Unified representation, properties and applications for fractional-order grey generating operators

Purpose The purpose of this paper is to explore data-driven approaches to determining the order and weight coefficients of grey sequence operators based on data characteristics. The absence of a unified structural framework for fractional-order grey sequence operators has hindered research and theoretical development concerning their underlying mechanisms. Choosing an appropriate accumulation order represents a key strategy for improving the modeling performance of uncertain systems. However, few studies have explored data-driven approaches to determining the order and weight coefficients of grey sequence operators based on data characteristics. Design/methodology/approach First, based on Morera’s theorem, the Gamma function is analytically continued, and theoretical proofs are provided for the unified expressions of classical grey generation operators, including fractional-order accumulation and differencing, reverse fractional-order accumulation and differencing, as well as their weighted variants. Next, from the perspective of matrix operators, the algebraic properties and convexity and/or concavity of these grey generation operators are investigated, and sufficient conditions for the weighted reverse fractional-order accumulation operator are derived under the principle of new information priority. Finally, numerical simulations are conducted to verify the performance of the proposed operators on four classic types of data characteristic sequences: increasing concave, increasing convex, decreasing concave and decreasing convex. Findings When the accumulation order satisfies r = 1, or the reverse accumulation order satisfies r = 2, the grey-generated sequence X(r) is a concave sequence. When the accumulation order satisfies 0 <r < 1, the fractional-order accumulated grey generation operator adheres to the new information priority principle; when r > 1, the fractional-order reverse accumulated grey generation operator satisfies this principle. Furthermore, when 0 < r ≤ 1 and λ > 1/r, or when r > 1 and λ > (n−1)/(n + r−2), the weighted fractional-order reverse accumulated grey generation operator also satisfies the new information priority principle. In Case 1, when r = 0.2209 and λ = 1.5799, the comprehensive error (MAPE) of the WFTDGM(1,1) model is 2.1599%, indicating significantly higher accuracy compared to the benchmark model. In Case 2, the MAPE value of the FTDGOM(1,1) model is 0.1774%, demonstrating the highest fitting accuracy relative to the benchmark models. Practical implications To address the uncertainties in system predictions related to power consumption in Ningxia and energy intensity in the United States, an empirical analysis was conducted using the FTDGM(1,1), WFTDGM(1,1) and FTDGOM(1,1) models. Originality/value The unified structural form of the fractional-order grey generation operator ensures intrinsic consistency between the cumulative and decumulative operations, which are inverse to each other. The algebraic properties of the fractional-order grey generation operator provide a theoretical foundation for enhancing the efficiency of grey system modeling. An empirical analysis of power consumption in Ningxia offers a scientific basis for decision-making in power and energy planning.