Hello!
I try to find a method for estimating smoothness (or roughtness) of a discreet signal.
The first obvious guess is standard deviation. However, it works only for stationary signals (as in fig. A). Comparing plots A and B, one intuitively sees that B is "smoother" than A, but standard deviation of these two signals is similar. Also standard deviation fails completely for non-function data, as in fig. C.
I have heard of a method where we treat signal as a curved tube and try to slip a metal bar (or wire) into that tube. The energy needed for such a move (and bar deformation) would be an estimate of signal's "roughness". But, I cannot find a comprehensive reference for that method.
Are there any other methods dedicated for such a problem? What is your practice?
Best regards,
Michal
[Skip to TL,DR: for quick suggestion]
Hi Michał!
Thank you for the fantastic question – it is one of those concepts that are very well defined in some contexts1 but do not seem to be so in others! Very briefly, smoothness is not defined in a uniform way for the purpose you mention, and you need to choose one of the measures of smoothness that seem to be used widely.
Let us take some issues out of the way first:
1. Figure C is not a signal because it is not a function. For point clouds we need completely different measures (suggestion at the end).
2. Stationarity usually refers to processes that generate signals, and not the signals themselves.
3. I assume the points are sampled uniformly, otherwise we have a whole different can of worms (can be solved by first making the sampling uniform, as in https://de.mathworks.com/help/signal/ug/resampling-nonuniformly-sampled-signals.html)
4. For Fig. A and Fig. B, let us subtract their mean value first. I think your intuition of smoothness does not depend on shifting them vertically (see picture below)? This makes talking about the signals a bit easier.
Now: As you mentioned, standard deviation is not a good candidate for the kind of smoothness you are looking for. It is the square root of signal variance which is exactly the same as the average power of the signals. We could design a signal to have the exact same power but with a higher frequency i.e. more rapid changes, but I feel you want high frequencies to mean less smooth.
Your bending energy intuition is a perfect one and can be calculated first by a cubic spline interpolation2. It is however a bit cumbersome.
TL,DR:
I suggest you could take the standard deviation of the slope of the signals: this measure fits with your intuition and seems to be widely used and cited. Look at Figure 2D here3:
Article Surface Roughness of Topography: A Multi-Scale Analysis of L...
Best
Suddha
____________________
[1] https://en.wikipedia.org/wiki/Smoothness: This is the popular smoothness from calculus. This smoothness is either/or or has a number: C2 means the function is continuously differentiable twice. You probably do not want this.
[2] https://math.stackexchange.com/questions/1369155/how-do-i-rate-smoothness-of-discretely-sampled-data-picture
[3] For citing: Grohmann, C. H., Smith, M. J., & Riccomini, C. (2010). Multiscale analysis of topographic surface roughness in the Midland Valley, Scotland. IEEE Transactions on Geoscience and Remote Sensing, 49(4), 1200-121. DOI: 10.1109/TGRS.2010.2053546
[4] For point cloud, one could define a "center" and convert from polar to cartesian, or use Fourier descriptors. Considerably more complex and needs new definitions for your case for center at least. http://fourier.eng.hmc.edu/e161/lectures/fd/node1.html
Hmmm...so this question appeared in "questions I might be able to answer," so I'll give it a quick shot. I've found that "signal" can be used to mean many things, but if you look at it as a question of model fit (generally how well a data fit the model), then the graphs are clearly not the same, and where as a linear equation would most likely best fit your first graph, it would fail horribly for either 2 or 3. When we talk about "fail," generally what we're talking about is the RMSE (root mean squared error). So RMSE is the standard deviation of the residuals and generally tells you how concentrated your data are around the line of best fit. (Some use MAE, which is the average of the absolute value of the residuals...and has benefits with regard to interpretability and is preferred when the error cost is linear and not quadratic).
Using a spline or some kind of polynomial equation would greatly improve RMSE/MAE (hence model fit) in your second and third examples. The other end of this is that you have to make sure you don't overfit your data, but this wouldn't be a concern if you are merely trying to imitate the line (which is what I'm taking you to mean when you say "signal").
Can one answer about how He is? where is he go? why he is here?
Suddha Sourav James Ades You gave me some useful hints, thank you for that!
- Badges
- Science topic
- Similar topics
- Biological Science
- Faunistics
- Ornithology
- birds