Superconvergent Nyström and degenerate kernel methods for eigenvalue problems

The Nyström and degenerate kernel methods, based on projections at Gauss points onto the space of (discontinuous) piecewise polynomials of degree ?r-1, for the approximate solution of eigenvalue problems for an integral operator with a smooth kernel, exhibit order 2r. We propose new superconvergent Nyström and degenerate kernel methods that improve this convergence order to 4r for eigenvalue approximation and to 3r for spectral subspace approximation in the case where the kernel is sufficiently smooth. Moreover for a simple eigenvalue, we show that by using an iteration technique, an eigenvector approximation of order 4r can be obtained. The methods introduced here are similar to that studied by Kulkarni in [10] and exhibit the same convergence orders, so a comparison with these methods is worked out in detail. Also, the error terms are analyzed and the obtained methods are numerically tested. Finally, these methods are extended to the case of discontinuous kernel along the diagonal and superconvergence results are also obtained.     

Learning management knowledge for manufacturing systems in the early stages using time series data

Acquiring knowledge in manufacturing systems in the early stages always has a challenging task due to the lack of sufficient data. This makes it hard for the derived management model to reach a reliable and stable level. Li and Lin (2006) developed a useful method that can deal with the problem of knowledge acquisition based on a small data set. However, this method assumes all data are collected at the same time, since they treat the data set as a source (from one population) of a priori knowledge for learning. In fact, instead of being a random data set, these collected data can be time dependent, that is, they tend to be a sequence of observations, occurring at different times. The consideration of this dependence property in the data will benefit the knowledge acquisition in the early stages by expanding the learning model from an independent model to a dependent model. This research expanded the intervalized kernel density estimator (IKDE) presented in Li and Lin (2006) to a more general form to improve the learning model in the early stages. The general model, called GIKDE, joints the concepts of time series and stochastic processes in order to deal with both independent and dependent data sets. The Virtual Sample Generation process based on GIKDE was also developed to produce extra information for expediting the learning. Results obtained from the application of the model to a control charts data, using a back-propagation neural network as the learning tool, show that this unique approach is an effective method of knowledge acquisition for a manufacturing system in the early stages.     

Approximating and learning by Lipschitz kernel on the sphere

This paper investigates some approximation properties and learning rates of Lipschitz kernel on the sphere. A perfect convergence rate on the shifts of Lipschitz kernel on the sphere, which is faster than O（n-1/2）, is obtained, where n is the number of parameters needed in the approximation. By means of the approximation, a learning rate of regularized least square algorithm with the Lipschitz kernel on the sphere is also deduced.     

Steady state bifurcations and exact multiplicity conditions via Carleman linearization

The problem of steady-state bifurcations of vector fields under parameter perturbation is resolved by a linear algebraic method. Exact multiplicity conditions for any steady state are obtained in terms of the system parameters. No reduction of the steady-state system to one equation is required. Instead the one-dimensional case is included as a subspace in this generalized framework. The key point that this paper highlights is that the order of the steady multiplicity at bifurcation can be determined by examining the dimension of the kernel of the successive Carleman linear operators for all cases of practical interest. In particular, the dimension of the kernel of any Carleman linear operator of order l, equals l if l is less than the multiplicity, μ. However, the μth order Carleman operator retains a (μ − 1)-dimensional kernel.     

Time scale symplectic systems without normality

We present a theory of the definiteness (nonnegativity and positivity) of a quadratic functional F over a bounded time scale. The results are given in terms of a time scale symplectic system (S), which is a time scale linear system that generalizes and unifies the linear Hamiltonian differential system and discrete symplectic system. The novelty of this paper resides in removing the assumption of normality in the characterization of the positivity of F, and in establishing equivalent conditions for the nonnegativity of F without any normality assumption. To reach this goal, a new notion of generalized focal points for conjoined bases (X,U) of (S) is introduced, results on the piecewise-constant kernel of X(t) are obtained, and various Picone-type identities are derived under the piecewise-constant kernel condition. The results of this paper generalize and unify recent ones in each of the discrete and continuous time setting, and constitute a keystone for further development in this theory.     

Financial market forecasting using a two-step kernel learning method for the support vector regression

In this paper, we propose a two-step kernel learning method based on the support vector regression (SVR) for financial time series forecasting. Given a number of candidate kernels, our method learns a sparse linear combination of these kernels so that the resulting kernel can be used to predict well on future data. The L ₁-norm regularization approach is used to achieve kernel learning. Since the regularization parameter must be carefully selected, to facilitate parameter tuning, we develop an efficient solution path algorithm that solves the optimal solutions for all possible values of the regularization parameter. Our kernel learning method has been applied to forecast the S&P500 and the NASDAQ market indices and showed promising results.     

