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The purpose of this paper is to present the terms of sequence operators and grey data mining.

The definitions of basic terms about sequence operators and grey data mining are presented one by one.

The reader could know the basic explanation about the important terms about sequence operators and grey data mining from this paper.

Many of the colleagues thought that unified definitions of key terms would be beneficial for both the readers and the authors.

It is a fundamental work to standardize all the definitions of terms for a new discipline. It is also propitious to spread the universal principles of grey system theory.

The purpose of this paper is to derive the analytical expression of fractional order reducing generation operator (or inverse accumulating generating operation) and study its properties.

This disaggregation method includes three main steps. First, by utilizing Gamma function expanded for integer factorial, this paper expands one order reducing generation operator into integer order reducing generation operator and fractional order reducing generation operator, and gives the analytical expression of fractional order reducing generation operator. Then, studies the commutative law and exponential law of fractional order reducing generation operator. Lastly, gives several examples of fractional order reducing generation operator and verifies the commutative law and exponential law of fractional order reducing generation operator.

The authors pull the analytical expression of fractional order reducing generation operator and verify that fractional order reducing generation operator satisfies commutative law and exponential law.

Expanding the reducing generation operator would help develop grey prediction model with fractional order operators and widen the application fields of grey prediction models.

The analytical expression of fractional order reducing generation operator, properties of commutative law and exponential law for fractional order reducing generation operator are first studied.

The author aims to study and predict the sustainability governance performances of firms using an advanced grey prediction model. The case implication of the prediction model is also studied considering select firms in the Indian context.

The author has proposed an advanced grey prediction model, the first-entry grey prediction model (FGM (1, 1)) for forecasting the sustainability governance performances of firms. The proposed model is tested using the periodic data of sustainability governance performances of 10 Indian firms.

The author observes that the majority of firms (6 out of 10) show dipping performances for sustainability governance for the future predicted period. This throws insights into the direction of improving good governance practices for Indian firms.

The idea and motivation for sustainability-focussed governance need a bi-directional focus from the side of managers that act as the agents and from the side of shareholders that act as the principals, as seen from an agency theory perspective for sustainability governance.

Sustainability governance culture can be inculcated to a firm at the strategic level by having a bi-directional focus from managers and shareholders, so as to enhance the social and environmental sustainability performances.

The governance performance evaluations for firms particularly in developing countries were not dated back more than a decade or two. Hence, the author implements a prediction model that can be best suited, when there are small periodic data sets available for prediction.

Electricity consumption prediction has been an important topic for its significant impact on electric policies. Due to various uncertain factors, the growth trends of electricity consumption in different cases are variable. However, the traditional grey model is based on a fixed structure which sometimes cannot match the trend of raw data. Consequently, the predictive accuracy is variable as cases change. To improve the model's adaptability and forecasting ability, a novel fractional discrete grey model with variable structure is proposed in this paper.

The novel model can be regarded as a homogenous or non-homogenous exponent predicting model by changing the structure. And it selects the appropriate structure depending on the characteristics of raw data. The introduction of fractional accumulation enhances the predicting ability of the novel model. And the relative fractional order r is calculated by the numerical iterative algorithm which is simple but effective.

Two cases of power load and electricity consumption in Jiangsu and Fujian are applied to assess the predicting accuracy of the novel grey model. Four widely-used grey models, three classical statistical models and the multi-layer artificial neural network model are taken into comparison. The results demonstrate that the novel grey model performs well in all cases, and is superior to the comparative eight models.

A fractional-order discrete grey model with an adaptable structure is proposed to solve the conflict between traditional grey models' fixed structures and variable development trends of raw data. In applications, the novel model has satisfied adaptability and predicting accuracy.

Image edge detection is an essential issue in image processing and computer vision. The purpose of this paper is to provide a novel and effective algorithm for image edge detection.

Because GM (1,1) model is a typical model for tendency analysis, GM (1,1) model can be used for detecting edge. Prediction image data are close to the original image data by reason of the data being smooth in the non‐edge zone of image. The principle of edge detection by GM (1,1) model is that the predicted value at an edge point will be an overestimate or underestimate owing to the data changing drastically in the edge zone of the image. First, the edge image information is obtained by a preprocessed image subtracting from prediction image via GM (1,1). Second, median filter is used to eliminate isolated point noise in edge information images, and discrete wavelet transform is used to extract the image edge. Finally, this paper verifies the proposed algorithm by experiment.

Experimental results show that the proposed algorithm has advantages such as precisely locating, abundant weak edge, and better anti‐noise performance.

The algorithm proposed in the paper can precisely detect the information of edge image, and get a clear image detail.

Grey system theory developed vigorously lays the foundation for image processing. Wavelet analysis in image processing has its characteristics. This paper combines grey prediction model with discrete wavelet transform (DWT) successfully and obtains a novel and effective algorithm for image edge detection.

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.

Due to high efficiency and low carbon of natural gas, the consumption of natural gas is increasing rapidly, and the prediction of natural gas consumption has become the focus. The purpose of this paper is to employ a prediction technique by combining grey prediction model and trigonometric residual modification for predicting average per capita natural gas consumption of households in China.

