6

1

I am looking for information on (formal) algebraic systems that can be used to transform time-series - in either a practical or academic context.

I hope that there exists (at least one) small, expressive, set of operators - ranging over (finite) time-series. I want to compare and contrast different systems with respect to algebraic completeness, and brevity of representation, of common time-series transformations in various domains.

I realise this question is broad - but hope it is not too vague for datascience.stackexchange. I welcome any pointers to relevant literature for specific scenarios, or the general subject.

**Edit... (Attempt to better explain what I meant by an algebraic system...)**

I was thinking about "abstract algebras" as discussed in Wikipedia:

http://en.wikipedia.org/wiki/Algebra#Abstract_algebra http://en.wikipedia.org/wiki/Abstract_algebra#Basic_concepts

Boolean Algebras are (very simple) algebras that range over Boolean values. A simple example of such an algebra would consist the values True and False and the operations AND, OR and NOT. One might argue this algebra is 'complete' as, from these two constants (free-variables) and three basic operations, arbitrary boolean functions can be constructed/described.

I am interested to discover algebras where the values are (time-domain) time-series. I'd like it to be possible to construct "arbitrary" functions, that map time-series to time-series, from a few operations which, individually, map time-series to time-series. I am open to liberal interpretations of "arbitrary". I would be especially interested in examples of these algebras where the operations consist 'higher-order functions' - where such operations have been developed for a specific domain.

1sounds more like something for stats or even maths stack exchange sites... – Spacedman – 2014-09-16T16:35:52.827

Thanks for those suggestions. I am considering a similar question for the maths site... but hope to discover insights from data-science experts first. The question straddles practical data science and theoretical maths. – aSteve – 2014-09-16T22:42:54.807

This is a really interesting question and I agree it has practical applications, but I also really think it belongs on Mathematics. It should get a lot more exposure there. – shadowtalker – 2014-10-17T17:00:43.200

1You can't be successful if you don't have a goal. What kind of insights do you want to get? You have to define that first. Otherwise: Just use element-wise addition and multiplication. You're done now! Or aren't you? If you understand why this doesn't do it - maybe you can make progress. – Gerenuk – 2014-11-16T16:06:42.517

@Gerenuk : This would be a reasonable response if I wanted to test a specific hypothesis about specific time series. I'm interested in the meta-problem... i.e. are there

anysystems that go beyond element-wise addition and multiplication? If so, what are they, how do they go further, and what sort of (interesting) transformations do they facilitate? – aSteve – 2014-11-16T19:38:34.5971@aSteve: I could make up many non-trivial algebras, too. It would be a very tedious process asking you every time why this particular algebra does not suit your needs. It's really much more fruitful if there is an notion of what you want to achieve. – Gerenuk – 2014-11-16T20:55:24.630

@Gerenuk: I'm trying to establish a taxonomy of algebras with existing (practical or academic) applications. Every significantly distinct algebra is of interest - as is the context in which it was deemed valuable. Ultimately, I would like to establish if an (eloquent) universal time-series algebra can be defined, and - if not - why not. What is the minimum adequate sets of operators and constants? – aSteve – 2014-11-16T23:13:09.507

I have no particular expertise in this field, but would not a set of shift and scaling, and perhaps copy, operators be enough to transform any time series into any other time series ? – image_doctor – 2015-03-16T21:30:19.597

That sort of set sounds plausible, to me. I'd like to find, either, an explicit set that is used in some practical situation.... or a specific theoretic set, ideally accompanied by an (informal?) "proof" that this set is "sufficient" - in, erm, any sense that that makes sense. – aSteve – 2015-03-17T22:13:20.233

It is interesting that time series form an abelian group. I am looking for an example of binary time series that form a group. Let A be the set of all binary time series. (for example, all spike trains between time t_1 and t_2) I am looking for an operation

, such that (A,) form a group. Is the set of all spike trains form an abelian group? – None – 2015-08-09T10:44:16.053