The optimal solution of multi-kernel regularization learning

In regularized kernel methods, the solution of a learning problem is found by minimizing a functional consisting of a empirical risk and a regularization term. In this paper, we study the existence of optimal solution of multi-kernel regularization learning. First, we ameliorate a previous conclusion about this problem given by Micchelli and Pontil, and prove that the optimal solution exists whenever the kernel set is a compact set. Second, we consider this problem for Gaussian kernels with variance σ∈(0,∞), and give some conditions under which the optimal solution exists.     

Well-posedness of Smoluchowski's coagulation equation for a class of homogeneous kernels

The uniqueness and existence of measure-valued solutions to Smoluchowski's coagulation equation are considered for a class of homogeneous kernels. Denoting by λ∈(-∞,2]?{0} the degree of homogeneity of the coagulation kernel a, measure-valued solutions are shown to be unique under the sole assumption that the moment of order λ of the initial datum is finite. A similar result was already available for the kernels a(x,y)=2, x+y and xy, and is extended here to a much wider class of kernels by a different approach. The uniqueness result presented herein also seems to improve previous results for several explicit kernels. Furthermore, a comparison principle and a contraction property are obtained for the constant kernel.     

A new approach to assess product lifetime performance for small data sets

Because of cost and time limit factors, the number of samples is usually small in the early stages of manufacturing systems, and the scarcity of actual data will cause problems in decision-making. In order to solve this problem, this paper constructs a counter-intuitive hypothesis testing method by choosing the maximal p-value based on a two-parameter Weibull distribution to enhance the estimate of a nonlinear and asymmetrical shape of product lifetime distribution. Further, we systematically generate virtual data to extend the small data set to improve learning robustness of product lifetime performance. This study provides simulated data sets and two practical examples to demonstrate that the proposed method is a more appropriate technique to increase estimation accuracy of product lifetime for normal or non-normal data with small sample sizes.     

A global in time existence and uniqueness result for an integrodifferential hyperbolic inverse problem with memory effect

We consider an inverse problem for a one-dimensional integrodifferential hyperbolic system, which comes from a simplified model of thermoelasticity. This inverse problem aims to identify the displacement u, the temperature η and the memory kernel k simultaneously from the weighted measurement data of temperature. By using the fixed point theorem in suitable Sobolev spaces, the global in time existence and uniqueness results of this inverse problem are obtained. Moreover, we prove that the solution to this inverse problem depends continuously on the noisy data in suitable Sobolev spaces. For this nonlinear inverse problem, our theoretical results guarantee the solvability for the proposed physical model and the well-posedness for small measurement time τ, which is quite different from general inverse problems.     

Mise of kernel estimates of a density and its derivatives

For an integer p ⩾ 0, Singh has considered a class of kernel estimators ƒ∼^(p) of the pth order derivative ƒ^(p) of a density ƒ and showed how specializations of some of the results there improve the corresponding existing results. In this paper these improved estimators are examined on a global measure of quality of an estimator, namely, the mean integrated square error (MISE) behavior. An upper bound, which can not be tightened any further for a wide class of kernels, is obtained for MISE (ƒ∼^(p)). The exact asymptotic value for the same is also obtained. Under two alternative conditions, weaker than those assumed for the two results mentioned above, convergence of MISE (ƒ∼^(p)) to zero is proved. Specializations of some of the results here improve the corresponding existing results by weakening the conditions, sharpening the rates of convergence or both.     

Learning mixtures of truncated basis functions from data

In this paper we investigate methods for learning hybrid Bayesian networks from data. First we utilize a kernel density estimate of the data in order to translate the data into a mixture of truncated basis functions (MoTBF) representation using a convex optimization technique. When utilizing a kernel density representation of the data, the estimation method relies on the specification of a kernel bandwidth. We show that in most cases the method is robust wrt. the choice of bandwidth, but for certain data sets the bandwidth has a strong impact on the result. Based on this observation, we propose an alternative learning method that relies on the cumulative distribution function of the data.Empirical results demonstrate the usefulness of the approaches: Even though the methods produce estimators that are slightly poorer than the state of the art (in terms of log-likelihood), they are significantly faster, and therefore indicate that the MoTBF framework can be used for inference and learning in reasonably sized domains. Furthermore, we show how a particular sub-class of MoTBF potentials (learnable by the proposed methods) can be exploited to significantly reduce complexity during inference.     

