arXiv subjects

Hong-Yan Zhang

Publications and source records attributed to Hong-Yan Zhang.

12 recordsLinked to original sources

arXiv2025

A Correction for the Paper "Symplectic geometry mode decomposition and its application to rotating machinery compound fault diagnosis"

The symplectic geometry mode decomposition (SGMD) is a powerful method for decomposing time series, which is based on the diagonal averaging principle (DAP) inherited from the singular spectrum analysis (SSA). Although the authors of SGMD method generalized the form of the trajectory matrix in SSA, the DAP is not updated simultaneously. In this work, we pointed out the limitations of the SGMD method and fixed the bugs with the pulling back theorem for computing the given component of time series from the corresponding component of trajectory matrix.

eess.SP↗

arXiv2025

High Order Expansion Method for Kuiper's $V_n$ Statistic in Goodness-of-fit Test

Kuiper's $V_n$ statistic, a measure for comparing the difference of ideal distribution and empirical distribution, is of great significance in the goodness-of-fit test. However, Kuiper's formulae for computing the cumulative distribution function, false positive probability and the upper tail quantile of $V_n$ can not be applied to the case of small sample capacity $n$ since the approximation error is $\mathcal{O}(n^{-1})$. In this work, our contributions lie in three perspectives: firstly the approximation error is reduced to $\mathcal{O}(n^{-(k+1)/2})$ where $k$ is the expansion order with the \textit{high order expansion} for the exponent of differential operator; secondly, a novel high order formula with approximation error $\mathcal{O}(n^{-3})$ is obtained by massive calculations; thirdly, the fixed-point algorithms are designed for solving the Kuiper pair of critical values and upper tail quantiles based on the novel formula. The high order expansion method for Kuiper's $V_n$-statistic is applicable for various applications where there are more than five samples of data. The principles, algorithms and code for the high order expansion method are attractive for the goodness-of-fit test.\\ \textbf{Keywords}: Goodness-of-fit Methods, Kuiper's statistic, Quantile estimation, Algorithm design, High order expansion (HOE)

math.ST↗

arXiv2025

Pulling Back Theorem for Generalizing the Diagonal Averaging Principle in Symplectic Geometry Mode Decomposition and Singular Spectrum Analysis

The symplectic geometry mode decomposition (SGMD) is a powerful method for analyzing time sequences. The SGMD is based on the upper conversion via embedding and down conversion via diagonal averaging principle (DAP) inherited from the singular spectrum analysis (SSA). However, there are two defects in the DAP: it just hold for the time delay $\tau=1$ in the trajectory matrix and it fails for the time sequence of type-1 with the form $X=\{x[n]\}^N_{n=1}$. In order to overcome these disadvantages, the inverse step for embedding is explored with binary Diophantine equation in number theory. The contributions of this work lie in three aspects: firstly, the pulling back theorem is proposed and proved, which state the general formula for converting the component of trajectory matrix to the component of time sequence for the general representation of time sequence and for any time delay $\tau\ge 1$; secondly a unified framework for decomposing both the deterministic and random time sequences into multiple modes is presented and explained; finally, the guidance of configuring the time delay is suggested, namely the time delay should be selected in a limited range via balancing the efficiency of matrix computation and accuracy of state estimation. It could be expected that the pulling back theorem will help the researchers and engineers to deepen the understanding of the theory and extend the applications of the SGMD and SSA in analyzing time sequences.

eess.SP↗

arXiv2024

Typical Algorithms for Estimating Hurst Exponent of Time Sequence: A Data Analyst's Perspective

The Hurst exponent is a significant metric for characterizing time sequences with long-term memory property and it arises in many fields. The available methods for estimating the Hurst exponent can be categorized into time-domain and spectrum-domain methods. Although there are various estimation methods for the Hurst exponent, there are still some disadvantages that should be overcome: firstly, the estimation methods are mathematics-oriented instead of engineering-oriented; secondly, the accuracy and effectiveness of the estimation algorithms are inadequately assessed; thirdly, the framework of classification for the estimation methods are insufficient; and lastly there is a lack of clear guidance for selecting proper estimation in practical problems involved in data analysis. The contributions of this paper lie in four aspects: 1) the optimal sequence partition method is proposed for designing the estimation algorithms for Hurst exponent; 2) the algorithmic pseudo-codes are adopted to describe the estimation algorithms, which improves the understandability and usability of the estimation methods and also reduces the difficulty of implementation with computer programming languages; 3) the performance assessment is carried for the typical estimation algorithms via the ideal time sequence with given Hurst exponent and the practical time sequence captured in applications; 4) the guidance for selecting proper algorithms for estimating the Hurst exponent is presented and discussed. It is expected that the systematic survey of available estimation algorithms could help the users to understand the principles and the assessment of the various estimation methods could help the users to select, implement and apply the estimation algorithms of interest in practical situations in an easy way.

