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QUESTION:Please explain INTERVENTION MODELING/DETECTION when I don't know a priori the timing and the duration of an event. Be as formal as you can. ANSWER:Early work restricted outlier detection to identification of three types of outliers in a time series. These outliers were represented as intervention variables of the forms: pulse, level shifts and seasonal pulses. The procedure for detecting the outlier variables is as follows. Develop the appropriate ARIMA model for the series. Test the hypothesis that there is an outlier via a series of regressions at each time period. Modify the residuals for any potential outlier and repeat the search until all pos sible outliers are discovered. These outliers can then be included as intervention variables in a multiple input B-J model. The noise model can be identified from the original series modified for the outliers. AFS has extended outlier detection to detecting the presence of local time trends. This option to the program provides a more complete method for the development of a model to forecast a univariate time series. The basic premise is that a univariate time series may not be homogoneous and, therefore, the modeling procedure should account for this. By homogeneous, we mean that the underlying noise process of a univariate time series is random about a constant mean. If a series is not homogeneous, then the process driving the series has undergone a change in structure and an ARIMA model i s not sufficient. The AUTOBOX heuristic that is in place checks the series for homogeneity and modifies the model if it finds any such changes in structure. The point is that it is necessary for the mean of the residuals to be close enough to zero so that it can be assumed to be zero for all intents and purposes. That requirement is necessary but it is not sufficient. The mean of the errors (residuals) must be near zero for all time slices or sections. This is a more stringent requirement for model adequacy and is at the heart of intervention detection. Note that some inferior forecasting programs use standardized residuals as the vehicle for identifying outliers. This is inadequate when the ARIMA model is non-null. Consider the case where the observed series exhibits a change in level at a particular point in time. If you try to identify outliers or interventions in this series via classical standardized residuals you get one outlier or one unusual value. The problem is that if you "fix" the bad observation at the identified time point, the subsequent value is identified as an outlier due to the recursive process. The simple-minded approach of utilizing standardized residuals is in effect identification of innovative outliers and not additive outliers. The logic behind the automatic intervention procedure has its roots in the technique proposed by Chang and Tiao (1983) and programmed by Bell (1983). It starts by developing an ARIMA model for the univariate time series (using the automatic ARIMA algorithm). A series of regressions on the residuals from the ARIMA model checks for any underlying changes in structure. If the series is found to be homogeneous, then the ARIMA model is used to forecast. If the series is found to be nonhomogeneous, then the va rious changes in structure are represented in a transfer function model by dummy (intervention) input variables and the ARIMA model becomes the tentative noise model. The program then estimates the transfer function-noise model and performs all of the diagnostic checks for sufficiency, necessity and invertibility. The model is updated as needed, and the diagnostic checking stage ends when all of the criteria for an acceptable model are met. The final step is to generate the forecast values. The user control s the level of detail that the output report is to contain, as well as some key options for modeling precision (lambda search and backcasting, for example). The user can also elect to have this process start with an examination of the original time series. This may be necessary for those cases where the series is overwhelmingly influenced by outlier variables. We now present a summary of the mathematical properties underlying this procedure. This is taken from the Downing and McLaughlin (1986) paper (with permission!). For purposes of this discussion, we present, in their notation, the following equation, which is the general ARIMA model: P(B) (N(t) - MEAN) = CONSTANT + T(B) A(t), (eq. 1) where N(t) = the discrete time series, MEAN = the average of time series, P(B) = the autoregressive factor(s), CONSTANT= the deterministic trend, T(B) = the moving average factor(s), A(t) = the noise series, and B = the backshift operator. Outliers can occur in many ways. They may be the result of a gross error, for example, a recording or transcript error. They may also occur by the effect of some exogenous intervention. These can be described by two different, but related, generating models discussed by Chang and Tiao (1983) and by Tsay (1986). They are termed the innovational outlier (IO) and additive outlier (AO) models. An additive outlier can be defined as, Y(t) = N(t) + W E(to) (eq. 2) while an innovational outlier is defined as, Y(t) = N(t) + [P(B)/T(B)] W E(to) (eq. 3) where Y = the observed time series, t in length W = the magnitude of the outlier, E (t ) = 1 if t = to, 0 if t <>to that is, E (t ) is a time indicator signifying the time occurrence t o of the outlier, and N is an unobservable outlier free time series that t follows the model given by (eq. 1). Expressing Equation (eq. 2) in terms of white noise series A in Equation (eq. 1), we find that for the AO model Y(t) = [T(B)/P(B)] A(t) + W E(to), (eq. 4) while for the the IO model Y(t) = [T(B)/P(B)][ A(t) + W E(to)], (eq. 5) Equation (eq. 4) indicates that the additive outlier appears as simply a level change in the t th observation and is described as a o "gross error" model by Tiao (1985). The innovational outlier represents an extraordinary shock at time period to since it influences observations Y(to), Y(to+1)..... through the memory of the system described by T(B)/P(B). The reader should note that the residual outlier analysis as conducted in the course of diagnostic checking is an AO type. Also note that AO and IO models are relatable. In other words, a single IO model is equivalent to a potentially infinite AO model and vice versa. To demonstrate this, we expand equation (eq.5) to Y(t) = [T(B)/P(B)] A(t) + [T(B)/P(B)] W E(to) , (eq. 6) and then express (eq. 6) in terms of (eq. 4) Y(t) = [T(B)/P(B)] A(t) + WW E(to) , (eq. 7) where WW = [T(B)/P(B)] W . Due to estimation considerations, the following discussion will be concerned with the additive outlier case only. Those interested in the estimation, testing, and subsequent adjustment for innovative outliers should read Tsay (1986). Note that while the above models indicate a single outlier, in practice several outliers may be present. The estimation of the AO can be obtained by forming II(B) = [T(B)/P(B)] (eq. 8) and calculating the residuals E(t) by E(t) = II(B) Y(t) (eq. 9) = II(B)[ [T(B)/P(B)] A(t) + W E(to) ] = A(t) + W II(B) E(to) . By least squares theory, the magnitude W of the additive outlier can be estimated by EST of W(to) = n*n II(B) E(to) (eq. 10) The variance of W(to) is given by: Var(W(to)) = n*n var(A) (eq. 11) where var(A) is the variance of the white noise process A(t) . Based on the above results, Chang and Tiao (1983) proposed the following test statistic for outlier detection: ç(to)= EST W(to) / n sqrt(var(A)). (eq. 12) If the null hypothesis of no outlier is true, then ç(to) has the standard normal distribution. Usually, in practice the true parameters II and åý are unknown, but consistent estimates exist. Even more important is the fact that to, the time of the outlier, is unknown, but every time point may be checked. In this case one uses the statistic: ç = max absolute value of ç(to) where to goes from 1 to n (eq. 13) and declares an outlier at time to if the maximum occurs at to and is greater than some critical value C. Chang and Tiao (1983) suggest values of 3.0, 3.5 and 4.0 for C. The outlier model given by Equation (eq. 4) indicates a pulse change in the series at time to. A step change can also be modeled simply by replacing E(to) with S(to) where: S(to) = 1 if t greater than to (eq. 14) 0 if not We note that (1-B)S(to) = E(to) . Using S(to) one can apply least squares to estimate the step change and perform the same tests of hypothesis reflected in Equations (eq. 12) and (eq. 13). In this way, significant pulse and/or step changes in the time series can be detected. A straightforward extension of this approach to transfer functions has also been introduced in this version of AUTOBOX. This, of course, implies that the outliers or interventions are not only identified on the basis of the noise filter but the form and nature of the individual transfer functions. |