arXiv subjects

Yiheng Wei

Publications and source records attributed to Yiheng Wei.

18 recordsLinked to original sources

arXiv2023

A Bayesian two-step multiple imputation approach based on mixed models for the missing in EMA data

Ecological Momentary Assessments (EMA) capture real-time thoughts and behaviors in natural settings, producing rich longitudinal data for statistical and physiological analyses. However, the robustness of these analyses can be compromised by the large amount of missing in EMA data sets. To address this, multiple imputation, a method that replaces missing values with several plausible alternatives, has become increasingly popular. In this paper, we introduce a two-step Bayesian multiple imputation framework which leverages the configuration of mixed models. We adopt the Random Intercept Linear Mixed model, the Mixed-effect Location Scale model which accounts for subject variance influenced by covariates and random effects, and the Shared Parameter Location Scale Mixed Effect model which links the missing data to the response variable through a random intercept logistic model, to complete the posterior distribution within the framework. In the simulation study and an application on data from a study on caregivers of dementia patients, we further adapt this two-step Bayesian multiple imputation strategy to handle simultaneous missing variables in EMA data sets and compare the effectiveness of multiple imputations across different mixed models. The analyses highlight the advantages of multiple imputations over single imputations. Furthermore, we propose two pivotal considerations in selecting the optimal mixed model for the two-step imputation: the influence of covariates as well as random effects on the within-variance, and the nature of missing data in relation to the response variable.

stat.ME↗

arXiv2023

Multi/Single-stage structured zero-gradient-sum approach for prescribed-time optimization

Prescribed-time convergence mechanism has become a prominent research focus in the current field of optimization and control due to its ability to precisely control the target completion time. The recently arisen prescribed-time algorithms for distributed optimization, currently necessitate multi-stage structures to achieve global convergence. This paper introduces two modified zero-gradient-sum algorithms, each based on a multi-stage and a single-stage structural frameworks established in this work. These algorithms are designed to achieve prescribed-time convergence and relax two common yet stringent conditions. This work also bridges the gap in current research on single-stage structured PTDO algorithm. The excellent convergence performance of the proposed algorithms is validated through a case study.

math.OC↗

arXiv2023

On some analytic properties of nabla tempered fractional calculus

Despite many applications regarding fractional calculus have been reported in literature, it is still unknown how to model some practical process. One major challenge in solving such a problem is that, the nonlocal property is needed while the infinite memory is undesired. Under this context, a new kind nabla fractional calculus accompanied by a tempered function is formulated. However, many properties of such fractional calculus needed to be discovered. From this, this paper gives particular emphasis to the topic. Some remarkable properties like the equivalence relation, the nabla Taylor formula, and the nabla Laplace transform for such nabla fractional calculus are developed and analyzed. It is believed that this work greatly enriches the mathematical theory of nabla tempered fractional calculus and provides high value and huge potential for further applications.

math.GM↗

arXiv2022

Some useful inequalities for nabla tempered fractional calculus

This paper gives particular emphasis to the nabla tempered fractional calculus, which involves the multiplication of the rising function kernel with tempered functions, and provides a more flexible alternative with considerable promise for practical applications. Some remarkable inequalities for such nabla fractional calculus are developed and analyzed, which greatly enrich the mathematical theory of nabla tempered fractional calculus. Numerical results confirm the validity of the developed properties once again, which also reveals that the introduction of tempered function provides high value and huge potential.

math.CA↗

arXiv2021

Finite-time and Fixed-time Convergence in Continuous-time Optimization

It is known that the gradient method can be viewed as a dynamic system where various iterative schemes can be designed as a part of the closed loop system with desirable properties. In this paper, the finite-time and fixed-time convergence in continuous-time optimization are mainly considered. By the advantage of sliding mode control, a finite-time gradient method is proposed, whose convergence time is dependent on initial conditions. To make the convergence time robust to initial conditions, two different designs of fixed-time gradient methods are then provided. One is designed using the property of sine function, whose convergence time is dependent on the frequency of a sine function. The other one is designed using the property of Mittag-Leffler function, whose convergence time is determined by the first positive zero of a Mittag-Leffler function. All the results are extended to more general cases and finally demonstrated by some dedicated simulation examples.

