Introduction

Consider some additional ideas and models that supplement the ones described in chapter 1. The call rate depends on many factors (see Figures 2.1 and 2.2) thus a multi-dimensional adaptive version of the Auto-Regression-Moving-Average (ARMA) model and software is applied (see chapter 1).

Figure 2.1: A fragment of the the DATAFILE 'call.data' including the number, the date, the call rate (repeated twice) , the real-valued external factor (repeated twice) and the indicators of eight Boolean external factors

Figure 2.2: Eight Boolean external factors related to call rate

However, the results show that by including the external factors directly into the ARMA model one does not improve the predictions, as usual. A reason is that it is difficult to estimate the delay time (see expression (1.22)) and the duration of SE influences, in the ARMA framework. Most of SE are rare (see Figure 2.1). That means the corresponding columns of date file are filled mostly by zeros. That is an additional difficulty.

Note, that most of the SE indicated in Figure 2.2 are predictable. For example, one can predict factor 6 ( public holidays, traditional celebrations) and factor 7 (last day for ordering) exactly. Other SE can be predicted approximately. Obviously, the knowledge of future values of external factors helps to predict the main ones. However, the software version we use does not exploit this possibility.

Therefore we consider call rates $\lambda(t)$ as a sum of two stochastic functions

$$\lambda(t)=z(t)+\nu(t).$$ (111)

Usually the rate $\lambda(t)$ has several components $\lambda(t)=(\lambda_1(t),...,\lambda_l(t))$ corresponding to different types of calls. In expression (2.51) $\nu(t)$ denotes a "stationary" component which is described by the ARMA model. A "non-stationary" component is denoted by $z(t)$. This way we separate the stationary part from the non-stationary one. The separation is not well defined, however, it often helps making actual predictions.That is important, because theoretical analysis of non-stationary stochastic functions is difficult.

The non-stationary component is defined by a local expert, as usual. The expert applies his knowledge while using the previous data and making the future predictions. Thus we call $z(t)$ as the "expert" component and $\nu(t)$ as the "statistical" one. The estimation of the statistical component $\nu(t)$ is investigated in chapter 1 while considering the ARMA model. Now we consider models to predict the expert component $z(t)$ by estimation of "scales". The scales express the differences between different events and time intervals.