Example of Structural Optimization Including External Factors

Predictions often depend on several factors In such a case multi-dimensional ARMA should be used. Denote by $\nu(Mt),\ t \ge p/M$ the main statistical component and by $\nu(Mt-i),\ i=1,..,p -1$ the external factors. In such a case we extend the traditional ARMA model this way

$$\nu(Mt)=\sum_{i=0}^{p-1} a_{i} \nu(Mt-i-1)+\sum_{j=0}^{q-1} b_j \epsilon(M(t-j-1))+\epsilon(Mt),$$ (58)

$$p/M \le t \le T0$$

Here $p$ is the number of AR (Auto-Regression) components and $q$ is the number of MA (Moving Average) ones. We say that continuous variables $a$ and $b$ define the state of the ARMA model (state variables). The discrete parameters $p$, $q$, and $t0$ define the structure (structural variables). The structural variable $t0$ defines the time when we start scanning the time series for the optimization of state variables $a,b$. Denote by $T0$ the scanning end . That means that first for the fixed structural variables $p,q,t0$ we minimize the squared deviation

$$\Delta_{00}(p,q,t0)=\sum_{t=t0}^{T0} \epsilon(Mt)^2,\ t0 \ge p/M$$ (59)

where $T0<T1<T$ and $\epsilon(Mt)^2$ is from expression (1.60) . Denote the optimal values $a=a(p,q,t0)$ and $b=b(p,q,t0)$.

At fixed $b$ the optimal values of $a=a^b$ are defined by a system of linear equations calculated from the condition that all the partial derivatives $\partial \Delta_{00}/\partial {a_i},\ i=0,...,p-1$ of the sum (1.60) are zero. Thus obtaining the least squares estimates $a_i=a_i^b,\ i=0,...,p-1$ at given $b$we have to solve $p$ linear equations with $p$ variables $a_{i},\ i=0,...,p-1$ (see chapter 1.3 for details).

The optimization of the discrete structural variables $p,q,t0$ is performed using a different data set that starts at $T0$ and ends at $T1$ while keeping the previous optimal values of the state variables $a=a(p,b,t0)$ and $b=b(p,q,t0)$ obtained by minimization of the sum $\Delta_{01}(p,q,t0)$ using the data from $t0$ to $T0$. Here the sum

$$\Delta_{01}(p,q,t0)=\sum_{t=T0}^{T1} \epsilon(Mt)^2,$$ (60)