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The purpose of this paper is to propose a new grey system model used for prediction.

It had been proven that the GM(1,1) model is a biased exponential model, the model is fit for non‐negative raw data, which accord with or basically accord with the exponential form and do not have a quick growth rate. Based on the results, an unbiased GM(1,1) model was proposed. With the method of transforming every datum of raw data sequence into its 2‐th root, a new data sequence from the raw data sequence can be produced. The new data sequence is used to establish an unbiased GM(1,1) model and statistical experiments and a practical example in load forecasting are given in the paper.

The results of statistical experiments and a practical example in load forecasting show the proposed method is effective in increasing the accuracy of the model.

The model exposed in the paper can be used for constructing models of prediction in many fields such as agriculture, electric power, IT, transportation, economics, management, etc.

The paper succeeds in proposing a modified unbiased GM(1,1) model that has high accuracy. The model is applied to the field of load forecasting and the results show the model is better than the unbiased GM(1,1) model. The model proposed has great theoretical and practical value.

The paper aims to analyze some properties of GM(1,1,β) model based on the principle that the grey GM(1,1) model parameters are grey and adjustable.

At first, according to the principle that grey GM(1,1) model parameters are grey and adjustable, and the GM(1,1,β) model with parameter packet is put forward. Second, some properties of the GM(1,1,β) model are discussed, and the applicable region of the GM(1,1,β) model is given based on the grey differential equation of the GM(1,1,β) model. At last, the background value coefficient's calculation formula and optimization algorithm of the GM(1,1,β) model are also given. A numeric example is also computed in the last part of the paper.

The result of the study shows that the application scope of the GM(1,1,β) model is (−8,+8).

The GM(1,1,β) model provides the theoretical basis for the GM(1,1) model's optimization and can hence forecast its precision.

The paper succeeds in realizing the GM(1,1) model's application scope (−2,+2) is broadened to (−8,+8).

The purpose of this paper is to enhance the forecast precision of GM(1,1) model using an improved artificial fish swarm algorithm.

An optimization model of GM(1,1) model about identifying the parameters is proposed, which takes the minimum of the average relative error as objective function and takes the development coefficient and grey action quantity as decision variables, then an improved artificial fish swarm algorithm is designed to solve the optimization model.

The results show that the proposed method may enhance the precision of GM(1,1) model, and have better performance than particle swarm optimization.

The method exposed in the paper can be used to optimize the parameters of GM(1,1) model, which is used frequently to solve the economic and management problem.

The paper succeeds in enhancing the forecast precision of GM(1,1) model using an improved artificial fish swarm algorithm.

The purpose of this paper is to study the parameters' properties of GM(n, h) model on the basis of multiple transformation and the relationship of GM(n, h) model and other grey models.

Multiple transformation property of parameters is important to construct a grey model. However, there is no research on the property of GM(n, h) model, therefore it is meaningful to study the relationship between GM(n, h) model and other grey models.

The multiple transformation property of parameters of GM(n, h) model is recognized. The parameters' value is dependent on multiple transformation value. The values of simulative and predicative are only dependent to the multiple transformation of the main variable and independent to other variables.

The properties of other grey models could be obtained by analyzing the property of GM(n, h) model.

It is a very useful result for constructing a grey model.

This paper discusses multiple transformation property of GM(n, h) model and the relationship between the GM(n, h) model and other grey models. These grey models are put into a common model and the affections that parameters' multiple transformation caused to the model are studied.

The purpose of this paper is to establish a new model for non‐equidistance sequence and research affine properties of the new model.

Generalized non‐equidistance GM(1,1) model is put forward based on generalized accumulated generating operation (AGO) theory, and particle swarm optimization is used to solve the parameters of the new model, then affine properties of the new model are researched based on matrix analysis.

The results are convincing: the simulation and prediction precisions of generalized non‐equidistance GM(1,1) model are raised greatly, and it is proved that the affine transformation sequence has the same simulative accuracy with the raw sequence for generalized non‐equidistance GM(1,1) model.

The method exposed in the paper can be used to model and predict for non‐equidistance sequence in the practical problem.

The paper succeeds in establishing a new non‐equidistance grey model and obtaining the affine properties of generalized non‐equidistance GM(1,1) model.

The purpose of this paper is to expand discrete GM (1,1) model and solve the problem of non‐equidistance grey prediction problem with integral interval or digital interval.