math.OC↗

arXiv2021

Consistent approximation of fractional order operators

Fractional order controllers become increasingly popular due to their versatility and superiority in various performance. However, the bottleneck in deploying these tools in practice is related to their analog or numerical implementation. Numerical approximations are usually employed in which the approximation of fractional differintegrator is the foundation. Generally, the following three identical equations always hold, i.e., $\frac{1}{s^α}\frac{1}{s^{1-α}} = \frac{1}{s}$, $s^α\frac{1}{s^α} = 1$ and $s^αs^{1-α} = s$. However, for the approximate models of fractional differintegrator $s^α$, $α\in(-1,0)\cup(0,1)$, there usually exist some conflicts on the mentioned equations, which might enlarge the approximation error or even cause fallacies in multiple orders occasion. To overcome the conflicts, this brief develops a piecewise approximate model and provides two procedures for designing the model parameters. The comparison with several existing methods shows that the proposed methods do not only satisfy the equalities but also achieve high approximation accuracy. From this, it is believed that this work can serve for simulation and realization of fractional order controllers more friendly.

math.NA↗

arXiv2020

Lyapunov functions for nabla discrete fractional order systems

This paper focuses on the fractional difference of Lyapunov functions related to Riemann-Liouville, Caputo and Grunwald-Letnikov definitions. A new way of building Lyapunov functions is introduced and then five inequalities are derived for each definition. With the help of the developed inequalities, the sufficient conditions can be obtained to guarantee the asymptotic stability of the nabla discrete fractional order nonlinear systems. Finally, three illustrative examples are presented to demonstrate the validity and feasibility of the proposed theoretical results.

math.DS↗

arXiv2020

Sufficient and necessary conditions for stabilizing singular fractional order systems with partially measurable state

This paper is concerned with the stabilization problem of singular fractional order systems with order $α\in(0,2)$. In addition to the sufficient and necessary condition for observer based control, a sufficient and necessary condition for output feedback control is proposed by adopting matrix variable decoupling technique. The developed results are more general and efficient than the existing works, especially for the output feedback case. Finally, two illustrative examples are given to verify the effectiveness and potential of the proposed approaches.

math.DS↗

arXiv2020

On the Leibniz rule and Laplace transform for fractional derivatives

Taylor series is a useful mathematical tool when describing and constructing a function. With the series representation, some properties of fractional calculus can be revealed clearly. This paper investigates two typical applications: Lebiniz rule and Laplace transform. It is analytically shown that the commonly used Leibniz rule cannot be applied for Caputo derivative. Similarly, the well-known Laplace transform of Riemann-Liouville derivative is doubtful for n-th continuously differentiable function. By the aid of this series representation, the exact formula of Caputo Leibniz rule and the explanation of Riemann-Liouville Laplace transform are presented. Finally, three illustrative examples are revisited to confirm the obtained results.

math.GM↗

arXiv2020

Infinite series representation of fractional calculus: theory and applications

This paper focuses on the equivalent expression of fractional integrals/derivatives with an infinite series. A universal framework for fractional Taylor series is developed by expanding an analytic function at the initial instant or the current time. The framework takes into account of the Riemann-Liouville definition, the Caputo definition, the constant order and the variable order. On this basis, some properties of fractional calculus are confirmed conveniently. An intuitive numerical approximation scheme via truncation is proposed subsequently. Finally, several illustrative examples are presented to validate the effectiveness and practicability of the obtained results.

math.GM↗

arXiv2020

Some fundamental properties on the sampling free nabla Laplace transform

Discrete fractional order systems have attracted more and more attention in recent years. Nabla Laplace transform is an important tool to deal with the problem of nabla discrete fractional order systems, but there is still much room for its development. In this paper, 14 lemmas are listed to conclude the existing properties and 14 theorems are developed to describe the innovative features. On one hand, these properties make the N-transform more effective and efficient. On the other hand, they enrich the discrete fractional order system theory

math.OC↗

arXiv2019

Analytical calculation of the inverse nabla Laplace transform

The inversion of nabla Laplace transform, corresponding to a causal sequence, is considered. Two classical methods, i.e., residual calculation method and partial fraction method are developed to perform the inverse nabla Laplace transform. For the first method, two alternative formulae are proposed when adopting the poles inside or outside of the contour, respectively. For the second method, a table on the transform pairs of those popular functions is carefully established. Besides illustrating the effectiveness of the developed methods with two illustrative examples, the applicability are further discussed in the fractional order case.

