There is Pearson's, which measures linear relationships. There is also Spearman's, which can detect monotonic relationships. I am wondering if there is a similar coefficient someone has come up with for measuring the "smoothness" of some data. By "smoothness," I don't mean differentiable, but more as the term is used in image processing, where close inputs yield close outputs.

My idea for such a metric: Given a reference point $(x_a,y_a)$ and a second data point $(x_b,y_b)$, let $w_{ab} = distance(x_a,x_b)$. Then the "smoothness" of the data is the weighted average of $distance(y_a,y_b)$ over all $a, b$ pairs with weights $w_{ab}$.

I'm wondering if a coefficient that measures something similar is already in widespread use. Seems to me it would be commonly used when a dataset is high-dimensional and one cannot simply graph the data to determine if nonlinear regression would be useful. Nothing is turning up in internet searches, but I suspect I am just not familiar with the academic terminology.

  • 2
    $\begingroup$ There are a great many. Quite a few of them are modeled on or inspired by the variogram. $\endgroup$
    – whuber
    Jan 19, 2019 at 1:45
  • 1
    $\begingroup$ Somewhat related method using a very different approach than correlation. Reshef, D. N., Reshef, Y. A., Finucane, H. K., Grossman, S. R., McVean, G., Turnbaugh, P. J., … Sabeti, P. C. (2011). Detecting Novel Associations in Large Data Sets. Science, 334(6062), 1518–1524. doi.org/10.1126/science.1205438 $\endgroup$
    – Alexis
    Jan 19, 2019 at 4:40


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.