Grey Number Sequence Research Articles

Performance reliability degradation analysis and lifetime prediction based on generalized normal grey cloud Bayesian Wiener process

AbstractThis study addresses the problem of performance reliability degradation and lifetime prediction in complex systems with high reliability but insufficient information. It comprehensively considers the effects of various uncertainties, such as measurement, cognitive, and stochastic uncertainties, and proposes a generalized normal grey cloud Bayesian Wiener process (GNGCBWP) stochastic degradation model. First, based on the historical degradation information of the system, the interval grey number sequence of the degradation increment is constructed, and the performance degradation path of the system is analyzed to determine the degradation model type. Then, combining the normal grey cloud and Bayesian estimation theory, the relevant definitions and theorems of generalized normal grey cloud Bayesian estimation (GNGCBE) are proposed. Finally, the parameter estimation results obtained using the GNGCBE method are substituted into the grey Wiener process degradation model to dynamically analyze performance reliability degradation and predict the remaining useful life (RUL) in real‐time. And the validity and scientificity of the proposed model is verified through a case study of laser equipment. Moreover, the comparison results of the models indicate that the proposed model has advantages in dynamic analysis of performance reliability degradation and real‐time RUL prediction for highly reliable complex systems with limited or poor degradation information. It can significantly reduce the impact of multiple uncertainties on the analysis results and improve the accuracy and reliability of the final prediction results.

A summary of grey forecasting models

PurposeThe purpose of this paper is to summarize progress of grey forecasting modelling, explain mechanism of grey forecasting modelling and classify exist grey forecasting models.Design/methodology/approachGeneral modelling process and mechanism of grey forecasting modelling is summarized and classification of grey forecasting models is done according to their differential equation structure. Grey forecasting models with linear structure are divided into continuous single variable grey forecasting models, discrete single variable grey forecasting models, continuous multiple variable grey forecasting models and discrete multiple variable grey forecasting models. The mechanism and traceability of these models are discussed. In addition, grey forecasting models with nonlinear structure, grey forecasting models with grey number sequences and grey forecasting models with multi-input and multi-output variables are further discussed.FindingsIt is clearly to explain differences between grey forecasting models with other forecasting models. Accumulation generation operation is the main difference between grey forecasting models and other models, and it is helpful to mining system developing law with limited data. A great majority of grey forecasting models are linear structure while grey forecasting models with nonlinear structure should be further studied.Practical implicationsMechanism and classification of grey forecasting models are very helpful to combine with suitable real applications.Originality/valueThe main contributions of this paper are to classify models according to models' structure are linear or nonlinear, to analyse relationships and differences of models in same class and to deconstruct mechanism of grey forecasting models.

Time-delay nonlinear model based on interval grey number and its application

In this paper, an optimization model is proposed to simulate and predict the current situation of smog. The model takes the interval grey number sequence with the known possibility function as the original data, and constructs a time-delay nonlinear multivariable grey model <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\text{MGM}(1,m \vert \tau,\gamma)$</tex> based on the new kernel and degree of greyness sequences considering its time-delay and nonlinearity. The time-delay parameter is determined by the maximum value of the grey time-delay absolute correlation degree, and the nonlinear parameter is determined by the minimum value of average relative error. In order to verify the feasibility of the model, this paper uses the smog related data of Nanjing city for simulation and prediction. Compared with the other four models, the new model has higher simulation and prediction accuracy.

Analysis on the related factors of China's technological innovation ability using greyness relational degree

PurposeIn order to make grey relational analysis applicable to the interval grey number, this paper discusses the model of grey relational degree of the interval grey number and uses it to analyze the related factors of China's technological innovation ability.Design/methodology/approachFirst, this paper gives the definitions of the lower bound domain, the value domain, the upper bound domain of interval grey number and the generalized measure and the generalized greyness of interval grey number. Then, based on the grey relational theory, this paper proposes the model of greyness relational degree of the interval grey number and analyzes its relationship with the classical grey relational degree. Finally, the model of greyness relational degree is applied to analyze the related factors of China's technological innovation ability.FindingsThe results show that the model of greyness relational degree has strict theoretical basis, convenient calculation and easy programming and can be applied to the grey number sequence, real number sequence and grey number and real number coexisting sequence. The relational order of the four related factors of China's technological innovation ability is research and development (R&D) expenditure, R&D personnel, university student number and public library number, and it is in line with the reality.Practical implicationsThe results show that the sequence values of greyness relational degree have large discreteness, and it is feasible and effective to analyze the related factors of China's technological innovation ability.Originality/valueThe paper succeeds in realizing both the model of greyness relational degree of interval grey number with unvalued information distribution and the order of related factors of China's technological innovation ability.