stat.ME↗

arXiv2024

Balanced Binary Tree Schemes for Computing Zernike Radial Polynomials

Zernike radial polynomials (ZRP) play a significant role in application areas such as optics design, imaging systems, and image processing systems. Currently, there are two kinds of numerical schemes for computing the ZRP automatically with computer programs: one is based on the definition in which the factorial operations may lead to the overflow problem and the high order derivatives are troublesome, and the other is based on recursion which is either unstable or with high computational complexity. In this paper, our emphasis is focused on exploring the balanced binary tree (BBT) schemes for computing the ZRP: firstly an elegant formulae for computation is established; secondly the recursive and iterative algorithms based-on BBT are proposed; thirdly the computational complexity of the algorithms are analyzed rigorously; finally the performance of BBT schemes by testing the running time is verified and validated. Theoretical analysis shows that the computational complexity of balanced binary tree recursive algorithm (BBRTA) and iterative algorithm are exponential and quadratic respectively, which coincides with the running time test very well. Experiments show that the time consumption is about $1\sim 10$ microseconds with different computation platforms for the balanced binary tree iterative algorithm (BBTIA), which is stable and efficient for real-time applications. In the sense of STEM education, the connection of the BBT and ZRP exhibits the beauty and applications of discrete mathematical structure behind the engineering problem, which is worthy of introducing to the college students, computer programmers and optics engineers.

math.NA↗

arXiv2024

Open Source Implementations of Numerical Algorithms for Computing the Complete Elliptic Integral of the First Kind

The complete elliptic integral of the first kind (CEI-1) plays a significant role in mathematics, physics and engineering. There is no simple formula for its computation, thus numerical algorithms are essential for coping with the practical problems involved. The commercial implementations for the numerical solutions, such as the functions \lstinline|ellipticK| and \lstinline|EllipticK| provided by MATLAB and Mathematica respectively, are based on $\mathcal{K}_{\mathrm{cs}}(m)$ instead of the usual form $K(k)$ such that $\mathcal{K}_{\mathrm{cs}}(k^2) =K(k)$ and $m=k^2$. It is necessary to develop open source implementations for the computation of the CEI-1 in order to avoid potential risks of using commercial software and possible limitations due to the unknown factors. In this paper, the infinite series method, arithmetic-geometric mean (AGM) method, Gauss-Chebyshev method and Gauss-Legendre methods are discussed in details with a top-down strategy. The four key algorithms for computing CEI-1 are designed, verified, validated and tested, which can be utilized in R\& D and be reused properly. Numerical results show that our open source implementations based on $K(k)$ are equivalent to the commercial implementation based on $\mathcal{K}_{\mathrm{cs}}(m)$. The general algorithms for computing orthogonal polynomials developed are significant byproducts in the sense of STEM education and scientific computation.

math.NA↗

arXiv2024

Even Order Explicit Symplectic Geometric Algorithms for Solving Quaternions in Guidance Navigation and Control via Diagonal Padé Approximation and Cayley Transform

Quaternion kinematical differential equation (QKDE) plays a key role in navigation, control and guidance systems. Although explicit symplectic geometric algorithms (ESGA) for this problem are available, there is a lack of a unified way for constructing high order symplectic difference schemes with configurable order parameter and the fractional interval sampling problem should be treated carefully. We present even order explicit symplectic geometric algorithms to solve the QKDE with diagonal Padé approximation via a four-step strategy. Firstly, the Padé-Cayley lemma is proved and used to simplify the symplectic Padé approximation for the linear Hamiltonian system with infinitesimal symplectic structure. Secondly, both parallel and alternative iterative methods are proposed to construct the symplectic difference schemes with even order accuracy. Thirdly, the symplecity, orthogonality and invertibility of the single-step transition matrices are proved rigorously. Finally, the explicit symplectic geometric algorithms are designed for both the linear time-invariant and linear time-varying QKDE. The maximum absolute error for solving the QKDE is $\mathcal{O}((t_f-t_0)τ^{2\ell})$ where $τ$ is the time step, $\ell$ is the order parameter and $[t_0,t_f]$ is the time span. The linear time complexity and constant space complexity of computation as well as the simple algorithmic structure show that our algorithms are appropriate for real-time applications in aeronautics, astronautics, robotics and so on. The performance of the proposed algorithms are verified and validated by mathematical analysis and numerical simulation.

eess.SY↗

arXiv2024

Fixed-Point Algorithms for Solving the Critical Value and Upper Tail Quantile of Kuiper's Statistics

Kuiper's statistic is a good measure for the difference of ideal distribution and empirical distribution in the goodness-of-fit test. However, it is a challenging problem to solve the critical value and upper tail quantile, or simply Kuiper pair, of Kuiper's statistics due to the difficulties of solving the nonlinear equation and reasonable approximation of infinite series. In this work, the contributions lie in three perspectives: firstly, the second order approximation for the infinite series of the cumulative distribution of the critical value is used to achieve higher precision; secondly, the principles and fixed-point algorithms for solving the Kuiper pair are presented with details; finally, finally, a mistake about the critical value $c^α_n$ for $(α, n)=(0.01,30)$ in Kuiper's distribution table has been labeled and corrected where $n$ is the sample capacity and $α$ is the upper tail quantile. The algorithms are verified and validated by comparing with the table provided by Kuiper. The methods and algorithms proposed are enlightening and worth of introducing to the college students, computer programmers, engineers, experimental psychologists and so on.

