Exponential weighting

The weighting consists of using a tapering coefficient $\lambda$ to minimize the weighted norm of the residuals:

$$\sum_{t=0}^T \lambda^{T-t}\varepsilon^2_{n,T}(t) \;.$$

It forces the algorithm to forget the ``old data'' in the minimization problem: it extends the adaptive properties of the algorithm. We can use the recursion results derived up to now, simply adapted to another choice of scalar product:

$$x.y=\sum_t \lambda^{T-t}x(t)y(t) \;.$$

Actually, only the covariances and correlation recursions are modified:

$$\begin{eqnarray} \Delta_{k+1,T}&=&\lambda\Delta_{k,T}+{\varepsilon_{k,T}(T).r_{k... ...{0,T}&=&R^r_{0,T}=\lambda R^{\varepsilon}_{0,T-1}+y^2(T) \nonumber\end{eqnarray}$$ (5)