A time-varying grey Riccati model based on interval grey numbers for China's clean energy generation predicting

PurposeIn order to accurately predict the uncertain and nonlinear characteristics of China's three clean energy generation, this paper presents a novel time-varying grey Riccati model (TGRM(1,1)) based on interval grey number sequences.Design/methodology/approachBy combining grey Verhulst model and a special kind of Riccati equation and introducing a time-varying parameter and random disturbance term the authors advance a TGRM(1,1) based on interval grey number sequences. Additionally, interval grey number sequences are converted into middle value sequences and trapezoid area sequences by using geometric characteristics. Then the predicted formula is obtained by using differential equation principle. Finally, the proposed model's predictive effect is evaluated by three numerical examples of China's clean energy generation.FindingsBased on the interval grey number sequences, the TGRM(1,1) is applied to predict the development trend of China's wind power generation, China's hydropower generation and China's nuclear power generation, respectively, to verify the effectiveness of the novel model. The results show that the proposed model has better simulated and predicted performance than compared models.Practical implicationsDue to the uncertain information and continuous changing of clean energy generation in the past decade, interval grey number sequences are introduced to characterize full information of the annual clean energy generation data. And the novel TGRM(1,1) is applied to predict upper and lower bound values of China's clean energy generation, which is significant to give directions for energy policy improvements and modifications.Originality/valueThe main contribution of this paper is to propose a novel TGRM(1,1) based on interval grey number sequences, which considers the changes of parameters over time by introducing a time-varying parameter and random disturbance term. In addition, the model introduces the Riccati equation into classic Verhulst, which has higher practicability and prediction accuracy.

The nonlinear time lag multivariable grey prediction model based on interval grey numbers and its application

The linear relationship of the original grey prediction model is too single, and the original grey prediction model does not consider the time delay of the effect of the current input parameters on the output parameters. In order to solve these problems, the interval grey number sequence is taken as the modelling sequence of the model, and the nonlinear parameter γ and the time-delay parameter τ are introduced into the multivariate grey prediction model, so as to construct the nonlinear time-delay multivariable grey prediction model for interval grey number. In view of the uncertain characteristics of the smog index data, this paper applies the improved model to the simulation and prediction of the smog index data. Compared with the original model, the results show that the prediction effect of the model proposed in this paper is superior to the original model in terms of its effectiveness and feasibility.

Forecasting smog in Beijing using a novel time-lag GM(1,N) model based on interval grey number sequences

PurposeSmog seriously affects the ecological environment and poses a threat to public health. Therefore, smog control has become a key task in China, which requires reliable prediction.Design/methodology/approachThis paper establishes a novel time-lag GM(1,N) model based on interval grey number sequences. Firstly, calculating kernel and degree of greyness of the interval grey number sequence respectively. Then, establishing the time-lag GM(1,N) model of kernel and degree of greyness sequences respectively to obtain their values after determining the time-lag parameters of two models. Finally, the upper and lower bounds of interval grey number sequences are obtained by restoring the values of kernel and degree of greyness.FindingsIn order to verify the validity and practicability of the model, the monthly concentrations of PM2.5, SO2 and NO2 in Beijing during August 2017 to September 2018 are selected to establish the time-lag GM(1,3) model for kernel and degree of greyness sequences respectively. Compared with three existing models, the proposed model in this paper has better simulation accuracy. Therefore, the novel model is applied to forecast monthly PM2.5 concentration for October to December 2018 in Beijing and provides a reference basis for the government to formulate smog control policies.Practical implicationsThe proposed model can simulate and forecast system characteristic data with the time-lag effect more accurately, which shows that the time-lag GM(1,N) model proposed in this paper is practical and effective.Originality/valueBased on interval grey number sequences, the traditional GM(1,N) model neglects the time-lag effect of driving terms, hence this paper introduces the time-lag parameters into driving terms of the traditional GM(1,N) model and proposes a novel time-lag GM(1,N) model.