stat.CO↗

arXiv2023

An Automatic Method for Generating Symbolic Expressions of Zernike Circular Polynomials

Zernike circular polynomials (ZCP) play a significant role in optics engineering. The symbolic expressions for ZCP are valuable for theoretic analysis and engineering designs. However, there are still two problems which remain open: firstly, there is a lack of sufficient mathematical formulas of the ZCP for optics designers; secondly the formulas for inter-conversion of Noll's single index and Born-Wolf's double indices of ZCP are neither uniquely determinate nor satisfactory. An automatic method for generating symbolic expressions for ZCP is proposed based on five essential factors: the new theorems for converting the single/double indices of the ZCP, the robust and effective numeric algorithms for computing key parameters of ZCP, the symbolic algorithms for generating mathematical expressions of ZCP, and meta-programming \& \LaTeX{} programming for generating the table of ZCP. The theorems, method, algorithms and system architecture proposed are beneficial to both optics design process, optics software, computer-output typesetting in publishing industry as well as STEM education.

cs.SC↗

arXiv2022

CDIO-CT collaborative strategy for solving complex STEM problems in system modeling and simulation: an illustration of solving the period of mathematical pendulum

The problem-project-oriented STEM education plays a significant role in training students' ability of innovation. Although the conceive-design-implement-operate (CDIO) approach and the computational thinking (CT) are hot topics in recent decade, there are still two deficiencies: the CDIO approach and CT are discussed separately and a general framework of coping with complex STEM problems in system modeling and simulation is missing. In this paper, a collaborative strategy based on the CDIO and CT is proposed for solving complex STEM problems in system modeling and simulation with a general framework, in which the CDIO is about ``how to do", CT is about ``how to think", and the project means ``what to do". As an illustration, the problem of solving the period of mathematical pendulum (MP) is discussed in detail. The most challenging task involved in the problem is to compute the complete elliptic integral of the first kind (CEI-1). In the philosophy of STEM education, all problems have more than one solutions. For computing the CEI-1, four methods are discussed with a top-down strategy, which includes the infinite series method, arithmetic-geometric mean (AGM) method, Gauss-Chebyshev method and Gauss-Legendre method. The algorithms involved can be utilized for R & D projects of interest and be reused according to the requirements encountered. The general framework for solving complex STEM problem in system modeling and simulation is worth recommending to the college students and instructors.

physics.ed-ph↗

arXiv2016

Symplectic Geometric Algorithm for Quaternion Kinematical Differential Equation

Solving quaternion kinematical differential equations is one of the most significant problems in the automation, navigation, aerospace and aeronautics literatures. Most existing approaches for this problem neither preserve the norm of quaternions nor avoid errors accumulated in the sense of long term time. We present symplectic geometric algorithms to deal with the quaternion kinematical differential equation by modeling its time-invariant and time-varying versions with Hamiltonian systems by adopting a three-step strategy. Firstly, a generalized Euler's formula for the autonomous quaternion kinematical differential equation are proved and used to construct symplectic single-step transition operators via the centered implicit Euler scheme for autonomous Hamiltonian system. Secondly, the symplecitiy, orthogonality and invertibility of the symplectic transition operators are proved rigorously. Finally, the main results obtained are generalized to design symplectic geometric algorithm for the time-varying quaternion kinematical differential equation which is a non-autonomous and nonlinear Hamiltonian system essentially. Our novel algorithms have simple algorithmic structures and low time complexity of computation, which are easy to be implemented with real-time techniques. The correctness and efficiencies of the proposed algorithms are verified and validated via numerical simulations.

eess.SY↗

arXiv2015

Study of $B \to K_0^(1430)K^{()}$ decays in QCD Factorization Approach

Within the QCD factorization approach, we calculate the branching fractions and $CP$ asymmetry parameters of 12 $B \to K_0^*(1430)K^{(*)}$ decay modes under the assumption that the scalar meson $K_0^*(1430)$ is the first excited state or the lowest lying ground state in the quark model. We find that the decay modes with the scalar meson emitted, have large branching fractions due to the enhancement of large chiral factor $r_χ^{K_0^*}$. The branching fractions of decays with the vector meson emitted, become much smaller owing to the smaller factor $r_χ^{K^*}$. Moreover, the annihilation type diagram will induce large uncertainties because of the extra free parameter dealing with the endpoint singularity. For the pure annihilation type decays, our predictions are smaller than that from PQCD approach by 2-3 orders of magnitudes. These results will be tested by the ongoing LHCb experiment, forthcoming Belle-II experiment and the proposing circular electron-positron collider.

hep-ph↗