Data-driven process monitoring is a crucial means to ensure the safe operation of industrial processes. Most industrial data exists in the form of time series. Due to the complexity of mechanism, noise interference, and other factors, industrial time series often exhibit characteristics such as low quality, strong dynamics, and nonstationary characteristics, which pose challenges to the establishment of monitoring models. Although researchers have proposed relevant methods addressing different characteristics, the intrinsic connections between these methods have hardly been explored. This paper reveals, for the first time, the common motivation underlying these methods: in industrial processes, merely knowing the presence or absence of a fault often fails to meet actual needs. It is necessary to deeply decompose the complex characteristics of time series to achieve a comprehensive perception of the process state. Innovatively adopting a decomposition perspective, this paper reviews existing multivariate statistical methods for modeling various complex characteristics of time series. By decomposing complex time series into multiple components with practical physical significance, the paper provides interpretable monitoring results. It summarizes and compares the core decomposition ideas of different modeling methods. Subsequently, the paper elucidates the construction and implications of monitoring statistics for various methods. Finally, this paper summarizes and forecasts the work on decomposing and modeling industrial time series, proposing future research directions.