Spatially nonhomogeneous equilibrium in a reaction-diffusion system with distributed delay

We consider a reaction-diffusion system with general time-delayed growth rate and kernel functions. The existence and stability of the positive spatially nonhomogeneous steady-state solution are obtained. Moreover, taking minimal time delay τ as the bifurcation parameter, Hopf bifurcation near the steady-state solution is proved to occur at a critical value τ=τ₀. Especially, the Hopf bifurcation is forward and the bifurcated periodic solutions are stable on the center manifold. The general results are applied to competitive and cooperative systems with weak or strong kernel function respectively.     

The learning rate of l 2-coefficient regularized classification with strong loss

In the present paper, we give an investigation on the learning rate of l2-coefficient regularized classification with strong loss and the data dependent kernel functional spaces. The results show that the learning rate is influenced by the strong convexity.     

Finite rank kernels for multi-task learning

Motivated by the importance of kernel-based methods for multi-task learning, we provide here a complete characterization of multi-task finite rank kernels in terms of the positivity of what we call its associated characteristic operator. Consequently, we are led to establishing that every continuous multi-task kernel, defined on a cube in an Euclidean space, not only can be uniformly approximated by multi-task polynomial kernels, but also can be extended as a multi-task kernel to all of the Euclidean space. Finally, we discuss the interpolation of multi-task kernels by multi-task finite rank kernels.     

Une generalisation du theoreme de Richardson sur l'existence de noyaux dans les graphes orientes

Richardson's theorem [4] asserts that if a digraph has no odd circuit, it possesses a kernel. We improve this theorem in the following manner: if each odd circuit has two short crossing chords, i.e. two chords of the type (X_i,X_i+2), (X_i+1,X_i+3), it possesses a kernel.     

On systems of linear functional equations of the second kind in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>

We consider a general system of functional equations of the second kind in L ₂ with a continuous linear operator T satisfying the condition that zero lies in the limit spectrum of the adjoint operator T*. We show that this condition holds for the operators of a wide class containing, in particular, all integral operators. The system under study is reduced by means of a unitary transformation to an equivalent system of linear integral equations of the second kind in L ₂ with Carleman matrix kernel of a special kind. By a linear continuous invertible change, this system is reduced to an equivalent integral equation of the second kind in L ₂ with quasidegenerate Carleman kernel. It is possible to apply various approximate methods of solution for such an equation.     

Solving the generalized regularized long wave equation on the basis of a reproducing kernel space

On the basis of a reproducing kernel space, an iterative algorithm for solving the generalized regularized long wave equation is presented. The analytical solution in the reproducing kernel space is shown in a series form and the approximate solution u_n is constructed by truncating the series to n terms. The convergence of u_n to the analytical solution is also proved. Results obtained by the proposed method imply that it can be considered as a simple and accurate method for solving such evolution equations.     

A self-adaptive global best harmony search algorithm for continuous optimization problems

This paper presents a self-adaptive global best harmony search (SGHS) algorithm for solving continuous optimization problems. In the proposed SGHS algorithm, a new improvisation scheme is developed so that the good information captured in the current global best solution can be well utilized to generate new harmonies. The harmony memory consideration rate (HMCR) and pitch adjustment rate (PAR) are dynamically adapted by the learning mechanisms proposed. The distance bandwidth (BW) is dynamically adjusted to favor exploration in the early stages and exploitation during the final stages of the search process. Extensive computational simulations and comparisons are carried out by employing a set of 16 benchmark problems from literature. The computational results show that the proposed SGHS algorithm is more effective in finding better solutions than the state-of-the-art harmony search (HS) variants.     

Speeding up chaos and limit cycles in evolutionary language and learning processes

Evolution of human language and learning processes have their foundation built on grammar that sets rules for construction of sentences and words. These forms of replicator–mutator (game dynamical with learning) dynamics remain however complex and sometime unpredictable because they involve children with some predispositions. In this paper, a system modeling evolutionary language and learning dynamics is investigated using the Crank–Nicholson numerical method together with the new differentiation with non‐singular kernel. Stability and convergence are comprehensively proven for the system. In order to seize the effects of the non‐singular kernel, an application to game dynamical with learning dynamics for a population with five languages is given together with numerical simulations. It happens that such dynamics, as functions of the learning accuracy μ, can exhibit unusual bifurcations and limit cycles followed by chaotic behaviors. This points out the existence of fickle and unpredictable variations of languages as time goes on, certainly due to the presence of learning errors. More interestingly, this chaos is shown to be dependent on the order of the non‐singular kernel derivative and speeds up as this derivative order decreases. Hence, can people use that order to control chaotic behaviors observed in game dynamical systems with learning! Copyright © 2016 John Wiley & Sons, Ltd.