Examination and prediction of fog and haze pollution using a Multi-variable Grey Model based on interval number sequences

In this study, a new Multivariable Grey Model (1,m) aimed at interval grey number sequences with known possibility functions is built using the kernel and degree of greyness under new definitions. Based on the new model, formulae are deduced to calculate and predict the upper and lower bounds of interval grey numbers. Since the grey system model and fog- and haze-prone weather have the same characteristics of uncertainty, this model was applied to simulate and predict the measurable indicators of fog and haze in Nanjing, China. We selected visibility data and particulate matter data with a diameter of 2.5 µm to build a new Multivariable Grey Model (1,2) with a new kernel and degree of greyness sequence. In addition, we established the traditional Multivariable Grey Model (1,2) with the original kernel and degree of greyness and the Auto-Regressive Integrated Moving Average Model (1,1,0). The results show that the new Multivariable Grey Model (1,2) has the best simulation and prediction effects among the three models, with average relative errors of simulation and prediction at 1.32% and 0.32%, respectively. To further verify the validity and feasibility of the proposed model, we added another real-world example to establish the three models mentioned above. The results prove that the proposed model has evidently superior performance to another two models.

Forecasting the multifactorial interval grey number sequences using grey relational model and GM (1, N) model based on effective information transformation

In the context of data eruption, the data often show a short-term pattern and change rapidly which makes it difficult to use a single real value to express. For this kind of small-sample and interval data, how to analyze and predict multi-factor sequences efficiently becomes a problem. By this means, grey system theory (GST) is developed in which the interval grey numbers, as a typical object of GST, characterize the range of data and the grey relational and prediction models analyze the relations of multiple grey numbers and forecast the future. However, traditional grey relative relational model has some limitations: the results obtained always show low resolution, and there are no extractions for the interval feature information from the interval grey number sequence. In this paper, the grey relational analysis model (GRA) based on effective information transformation of interval grey numbers is established, which contains comprehensive information of area differences and slope variances and optimizes the resolution of traditional grey degree. Then, according to the relational results, the multivariable GM model (GM (1, N)) is proposed to forecast the interval grey number sequence. To verify the effectiveness of this novel model, it is established to analyze the relationship between the degree of traffic congestion and its relevant factors in the Yangtze River Delta of China and predict the development of urban traffic congestion degrees in this area over the next 5 years. In addition, some traditional statistical methods (principal component analysis, multiple linear regression models and curve regression models) are established for comparisons. The results show high performances of the novel GRA model and GM (1, N) model, which means the models proposed in this paper are suitable for interval grey numbers from regional data. The strengths which recommend the use of this novel method lie in its high recognition mechanism and multi-angle information transformation for interval grey numbers as well as its characteristic of timeliness in information processing.

A Novel Linear Time-Varying GM(1,N) Model for Forecasting Haze: A Case Study of Beijing, China

Haze is the greatest challenge facing China’s sustainable development, and it seriously affects China’s economy, society, ecology and human health. Based on the uncertainty and suddenness of haze, this paper proposes a novel linear time-varying grey model (GM)(1,N) based on interval grey number sequences. Because the original GM(1,N) model based on interval grey number sequences has constant parameters, it neglects the dynamic change characteristics of parameters over time. Therefore, this novel linear time-varying GM(1,N) model, based on interval grey number sequences, is established on the basis of the original GM(1,N) model by introducing a linear time polynomial. To verify the validity and practicability of this model, this paper selects the data of PM10, SO2 and NO2 concentrations in Beijing, China, from 2008 to 2018, to establish a linear time-varying GM(1,3) model based on interval grey number sequences, and the prediction results are compared with the original GM(1,3) model. The result indicates that the prediction effect of the novel model is better than that of the original model. Finally, this model is applied to forecast PM10 concentration for 2019 to 2021 in Beijing, and the forecast is made to provide a reference for the government to carry out haze control.

