Geometric interpretation

The operators ${\bf R_n}$ and ${\bf R_s}$ are the noise and signal resolution operators. They describe how well the predictions match the noise and signal Menke (1989). In the following equations, we consider that  

$${\bf R_nn}={\bf n}$$ (114)

and  

$${\bf R_ss}={\bf s},$$ (115)

meaning that each component of the data has been predicted. These equalities will help us to build a comprehensive geometric interpretation for the different operators. Based on equations () and (), we have for the data vector d the following equalities:

$$\begin{eqnarray} {\bf R_sd} &=& {\bf R_sn}+{\bf s} \nonumber \\ {\bf R_nd} &=& {\bf R_ns}+{\bf n},\end{eqnarray}$$ (116)

and

 

geom11 Figure 1 A geometric interpretation of the noise filter when n and s are not orthogonal.

 

geom21 Figure 2 A geometric interpretation of the noise filter when n and s are orthogonal.

$$\begin{eqnarray} {\bf \overline{R_s}d} &=& {\bf \overline{R_s}n} \nonumber \\ {\bf \overline{R_n}d} &=& {\bf \overline{R_n}s}.\end{eqnarray}$$ (117)

In the following equations, we prove that $\Vert{\bf \overline{R_n}s}\Vert^2+\Vert{\bf R_ns}+{\bf n}\Vert^2=\Vert{\bf d}\Vert^2$:

$$\begin{eqnarray} \Vert{\bf \overline{R_n}s}\Vert^2+\Vert{\bf R_ns}+{\bf n}\Vert... ...+\Vert{\bf R_ns}+{\bf n}\Vert^2&=& \Vert{\bf d}\Vert^2. \nonumber\end{eqnarray}$$ (118)

Similarly, we have $\Vert{\bf \overline{R_s}n}\Vert^2+\Vert{\bf R_sn}+{\bf s}\Vert^2=\Vert{\bf d}\Vert^2$. If we use equations () and (), the last two equalities can be written as follows:

$$\begin{eqnarray} \Vert{\bf \overline{R_n}d}\Vert^2+\Vert{\bf R_nd}\Vert^2 &=& \V... ...ine{R_s}d}\Vert^2+\Vert{\bf R_sd}\Vert^2 &=& \Vert{\bf d}\Vert^2.\end{eqnarray}$$ (119)

Hence, ${\bf \overline{R_n}d}$, ${\bf R_nd}$ and d form a right triangle with hypotenuse d and legs ${\bf \overline{R_n}d}$and ${\bf R_nd}$, as depicted in Figure ; similarly, ${\bf \overline{R_s}d}$, ${\bf R_sd}$ and d form a right triangle with hypotenuse d and legs ${\bf \overline{R_s}d}$and ${\bf R_sd}$.If n and s are orthogonal, s is in the null space of ${\bf R_n}$ and ${\bf \overline{R_n}d=\overline{R_n}s=s}$(Figure ). Similarly, n is in the null space of ${\bf R_s}$ and ${\bf \overline{R_s}d=\overline{R_s}n=n}$.