Statistical signification of $\theta_{1,k,T}$

The angle $\theta_{1,k,T}$ has a statistical interpretation. Effectively, with the definition of P_1,k,T, we see that:

$$\sin^2\theta_{1,k,T}=(A'_{1,k,T}\pi_T)'(A'_{1,k,T}A_{1,k,T})^{-1}(A'_{1,k,T}\pi_T) \;.$$

Notice first that $A'_{1,k,T}\pi_T=(y(T-1),\cdots,y(T-k))'$ . Moreover A'_1,k,TA_1,k,T contains the lags $0,\cdots,k-1$ of the autocorrelation of the data: thus, it is the covariance matrix R^y_k,T of the data (up to time T). So, the value $\exp(-0.5\sin^2\theta_{1,k,T})$ can be compared to the probability of the sequence $\tilde{y}=(y(T-1),\cdots,y(T-k))'$ according to the covariance matrix R^y_k,T:

$$\mbox{Proba}(\tilde{y})={1\over\vert\vert 2\pi R^y_{k,T}\ver... ...ert\vert^{{n\over 2}}}\exp(-{1\over 2}\sin^2\theta_{1,k,T}) \;.$$

A small value of $\theta_{1,k,T}$ means that the previous samples don't deviate from the general statistics of the data. On the contrary, a sudden burst of noise, or a new strong seismic arrival, will produce a large value of $\sin^2\theta_{1,k,T}$. This occurrence will also perturb strongly the covariances $R^{\varepsilon}_{k,T}$, R^r_k,T, and the correlation $\Delta_{k+1,T}$, whose updatings involve a division by $\cos^2\theta_{1,k,T}$ (small in that case). So the variable $\sin^2\theta_{1,k,T}$ can be assimilated to a likelihood variable, and used for detection of unexpected events.