Estimation Procedures

While estimating the scales $s_i$ , we consider three direct estimation procedures and also the method of least squares.

The first direct estimation procedure is the empirical one. The scales $s$ describing the impact of all sorts of events including the partial and the multiple ones are estimated as the relations (2.53) or (2.56) of the call rates in the pair of periods $p(i-1)$ and $p(i)$ which precede the present period $i-1$ and the next one $i$. The numerator $z_{p(i)}$ is the call rate of the period preceding the next one and the denominator $z_{p(i-1)}$ is the call rate of the period preceding the present one.

Using the empirical procedure partial and multiple special events are considered as different SE. In this case scales $s$ are defined separately for each of them. Therefore the empirical procedure is convenient if the data is available for all the types of periods including the partial and multiple ones. In such a case estimation of the delays $d$ and durations $\tau$ of DSE is not needed because it is supposed that an identical situationcan be found examining the past observations. We need a lot of data for that.

The second direct procedure is the subjective one. This means that all the unknown parameters of the expert model such as the scales $s$, the delay times $d$ and the duration times $\tau$ are defined in accordance with the expert opinion. This is a reasonable way starting a new system when no data is available.

The third direct procedure is a mixture of the empirical and the subjective ones. We use empirical estimation if the corresponding data is available. We use subjective estimation otherwise.

Using the method of least squares we search for such scales $s$, such delay and duration times $d,t$ which minimize the sum of squared differences between the call rates predicted by the expert model and the observed call rates $z_{ij}$.

All four procedures are applied to different time periods: hours, days, weeks and seasons. First, the scalar case when we predict a single number, an average call rate of the next period, will be considered. In the vector case, a vector which components define call rates of different parts of the next period is regarded.