identification methods are based on classical least-squares (LS) criterion due to its mathematical tractability, computational simplicity and optimality under Gaussian assumption. However, the performance of the LS based methods deteriorates seriously in non-Gaussian situations, especially when responses are corrupted by impulsive noise (outliers) which are frequently encountered in real test cases. To deal with this problem, a maximum correntropy criterion based recursive modal identification method is proposed in this paper, and a class of heavy-tailed alpha-stable distribution is adopted to model non-Gaussian impulsive noise. The effect of the kernel bandwidth parameter in proposed method is discussed and a rational value is given for the model structure selection. Finally, the proposed method is comparatively assessed against its LS counterpart via a numerical example and a laboratory time-varying structure experiment. The comparisons have illustrated the advantages of the proposed method on the modal frequency estimates in terms of estimation accuracy and robustness under the non-Gaussian impulsive noise.

In this work we introduce a novel algorithm for transient rendering in participating media. Our method is consistent, robust, and is able to generate animations of time-resolved light transport featuring complex caustic light paths in media. We base our method on the observation that the spatial continuity provides an increased coverage of the temporal domain, and generalize photon beams to transient-state. We extend the beam steady-state radiance estimates to include the temporal domain. Then, we develop a progressive version of spatio-temporal density estimations, that converges to the correct solution with finite memory requirements by iteratively averaging several realizations of independent renders with a progressively reduced kernel bandwidth. We derive the optimal convergence rates accounting for space and time kernels, and demonstrate our method against previous consistent transient rendering methods for participating media.

As an important service for 5G communications, ultra-reliable low-latency communications (URLLC) support emerging mission-critical applications, such as factory automation and autonomous driving. For such applications, the probability of failing to successfully transmit URLLC packets should be below a certain threshold. However, in the case of limited knowledge of the channel distribution, achieving such a reliability target requires precise channel modeling. In this paper, we study applying a non-parametric statistical learning approach (i.e. kernel density estimation (KDE)) to estimate the information of the wireless transmission environment (i.e. the probability density function of the channel distribution). Based on the estimated cumulative distribution function, a transmission rate control technique has been developed and the corresponding reliability has been investigated using two measures representing the average performance and the confidence level. Moreover, this paper compares the performance of KDE and traditional empirical estimation scheme. The results show that KDE achieves a high level of confidence in guaranteeing the reliability constraint despite of the limited number of training data when choosing a suitable kernel bandwidth.

It is a common practice to evaluate probability density function or matter spatial density function from statistical samples. Kernel density estimation is a frequently used method, but to select an optimal bandwidth of kernel estimation, which is completely based on data samples, is a long-term issue that has not been well settled so far. There exist analytic formulae of optimal kernel bandwidth, but they cannot be applied directly to data samples, since they depend on the unknown underlying density functions from which the samples are drawn. In this work, we devise an approach to pick out the totally data-based optimal bandwidth. First, we derive correction formulae for the analytic formulae of optimal bandwidth to compute the roughness of the sample's density function. Then substitute the correction formulae into the analytic formulae for optimal bandwidth, and through iteration, we obtain the sample's optimal bandwidth. Compared with analytic formulae, our approach gives very good results, with relative differences from the analytic formulae being only 2%-3% for a sample size larger than 10^4. This approach can also be generalized easily to cases of variable kernel estimations., 7 pages, 8 figures

In this paper, we present a novel proxy-based method of the adaptive haptic rendering of a variable density 3D point cloud data at different levels of detail without pre-computing the mesh structure. We also incorporate features like rotation, translation, and friction to provide a better realistic experience to the user. A proxy-based rendering technique is used to avoid the pop-through problem while rendering thin parts of the object. Instead of a point proxy, a spherical proxy of a variable radius is used, which avoids the sinking of proxy during the haptic interaction of sparse data. The radius of the proxy is adaptively varied depending upon the local density of the point data using kernel bandwidth estimation. During the interaction, the proxy moves in small steps tangentially over the point cloud such that the new position always minimizes the distance between the proxy and the haptic interaction point (HIP). The raw point cloud data re-sampled in a regular 3D lattice of voxels are loaded to the haptic space after proper smoothing to avoid aliasing effects. The rendering technique is validated with several subjects, and it is observed that this functionality supplements the user's experience by allowing the user to interact with an object at multiple resolutions., Accepted to World Haptics Conference (WHC), 2013

