Haiyan Shi

Engineering systems generally exhibit continuous state degradation in practical operation, and reliability evaluation is often challenged by data scarcity and uncertain component weights. Existing studies on weighted k-out-of-n systems mainly focus on deterministic or discrete-state models, while rarely addressing reliability modeling and component importance assessment for continuous-state systems with uncertain information under data shortage, which constitutes a clear research gap. This paper first defines the connotation of uncertain continuous state as the continuous degradation process of system performance affected by ambiguous parameter information and insufficient historical data. On this basis, a reliability modeling framework is established for a weighted k-out-of-n system with uncertain continuous states. Adopting stochastic reliability theory, this paper comparatively investigates system reliability under constant component weights and uncertain variable component weights, and further adopts the Birnbaum importance measure to quantify component importance under variable weight uncertainty. To reduce computational complexity and improve solution efficiency, a binary search-based numerical algorithm is developed to solve the established model. A distributed solar power generation system is employed as a practical case to validate the feasibility and applicability of the proposed model and algorithm. The presented approach effectively fills the limitation of existing discrete and deterministic models, and provides a novel theoretical reference for reliability analysis and key component identification of continuous degradation engineering systems.

As a powerful tool for characterizing the time-evolution behavior of dynamic systems with delay characteristics, the parameter estimation problem for uncertain delay differential equations has always been a research hotspot in the field of uncertain statistics. In order to eliminate the impact of outliers on the relevant results during parameter estimation, this paper proposes the improved uncertain maximum likelihood estimation and the generalized improved uncertain maximum likelihood estimation for uncertain delay differential equations based on symmetric statistical invariants, named residuals. After that, a numerical algorithm is also designed to solve the numerical solutions of the corresponding estimators. Finally, two numerical examples and an empirical study on stock price modeling are provided to illustrate the effectiveness of the methods proposed in this paper.

In uncertain dynamical systems, there is a class of parameter estimation problems for fractional differential equations (UFDEs) that are important and related research is scarce. Given this, the parameter estimation problem for Caputo type of UFDEs under the initial conditions is proposed. The least squares estimation method is a method that minimizes the sum of squared errors between the obtained data and the actual data. The estimation of unknown parameters in the UFDEs are obtained by the least squares estimation method based on limited observed values, and the rationality of the parameters is tested following uncertain hypothesis testing. A specific algorithm for parameter estimation is provided, and the effectiveness of the proposed model is demonstrated by two examples in the financial markets: Alibaba's stock price and the exchange rate between the RMB and Euro.

A class of complex systems can be structurally decomposed into several typical multi-state weighted k/n systems. In view of this, this paper proposes a reliability model to evaluate the reliability of such systems. The system reliability evaluation process is divided into two critical stages: (1) Primary stage: Structural reliability analysis (system-level) and (2) Secondary stage: Subsystem reliability verification (component-level). In practical applications, when there is a lack of operational data, expert experience is needed to evaluate the states and availability weights of components, which is a typical application scenario for uncertainty theory. Under the framework of uncertainty theory, two-stage reliability measures and importance measures are defined, and corresponding calculation formulas are derived. To improve computational efficiency, a binary search algorithm is proposed. Finally, an application case of SDN was proposed to demonstrate the effectiveness of the theory and method. This study provides an idea for the reliability modeling of complex systems (multiple pipeline transmissions, power systems, distributed computing, etc.) in the absence of operational data.

This paper proposes a two-stage weighted k-out-of-n (W-k/n) system reliability model based on uncertainty theory to address the problem of traditional W-k/n system theory being difficult to apply to the reliability evaluation of complex systems with two-layer structures. This model effectively solves the evaluation problem in the case of insufficient data by decomposing the reliability assessment of complex systems into two stages: system structure reliability and subsystem reliability. In the model, the system state and weights are described by uncertain variables that follow a certain distribution. Under the framework of uncertainty theory, two-stage reliability measures and importance measures are defined, and corresponding calculation formulas are derived. In order to reduce computational complexity, a corresponding numerical solution method including traverse search algorithm and binary search algorithm is proposed. The effectiveness of the method is also demonstrated through a project management case of electronic consumer goods. This study provides new theoretical ideas and methodological support for reliability modeling of complex systems under data loss conditions.

The uncertain moving average model is a powerful tool to study the time series in which the data are affected by the previous disturbance terms under uncertain environments. However, the influence of uncertain disturbance terms is often ignored in the relevant statistical inference studies. In order to solve this problem, this paper constructs a symmetric statistical invariant, normal uncertain variable, by sorting out the uncertain moving average model and combining the uncertain disturbance terms, and then applies the least square principle to the parameter estimation of the symmetric statistical invariant to determine the unknown parameters and uncertain disturbance terms in the uncertain moving average model. In addition, a numerical algorithm is designed to calculate the corresponding estimators, and the problems related to hypothesis testing and forecast are also studied. Finally, a numerical example is given to illustrate the proposed method.

In the study of uncertain autoregressive models, how to estimate the unknown parameters and uncertain disturbance term in the models is always a key problem. In view of this, this paper proposes a statistical inference method based on the principle of least squares to determine the unknown parameters and uncertain disturbance term in an uncertain autoregressive model, and designs a numerical algorithm to calculate the numerical solutions of the corresponding estimators. Then, the uncertain hypothesis test is used to verify the applicability of the estimated uncertain autoregressive model, and point forecast and interval forecast are also made for the time series of future moments. Finally, a case study of the Consumer Price Index for all items in U.S. cities is provided to illustrate the effectiveness of the approach proposed in this paper.

In the field of statistics, uncertain regression analysis occupies an important position. It can thoroughly analyze data sets contained in complex uncertainties, aiming to quantify and reveal the intricate relationships between variables. It is worth noting that the traditional least squares method only takes into account the reduction in the deviations between predictions and observations, and fails to fully consider the inherent characteristics of the correlation uncertainty distributions under the uncertain regression framework. In light of this, this paper constructs a statistical invariant with symmetric uncertainty distribution based on the observations and the disturbance term. It also proposes the least squares estimation of unknown parameters and disturbance term in the uncertain regression model based on the least squares principle and, combined with the mathematical properties of the normal uncertainty distribution, gives a numerical algorithm for solving specific estimates. Finally, in order to verify the effectiveness of the least squares estimation method proposed in this paper, we also design two numerical examples and an empirical study of forecasting of electrical power output.