An Approach for the Identification and Estimation of Outliers in a Time Series with a Nonstationary Mean

We address the problem of identifying and estimating outliers in a time series with a nonstationary mean. We first apply a moving linear model to decompose the time series into a constrained component and a remaining component that is a stationary time series containing the outliers. We then propose an approach of identifying and estimating the outliers. The proposed approach is based on the maximum likelihood method for an autoregression (AR) model with outliers in variables and has the following features. (1) The used model has a simple structure that is easy to understand. (2) The proposed approach allows for the estimation of not only the positions but also the magnitudes of the outliers. As an illustrative example, we apply the proposed approach to the analysis of the index of industrial production in Japan. The results show the high performance of the approach.

I. Introduction

Time series modeling methods are widely applied in multiple research fields, such as in the extraction of a signal from earthquake data [6] and business cycle analysis [10]. Anomaly detection has recently become an important issue in many areas of research, and especially in time series analysis [9]. Anomaly detection in time series analysis means to discover unexpected values or rapid fluctuations in the time series data. In this paper, we regard the general subject of anomaly detection as the identification and estimation of outliers in a time series with a nonstationary mean. A difficulty is that the objective time series has a nonstationary mean whereas typically adopted approaches can usually be applied only to time series with a stationary structure. We thus apply the moving linear model approach proposed by [7] to decompose the objective time series into a constrained component, which corresponds to the nonstationary mean, and a remaining component, which corresponds to the stationary term containing the outliers. We then propose an approach of identifying and estimating the outliers from the remaining component.