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The purpose of this paper is to propose a new method based on nonlinear least squares (NLS) for solving the parameters of nonlinear grey Bernoulli model (NGBM(1,1)) and to verify the proposed model using the case of employee demand prediction of high-tech enterprises in China.

First of all, minimising the square sum of fitting error of grey differential equation of NGBM(1,1) is taken as the optimisation target and the parameters of classic grey model (GM(1,1)) are set as the initial value of parameter vector. Afterwards, the structural parameters and power exponents are solved by using the Gauss-Newton iteration algorithm so as to calculate the parameters of NGBM(1,1) under given rules for ceasing the algorithm. Finally, by taking the employee demand of high-tech enterprises in the state-level high-tech industrial development zone in China as examples, the validity of the new method is verified.

The results show that the parameter estimation algorithm based on the NLS method can effectively identify the power exponents of NGBM(1,1) and therefore can favourably adapt to the nonlinear fluctuations of sequences. In addition, the algorithm is superior to the GM(1,1) model, grey Verhulst model, and Quadratic-Exponential smoothing algorithm in terms of the simulation and prediction accuracy.

Under the framework of solving parameters based on NLS, various aspects of NGBM(1,1) remain to be further investigated including background value, initial condition and variable structural modelling methods.

The parameter estimation algorithm based on NLS can effectively identify the power exponent of NGBM(1,1) and therefore it can favourably adapt to the nonlinear fluctuation of sequences.

According to the basic principle of NLS, a new method for solving the parameters of NGBM(1,1) is proposed by using the Gauss-Newton iteration algorithm. Moreover, by conducting the modelling case about employees demand in high-tech enterprises in China, the effectiveness and superiority of the new method are verified.

The purpose of this paper is to find out which algorithm, among Genetic Algorithm (GA), Particle Swarm Optimizer (PSO), the novel Grey Wolf Optimizer (GWO) and the novel Ant Lion Optimizer (ALO), is the best to obtain the optimal value of the nonlinear parameter γ of nonlinear grey Bernoulli model (NGBM(1,1)) under different situations.

The optimization of γ has been attributed to a nonlinear programming problem at first. The convergence, convergence rate, time consuming and stability of GA, PSO, GWO and ALO are compared in the numerical experiments, and in each subcase the criteria are set to be the same. Over 10,000 iterations have been run on the same environment in order to guarantee the reliability of the results.

All the selected algorithms can converge to the same optimal value with sufficient iterations. But the best algorithm should be chose under different situations.

The optimal value of γ seems to exist uniquely due to the empirical results. And there does not exist a best algorithm for all the cases. The researchers and commercial software developers should choose a proper algorithm due to different cases.

The performance of GA, PSO, GWO and ALO to compute the optimal γ of NGBM(1,1) has been compared for the first time. And it is the original work which uses the GWO and ALO to optimize the NGBM(1,1).

A new method for forecasting wind turbine capacity of China is proposed through grey modelling technique.

First of all, the concepts of discrete grey model are introduced into the NGBM(1,1) model to reduce the discretization error from the differential equation to its discrete forms. Then incorporating the conformable fractional accumulation into the discrete NGBM(1,1) model is carried out to further improve the predictive performance. Finally, in order to effectively seek the emerging coefficients, namely, fractional order and nonlinear coefficient, the whale optimization algorithm (WOA) is employed to determine the emerging coefficients.

The empirical results show that the newly proposed model has a better prediction performance compared to benchmark models; the wind turbine capacity from 2019 to 2021 is expected to reach 275954.42 Megawatts in 2021. According to the forecasts, policy suggestions are provided for policy-makers.

By combing the fractional accumulation and the concepts of discrete grey model, a new method to improve the prediction performance of the NGBM(1,1) model is proposed. The newly proposed model is firstly applied to predict wind turbine capacity of China.

This study aims to achieve long-term prediction on a specific monotonic data series of atmospheric corrosion rate vs time.

This paper presents a new method, used to the collected corrosion data of carbon steel provided by the China Gateway to Corrosion and Protection, that combines non-linear gray Bernoulli model (NGBM(1,1) with genetic algorithm to attain the purpose of this study.

