Maximum Likelihood Estimation for Uncertain Moving Average Model Under Imprecise Observations

Time series analysis serves as a practical way to investigate data that changes over time, and has great potential for development in fields such as weather prediction and financial engineering. However, due to the imperfections in technology, it is not always possible to obtain precise observational data. Therefore, modeling with uncertain time series is more suitable. Choosing appropriate parameter estimation methods is a critical aspect of the modeling process for uncertain time series. This paper proposes using maximum likelihood estimation for the parameter estimation of the first-order uncertain moving average (UMA) model, and for predicting future values and calculating the confidence intervals. Subsequently, the effectiveness of this method is demonstrated through numerical examples. Firstly, we employ an iterative method to convert the first-order UMA model into an uncertain autoregressive (UAR) model. Secondly, the maximum likelihood method is used to estimate the unknown parameters of the UMA model. Additionally, for this model where the observations are linear uncertain variables, a specific maximum likelihood estimation approach is provided. Thirdly, following the estimation values obtained, future data predictions and confidence intervals are calculated. Furthermore, when the observations are linear uncertain variables, corresponding confidence intervals are also provided. Finally, two practical cases are presented to demonstrate the practicality of the method. Moreover, contrasted with the least squares method, the results indicate that this method can enhance the accuracy of predictions.