Discrete GM (1,1) model can be utilized to simulate exponential sequence without errors, but it can't be utilized to simulate non‐equidistance data sequence. This paper applied optimization theories to establish generalized discrete GM (1,1) model. First, this paper established the time response of simulation sequence directly. Second, this paper established the steps of non‐equidistance data sequence. Finally, this paper utilized examples to test the method put forward.

The results indicate the generalized discrete GM (1,1) (GDGM) model can perfectly simulate non‐equidistance exponential series. Discrete GM (1,1) model is only the special form of GDGM model.

Though grey forecasting models are widely used, most of the forecasting models are based on the equal distance sequence. Due to many reasons, the raw data available usually is incomplete. There are mainly four reasons which caused non‐equidistance sequence. So generalized discrete GM (1,1) model can be utilized to simulate non‐equidistance sequence and has great application values.

The paper succeeds in establishing a generalized discrete GM (1,1) model which can be utilized to solve non‐equidistance data sequence forecasting. The GDGM model can be solved by MATLAB or other corresponding software.

The purpose of this paper is to provide a new recursive GM (1,1) model based on forgetting factor and apply it to the modern weapon and equipment system.

In order to distinguish the contribution of new and old data to the grey prediction model with new information, the authors add forgetting factor to the objective function. The purpose of the above is to realize the dynamic weighting of new and old modeling data, and to gradually forget the old information. Second, the recursive estimation algorithm of grey prediction model parameters is given, and the new information is added in real time to improve the prediction accuracy of the model.

It is shown that the recursive GM (1,1) model based on forgetting factor can achieve both high effectiveness and high efficiency.

The paper succeeds in proposing a recursive GM (1,1) model based on forgetting factor, which has high accuracy. The model is applied to the field of modern weapon and equipment system and the result the model is better than the GM(1,1) model. The experimental results show the effectiveness and the efficiency of the prosed method.

The purpose of this paper is to establish a random simulation method to compare the forecasting performance between grey prediction models, and between grey model and other kinds of prediction models. Then, the different performance of three grey models and linear regression prediction model is studied, based on the proposed method.

A random simulation method was proposed to test the modelling accuracy of grey prediction model. This method was enlightened by Monte Carlo simulation method. It regarded a class of sequences as population, and selected a large sample from population though random sampling. Then, sample sequences were modeled by grey prediction model. Through modeling error calculation, the average error of grey model for the sample was obtained. Finally, the grey model accuracy for this kind of problem was acquired by statistical inference testing model. Through the statistical significant test method, the modeling accuracy of grey models for the same problem can be compared. Also, accuracy difference between grey prediction model and regression analysis, support vector machine, neural network, and other forecasting methods can be also compared.

Though random simulation experiments, the following conclusion was obtained. First, grey model can be applied to the long sequence whose growth rate was less than 20 per cent, and the short sequence whose growth rate was less than 50 per cent. Second, GM(1,1) cannot be applied to a long sequence with high growth. Third, growth rate was a more important factor than growth length on modeling accuracy of GM(1,1). Fourth, when the sequence length was short, accuracy of GM(1,1) model was higher than linear regression. While the length of the sequence was more than 15, and the growth rate in [0‐10 per cent], two kinds of modeling error was not significantly different.

The method proposed in the paper can be used to compare the performance of different prediction models, and to select appropriate model for a prediction problem.

The paper succeeded in establishing an accuracy test method for grey models and other prediction models. It will standardize the grey modelling and contribute to application of grey models.

The purpose of this paper is to apply grey system theory to population system and project China's population.

The paper applies the GM(1,1) model to China's population projections. Two key aspects of the method are crucial for obtaining best accuracy of prediction. They are the choice of the length for the original data to be used in the model and the adoption of the GM(1,1) metabolic model in prediction. The former determines what initial data to be used while the latter describes an iteration process on how to proceed to predict.

The results show that in 2015 China's population will reach 1.37 billion and in 2050 it will be between 1.42 and 1.48 billion, which is in accordance with the latest projections from the UN. The findings show the GM(1,1) metabolic model is an effective mathematical means in population projections.

The paper suggests that GM(1,1) metabolic model can provide an effective simulation model for complicated systems with uncertainty and can be used in many fields.

The paper provides useful advice for the department of population.

Most population projections have been based on assumptions about fertility, mortality, and migration. The paper considers the population system as a grey system and introduces the GM(1,1) metabolic model to population projections.

Smog 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.

This 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.

In 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.

The 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.

Based 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.