I have two time series a and b, which I want to compare. Due to their range difference I normalize them first.
a(i), b(i) are natural numbers for i=1,...,N
two different normalizations:
( mean and std_dev both refer to the whole time series )
1) a'(i) := a(i) / mean( a )
goal: mean( a' ) = 1
2) a'(i) := [ a(i) - mean( a ) ] / std_dev( a )
goal: usual normalization
what confuses me is how do the meanings after those transformations differ?
does the first transformation make any sense at all?
Best Answer
The first one will make two series indistinguishable, provided they are proportional to one another, i.e., $a_i = \lambda b_i$ for all $i$.
The second one will make two series indistinguishable, provided they are linear combinations of one another, i.e., $a_i = \lambda b_i + \mu$ for all $i$.
The first will set mean to one and the second will set mean to zero and variance to one; I don't think there is much to be said about the behavior of structurally different series under these normalizations.