# Running window design for irregular or nonuniform time series

I have to deal with multiple time series $$X_n$$ that are non-uniformly or irregularly sampled at increasing times $$\Theta=\{t_k\}_{k\in \mathbb{Z}}$$ ($$t_k). In case this could help, this is an inherent unknown jitter: $$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.)
• Don't know about NaNs but this paper seems to deal with the described jitters (section 4). Jun 1 at 20:53
• is it possible that the unit interval you're referring to is $t_k \in [t_{k-1}, t_{k-1}+1[$? Jun 2 at 13:18
• I was unclear, and forgot the braces around the subscript Jun 2 at 13:40