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Suppose I have two series of numbers and they don't have to be time series.

$S_1=$ {9,8,1 ...}

$S_2=$ {7,9,2 ...}

What are the methods to compare these two series and make a conclusion such that these two series are statistically similar, they have similar patterns or they are completely different?

I'm looking for pattern similarity, for example do they both draw an S curve, or they are linear, and I want to ignore time shifts.

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The issue with this question is that "statistically similar" is quite ambiguous. There are a lot of things that can be similar by one criteria, but different by others. For example, if series A is just series B with each element increased by 10, then the two series are identical if you're looking at how they move, but very different if you're looking for things that move around the same mean.

Another example is if you take a time series A and then shift that time series by a few periods to make A', then the two series will be identical if you look at them as "leaders/laggers", but often when using simple measures like correlation it will tell you that these two series are completely different.

So long story short there is no general method to say "these two series are similar" because your definition of "similar" may be very different from somebody else's. You'll need to refine what you're looking for before you can start making comparisons.

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  • $\begingroup$ I updated question. $\endgroup$
    – metdos
    Commented Feb 27, 2012 at 19:45
  • $\begingroup$ So in other words, you're looking for things that look the same on a graph? $\endgroup$
    – robbrit
    Commented Feb 28, 2012 at 17:18
  • $\begingroup$ Yes, we can say that. $\endgroup$
    – metdos
    Commented Feb 28, 2012 at 20:04
  • $\begingroup$ Unfortunately that's not an easy problem to solve. You might want to look into image pattern recognition in this case, since often "similarity" with respect to images is very different by other standards (ie. Gaussian vs. Cauchy). $\endgroup$
    – robbrit
    Commented Feb 28, 2012 at 20:14
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Agree, but you do need to start somewhere. You can look at trend, autocorrelation, variation around the trend, whether or not each sequence is stationary, compare the distributions of values, estimate and compare a Markov chain model for each sequence, compare extreme values, etc. Sounds like a great open-ended project for some undergrads.

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