Similarity analysis of DNA sequences based on the EMD method

DNA sequences can be translated into 2D graphs and into numerical sequences; we call the numerical sequences nonlinear signal sequences. We can use the empirical mode decomposition (EMD) method to divide nonlinear signal sequences into a group of well-behaved intrinsic mode functions (IMFs) and a residue, so that we can compare the similarities among DNA sequences conveniently and intuitively. This work tests the method’s suitability by using the mitochondria of four different species.     

Introducing <Emphasis FontCategory="NonProportional" Type="Bold">libeemd</Emphasis>: a program package for performing the ensemble empirical mode decomposition

The ensemble empirical mode decomposition (EEMD) and its complete variant (CEEMDAN) are adaptive, noise-assisted data analysis methods that improve on the ordinary empirical mode decomposition (EMD). All these methods decompose possibly nonlinear and/or nonstationary time series data into a finite amount of components separated by instantaneous frequencies. This decomposition provides a powerful method to look into the different processes behind a given time series data, and provides a way to separate short time-scale events from a general trend. We present a free software implementation of EMD, EEMD and CEEMDAN and give an overview of the EMD methodology and the algorithms used in the decomposition. We release our implementation, libeemd, with the aim of providing a user-friendly, fast, stable, well-documented and easily extensible EEMD library for anyone interested in using (E)EMD in the analysis of time series data. While written in C for numerical efficiency, our implementation includes interfaces to the Python and R languages, and interfaces to other languages are straightforward.     

A PDE based and interpolation-free framework for modeling the sifting process in a continuous domain

The empirical mode decomposition (EMD) is a powerful tool in signal processing. Despite its algorithmic origin making its theoretical analysis and formulation very difficult, a few recent works has contributed to its theoretical framework. Herein, the former local mean is formulated in a more convenient way by introducing operators to calculate local upper and lower envelopes. This enables the use of differential calculus and other classical calculations on the new local mean. Based on its more accurate formulation, a partial differential equation (PDE) consistency result is provided to approximate the sifting process iterations, without any envelope interpolation. In addition, a new stopping criterion based on the introduced local mean is proposed. This new criterion is a local measure and resolves the null integral conservative property of the previous derived PDE, which made any signal having a null integral be a PDE-based mode. Moreover, the δ inner model parameter is now linked to the signal intrinsic properties, providing to the latter a physical meaning and making the proposed model keep the auto-adaptive property of the EMD. New decomposition modes are now analytically and fully characterized, and also interpolation free. Finally, properties of the interpolation free PDE model are presented. Results obtained with our proposed approach by explicit computations thanks to the eigendecomposition of the Laplacian operator, and also by numerical resolution of the derived PDE, show noticeable improvements for both stationary and non stationary signals, in comparison to the former EMD algorithm.     

A meshless method for nonhomogeneous polyharmonic problems using method of fundamental solution coupled with quasi-interpolation technique

This paper proposes a meshless method based on coupling the method of fundamental solutions (MFS) with quasi-interpolation for the solution of nonhomogeneous polyharmonic problems. The original problems are transformed to homogeneous problems by subtracting a particular solution of the governing differential equation. The particular solution is approximated by quasi-interpolation and the corresponding homogeneous problem is solved using the MFS. By applying quasi-interpolation, problems connected with interpolation can be avoided. The error analysis and convergence study of this meshless method are given for solving the boundary value problems of nonhomogeneous harmonic and biharmonic equations. Numerical examples are also presented to show the efficiency of the method.     

Quasi-interpolations with interpolation property

Interpolation and quasi-interpolation are very important methods for function approximation. But both of them have their respective disadvantages. The interpolation function can fit the given sample points, but some oscillation may arise as in the case of the Lagrange interpolation. The quasi-interpolation function, for example, the Bernstein quasi-interpolation function, satisfies the convergence condition, but does not keep fitting the given sample points. In this paper, we present a method to construct a quasi-interpolation operator with certain interpolation property.     

A shape-preserving quasi-interpolation operator satisfying quadratic polynomial reproduction property to scattered data

In this paper, we construct a univariate quasi-interpolation operator to non-uniformly distributed data by cubic multiquadric functions. This operator is practical, as it does not require derivatives of the being approximated function at endpoints. Furthermore, it possesses univariate quadratic polynomial reproduction property, strict convexity-preserving and shape-preserving of order 3 properties, and a higher convergence rate. Finally, some numerical experiments are shown to compare the approximation capacity of our quasi-interpolation operator with that of Wu and Schaback’s quasi-interpolation scheme.     

