Time series prediction method for multi-source observation data

Traditional time series analysis focuses on modeling and predicting time series data obtained from a single observation source. However, in a confrontational environment, one party often creates biased data to interfere with the other party's predictions. To mitigate the observation error inherent in single-source data, there is a growing emphasis on the importance of utilizing and studying multi-source observation data, which motivates this paper to propose methods for analyzing and predicting this kind of data. First, this paper proves that the accuracy of the estimation can be improved by introducing multiple sets of time-invariant data with bias. Subsequently, when this approach is extended to include time-varying data, these findings continue to hold. For time-varying data with bias, this paper proposes a comprehensive prediction procedure to provide the uncertainty distribution for the forthcoming moment. Finally, three examples are proposed to illustrate the effectiveness and efficiency of the methods.