Fractional derivative multivariable grey model for nonstationary sequence and its application

Abstract

Most of the existing multivariable grey models are based on the 1-order derivative and 1-order accumulation, which makes the parameters unable to be adjusted according to the data characteristics of the actual problems. The results about fractional derivative multivariable grey models are very few at present. In this paper, a multivariable Caputo fractional derivative grey model with convolution integral CFGMC(q, N) is proposed. First, the Caputo fractional difference is used to discretize the model, and the least square method is used to solve the parameters. The orders of accumulations and differential equations are determined by using particle swarm optimization (PSO). Then, the analytical solution of the model is obtained by using the Laplace transform, and the convergence and divergence of series in analytical solutions are also discussed. Finally, the CFGMC(q, N) model is used to predict the municipal solid waste (MSW). Compared with other competition models, the model has the best prediction effect. This study enriches the model form of the multivariable grey model, expands the scope of application, and provides a new idea for the development of fractional derivative grey model.

An optimized continuous fractional grey model for forecasting of the time dependent real world cases

“…Traditional grey models are defined with first order whitening differential equations and they are characterized by their simplicity. However, if the case data is a disordered sequence, the characteristic features of the sequence may not exactly find out by first order accumulative generation operation (AGO) [40]. In addition, first order derivative models are ideal memory models, which are not suitable for describing irregular phenomena.…”

“…As a result of this, the parameters of the model may not be compatible according to the data characteristics of the actual problem for a sequence with large data fluctuation. Therefore, fractional accumulation generating operation and fractional derivative should be introduced into the grey model to overcome this problem [40].…”

“…A fractional grey model FANGM(1,1,k,c) is proposed by [35]. Mao et al introduced the fractional grey model FGM(q,1) [20], a power-driven fractional accumulated grey model named PFAGM is proposed by [42], Yuxiao et al [40] proposed the multivariable Caputo fractional derivative grey model with convolution integral named CFGMC(q,N).…”

Prediction of Coalbed Methane Production in China Based on an Optimized Grey System Model

“…In the past 30 years, scholars have made a series of achievements on the grey forecasting models, and their development can be briefly summed up from three dimensions. First, the model structure has gone through a model suitable for modeling an approximately homogeneous exponential sequence, a model suitable for modeling approximately nonhomogeneous exponential sequence, − and adaptive models with intelligent adjustable structure. , Second, the modeling object has expanded from the single real-number sequence to various modeling sequences such as interval grey number sequence, discrete grey number sequence and grey isomerism data sequence. − Third, in terms of parameter optimization, some models have been upgraded in the optimization of the initial value, , the background value , and the order. , These studies enriched and improved the system of grey forecasting model, promoted the wide application of grey forecasting model, − and laid a theoretical foundation for successfully solving various practical problems.…”

Generalized fractional grey system models: Memory effects perspective

“…Yang (2016) [11] Continuous fractional-order grey model GM(q,1)/GM(q,N) -Yang (2018) [31] Interval grey modelling based on fractional calculus Interval GM(1,1) -Wu (2018) [32] The GMC( [40] Fractional derivative multivariable grey model CFGMC(q,N) Gao(2020) [41] fractional grey Riccati model FGRM(1,1) -Mao (2020) [29] Fractional grey model based on nonsingular exponential kernel EFGM(q,1)…”