## 1 Introduction  
With the rapid development of the economy, the output of coal resources is increasing year by year. Large-scale coal mining has also brought a series of serious social and environmental problems [1], such as land subsidence, building damage, ground fissures, etc. [2; 3; 4], which have attracted worldwide attention. Therefore, the accurate prediction of surface movement and deformation is of great significance to the exploitation of coal resources. In the field of mining subsidence research, the probability integral method (PIM) is the most commonly used parameter inversion method for mining subsidence prediction in China. Its theoretical basis is the random medium theory. This theoretical framework was first introduced to the study of mining subsidence by the Polish scholar [PERSON] in the 1950s [5]. Subsequently, Chinese scholars, such as [PERSON] and [PERSON], further developed this theory into what is now known as the probability integral method [6]. Currently, the probability integral method stands as one of the more mature and most widely utilized techniques in China for mining subsidence prediction. Over time, the inversion methods employed for estimating the parameters associated with the probability integral method have evolved significantly. This evolution has progressed from linear approximation methods to experimental design approaches and ultimately to optimization and intelligent algorithms. Due to the complex nature of the probability integral method function, which encompasses numerous unknown parameters and does not adhere to linear requirements, conventional linear approximation methods often face convergence challenges, even when appropriate initial values are selected [7; 8; 9; 10]. On the other hand, the experimental design approach tends to necessitate a substantial number of trials due to the multitude of parameters involved, resulting in increased workload and reduced parameter inversion speed [11]. Consequently, many researchers have sought to employ optimization algorithms and intelligent algorithms in the parameter inversion process. For instance, [PERSON] et al. [12] successfully applied a genetic algorithm to the probability integral method for parameter inversion, demonstrating notable advantages in terms of accuracy and reliability. Subsequent investigations explored the use of the modular vector method [13; 14; 15], particle swarm algorithms [16; 17; 18; 19], simulated annealing algorithms [20; 21], and others [22; 23; 24; 25; 26; 27] for parameter inversion within the probability integral method framework, all yielding highly satisfactory results. In a comparative analysis of parameter inversion outcomes using various intelligent algorithms, [PERSON] et al. [28] confirmed that, with an appropriate choice of initial exploration values, the modular vector method excels in accuracy and reliability when contrasted with other algorithms. Nevertheless, it is important to note that different selections of initial exploration values may lead to cases where the fitting error is minimal, yet, the results of parameter inversion exhibit substantial discrepancies. To mitigate the risk of erroneous initial exploration value selection leading to the confinement of the modular vector method's inversion results within local extremities, this paper introduces an improved algorithm, which amalgamates the modular vector method with the pattern recognition approach [29; 30] in mining subsidence analysis. The algorithm leverages geological data from the mining area to determine the initial exploration values of the modular vector method. This study first verifies the accuracy and robustness of the improved modular vector method in the inversion of anticipated parameters within the probability integral method through simulation experiments conducted under geological conditions characteristic of shallow-buried coal seams [31]. Subsequently, an illustrative analysis is conducted, focusing on a specific working face within the Shendong mining area.