Novelty and outlier detection are both used for anomaly detection. This paper works through the method of One-Class support vector machine (SVM) which could estimate the contour of initial observations and can be applied in the problems of anomaly detection. In this paper, the experiments on both artificial and real-world data sets are performed to demonstrate the importance of kernel and kernel parameter choice and the corresponding sensitivity of the algorithm. Due to the desire for general regulations of kernel parameter selection, some general methods of selecting the kernel bandwidth parameter of RBF kernel are therefore investigated, including median heuristic method and Bayesian kernel learning method. Then, the experiments based on these methods are conducted to observe their effect. Consequently, some new discoveries together with current issues on these methods, as well as some applicable situations of these methods, are found in this paper. Furthermore, this paper also confirms the effectiveness of RBF kernel and that the initial value of its bandwidth parameter can be set by those two methods.

Unmanned aerial vehicles using light detection and ranging (UAV LiDAR) with high spatial resolution have shown great potential in forest applications because they can capture vertical structures of forests. Individual tree segmentation is the foundation of many forest research works and applications. The tradition fixed bandwidth mean shift has been applied to individual tree segmentation and proved to be robust in tree segmentation. However, the fixed bandwidth-based segmentation methods are not suitable for various crown sizes, resulting in omission or commission errors. Therefore, to increase tree-segmentation accuracy, we propose a self-adaptive bandwidth estimation method to estimate the optimal kernel bandwidth automatically without any prior knowledge of crown size. First, from the global maximum point, we divide the three-dimensional (3D) space into a set of angular sectors, for each of which a canopy surface is simulated and the potential tree crown boundaries are identified to estimate average crown width as the kernel bandwidth. Afterwards, we use a mean shift with the automatically estimated kernel bandwidth to extract individual tree points. The method is iteratively implemented within a given area until all trees are segmented. The proposed method was tested on the 7 plots acquired by a Velodyne 16E LiDAR system, including 3 simple plots and 4 complex plots, and 95% and 80% of trees were correctly segmented, respectively. Comparative experiments show that our method contributes to the improvement of both segmentation accuracy and computational efficiency.

The reconstruction of smooth density fields from scattered data points is a procedure that has multiple applications in a variety of disciplines, including Lagrangian (particle-based) models of solute transport in fluids. In random walk particle tracking (RWPT) simulations, particle density is directly linked to solute concentrations, which is normally the main variable of interest, not just for visualization and post-processing of the results, but also for the computation of non-linear processes, such as chemical reactions. Previous works have shown the improved nature of kernel density estimation (KDE) over other methods such as binning, in terms of its ability to accurately estimate the “true” particle density relying on a limited amount of information. Here, we develop a grid-projected KDE methodology to determine particle densities by applying kernel smoothing on a pilot binning; this may be seen as a “hybrid” approach between binning and KDE. The kernel bandwidth is optimized locally. Through simple implementation examples, we elucidate several appealing aspects of the proposed approach, including its computational efficiency and the possibility to account for typical boundary conditions, which would otherwise be cumbersome in conventional KDE. © 2019 Elsevier Ltd

This paper explores the impact of different distance metrics on collinearity in local regression models such as geographically weighted regression. Using a case study of house price data collected in Hà Nội, Vietnam, and by fully varying both power and rotation parameters to create different Minkowski distances, the analysis shows that local collinearity can be both negatively and positively affected by distance metric choice. The Minkowski distance that maximised collinearity in a geographically weighted regression was approximate to a Manhattan distance with (power = 0.70) with a rotation of 30°, and that which minimised collinearity was parameterised with power = 0.05 and a rotation of 70°. The results indicate that distance metric choice can provide a useful extra tuning component to address local collinearity issues in spatially varying coefficient modelling and that understanding the interaction of distance metric and collinearity can provide insight into the nature and structure of the data relationships. The discussion considers first, the exploration and selection of different distance metrics to minimise collinearity as an alternative to localised ridge regression, lasso and elastic net approaches. Second, it discusses the how distance metric choice could extend the methods that additionally optimise local model fit (lasso and elastic net) by selecting a distance metric that further helped minimise local collinearity. Third, it identifies the need to investigate the relationship between kernel bandwidth, distance metrics and collinearity as an area of further work.

