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I know similar questions have been asked for time series data. But my question is a little bit different.

Consider that we have input dataset $X \in R^{N \times M}$, where $M$ is the dimension of inputs and $N$ is the number of samples. The output can be more than one dimension. Therefore, the output of our dataset is $Y \in R^{N \times T}$, where $T$ is the dimension of outputs. The problem refers to multitarget regression in the literature.

My question: Is there any criteria to measure the smoothness or the roughness of this dataset?

My definition for smoothness: Small change on inputs does not change outputs significantly.

Also, if there is a method for just $T=1$, I would like to know.

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Curvature based penalty terms have been suggested for regularising RBF and MLP neural networks, see e.g. Bishop (equation 8). Note however your mileage may vary (hint: what contsraints apply to the gaps between datapoints?).

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  • $\begingroup$ Is curvature-based a method to smooth the dataset or a method to measure the smoothness? $\endgroup$ – mohammad Mar 4 '20 at 8:47
  • $\begingroup$ It is a measure of the smoothness of a fit to the dataset, but that might include a model that exactly interpolates the dataset, which is probably as good as any method of estimating the smoothness of the data itself. $\endgroup$ – Dikran Marsupial Mar 4 '20 at 9:20
  • $\begingroup$ Thanks. Is there any package or function in r, python or Matlab to measure this criterion? $\endgroup$ – mohammad Mar 4 '20 at 9:26
  • $\begingroup$ Not that I know of. If there is a package that implements spline fitting, it may have something similar though. $\endgroup$ – Dikran Marsupial Mar 4 '20 at 9:34

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