# 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?

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 are your series stationary --i.e. is mean(a) constant over time? – user603 Jul 13 '12 at 9:52 no, the series' are highly volatile. there is nothing constant over time. – Яaffael1984 Jul 13 '12 at 9:56 I believe that your series can jump all over the place, but have a (near) constant mean. That's why people detrend or difference time series. – Wayne Jul 13 '12 at 12:49

## 1 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.

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 the series are neither proportional nor linear combinations. They are individual time series driven by a similar basic force. So I end up with two different normalizations for one series. Which normalization does make more "sense" if I want to compare? – Яaffael1984 Jul 13 '12 at 10:02 I'm just describing the normalizations in terms of their properties, if that makes sense. To compare, you would have to apply the same normalization to both time series. And which one to use depends solely on what you mean by "the same basic force". If you mean they are correlated, then apply the second normalization. – Peteris Jul 13 '12 at 10:08 Hi, couldn't you index both series? Let's say the first observation in each series is 100. Then you have them on the same scale, and you can compare them. – Pantera Jul 21 '12 at 18:11