Method of Least Squares

Denote by $\psi_{ij}(s,d,\tau)$ the call rate prediction for the part $j$ of the period $i$ using the data up to the period $i-1$ and the expert model with fixed parameters $s,d,\tau$ that determine the impact scales $s$ and the delay and duration $d,\tau$ of all the special events, including DSE and PSE. We minimize the sum

$$\min_{s,d,\tau} \sum_{i=T_0}^T \sum_{j=1}^J (z_{ij} - \psi_{ij}(s,d,\tau))^2$$ (147)

Here $T$ is the end and $T_0$ is the beginning of the "learning" set. Defining the expression (2.87) as a function 'fi' in the file 'fi.C' we may directly apply various global and local optimization methods in the same way as they are used estimating the ARMA parameters (see section 1).