As a relatively recent development, spatial smoothing methods have been introduced to identify seismic patterns. Among the methods developed to model the spatial variation, methods based on utilization of 3-D Gaussian isotropic kernels have a wide reception. The most important question remaining to be answered in the application of these methods is the determination of the optimum kernel bandwidth. At the present, researchers’ efforts to clarify the subject have still not yet finalized, this study aims to provide insightful knowledge for future efforts. In this study, for the region bounded by 27°–33° longitudes and 39°–41° latitudes, where the western section of the famous Northern Anatolian fault lies, smoothing techniques are implemented to determine the optimum smoothing kernel bandwidth for point sources. The influence of the modeling of seismicity through the computation of the optimum smoothing kernel bandwidth is examined. In addition, the sensitivity of each smoothing technique to the seismic patterns, whether densely clustered or scarcely populated, is investigated. In the end, the smoothing method based on optimum neighbor number is identified as highly sensitive to the density of seismicity and seismic clusters, and better in modeling high seismicity compared to the model based on single optimum smoothing distance used for the entire region of interest.

In the present investigation two smart prediction tools, namely the general regression neural network (GRNN) and multiple regression analysis (MRA) models were developed to predict and compare some of the key machinability aspects like average kerf width, average surface roughness and material removal rate in the wire electrical discharge machining process of titanium grade 6. Pulse-on time, pulse-off time, wire feed and wire tension were considered as machining variables to develop the predictive model. In order to curtail cross-validation error in GRNN, optimized kernel bandwidth was utilized using the grid search method. The neural network and regression models were trained, validated and tested with measured data. A mathematical model was developed using multiple regression analysis. The ANOVA test was also conducted to determine the significant parameters affecting the responses. The results indicated that the predicted responses lie within ± 5% and ± 10% error for GRNN and MRA, respectively, which suggests that the GRNN model is more reliable and adequate than the regression model. A comparative study with previous research work was also done to confirm the novelty along with application potential of the proposed model.

Recently, as location-based social networks (LBSNs) rapidly grow, general users utilize point-of-interest recommender systems to discover attractive locations. Most existing POI recommendation algorithms always employ the check-in data and rich contextual information (e.g., geographical information and users social network information) of users to learn their preference on POIs. Unfortunately, these studies generally suffer from two major limitations: (1) when modeling geographical influence, users personalized behavior differences are ignored; (2) when modeling the users social influence, the implicit social influence is seldom exploited. In this paper, we propose a novel POI recommendation approach called GeoEISo. GeoEISo achieves three key goals in this work. (1) We develop a kernel estimation method with a self-adaptive kernel bandwidth to model the geographical influence between POIs. (2) We use the Gaussian radial basis kernel function based support vector regression (SVR) model to predict explicit trust values between users, and then devise a novel trust-based recommendation model to simultaneously incorporate both the explicit and implicit social trust information into the process of POI recommendation. (3) We develop a unified geo-social framework which combines users preference on a POI with the geographical influence as well as social correlations. Experimental results on two real-world datasets collected from Foursquare show that GeoEISo provides significantly superior performances compared to other state-of-the-art POI recommendation models.

In this paper, we employ a new multiresolution upscaling method for the data in porous media. The method employs an adaptive bandwidth of the kernel function as the multiresolution upscaling operator. We have implemented a hierarchical multiresolution upscaling strategy, which is based on maintaining the important fine scale model structures of the reservoir. In this method, bandwidth is a function of the variability of cell, which controls the upscaling process. In areas with a high variability of the considered feature, small bandwidth leads to maintaining the fine scale structures; and in homogeneous reservoir areas, large bandwidths will lead to coarse-sized cells. In this paper, we have used this proposed method for simultaneous upscaling of two features: density and neutron logs. Determination of optimal bandwidth or threshold for each feature is an important challenge in the bandwidth method. Due to different variabilities of two properties or features, simultaneous upscaling should be performed to capture the variability of both features in the model. The concept of simultaneous upscaling provides a common coarse model for two properties. To prepare the SUTF model, two minimum and maximum bandwidth approaches are used. The upscaling algorithm has been applied for density and neutron logs. Although the fine scale model contains 2290 cells, the number of coarsened cells in SUTF model based on the minimum bandwidth and maximum bandwidth approaches are 402 and 110, respectively. In addition to this new proposed method, the method of wavelet transform has also been used as a well-known method for upscaling, and finally, the results of the two methods are compared. By comparing the results of the two methods, we observe that the upscaling error in the kernel bandwidth method is less than that in the wavelet transform method. The advantage of the kernel bandwidth method over the wavelet transform is the reduction of the loss of the main information as well as the reduction of the numerical dispersion in the new proposed method compared to the wavelet transform method. These benefits have appeared at the upscaling error that is less in the proposed method.