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I recently have encountered very short time signals (from three to 12 samples, typically). They appeared for instance in small time frames in engine simulation that should be extrapolated in a robust manner, or gaz measurements taken for instance at $t_0 = 0$ (seconds), $t_1 = 5$, $t_2 = 10$, $t_3 = 20$, $t_4 = 50$, $t_5 = 100$. The idea behind this time sampling is that the signal should rise or fall, and may reach some steady-state. Such data cumulates caveats:

  1. Very short,
  2. Sometimes corrupted by one missing data, or one potential outlier,
  3. Uneven sampling.

Traditional tools (Fourier, correlation, fitting) appear of limited use, and are probably unusable. Yet I would like to:

  • describe those signals with few robust parameters,
  • estimate offsets and delays between two signals,
  • cluster them with shape similarity.

I for instance would like to detect "V-shaped" short signals that first go down then up, obtain a location estimator for the wedge, and separate them from monotonous or down-then-flat signals.

So far, I am performing tedious 3- or 4-point linear or parabolic fits, estimate residues and dispersion to select "a best model", with standard quadratic or robust quantile estimation. My questions are:

  • Is there recommended literature on very-short signal processing?
  • Would users share their best-practice experience on such signals, that ought to be processed?
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  • $\begingroup$ What about some moments: variance, skewness, kurtosis etc? I imagine a V shape vs a ramp would have different skewness. $\endgroup$ – geometrikal Sep 12 '15 at 11:52
  • $\begingroup$ To be honest, I have not tried moments for to bad reasons. First I am reluctant to do statistics on five points. Second I believe moments can become noise-sensitive quite easily, even with longer signals, especially when the underlying distribution is non-unimodal. But I should try. $\endgroup$ – Laurent Duval Sep 12 '15 at 15:01
  • $\begingroup$ What about resample to a fixed-sized vector and do a bit of PCA / k-means, or machine learning if you got some training data. $\endgroup$ – geometrikal Sep 13 '15 at 2:56
  • $\begingroup$ This idea is meaningful. Still, I wonder about how to resample a 5-sample series, so that the apriori injected in the resampling does not swallow the features. All my image or signal processing knowledge on 50+ sample data seems to collapse on much smaller series. $\endgroup$ – Laurent Duval Sep 13 '15 at 13:41
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Though problem! However, signal processing might have a tool at hand: It's called compressive sensing, and reduces the number of samples you need to sub-Nyquist-rate levels.

It's a bit nonsensical to derive the math behind compressive sensing for signals that are a sum of sinusoids (that being the only thing that I could do without literature) if that doesn't describe your signals overly well. However, the idea is:

If you want to know about everything happening with less than a given frequency over a specific time, you'll need all of the samples at a constant rate (typically, twice the highest frequency for real-valued samples). Then you can, for example, do a base transform to get the samples from time domain to frequency domain using the DFT.

For the following, assume that you could, if you had the "full, long, evenly-sampled" signal $\mathbf S \in \mathbb R^N$, define a base transform $\mathbf T$ to transform the time samples vector to a result vector giving you the information you want $\mathbf Y = \mathbb R^N$. That base transform would then simply be a matrix:

$$ \mathbf Y = \mathbf S^T \mathbf T$$

If what you want is sparse, ie. the dimensionality of the result vector would be much smaller than the observed samples vector, the transform matrix wouldn't be a square full-rank base transform, but a matrix that would have $N$ rows, but fewer columns.

The point here is that most of the values in $\mathbf T$ would be pretty small -- setting those to zero will incur an error, that "feels" like noise, but given a nonzero upper limit for allowable distortion, you can find a sparse matrix that still can allow for reliable conclusions about $\mathbf Y$.

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  • $\begingroup$ My belief is that compressive sensing is a bit out of topic here. We are in a case where data is acquired without sampling consideration. So it is "too late". $\endgroup$ – Laurent Duval Sep 12 '15 at 15:05
  • $\begingroup$ @LaurentDuval, well, yes, but the idea find a vector space reprentation of the things you want to observe and then find upper limits for the error introduced still applies. $\endgroup$ – Marcus Müller Sep 12 '15 at 15:07

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