Approximate approximations from scattered data

The aim of this paper is to extend the approximate quasi-interpolation on a uniform grid by dilated shifts of a smooth and rapidly decaying function to scattered data quasi-interpolation. It is shown that high order approximation of smooth functions up to some prescribed accuracy is possible, if the basis functions, which are centered at the scattered nodes, are multiplied by suitable polynomials such that their sum is an approximate partition of unity. For Gaussian functions we propose a method to construct the approximate partition of unity and describe an application of the new quasi-interpolation approach to the cubature of multi-dimensional integral operators.     

基于MQ拟插值函数逼近的非线性动力系统数值求解

借助多重二次曲面（multi quadrics,MQ）拟插值函数具有较好精确性和稳定性的优势,研究了基于MQ拟插值函数和4阶Runge-Kutta法相结合的方法,构造了求解带有初值问题的非线性动力系统的数值解法,分析了该方法与已有主要方法的优缺点,并给出了相应的数值算例、误差估计.结果表明该方法计算量小、能很好地逼近非线性动力系统的解析解.     

On the various kinds of synchronization in delayed Duffing-Van der Pol system

A detailed investigation is performed about the various zones of stability for delayed Duffing-Van der Pol system, which in term shows the specific role of delay in the formation of the attractor. Coupling of two such similar systems gives rise to rich dynamics if the coupling also belongs to a delayed class. This gives rise to varieties of synchronization channels – such as anticipatory synchronization and retarded synchronization. As the coefficient of delayed term changes value, it is observed that anticipatory synchronization makes way to phase synchronization. The onset of these various mechanisms are tested by the computation of similarity function and probability of recurrence. In the study of phase synchronization the machinery of empirical mode decomposition (EMD) analysis is adapted and lastly maximal Lyapunov exponent is computed as a verifying criterion.     

Generalized Strang–Fix condition for scattered data quasi-interpolation

Quasi-interpolation is very useful in the study of the approximation theory and its applications, since the method can yield solutions directly and does not require solving any linear system of equations. However, quasi-interpolation is usually discussed only for gridded data in the literature. In this paper we shall introduce a generalized Strang–Fix condition, which is related to nonstationary quasi-interpolation. Based on the discussion of the generalized Strang–Fix condition we shall generalize our quasi-interpolation scheme for multivariate scattered data, too. AMS subject classification 41A63, 41A25, 65D10Zong Min Wu: Supported by NSFC No. 19971017 and NOYG No. 10125102.     

A multiquadric quasi-interpolation with linear reproducing and preserving monotonicity

In this paper, we develop a multiquadric (MQ) quasi-interpolation which has the properties of linear reproducing and preserving monotonicity. Moreover, we give its approximation error by theoretic analysis and illustrate the effect by means of two examples. One of the examples is to approach the linear combination of two sine functions with different frequencies. Another is to approximate a function with discontinuity. From the results of the examples, we believe that the present MQ quasi-interpolation is feasible.     

Residual error estimate for BEM-based FEM on polygonal meshes

A conforming finite element method on polygonal meshes is reviewed which handles hanging nodes naturally. Trial functions are defined to fulfil the homogeneous PDE locally and they are treated by means of local boundary integral equations. Using a quasi-interpolation operator of Clément type a residual-based error estimate is obtained. This a posteriori estimator can be used to rate the accuracy of the approximation over polygonal elements or it can be applied to an adaptive BEM-based FEM. The numerical experiments confirm our results and show optimal convergence for the adaptive strategy on general meshes.     

Short-term prediction of wind power using EMD and chaotic theory

Due to the strong non-linear, complexity and non-stationary characteristics of wind farm power, a hybrid prediction model with empirical mode decomposition (EMD), chaotic theory, and grey theory is constructed. The EMD is used to decompose the wind farm power into several intrinsic mode function (IMF) components and one residual component. The grey forecasting model is used to predict the residual component. For the IMF components, identify their characteristics, if it is chaotic time series use largest Lyapunov exponent prediction method to predict. If not, use grey forecasting model to predict. Prediction results of residual component and all IMF components are aggregated to produce the ultimate predicted result for wind farm power. The ultimate predicted result shows that the proposed method has good prediction accuracy, can be used for short-term prediction of wind farm power.     