Results of the experiments showed that the present study’s method is more accurate than other algorithms. In particular, the mean absolute percentage error (MAPE) and the root mean square error (RMSE) of the proposed method in data sets are 9.15 per cent and 1.23 µm/a, respectively. Furthermore, this study illustrates that model parameter can be used to evaluate the similarity of curve tendency between two carbon steel data sets.

Corrosion data are part of a typical small-sample data set, and these also belong to a gray system because corrosion has a clear outcome and an uncertainly occurrence mechanism. In this work, a new gray forecast model was proposed to achieve the goal of long-term prediction of carbon steel in China.

Because of the high volatility with unstable data patterns in the real world, the ability of forecasting price indices is notoriously embarrassing and represents a major challenge with traditional time series mechanisms; especially, most of the traditional approaches are weak to forecast future predictions in the high volatile and unbalanced frameworks under the global and local financial depressions. The purpose of this paper is to propose a new statistical approach for portfolio selection and stock market forecasting to assist investors as well as stock brokers to predict the future behaviors.

This study mainly takes an attempt to understand the trends, behavioral patterns and predict the future estimations under the new proposed frame for the Colombo Stock Exchange (CSE), Sri Lanka. The methodology of this study is carried out under the two main phases. In the first phase, constructed a new portfolio mechanism based on k-means clustering. In the second stage, proposed a nonlinear forecasting methodology based on grey mechanism for forecasting stock market indices under the high-volatile fluctuations. The autoregressive integrated moving average (ARIMA) predictions are used as comparison mode.

Initially, the k-mean clustering was applied to pick out the profitable sectors running under the CSE and results indicated that BFI is more significant than other 20 sectors. Second, the MAE, MAPE and MAD model comparison results clearly suggested that, the newly proposed nonlinear grey Bernoulli model (NGBM) is more appropriate than traditional ARIMA methods to forecast stock price indices under the non-stationary market conditions.

Because of the flexible nonlinear modeling capability, proposed novel concepts are more suitable for applying in various areas in the field of financial, economic, military, geological and agricultural systems for pattern recognition, classification, time series forecasting, etc.

For the large sample of data forecasting under the normality assumptions, the traditional time series methodologies are more suitable than grey methodologies. However, the NGBM is better both in model building and ex post testing stagers under the s-distributed data patterns with limited data forecastings.

This paper aims to develop a novel grey Bernoulli model with memory characteristics, which is designed to dynamically choose the optimal memory kernel function and the length of memory dependence period, ultimately enhancing the model's predictive accuracy.

This paper enhances the traditional grey Bernoulli model by introducing memory-dependent derivatives, resulting in a novel memory-dependent derivative grey model. Additionally, fractional-order accumulation is employed for preprocessing the original data. The length of the memory dependence period for memory-dependent derivatives is determined through grey correlation analysis. Furthermore, the whale optimization algorithm is utilized to optimize the cumulative order, power index and memory kernel function index of the model, enabling adaptability to diverse scenarios.

The selection of appropriate memory kernel functions and memory dependency lengths will improve model prediction performance. The model can adaptively select the memory kernel function and memory dependence length, and the performance of the model is better than other comparison models.

The model presented in this article has some limitations. The grey model is itself suitable for small sample data, and memory-dependent derivatives mainly consider the memory effect on a fixed length. Therefore, this model is mainly applicable to data prediction with short-term memory effect and has certain limitations on time series of long-term memory.

In practical systems, memory effects typically exhibit a decaying pattern, which is effectively characterized by the memory kernel function. The model in this study skillfully determines the appropriate kernel functions and memory dependency lengths to capture these memory effects, enhancing its alignment with real-world scenarios.

Based on the memory-dependent derivative method, a memory-dependent derivative grey Bernoulli model that more accurately reflects the actual memory effect is constructed and applied to power generation forecasting in China, South Korea and India.

The widespread application of traditional grey model (GM) in different academic fields such as electrical engineering, education, mechanical engineering and agriculture provided the authors with an incentive to conduct the present empirical research in an accounting field, in particular, auditing practice. In this regard, the purpose of this paper is to employ the nonlinear type of the original GM to forecast the drastically changed data on audit reports, primarily due to the fact that the linear nature of GM is unable to forecast nonlinear data precisely. In essence, this paper adds value to the strand of audit report literature by examining the impact of different financial ratios on auditors’ opinion and then forecasting audit reports by employing GMs.

