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Let $(t_k), k \in \mathbb{N}$, be an increasing sequence of real numbers ($t_{k-1} < t_k$) and $(X_{t_k}$) be a sequence of real numbers indexed by $(t_k)$. Such a sequence is sometimes called a time series.

The idea is that this series represents a sequence of measurements of some sort, like, for example, the average temperature of some location at time $t_k$.

The analysis of time series is an established area of statistics. In concrete applications, for example in climate science, there are two common problems when applying statistical algorithms to time series:

  1. The time series are finite, which produces artefacts in statistical algorithms that are designed for infinite time series. This problem is well known and there exist several approches to handle it.

  2. The times series are uneven spaced, that is $t_k - t_{k-1}$ is not independent of $k$.

I don't know of any textbook, algorithm or paper that explicitly addresses the latter problem. My question is therefore: Is this not a problem, is the solution trivial or, if not, are there any treatments?

Of course it is possible to interpolate missing values to generate a time series with an even time spacing $\min_k (t_k - t_{k-1})$, but it seems to me that this is not a solution, because algorithms like the fast fourier transform, nonlinear regression analysis or wavelet transforms would produce artefacts that depend on the kind of interpolation (linear, qubic splines, whatever). And therefore an explicit explanation of why the kind of interpolation one uses does not produce any artefacts in the analysis of the time series seems to be warranted to me, but I have never seen one in the literature.

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3 Answers

Unfortunately the problem is not trivial. Right now, there is virtually no theory for analyzing unevenly-spaced time series in their unaltered form. I have been working extensively on the problem over the past year and have typed up some notes that might be helpful (they can be found at http://www.eckner.com/research.html)

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Thanks! Keep me up to date about what you find out this year :-) –  Tim van Beek Jan 3 '11 at 8:47
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Fourier transforms depend upon the fact that the modeled signal are going to be infinite in time-span and time-extent. While it is possible to get a very good example of a time-limited signal by using a finite set of Fourier coefficients, the finite-fourier-coefficient-approximation always ends up with "ringing artefacts" at any high-frequency edges beyond the bandwidth-limited approximation. These artifacts arise from the fact that Fourier decomposition using the "infinite-time-extent" sine-wave as its base-component.

This type of problem in representing "limited-time-span" signals is what led to the concepts of "wavelets" and wavelet-transforms, using such limited-time-span base components such as the Haar wavelet. This is a slightly different problem from having non-equally-spaced-in-time samples extracted from a time series, but even then in these cases, there is the assumption that the underlying time series is continuous over time or is composed of the superposition of multiple discrete events occuring as Bernoulli or Poisson processes over time with some convolution of the discrete events by a smoothing factor (volcano eruption or geyser spouting, with the effluent "smoothed out" by prevailing winds or water currents).

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All the algorithms that I know assume an even spacing of the time series. While it is true that these algorithms are developed from the assumption of a continuous process, maybe with a certain class of stochastic processes in mind, which may be invalid or only approximatly valid, I'm looking for a solution to the very down-to-earth practical problem that I cannot use any code I know with an uneven spaced time series, I have to convert it to an even spaced first. And I don't know how this transformation will affect the ensuing analysis. –  Tim van Beek Dec 10 '10 at 14:46
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... This is a problem, there is no trivial solution, just because you cannot and must not solve both problems (the interpolation problem and your problem of interest) separately: you must solve them jointly.

However, if adapting the algorithm of interest to uneven spaced times or solving both problem jointly is too difficult, you may consider resorting to "Poincaré-Jaynes-Bretthorst" interpolation that can be easily adapted to handle uneven spaced times. Please see my question

uneven spaced time series

for references.

"Poincaré-Jaynes-Bretthorst" interpolation is in some extent exact: it is necessary (according to Poincaré) and also sufficient, provided that you equip yourself with Jaynes' Principle of Maximum Entropy. Essentially, you "just" need to choose the order of the derivative to constrain (that makes no big difference in some cases.)

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@Pascal-Orosco, you accidentally pasted the wrong pointer for your question. The correct pointer is mathoverflow.net/questions/47675/… for mathoverflow.net/questions/47675/ You have a very interesting question. –  sleepless in beantown Dec 11 '10 at 6:19
    
@sleepless: sorry for the wrong link. I also believe that my question is important. I will give it a second chance by extracting the problem from Bretthorst' paper... –  Pascal Orosco Dec 11 '10 at 13:09
    
Thanx, I'll need to reserve some time to look into that. –  Tim van Beek Dec 13 '10 at 15:57
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