Quasi-interpolation for linear functional data

Quasi-interpolation has been studied extensively in the literature. However, most studies of quasi-interpolation are usually only for discrete function values (or a finite linear combination of discrete function values). Note that in practical applications, more commonly, we can sample the linear functional data (the discrete values of the right-hand side of some differential equations) rather than the discrete function values (e.g., remote sensing, seismic data, etc). Therefore, it is more meaningful to study quasi-interpolation for the linear functional data. The main result of this paper is to propose such a quasi-interpolation scheme. Error estimate of the scheme is also given in the paper. Based on the error estimate, one can find a quasi-interpolant that provides an optimal approximation order with respect to the smoothness of the right-hand side of the differential equation. The scheme can be applied in many situations such as the numerical solution of the differential equation, construction of the Lyapunov function and so on. Respective examples are presented in the end of this paper.     

Bivariate hierarchical Hermite spline quasi-interpolation

Spline quasi-interpolation (QI) is a general and powerful approach for the construction of low cost and accurate approximations of a given function. In order to provide an efficient adaptive approximation scheme in the bivariate setting, we consider quasi-interpolation in hierarchical spline spaces. In particular, we study and experiment the features of the hierarchical extension of the tensor-product formulation of the Hermite BS quasi-interpolation scheme. The convergence properties of this hierarchical operator, suitably defined in terms of truncated hierarchical B-spline bases, are analyzed. A selection of numerical examples is presented to compare the performances of the hierarchical and tensor-product versions of the scheme.     

A kind of multiquadric quasi-interpolation operator satisfying any degree polynomial reproduction property to scattered data

In this paper, by virtue of using the linear combinations of the shifts of f(x) to approximate the derivatives of f(x) and Waldron’s superposition idea (2009), we modify a multiquadric quasi-interpolation with the property of linear reproducing to scattered data on one-dimensional space, such that a kind of quasi-interpolation operator L_r+1f has the property of r+1(r∈Z,r≥0) degree polynomial reproducing and converges up to a rate of r+2. There is no demand for the derivatives of f in the proposed quasi-interpolation L_r+1f, so it does not increase the orders of smoothness of f. Finally, some numerical experiments are shown to compare the approximation capacity of our quasi-interpolation operators with that of Wu-Schaback’s quasi-interpolation scheme and Feng-Li’s quasi-interpolation scheme.     

Local Spectral Envelope: An Approach Using Dyadic Tree-Based Adaptive Segmentation

The concept of the spectral envelope was introduced as a statistical basis for the frequency domain analysis and scaling of qualitative-valued time series. A major focus of this research was the analysis of DNA sequences. A common problem in analyzing long DNA sequence data is to identify coding sequences that are dispersed throughout the DNA and separated by regions of noncoding. Even within short subsequences of DNA, one encounters local behavior. To address this problem of local behavior in categorical-valued time series, we explore using the spectral envelope in conjunction with a dyadic tree-based adaptive segmentation method for analyzing piecewise stationary processes.     

Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool

The EMD algorithm is a technique that aims to decompose into their building blocks functions that are the superposition of a (reasonably) small number of components, well separated in the time–frequency plane, each of which can be viewed as approximately harmonic locally, with slowly varying amplitudes and frequencies. The EMD has already shown its usefulness in a wide range of applications including meteorology, structural stability analysis, medical studies. On the other hand, the EMD algorithm contains heuristic and ad hoc elements that make it hard to analyze mathematically.In this paper we describe a method that captures the flavor and philosophy of the EMD approach, albeit using a different approach in constructing the components. The proposed method is a combination of wavelet analysis and reallocation method. We introduce a precise mathematical definition for a class of functions that can be viewed as a superposition of a reasonably small number of approximately harmonic components, and we prove that our method does indeed succeed in decomposing arbitrary functions in this class. We provide several examples, for simulated as well as real data.     

Generator,multiquadric generator,quasi-interpolation and multiquadric quasi-interpolation

The aim of this survey paper is to propose a new concept “generator”. In fact, generator is a single function that can generate the basis as well as the whole function space. It is a more fundamental concept than basis. Various properties of generator are also discussed. Moreover, a special generator named multiquadric function is introduced. Based on the multiquadric generator, the multiquadric quasi-interpolation scheme is constructed, and furthermore, the properties of this kind of quasi-interpolation are discussed to show its better capacity and stability in approximating the high order derivatives.