The grey forecasting model is known as a system containing uncertain information presented by grey numbers, equations and matrices. The grey forecasting model is employed by using a differential equation in an uncertain system with limited data set which is suitable for smoothing discrete data. In addition, the analyses are conducted by applying a sample of top 50 listed companies on the Tehran Stock Exchange during 2011-2016.

The findings suggest that audit reports are most influenced by the current ratio and conversely, least influenced by the ratio of working capital turnover. Moreover, the authors argue that the Nash nonlinear grey Bernoulli model is more precise than the nonlinear grey Bernoulli model and GM in forecasting audit reports.

The current study may give more strength to stakeholders in order to analyse and forecast audit report.

Although the Nash nonlinear grey Bernoulli model (NNGBM(1, 1)) is incomparable with respect to its flexibility over traditional grey models, errors are still inevitable in forecasting. The purpose of this paper is to propose a Fourier residual modified Nash nonlinear grey Bernoulli model (FNNGBM(1, 1)) and use it to forecast the nonlinear time series of the international trade of Chinese high-tech products.

A Fourier series is used to modify the forecasting residual of the NNGBM(1, 1) model, so as to improve its forecasting ability. The parameters optimization of FNNGBM(1, 1) is formulated as a combinatorial optimization problem and is solved collectively using the concept of Nash equilibrium.

The simulation and practical application to fluctuation data both prove that FNNGBM(1, 1) could offer a more precise forecast than NNGBM(1, 1) and the Fourier residual GM(1, 1) (FGM(1, 1)). The import/export data of Chinese high-tech products will maintain rapid growth, with corresponding trade balance enlargement; however, there will be a concomitant decrease in the trade specialization coefficient.

This study is deliberately general in its scope and outlook: its focus is mainly on the overall condition of Chinese high-tech products trade. Future research is recommended to analyze specific industrial trade sectors and extraneous influencing factors.

An effective method is proposed to enhance the accuracy of NNGBM(1, 1) model in forecasting a small sample, nonlinear time series.

This paper intends to construct a new adaptive grey seasonal model (AGSM) to promote the application of the grey forecasting model in quarterly GDP.

Firstly, this paper constructs a new accumulation operation that embodies the new information priority by using a hyperparameter. Then, a new AGSM is constructed by using a new grey action quantity, nonlinear Bernoulli operator, discretization operation, moving average trend elimination method and the proposed new accumulation operation. Subsequently, the marine predators algorithm is used to quickly obtain the hyperparameters used to build the AGSM. Finally, comparative analysis experiments and ablation experiments based on China's quarterly GDP confirm the validity of the proposed model.

AGSM can be degraded to some classical grey prediction models by replacing its own structural parameters. The proposed accumulation operation satisfies the new information priority rule. In the comparative analysis experiments, AGSM shows better prediction performance than other competitive algorithms, and the proposed accumulation operation is also better than the existing accumulation operations. Ablation experiments show that each component in the AGSM is effective in enhancing the predictive performance of the model.

A new AGSM with new information priority accumulation operation is proposed.

With the massive use of fossil energy polluting the natural environment, clean energy has gradually become the focus of future energy development. The purpose of this article is to propose a new hybrid forecasting model to forecast the production and consumption of clean energy.

Firstly, the memory characteristics of the production and consumption of clean energy were analyzed by the rescaled range analysis (R/S) method. Secondly, the original series was decomposed into several components and residuals with different characteristics by the ensemble empirical mode decomposition (EEMD) algorithm, and the residuals were predicted by the fractional derivative grey Bernoulli model [FDGBM (p, 1)]. The other components were predicted using artificial intelligence (AI) models (least square support vector regression [LSSVR] and artificial neural network [ANN]). Finally, the fitting values of each part were added to get the predicted value of the original series.

This study found that clean energy had memory characteristics. The hybrid models EEMD–FDGBM (p, 1)–LSSVR and EEMD–FDGBM (p, 1)–ANN were significantly higher than other models in the prediction of clean energy production and consumption.

Consider that clean energy has complex nonlinear and memory characteristics. In this paper, the EEMD method combined the FDGBM (P, 1) and AI models to establish hybrid models to predict the consumption and output of clean energy.