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

The paper addresses the problem of estimating the parameters of a fractional‑order Grey Model (GM(1,1)), a forecasting tool that is particularly suited for small, incomplete data sets. While the traditional Grey Model uses integer‑order accumulation, the fractional‑order version introduces an r‑order accumulation operator that reduces perturbations and improves stability, but it also makes the model’s two key parameters— the development coefficient a and the background coefficient b—more critical to prediction accuracy. Conventional parameter estimation methods such as the Least Squares Method (LSM) minimize the sum of squared deviations rather than the average relative error, which can lead to sub‑optimal parameter values, especially when data contain outliers or are not normally distributed.

To overcome these limitations, the authors propose using Adaptive Dynamic Cat Swarm Optimization (ADCSO), an evolutionary algorithm inspired by cat behavior. ADCSO combines two complementary modes: a “seeking” mode that generates multiple candidate solutions around a cat’s current position and selects the best based on a probability proportional to fitness, and a “tracing” mode that updates a cat’s velocity toward its personal best and the global best using adaptive inertia weight and acceleration coefficients. The algorithm dynamically partitions the swarm into seeking and tracing groups according to a mixture ratio, allowing simultaneous global exploration and rapid local exploitation. The fitness function is defined as the mean absolute error between observed data X(k) and model predictions X̂(k), turning the parameter estimation into a nonlinear optimization problem.

The methodology is evaluated on two real‑world data sets: (1) container throughput of Wuhan Port from 2011 to 2015, and (2) marine capture production of Zhejiang Province from 2007 to 2013. For each data set the authors test three fractional orders (r = 0.25, 0.5, 0.75) and compare ADCSO against LSM and Particle Swarm Optimization (PSO). Each algorithm is run ten times, and average errors are reported. ADCSO consistently yields the lowest average errors across all orders and both data sets. For the Wuhan Port case the optimal parameters found are r = 0.21, a = 0.015, b = 212 927, while for the Zhejiang marine data the optimal values are r = 0.06, a = ‑0.0409, b = 3 788. Convergence plots show that ADCSO reaches near‑optimal solutions faster and with less oscillation than PSO, demonstrating superior robustness against premature convergence.

The authors discuss the algorithm’s computational complexity, noting that with a population size of 40 and a maximum of 300 iterations, ADCSO’s runtime is comparable to PSO while delivering better accuracy. They also highlight the simplicity of the algorithm’s parameter settings (search memory size, mixture ratio, etc.) and its adaptability to other grey‑model variants. Limitations include the need for further sensitivity analysis of hyper‑parameters and validation on larger, multivariate data sets.

In conclusion, Adaptive Dynamic Cat Swarm Optimization provides an effective, stable, and fast approach for estimating the parameters of fractional‑order Grey Models. It outperforms traditional LSM and standard PSO in terms of prediction error reduction and convergence speed, making it a promising tool for small‑sample forecasting in various engineering and economic domains. Future work may extend ADCSO to multi‑step forecasting, multi‑variable grey models, and hybrid frameworks that combine grey theory with other machine‑learning techniques.