Method of Least Squares

Denote by $\psi_i(s,d,\tau)$ the call rate prediction by an expert model with fixed parameters $s,d,\tau$ for the next period $i$ using some data up to the period $i-1$. The parameters of the model include scales $s$ and delay and duration times $d,\tau$ We minimize the sum

$$\min_{s,d,\tau} \sum_{i=T_0}^T(z_i- \psi_i(s,d,\tau))^2$$ (127)

Here $T$ is the end and $T_0$ is the beginning of the "learning" set.

Defining the expression (2.67) 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). In the case (2.67), parameters $b$ of the ARMA model are replaced by parameters $s,d,\tau$ of the expert model EM. However , the optimization problem (2.67) is very difficult because the number of variables is great and the objective function is multi-modal.