math.GM↗

arXiv2019

Convolutional neural networks with fractional order gradient method

This paper proposes a fractional order gradient method for the backward propagation of convolutional neural networks. To overcome the problem that fractional order gradient method cannot converge to real extreme point, a simplified fractional order gradient method is designed based on Caputo's definition. The parameters within layers are updated by the designed gradient method, but the propagations between layers still use integer order gradients, and thus the complicated derivatives of composite functions are avoided and the chain rule will be kept. By connecting every layers in series and adding loss functions, the proposed convolutional neural networks can be trained smoothly according to various tasks. Some practical experiments are carried out in order to demonstrate fast convergence, high accuracy and ability to escape local optimal point at last.

math.OC↗

arXiv2019

Modelling and simulation of nabla fractional dynamic systems with nonzero initial conditions

The paper focuses on the numerical approximation of nabla fractional order systems with the conditions of nonzero initial instant and nonzero initial state. First, the inverse nabla Laplace transform is developed and the equivalent infinite dimensional frequency distributed models of discrete fractional order system are introduced. Then, resorting the nabla Laplace transform, the rationality of the finite dimensional frequency distributed model approaching the infinite one is illuminated. Based on this, an original algorithm to estimate the parameters of the approximate model is proposed with the help of vector fitting method. Additionally, the applicable object is extended from a sum operator to a general system. Three numerical examples are performed to illustrate the applicability and flexibility of the introduced methodology.

eess.SP↗

arXiv2019

A novel perspective to gradient method: the fractional order approach

In this paper, we give some new thoughts about the classical gradient method (GM) and recall the proposed fractional order gradient method (FOGM). It is proven that the proposed FOGM holds a super convergence capacity and a faster convergence rate around the extreme point than the conventional GM. The property of asymptotic convergence of conventional GM and FOGM is also discussed. To achieve both a super convergence capability and an even faster convergence rate, a novel switching FOGM is proposed. Moreover, we extend the obtained conclusion to a more general case by introducing the concept of p-order Lipschitz continuous gradient and p-order strong convex. Numerous simulation examples are provided to validate the effectiveness of proposed methods.

math.OC↗

arXiv2019

A universal framework of GKYP lemma for singular fractional order systems

The well-known GKYP is widely used in system analysis, but for singular systems, especially singular fractional order systems, there is no corresponding theory, for which many control problems for this type of system can not be optimized in the limited frequency ranges. In this paper, a universal framework of finite frequency band GKYP lemma for singular fractional order systems is established. Then the bounded real lemma in the sense of L is derived for different frequency ranges. Furthermore, the corresponding controller is designed to improve the L performance index of singular fractional order systems. Three illustrative examples are given to demonstrate the correctness and effectiveness of the theoretical results.

eess.SY↗

arXiv2019

Adaptive backstepping control for FOS with nonsmooth nonlinearities

This paper proposes an original solution to input saturation and dead zone of fractional order system. To overcome these nonsmooth nonlinearities, the control input is decomposed into two independent parts by introducing an intermediate variable, and thus the problem of dead zone and saturation transforms into the problem of disturbance and saturation afterwards. With the procedure of fractional order adaptive backstepping controller design, the bound of disturbance is estimated, and saturation is compensated by the virtual signal of an auxiliary system as well. In spite of the existence of nonsmooth nonlinearities, the output is guaranteed to track the reference signal asymptotically on the basis of our proposed method. Some simulation studies are carried out in order to demonstrate the effectiveness of method at last.

math.OC↗

arXiv2018

Time-domain response of nabla discrete fractional order systems

This paper investigates the time--domain response of nabla discrete fractional order systems by exploring several useful properties of the nabla discrete Laplace transform and the discrete Mittag--Leffler function. In particular, we establish two fundamental properties of a nabla discrete fractional order system with nonzero initial instant: i) the existence and uniqueness of the system time--domain response; and ii) the dynamic behavior of the zero input response. Finally, one numerical example is provided to show the validity of the theoretical results.

math.DS↗