## 1 Introduction
### Literature Review  
Extensive research has been conducted on how to predict the tamping maintenance demand using methods for modeling track geometry deterioration. The types of models studied for this purpose can be divided into three categories: deterministic models, machine learning models, and probabilistic models [6; 7; 8]. Deterministic models use statistical analysis methods, such as regression analysis, to analyze the functional relationship between the evolution of track geometry and service time, accumulated traffic load, and other factors of influence [9; 10]. [PERSON] and [PERSON] [11] established linear deterioration models for individual track sections that describe the change in the standard deviation of the longitudinal level over time or with the cumulative change in total traffic tonnage between two successive tamping operations. [PERSON] and [PERSON] [12], [PERSON] and [PERSON] [13], and others identified multivariate linear regression relationships between the standard deviation of the longitudinal level and factors such as the average annual total traffic tonnage, service time, and track structure. Some researchers believe that the track geometry deterioration pattern between two successive tamping operations exhibits nonlinear characteristics. Based on methods such as exponential and multistage linear fitting, nonlinear prediction models have been developed for tamping maintenance cycles [14; 15]. In recent years, machine learning methods have received increased attention because of their usefulness in modeling the deterioration of infrastructure conditions [16]. Artificial neural networks (ANNs), support vector machines (SVMs), and ensemble learning are machine learning techniques commonly adopted in the field of track geometry condition prediction [7; 17; 18; 19; 20]. These studies mainly sought to establish models for the relationship between the future and current states of track geometry conditions as functions of various factors for use in the short-term prediction of track geometry conditions. For long-life assets, such as tracks, tunnels, and bridges, machine learning techniques commonly use inspection and monitoring data along with data on related influencing factors as inputs for short- to medium-term condition prediction [16; 21; 22]. Because it is difficult to collect sufficient maintenance cycle or lifespan data to support model training, these techniques are not yet capable of supporting application scenarios, such as risk assessment and whole life-cycle maintenance cost prediction [23]. Moreover, machine learning prediction models previously established for the deterioration patterns of track geometry conditions are essentially regression models. Without sufficient consideration of all of the input factors influencing the deterioration process, they may not be able to accurately capture the uncertainty of the deterioration process, thereby failing to predict track geometry conditions with high accuracy [23]. Probabilistic models use stochastic process methods or probability distribution functions to predict the deterioration patterns of track geometry conditions, considering the uncertainty of the deterioration process. The advantage of these methods is that, on the one hand, they can predict the track geometry condition or state rate at any time during the entire life cycle, while on the other hand, they can output indicators, such as life expectancy, that support maintenance planning for a long period. A representative method is the Markov stochastic process, which discretizes the assessment of track geometry conditions into different state levels and describes the deterioration process of the transitions between these states [24; 25].