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
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:
Figure 1 A geometric interpretation of the
noise filter when n and s are not orthogonal.
Figure 2 A geometric interpretation of the
noise filter when n and s are orthogonal.
In the following equations, we prove that :
Similarly, we have . If we use equations () and (),
the last two equalities can be written as follows: