Detects and determines outliers and simultaneously estimates the model parameters in a time series whose underlying outlier free series follows a general seasonal or nonseasonal ARMA model.

Consider a univariate time series {Yt} that can be described by the following multiplicative seasonal ARIMA model of order (p, 0, q) x (0, d, 0)s:

In general, {Yt} is not directly observable due to the influence of outliers. Chen and Liu (1993) distinguish between four types of outliers: innovational outliers (IO), additive outliers (AO), temporary changes (TC) and level shifts (LS). If an outlier occurs as the last observation of the series, then Chen and Liu’s algorithm is unable to determine the outlier’s classification. In TS_OUTLIER_IDENTIFICATION, such an outlier is called a UI (unable to identify) and is treated as an innovational outlier.

In order to take the effects of multiple outliers occurring at time points t1, t2, …, tm into account, Chen and Liu consider the following model:

Here, {Yt*} is the observed outlier contaminated series, and ωj and Lj(B) denote the magnitude and dynamic pattern of outlier j, respectively. It(tj) is an indicator function that determines the temporal course of the outlier effect, Itj(tj) = 1, It(tj) = 0 otherwise. Note that Lj(B) operates on It via BkIt = It-k, k = 0, 1, … .

The last formula shows that the outlier free series {Yt} can be obtained from the original series {Yt*} by removing all occurring outlier effects:

TS_OUTLIER_IDENTIFICATION is an implementation of Chen and Liu’s algorithm. It determines the coefficients inφ (B), θ (B), and the outlier effects in the model for the observed series jointly in three stages. The magnitude of the outlier effects is determined by least squares estimates. Outlier detection itself is realized by examination of the maximum value of the standardized statistics of the outlier effects. For a detailed description, see Chen and Liu’s original paper (1993).

Intermediate and final estimates for the coefficients in φ (B) and θ (B) are computed by routines NSPE and NSLSE. If the roots of φ (B) or θ (B) lie on or within the unit circle, then the algorithm stops with an appropriate error message. In this case, different values for p and q should be tried.