# Is there a correlation coefficent for "smooth" functions?

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.

• There are a great many. Quite a few of them are modeled on or inspired by the variogram.
– whuber
Jan 19, 2019 at 1:45
• 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 Jan 19, 2019 at 4:40