A new approach for detecting gradual changes in non-stationary time series with seasonal effects
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Online ISSN 2005-2863 Print ISSN 1226-3192
RESEARCH ARTICLE
A new approach for detecting gradual changes in non‑stationary time series with seasonal effects Guebin Choi1 Received: 9 August 2020 / Accepted: 12 September 2020 © Korean Statistical Society 2020
Abstract This paper proposes a new method of detecting the gradual changes of time series when the changes in time series are mixed with seasonality. The key of the proposed method is to express the desired time-varying feature while removing the unwanted time-varying feature of seasonal effects through two-stage procedures. Asymptotic properties of the proposed methods are studied, and simulation results are presented. In addition, models with multiple changes have been studied. Furthermore, to demonstrate the usefulness of the proposed method, real data analysis with the number of Korean traveling to Japan is presented. Keywords Change point detection · Gradual changes · Non-stationary time series · Two-stage filter
1 Introduction Stationarity is a very important assumption when analyzing time series. Most time series, however, do not satisfy the stationary condition. In other words, probabilistic properties, such as central tendency and dispersion, often change over time. Therefore, finding these change points in a time series is important for understanding the non-stationary time series. Traditionally, the problem of finding change points has mainly dealt with abrupt changes, i.e., some parameters suddenly changing at some time point. However, Vogt and Dette (2015a) noted that the features of data often change gradually rather than abruptly, and suggested a detection method to capture such changes. It is more useful to detect gradual changes than abrupt changes. The reasons are as follows: First, it is often more practical to find smooth changes in real data. Vogt and Dette (2015a) have pointed out that it is crucial to locate the smooth change points of the Asian financial crisis in 1997, of certain climate processes, or of epileptic seizure of * Guebin Choi [email protected] 1
Artificial Intelligence Lab, LG Electronics Inc, Seoul 06763, Korea
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Fig. 1 A realization of Xt in Example 1
EGG data. Another reason is that even if some parameter of the model describing data abruptly changes, its effect may appear gradually. The current study focuses on the second reason. To clarify our motives, we consider the example below. Example 1 We consider a non-stationary time series Xt = 𝜃t Xt−12 + 𝜖t with { 0.5 t ≤ 250 𝜃t = 1 250 < t < 500.
Here, 𝜖t ∼ N(0, 1) . Note that Xt is stationary process when time point t ≤ 250 , and changes to non-stationary process when t > 250 where |𝜃t | ≥ 1 . Observe that although the parameter 𝜃t changes abruptly at t = 250 , variance of Xt is increasing gradually after the time point 250. Figure 1 shows a realization of Xt . It seems that the abrupt change of 𝜃t leads the gradual change of variance. iid
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