How to normalize two time series for comparison? 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?
 A: 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.
A: one thing i've found is how you will compare these two eventually, and how the normalized time series will affect the behavior of that comparison method. Let's say I have a set of original valued time series, and the values range from around 70 to 200. When I use your second normalization method, it has a mean of 0 and sd of 1. Even though the time series is exactly the same if you plot it, but notice that in the normalized time series, the values only range from -3 to 3. So what I found out is that, when I use k means clustering to cluster the original valued time series, the result is very good, around 88% accuracy. But the same clustering task on the new, normalized time series, has a lower accuracy of 66%. So to sum up, the different properties of the numbers before and after the normalization may affect the behavior of the algorithm that you're using to compare them. Beware.
