General LSL algorithm

We are interested in the prediction of a time series x_T from a related process y_T. We want to minimize:

$$\sum_{t=0}^T (\varepsilon^x_{n,T}(t))^2 = \sum_{t=0}^T [x(t)-f_{n,T}(1)y(t-1)-\cdots-f_{n,T}(n)y(t-n)]^2\;.$$

Once again, we are interested only in the residual $\varepsilon^x_{n,T}(T)$, so that the residuals are affected only by past values, and not by future values. This minimization problem leads to:

If we are interested in the residuals $\varepsilon^x_{k,T}$ themselves, then we have to find the orthogonal complement of x_T on the space $Y_{1,k,T}=(Zy_T,\cdots,Z^ky_T)$. So, the idea is to use the backward residuals of the self-prediction of y_T to form an orthogonal basis of the space Y_1,k,T, and then project the vector x_T onto this particular basis. Consequently, we will lead two recursions at the same time: one to compute the variables $\varepsilon_{k,T}$ and r_k,T of the self-prediction of y_T, and the other to compute the residuals $\varepsilon^x_{k,T}$ of the joint prediction of x_T from y_T.

The order-updating of the residuals $\varepsilon^x_{k,T}$ is very similar to the updating of the residuals $\varepsilon_{k,T}$. Effectively,

$$\begin{eqnarray} \varepsilon^x_{k+1,T}&=&x_T - P_{1,k+1,T}.x_T \nonumber \\ &=&... ...on^x_{k,T} - {r'_{k,T-1}x_T\over r'_{k,T-1}r_{k,T-1}}r_{k,T-1} \;.\end{eqnarray}$$ (6)

We can define the crosscorrelation $\Delta^x_{k+1,T}$ and the reflection coefficient K^x_p+1,T:

$$\Delta^x_{k+1,T}=r'_{k,T-1}x_T\;, \mbox{\hspace{1.0cm}and\hspace{1.0cm}} K^x_{p+1,T}={\Delta^x_{k+1,T}\over R^r_{k,T-1}} \;.$$

The time-updating is similar to equation (5). I use here an equation derived from Lee et al. (1981), saying that for any vectors u_T and v_T, we have the relationship:  

$$u'_T(P_{1,k,T}^{\perp}v_T)-u'_{T-1}(P_{1,k,T-1}^{\perp}v_{T-... ...\pi'_T(P_{1,k,T}^{\perp}v_T)].{1\over \cos^2\theta_{1,k,T}} \;.$$ (7)

This equation, used with $(u_T,v_T)\!=\!(y_T,Z^{k+1}y_T)$, leads to the time-updating of $\Delta_{k+1,T}$ (equation (5)) in the self-prediction of y_T. Here, using the equation (8) with u_T=x_T and v_T=Z^k+1y_T, we get the time-update recursion:  

$$\Delta^x_{k+1,T}=\Delta^x_{k+1,T-1} + {\varepsilon^x_{k,T}(T)r_{k,T-1}(T)\over\cos^2\theta_{1,k,T}} \;.$$ (8)

The use of exponential tapering also gives a recursion similar to equation (6):

$$\Delta_{k+1,T}^x=\lambda\Delta_{k+1,T-1}^x+{\varepsilon^x_{k,T}(T)r_{k,T-1}(T)\over\cos^2\theta_{1,k,T}} \;.$$

In conclusion, the recursions (7) and (9), joined to the previous recursions (3), (4), and (5) of the basic LSL algorithm, give us the entire set of recursions necessary for the general LSL algorithm.