## Looking for algebras designed to transform time series

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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:

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 any systems 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 – 2014-11-16T19:38:34.597

1@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

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It seems like you are overthinking this and getting caught up in your own head a little.

We can define a restricted set of time series as those which are well behaved (non-asymptotic) and single valued. These time series then form a time series algebra which have additive, multiplicative, commutative, and associative properties. There is an identity element (straight line with value 1) and a zero element (straight line with value 0). We even have a well defined inverse member for all time series not crossing zero. I believe this even forms an Abelian group.

Further, previous posters are correct that this group can be perfectly represented by various orthogonal vector spaces such as polynomials and Fourier space. This is true whether we are talking about Lagrange polynomials, Fourier series, or Gauss-Lobatto-Legendre polynomials.

The point then becomes... where does this get you? You got caught up in the suggestion to employ Fourier series in that you didn't want to remain in frequency space, but this is entirely unnecessary. You can, for instance, Fourier transform into frequency space, apply a high frequency filter, and then transform back into the time domain. This is incredibly effective in removing noise (although band pass filters tend to work better).

Stepping back from the question a little, it seems like you might really be asking: Given typical time series and typical time series operations (smoothing, de-seasoning, de-trending, etc) do the time series and their typical operations constitute a group with a well defined algebra? The answer is, yes. All of these operations just involve finding other time series which can be added, subtracted, multiplied, or divided by the original time series in order to yield a more physically tractable (interpretable) representation.

Thank you for this considered reply - especially the 'Stepping back...' paragraph. I'm aware that various operations on time series constitute groups. The rub is "finding other time series". Rather than rely upon serendipity, I'd like to establish how to find these "other time series" by constructing candidates systematically. I expect the effective strategies to be domain-specific. I am focused on time domain over frequency domain approaches because the time series which interest me most, right now, are finite length and non-cyclical. – aSteve – 2015-07-11T15:47:36.983

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The most direct and obvious transformation is from time domain to frequency domain. Possible methods include Fourier transform and wavelet transform. After the transform the signal is represented by a function of frequency-domain elements which can be operated on using ordinary algebra.

It's also possible to model a time series as a trajectory of a dynamical system in a state space (see: http://onlinelibrary.wiley.com/doi/10.1111/j.1467-9892.1980.tb00300.x/abstract, and http://www3.stat.sinica.edu.tw/statistica/oldpdf/A2n16.pdf). Dynamical systems can be modeled symbolically at a course-grain level (see: http://en.wikipedia.org/wiki/Symbolic_dynamics and http://www.math.washington.edu/SymbolicDynamics/) Symbolic dynamics draws on linear algebra.

Thank you for this interesting answer - it isn't what I anticipated. Time/frequency domain (Fourier/Wavelet) transforms operate on individual time-series, a very restrictive algebra. "Dynamic Systems" - on the other hand - focus on establishing models for time series - as opposed to establishing algebras ranging over time series. I hoped to discover sets of operators ranging over time-series that are 'somehow adequate'. – aSteve – 2014-09-17T09:59:14.570

Then I'm not sure what you mean by "algebra over time-domain time series". FWIW, you can operate algebraically in frequency domain to express relations over many time-series equivalents. A simple example: formulas for musical scales, incl. harmonics. Also, I don't know why you'd want to restrict yourself to time domain, given that frequency domain contains all the same information. – MrMeritology – 2014-09-17T12:34:54.860

I didn't intend to 'knock' exploiting the frequency domain... an approach that often yields seemingly 'magical' results. This question is focused exclusively on functions transforming time-domain time-series to time-domain time-series. I've added some details about abstract algebras (and a "Boolean Algebra" analogy) which, I hope, makes my question clearer. I am not excluding the possibility that operations could be defined using the Fourier transform... for example... but I don't want frequency domain representations as "outputs" from the operations which make up the algebra. – aSteve – 2014-09-17T13:32:40.867