I have to deal with multiple time series $X_n$ that are non-uniformly or irregularly sampled at increasing times $\Theta=\{t_k\}_{k\in \mathbb{Z}}$ ($ t_k<t_{k+1}$). In case this could help, this is an inherent unknown jitter: $t_k$ represents a real number from a unit-interval $[t_k,t_{k+1}[$. Moreover, there can be additional gaps or sampling drop-outs: some $t_k$ might be undefined (NaN) for a given $X_n$.

I am already using non-uniform schemes to compute finite differences and Fourier transformations. Now I am wondering about windows. I want to use both causal and acausal ones, on running windows of same duration $T$ across different time series $X_n$. I will typically have short frames, from five to twenty/thirty samples inside a $T$ period. The kind of problems I am facing is for instance, for a maximum five-sample frame, I may have all samples $\{t_0,t_1,t_2,t_3,t_4\}$ for $X_1$, and $\{t_1,t_2,t_4\}$ for $X_2$ because of drop-outs. There are several choices to fit the same continuous window formula through these samples, and adjust them for unit-total weight.

So far, I am using causal exponential windows, with maximum at $t_4$, and a standard normalizing for unit-weight. But I may need more clever techniques later.

I am wondering:

  • whether there exist best practice on choosing the appropriate end-points of the windows,
  • about reference papers of the impact of the above choices on features in the Fourier domain, especially for short windows,
  • if you use tricks to incorporate the uncertainty related to the unknown "jitter" since the exact time location is unknown in the interval $[t_k,t_{k+1}[$.
  • bonus: in case it may simplify answers, the $t_k$ could be a subset of relative integers (-3, 2, 78, 11, 23, etc.)
  • 2
    $\begingroup$ Don't know about NaNs but this paper seems to deal with the described jitters (section 4). $\endgroup$ Jun 1 at 20:53
  • 1
    $\begingroup$ is it possible that the unit interval you're referring to is $t_k \in [t_{k-1}, t_{k-1}+1[$? $\endgroup$ Jun 2 at 13:18
  • $\begingroup$ I was unclear, and forgot the braces around the subscript $\endgroup$ Jun 2 at 13:40

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