A Bayesian Approach for Detecting Outliers in ARMA Time Series

The presence of outliers in time series can seriously affect the model specification and parameter estimation. To avoid these adverse effects, it is essential to detect these outliers and remove them from time series. By the Bayesian statistical theory, this article proposes a method for simultaneously detecting the additive outlier (AO) and innovative outlier (IO) in an autoregressive moving-average (ARMA) time series. Firstly, an approximate calculation method of the joint probability density function of the ARMA time series is given. Then, considering the situation that AO and IO may present at the same time in an ARMA time series, a model for detecting outliers with the classification variables is constructed. By this model, this article transforms the problem of detecting outliers into a multiple hypothesis testing. Thirdly, the posterior probabilities of the multiple hypotheses are calculated with a Gibbs sampling, and based on the principle of Bayesian statistical inference, the locations and types of outliers can be obtained. What’s more, the abnormal magnitude of every outlier also can be calculated by the Gibbs samples. At last, the new method is tested by some experiments and compared with other methods existing. It has been proved that the new approach can simultaneously detect the AO and IO successfully and performs better in terms of detecting the outlier which is both AO and IO, and but cannot be recognized by other methods existing. Key-words: ARMA model; Additive outlier (AO); Innovative outlier (IO); Classification variable; Bayesian hypothesis test; Gibbs sampling