## Geometric interpretation

The operators and 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
 (33)
and
 (34)
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 (33) and (34), we have for the data vector d the following equalities:
 (35)
and
 (36)
In the following equations, we prove that :
 (37)
Similarly, we have . If we use equations (35) and (36), the last two equalities can be written as follows:
 (38)
Hence, , and d form a right triangle with hypotenuse d and legs and , as depicted in Figure 13; similarly, , and d form a right triangle with hypotenuse d and legs and .If n and s are orthogonal, s is in the null space of and (Figure 14). Similarly, n is in the null space of and .

 geom11 Figure 13 A geometric interpretation of the noise filter when n and s are not orthogonal.

 geom21 Figure 14 A geometric interpretation of the noise filter when n and s are orthogonal.