MGM(1,m) model based on interval grey number sequence and its applications

PurposeAiming at the traditional multivariate grey forecasting model only considers the modelling of real numbers; therefore, the purpose of this paper is to construct an MGM(1, m) model based on the interval grey number sequences according to the grey modelling theory.Design/methodology/approachFirst, the multivariable grey number sequences are transformed into the kernel and grey radius sequences which are two feature sequences of interval grey number sequences. Then the MGM(1, m) model for kernel sequences and grey radius sequences are established, respectively. Finally, the simulation and prediction of the upper and lower bounds of the interval grey number sequences are realized by the reductive calculation of the predicted values of the kernel and grey radius.FindingsThe model is applied to the prediction of visibility and relative humidity, the identification factors of the haze. The results show that the model has high accuracy on the simulation and prediction of multivariable grey number sequences, which is reasonable and practical.Originality/valueThe main contribution of this paper is to propose a method to simulate and forecast the multivariable grey number sequence that is to establish the prediction models for the whitening sequences of multivariable grey number sequences which are kernel and grey radius sequences and extend the possibility boundary of kernel by grey radius. The model can reflect the development trend of multivariable grey number sequence accurately. When the grey information is continuously complemented, the multivariable grey number prediction model is transformed into the traditional MGM(1, m) model. Therefore, the MGM(1, m) model based on interval grey number sequence is the generalisation and expansion of the traditional MGM(1, m) model.

Interval grey number sequence prediction by using non-homogenous exponential discrete grey forecasting model

This paper aims to study a new grey prediction approach and its solution for forecasting the main system variable whose accurate value could not be collected while the potential value set could be defined. Based on the traditional non-homogenous discrete grey forecasting model (NDGM), the interval grey number and its algebra operations are redefined and combined with the NDGM model to construct a new interval grey number sequence prediction approach. The solving principle of the model is analyzed, the new accuracy evaluation indices, i.e. mean absolute percentage error of mean value sequence (MAPEM) and mean percent of interval sequence simulating value set covered (MPSVSC), are defined and, the procedure of the interval grey number sequence based the NDGM (IG-NDGM) is given out. Finally, a numerical case is used to test the modelling accuracy of the proposed model. Results show that the proposed approach could solve the interval grey number sequence prediction problem and it is much better than the traditional DGM(1,1) model and GM(1,1) model.

Prediction Model of Interval Grey Numbers with a Real Parameter and Its Application

Grey prediction models have become common methods which are widely employed to solve the problems with “small examples and poor information.” However, modeling objects of existing grey prediction models are limited to the homogenous data sequences which only contain the same data type. This paper studies the methodology of building prediction models of interval grey numbers that are grey heterogeneous data sequence, with a real parameter. Firstly, the position of the real parameter in an interval grey number sequence is discussed, and the real number is expanded into an interval grey number by adopting the method of grey generation. On this basis, a prediction model of interval grey number with a real parameter is deduced and built. Finally, this novel model is successfully applied to forecast the concentration of organic pollutant DDT in the atmosphere. The analysis and research results in this paper extend the object of grey prediction from homogenous data sequence to grey heterogeneous data sequence. Those research findings are of positive significance in terms of enriching and improving the theory system of grey prediction models.

Prediction Modeling Method of Interval Grey Number Based on Different Type Whitenization Weight Functions

The modeling objects of existing prediction models for interval grey number are limited to the interval grey number sequences with unknown or the same type of whitenization weight function. Therefore, the existing methods are useless when the types of whitenization weight function of interval grey number in the modeling sequence are heterogeneous. On the basis of the existing prediction models for interval grey number and according to the axiom of undecreased degree of greyness and grey number, the present paper build a prediction model for interval grey number based on different types of whitenization weight functions through expanding the calculation of "kernel and grey degree" of the interval grey number. At last, this model was applied in forecasting the demand for emergency materials in disaster. The research results are significant for enriching and perfecting the grey prediction model theory system, and extending the applied scope of grey models and promoting the effective association of the grey theory and the practical issues.

IN‐GM(0, N) cost forecasting model of commercial aircraft based on interval grey numbers

PurposeThe purpose of this paper is to construct a novel GM(0, N) model based on grey number sequences and to solve the problem of cost forecasting of commercial aircraft which puzzled managers, especially the factors of cost with uncertain information.Design/methodology/approachBased on the definition of traditional GM(0, N) model, the paper considers the limited information and knowledge, and the algorithm of grey numbers with greyness and kernel was designed. A novel GM(0, N) model based on grey number sequences, named the IN‐GM(0, N) model, is proposed according to the definition of the grey numbers algorithm. The steps of the proposed model are then given. Finally, a case of domestic commercial aircrafts is developed as an example, based on information gathering and model calculating.FindingsThe results of this research indicate that the IN‐GM(0,N) model is effective in cost calculating, providing reliable technical support for cost estimation of large‐scale complex equipment including commercial aircraft.Practical implicationsCost forecasting of commercial aircraft can be solved effectively and the model can also be utilized to predicate other products.Originality/valueThe paper succeeds in constructing a novel grey forecasting model. This work contributes significantly to improving grey forecasting theory and to undoubtedly propose more novel grey forecasting models.

The absolute degree of grey incidence for grey sequence base on standard grey interval number operation

PurposeThis paper attempts to extend classic absolute degree of grey incidence so that the extended model can be used for grey number sequences.Design/methodology/approachThe classic absolute degree of grey incidence was extended according to basic principles of grey incidence analysis. First, modelling methods and theories of the classic grey incidences were summarized. Then, the zeroing starting operator in grey incidence was extended for grey sequence. Third, the parameters in classic incidence degree were redefined, and an absolute degree of grey incidence for grey number sequences was proposed. The degree can not only be applied to grey number sequence, but also contains the uncertain information of analysis result. Fourth, two non‐linear programming models were constructed to estimate the grey coverage interval of absolute degree of incidences. Finally, the comparison method of grey numbers was introduced for sorting the different absolute degrees of incidences.FindingsA theoretically feasible absolute degree of grey incidence was constructed for grey sequence. A case study showed that the proposed incidence degree was an effective method for grey sequence incidence analysis.Practical implicationsThe method exposed in the paper can be used for grey sequences clustering, grey decision making, multi‐attribute decision making theory, uncertain target recognition and other related fields.Originality/valueThe paper succeeded in establishing an incidence analysis model for grey sequences which was still a research gap in grey system theory.

A novel grey relational model based on grey number sequences

PurposeThis paper aims to construct a novel grey relational model based on grey number sequences and to solve the problems which exist in traditional grey relational models, in which the uncertain information cannot be described.Design/methodology/approachBased on the definition of traditional grey relational models, considering the limited information and knowledge, the algorithm of grey numbers was combined with the grey relational model. A general formula of grey operations and grey distance is defined. A novel grey relational model based on grey number sequences, named grey geometrical relational model, is proposed according to the definition of grey distance. Finally, several properties including parallel, multiple and order‐keeping about the proposed model are discussed.FindingsThe results show that the novel grey relational model satisfies the properties properly. It is useful to calculate the relational degree of two different grey number sequences. And the process of calculating is easier than traditional grey relational models.Practical implicationsThe method exposed in the paper can be used to calculate every two sequences. The method can also be used to rank sequences of more than two.Originality/valueThe paper succeeds in constructing a novel grey relational model. The properties of novel model are studied and it is a new development in grey systems theory undoubtedly.

Prediction model of interval grey number based on DGM (1, 1)

In grey system theory, the studies in the field of grey prediction model are focused on real number sequences, rather than grey number ones. Hereby, a prediction model based on interval grey number sequences is proposed. By mining the geometric features of interval grey number sequences on a two-dimensional surface, all the interval grey numbers are converted into real numbers by means of certain algorithm, and then the prediction model is established based on those real number sequences. The entire process avoids the algebraic operations of grey number, and the prediction problem of interval grey number is usefully solved. Ultimately, through an example’s program simulation, the validity and practicability of this novel model are verified.