Summary: Any stationary time-series can be decomposed by means of an optimization operator, called the \(\zeta\)-optimator, into several components (the time-series) \(\{V^ i_ t\}\), \(i=1,2,...,p\), such that the first component \(\{V_ t^ 1\}\), \(t=1,2,...,\nu\), is a smooth process having a larger autocorrelation in comparison with the original process \(\{Y_ t\}\), i.e. \(\rho_{v1}>\rho_ y\). Usually only a few such components are sufficient for approximating the time-series with good accuracy. The \(\zeta\)-optimator involves a shape parameter \(\alpha\), so the decomposition is unique provided that \(\alpha\) is fixed. Since the component \(\{V_ t^ 1\}\) involves much of the useful information it can be used for computing predictors for control purposes.
Thus, given the observations \(Y_{\nu},Y_{\nu -1},Y_{\nu -2},...\), a predictor of \(Y_{\nu +1}\) is \(\rho_{v1}V^ 1_{\nu}(q)\) where \[ V^ 1_{\nu}(q)=qY_{\nu}+q(1-q)Y_{\nu -1}+q(1-q)^ 2Y_{\nu - 2},..., \] the weights \(q(1-q)^ r\), \(r=0,1,2,..\). decreasing rapidly as \(q=q(\alpha)\in (0,1)\). Further, one may chose q rather than choosing \(\alpha\), since q(\(\alpha)\) is a one-one mapping. Once q is fixed, the predictor \(\rho_{v1}V^ 1_{\nu}(q)\) is obtained in a straightforward way by using the formula above. It is shown that \(\rho_{v1}V^ 1_{\nu}(q)\) converges to the best predictor as \(\alpha\) \(\to 0\). Some examples are worked out, illustrating both the decomposition and the forecasting procedures.