BUS606: Operations and Supply Chain Management

The irregular component in any decomposition model represents variations related to unpredictable events of all kinds. Most irregular values have a stable pattern, but some extreme values or outliers may be present. Outliers can often be traced to identifiable causes, for example strikes, droughts, floods, data processing errors. Some outliers are the result of displacement of activity from one month to the other.

Fig. 3 – Irregular component of Sales by Canadian Department Stores.

Fig. 3 displays the irregular component of Sales by Canadian Department Stores, which comprises extreme values, namely in 1994, 1998, 1999 and Jan 2000. Most of these outliers have to do with the closure of some department stores and the entry of a large department store in the Canadian market.

As illustrated by Fig. 3, the values of the irregular component may be very informative, as they quantify the effect of events known to have happened.

Note that it is much easier to locate outliers in the irregular component than in the raw series because the presence of seasonality hides the irregular fluctuations.

The irregulars are most commonly assumed to follow a white noise process defined by

$E\left(u_{t}\right)=0, E\left(u_{t}^{2}\right)=\sigma_{u}^{2}$

If $\sigma_{u}^{2}$ is assumed constant (homoscedastic condition), $u_{t}$ is referred to as white noise in the strict sense. If $\sigma_{u}^{2}$ is finite but not constant (heteroscedastic condition), $u_{t}$ is called white noise in the weak sense.

For inferential purposes, the irregular component is often assumed to be normally distributed and not correlated, which implies independence. Hence, $u_{t} \sim \operatorname{NID}\left(0, \sigma_{u}^{2}\right)$.

There are different models proposed for the presence of outliers depending on how they impact the series under question. If the effect is transitory, the outlier is said to be additive; and if permanent, to be multiplicative.

Box and Tiao (1975) introduced the following intervention model to deal with different types of outliers,

$y_{t}=\Sigma_{j=0}^{\infty} \quad h_{t-j} x_{t-j}+\eta_{t}=\Sigma_{j=0}^{\infty} b_{j} B^{j} x_{j}+\eta_{t}=H(B) x_{t}+\eta_{t}$                                          (25)

where the observed series $\left\{y_{t}\right\}$ consists of an input series $\left\{x_{t}\right\}$ considered a deterministic function of time and a stationary process $\left\{n_{t}\right\}$ of zero mean and non-correlated with $\left\{x_{t}\right\}$. In such a case the mean of $\left\{y_{t}\right\}$ is given by the deterministic function $\sum_{j=0}^{\infty} h_{t-j} x_{t-j}$. The type of function assumed for $\left\{x_{t}\right\}$ and weights $\left\{h_{j}\right\}$ depend on the characteristic of the outlier or unusual event and its impact on the series.