The GM(1,1) model is utilised to obtain the tendency term, then the generalised trigonometric model is used to catch the periodic phenomenon from the residual data of GM(1,1) model for improving predicting accuracy.

The case verified the view of Xie and Liu: “When the value of a is less, DGM model and GM(1,1) model can substitute each other.” The combination of the GM(1,1) and the trigonometric residual modification technique can observably improve the predicting accuracy of average per capita natural gas consumption of households in China. The mean absolute percentage errors of GM(1,1) model, DGM(1,1), unbiased grey forecasting model, and TGM model in ex post testing stage (from 2013 to 2015) are 32.5510, 33.5985, 36.9980, and 5.2996 per cent, respectively. The TGM model is suitable for the prediction of average per capita natural gas consumption of households in China.

According to the historical data of average per capita natural gas consumption of households in China, the authors construct GM(1,1) model, DGM(1,1) model, unbiased grey forecasting model, and GM(1,1) model with trigonometric residual modification. The accuracy of TGM is the best. TGM helps to improve the accuracy of GM(1,1).

This paper gives a successful practical application of grey model GM(1,1) with the trigonometric residual modification, where the cyclic variations exist in the residual series. The case demonstrates the effectiveness of trigonometric grey prediction model, which is helpful to understand the modeling mechanism of trigonometric grey prediction model.

The global warming, caused by the anthropogenic greenhouse gases, has been one of the major worldwide issues over the last decades. Among them, carbon dioxide (CO₂) is the most important one and is responsible for more than the two-third of the greenhouse effect. Currently, greenhouse gas emissions and CO₂ emissions – the root cause of the global warming – in particular are being examined closely in the fields of science and they also have been put on the agenda of the political leaders. The purpose of this paper is to predict the energy-related CO₂ emissions through using different discrete grey models (DGMs) in Turkey and total Europe and Eurasia region.

The proposed DGMs will be applied to predict CO₂ emissions in Turkey and total Europe and Eurasia region from 2015 to 2030 using data set between 1965 and 2014. In the first stage of the study, DGMs without rolling mechanism (RM) will be used. In the second stage, DGMs with RM are constructed where the length of the rolling horizons of the respected models is optimised.

In the first stage, estimated values show that non-homogeneous DGM is the best method to predict Turkey’s energy-related CO₂ emissions whereas DGM is the best method to predict the energy-related CO₂ emissions for total Europe and Eurasia region. According to the results in the second stage, NDGM with RM (k=26) is the best method for Turkey while optimised DGM with RM (k=4) delivers most reliable estimates for total Europe and Eurasia region.

This study illustrates the effect of different DGM approaches on the estimation performance for the Turkish energy-related CO₂ emission data.

When performing system-level developmental testing, time and expenses generally warrant a small sample size for failure data. Upon failure discovery, redesigns and/or corrective actions can be implemented to improve system reliability. Current methods for estimating discrete (one-shot) reliability growth, namely the Crow (AMSAA) growth model, stipulate that parameter estimates have a great level of uncertainty when dealing with small sample sizes. The purpose of this paper is to present an application of a modified GM(1,1) model for handling system-level testing constrained by small sample sizes.

The paper presents a methodology for incorporating failure data into a modified GM(1,1) model for systems with failures following a poly-Weibull distribution. Notional failure data are generated for complex systems and characterization of reliability growth parameters is performed via both the traditional AMSAA model and the GM(1,1) model for purposes of comparing and assessing performance.

The modified GM(1,1) model requires less complex computational effort and provides a more accurate prediction of reliability growth model parameters for small sample sizes and multiple failure modes when compared to the AMSAA model. It is especially superior to the AMSAA model in later stages of testing.

This research identifies cost-effective methods for developing more accurate reliability growth parameter estimates than those currently used.

The purpose of this paper is to unify the expression of fractional grey accumulating generation operator and the reducing generation operator, and build the FDGM(1,1) model with the unified fractional grey generation operator.

By systematically studying the properties of the fractional accumulating operator and the reducing operator, and analyzing the sensitivity of the order value, a unified expression of the fractional operators is given. The FDGM(1,1) model with the unified fractional grey generation operator is established. The relationship between the order value and the modeling error distribution is studied.

The expression of the fractional accumulating generation operator and the reducing generation operator can be unified to a simple expression. For −1<r < 1, the fractional grey generation operator satisfies the principle of new information priority. The DGM(1,1) model is a special case of the FDGM(1,1) model with r = 1.

The sensitivity of the unified operator is verified through random numerical simulation method, and the theoretical proof was not yet possible.

The FDGM(1,1) model has a higher modeling accuracy and modeling adaptability than the DGM(1,1) by optimizing the order.

The expression of the fractional accumulating generation operator and the reducing generation operator is firstly unified. The FDGM(1,1) model with the unified fractional grey generation operator is firstly established. The unification of the fractional accumulating operator and the reducing operator improved the theoretical basis of grey generation operator.