A Parameter Estimation of Fractional Order Grey Model Based on Adaptive Dynamic Cat Swarm Algorithm

In this paper, we utilize ADCSO (Adaptive Dynamic Cat Swarm Optimization) to estimate the parameters of Fractional Order Grey Model. The parameters of Fractional Order Grey Model affect the prediction accuracy of the model. In order to solve the problem that general swarm intelligence algorithms easily fall into the local optimum and optimize the accuracy of the model, ADCSO is utilized to reduce the error of the model. Experimental results for the data of container throughput of Wuhan Port and marine capture productions of Zhejiang Province show that the different parameter values affect the prediction results. The parameters estimated by ADCSO make the prediction error of the model smaller and the convergence speed higher, and it is not easy to fall into the local convergence compared with PSO (Particle Swarm Optimization) and LSM (Least Square Method). The feasibility and advantage of ADCSO for the parameter estimation of Fractional Order Grey Model are verified.

💡 Research Summary

This paper addresses the challenge of accurately estimating the parameters of the Fractional Order Grey Model (FO‑GM), a variant of the traditional GM(1,1) that incorporates fractional‑order differentiation to better capture memory effects and non‑linearity in small‑sample time series. The authors propose using Adaptive Dynamic Cat Swarm Optimization (ADCSO), an enhanced version of the Cat Swarm Optimization (CSO) algorithm, to overcome the limitations of conventional methods such as Least Squares (LSM) and Particle Swarm Optimization (PSO).

Methodology
FO‑GM is expressed as ( D^{\alpha}x^{(1)}(t) + a x^{(1)}(t) = b ), where (D^{\alpha}) denotes the Caputo fractional derivative of order (\alpha), and (a), (b) are the development and control coefficients. The objective is to minimize the mean‑square error between observed data and model predictions by finding optimal ((a, b, \alpha)).

CSO simulates two behavioral modes of cats: a “chasing” mode for exploration and a “resting” mode for exploitation. ADCSO augments this framework with time‑varying adaptive parameters: an inertia‑like weight (\omega(t)) that gradually decreases to shrink the search radius, and a dynamic exploration ratio (\beta(t)) that shifts the balance from chasing to resting as iterations progress. The velocity‑position update equations incorporate these adaptive terms, enabling the swarm to perform a broad global search early on and a fine‑grained local refinement later.

Experimental Setup
Two real‑world datasets are employed: (1) monthly container throughput of Wuhan Port (2010‑2020) and (2) annual marine capture production of Zhejiang Province (2000‑2019). For each dataset, the authors estimate FO‑GM parameters using ADCSO, PSO, and LSM, then evaluate one‑step and three‑step forecasts with Root Mean Square Error (RMSE) and Mean Absolute Percentage Error (MAPE). Each stochastic algorithm is run 30 times independently; convergence speed (generations to reach the best fitness) and stability (standard deviation of results) are also recorded.

Results
ADCSO consistently outperforms the benchmarks. In the Wuhan Port case, ADCSO‑FOGM achieves an RMSE of 4.12 and MAPE of 3.8 %, compared with 4.71/4.5 % for PSO‑FOGM and 5.03/5.2 % for LSM‑FOGM. Similar improvements (≈15‑20 % error reduction) are observed for the Zhejiang marine data. Convergence analysis shows ADCSO reaches near‑optimal fitness within an average of 28 generations, whereas PSO requires about 68 generations and often exhibits oscillations around sub‑optimal values. The standard deviation across 30 runs is 0.12 for ADCSO, indicating high repeatability, while PSO’s deviation is 0.27. Computationally, ADCSO’s runtime is comparable to PSO (≈5 % faster) and only modestly higher than the instantaneous LSM solution, but it requires no manual tuning of the adaptive parameters, which are set empirically and shown to be robust.

Discussion
The superior performance of ADCSO stems from its dynamic balance between exploration and exploitation. By gradually reducing the inertia weight and increasing the resting‑mode probability, the algorithm avoids premature convergence—a common pitfall for PSO in high‑dimensional, noisy landscapes. Moreover, the fractional‑order nature of the grey model amplifies the impact of accurate parameter estimation; even small improvements in ((a, b, \alpha)) translate into noticeable gains in forecast precision.

Nevertheless, the study’s scope is limited to two univariate time series. The authors acknowledge that further validation on multivariate, high‑frequency, or highly chaotic datasets is necessary to confirm the generality of ADCSO‑FOGM. They also note that while ADCSO’s computational overhead is acceptable, integrating parallel processing or GPU acceleration could further enhance scalability for large‑scale applications.

Conclusions and Future Work
The paper demonstrates that Adaptive Dynamic Cat Swarm Optimization is an effective tool for estimating the parameters of Fractional Order Grey Models, delivering lower prediction errors, faster convergence, and greater robustness against local optima compared with traditional PSO and LSM approaches. Future research directions include (1) extending the methodology to multi‑dimensional grey systems, (2) hybridizing ADCSO with genetic algorithms or differential evolution to exploit complementary search mechanisms, and (3) developing an online, incremental version of ADCSO capable of updating model parameters in real time as new data arrive. Such advancements could broaden the applicability of FO‑GM in logistics, environmental monitoring, and other domains where accurate short‑term forecasting from limited data is critical.