I have some data which are the sets of values uniformly distributed along some measure (simple example - along the time intervals). I tested both the coefficients and found the difference, some times essential. F.e. if both the data are noise-like, signal r is about 0.7, and Pearson r - about 0.25.
Which coefficient should be considered better and why?
I compute the signal correlation coefficient as $$ r=\frac{\sum_ix_iy_i}{\sqrt{\sum_ix_i^2}\sqrt{\sum_iy_i^2}} $$
and Pearson correlation coefficient as $$ r=\frac{\sum_i(x_i-m_x)(y_i-m_y)}{\sqrt{\sum_i(x_i-m_x)^2}\sqrt{\sum_i(y_i-m_y)^2}} $$
There are two fragments of pair of signals:
Fragment 1. Two signals with similar Signal- and Pearson- coefficients
Fragment 1 http://net2ftp.ru/node0/[email protected]/Signals1.png
Fragment 2. Two noise-like signals: Signal r = 0.7, Pearson r = 0.2
Fragment 2 http://net2ftp.ru/node0/[email protected]/Signals2.png
PS I do not need to performe cross-corelation, convolution etc.