Purpose – The purpose of this paper is to introduce the elementary concepts and fundamental principles of grey systems and the main components of grey systems theory. Also to discuss the astonishing progress that grey systems theory has made in the world of learning and its wide‐ranging applications in the entire spectrum of science.Design/methodology/approach – The characteristics of unascertained systems including incomplete information and inaccuracies in data are analysed and four uncertain theories: probability statistics, fuzzy mathematics, grey system and rough set theory are compared. The scientific principle of simplicity and how precise models suffer from inaccuracies are also shown.Findings – The four uncertain theories, probability statistics, fuzzy mathematics, grey system and rough set theory are examined with different research objects, different basic sets, different methods and procedures, different data requirements, different emphasis, different objectives and different characteristics....

A brief introduction to grey systems theory

– The purpose of this paper is to summarize the progress in grey system research during 2000-2015, so as to present some important new concepts, models, methods and a new framework of grey system theory. , – The new thinking, new models and new methods of grey system theory and their applications are presented in this paper. It includes algorithm rules of grey numbers based on the “kernel” and the degree of greyness of grey numbers, the concept of general grey numbers, the synthesis axiom of degree of greyness of grey numbers and their operations; the general form of buffer operators of grey sequence operators; the four basic models of grey model GM(1,1), such as even GM, original difference GM, even difference GM, discrete GM and the suitable sequence type of each basic model, and suitable range of most used grey forecasting models; the similarity degree of grey incidences, the closeness degree of grey incidences and the three-dimensional absolute degree of grey incidence of grey incidence analysis models; the grey cluster model based on center-point and end-point mixed triangular whitenization functions; the multi-attribute intelligent grey target decision model, the two stages decision model with grey synthetic measure of grey decision models; grey game models, grey input-output models of grey combined models; and the problems of robust stability for grey stochastic time-delay systems of neutral type, distributed-delay type and neutral distributed-delay type of grey control, etc. And the new framework of grey system theory is given as well. , – The problems which remain for further studying are discussed at the end of each section. The reader could know the general picture of research and developing trend of grey system theory from this paper. , – A lot of successful practical applications of the new models to solve various problems have been found in many different areas of natural science, social science and engineering, including spaceflight, civil aviation, information, metallurgy, machinery, petroleum, chemical industry, electrical power, electronics, light industries, energy resources, transportation, medicine, health, agriculture, forestry, geography, hydrology, seismology, meteorology, environment protection, architecture, behavioral science, management science, law, education, military science, etc. These practical applications have brought forward definite and noticeable social and economic benefits. It demonstrates a wide range of applicability of grey system theory, especially in the situation where the available information is incomplete and the collected data are inaccurate. , – The reader is given a general picture of grey systems theory as a new model system and a new framework for studying problems where partial information is known; especially for uncertain systems with few data points and poor information. The problems remaining for further studying are identified at the end of each section.

New progress of Grey System Theory in the new millennium

The scientific background that grey systems theory comes into being, the astonishing progress that grey systems theory has made in the world of learning and its wide-ranging applications in the entire spectrum of science, and the characteristics of unascertained systems include incomplete information and inaccuracies in data are presented in this paper. The scientific principle of simplicity and how precise models suffer from inaccuracies are also shown. We compared grey systems with other kinds of uncertainty models such as stochastic probability, rough set theory, and fuzzy mathematics. Finally, the elementary concepts and fundamental principles of grey systems, and main components of grey systems theory are introduced briefly.

A grey-based rough approximation model for interval data processing

To smooth the randomness, a grey forecasting model is formulated using the data of accumulating generation operator (AGO) rather than original data Then the inverse accumulating generation operator (IAGO) is applied to find the predicted values of original data It is proved that the errors from IAGO are affected by the order number of AGO To achieve an accurate prediction, GM(2,1), which stands for one-variable and second-order differential equation, has been improved by means of fractional order AGO Finally, four real data sets are imported for comparing the performance of the developed GM(2,1) with several other grey models, such as traditional GM(2,1) and GM(1,1) The simulation results show that optimized GM(2,1) has higher performances not only on model fitting but also on forecasting

Using fractional order accumulation to reduce errors from inverse accumulated generating operator of grey model

Changing the linear interpolation coefficient of the background value and the boundary value in GM(1,1) model is a way to improve the precision of GM(1,1) model.For a sequence satisfying the conditions of GM(1,1) modeling,a new method of improving the precision of GM(1,1) model by using the particle swarm optimization is developed.The computation of an example shows that this method may enhance the precision of GM(1,1) model.

Improving the Precision of GM(1,1) Model by Using Particle Swarm Optimization

The differences and relations between grey sets and rough sets are discussed. Based on the concept of a rough grey set, its entropy is studied. Moreover, the grey inference resulting from a rough grey set is pointed out. The relative propositions and theorems are obtained.

Rough grey sets

Association grey knowledge (in brief spoken of as grey knowledge) is introduced. By making use of difference information entropy, the grey degree which measures the grey characteristics of grey knowledge is given. Some theorems and propositions related to the grey degree and grey characteristics of grey knowledge are obtained. The bigger (or smaller) is the grey degree of grey knowledge, the greater (or lesser) is its grey characteristics, vice versa. It is pointed out that grey knowledge may extend or be transmuted white knowledge.

Measuring grey characteristics of association grey knowledge

How to choose the distinguishing coefficient will affect the distinguishing information degree or grey relational order of relational analysis.Optimization method of the distinguishing coefficient is set up,on the basis of the integration of genetic algorithm(GA)and the difference information entropy in grey theory.Lastly,a sample is shown.

Genetic Algorithm-Based Optimization of the Distinguishing Coefficient in Relational Analysis

When GM(1,1) is used in the life prediction of mechanical products, the experiment cost is saved and the experimental period is shortened. The research on the whitening differential equation of GM(1,1) indicates that both the background value and the initial value of GM(1,1) have important effect on the prediction accuracy. In this paper, a new PSO-GM(1,1, lambda, b ) optimization model has been proposed to improve the prediction accuracy of GM(1,1). Firstly GM(1,1) has been improved with the difference scheme and made use of the parameter lambda to correct the background value; secondly the parameter b has been used to correct the initial value; lastly because of the nonlinear traits between the parameters lambda , b and the prediction errors, particle swarm optimization(PSO) has been used to solve the best value of lambda, b according to the criterion of minimizing the absolute value of mean relative error, then a new PSO-GM(1,1, lambda, b ) optimization model has been constructed. The correction deduction and PSO calculation lambda , b show that the prediction accuracy of the PSO-GM (1,1, lambda, b) is much higher than that of the GM (1, 1). The two practical examples about the life prediction of mechanical products show that the optimization method of PSO-GM(1,1, lambda, b ) is remarkable.

Zhang Qi-shan

Papers

Improving the Precision of GM(1,1) Model by Using Particle Swarm Optimization

Rough grey sets

Measuring grey characteristics of association grey knowledge

Genetic Algorithm-Based Optimization of the Distinguishing Coefficient in Relational Analysis

Life prediction of mechanical products of GM(1,1) based on particle swarm optimization