Types of data interpolation

Question

Consider the following two "types" of interpolation. In one case, our model passes through all observed data, in the other one, it doesn't. Do these types of interpolations have a name?

If I recall correctly:

Some authors would say that in (a) the model "fully interpolates the data"
In classification, I believe some authors may say that model (a) "fully shatters the data".
I think it's also common to say that model (a) "memorizes" the data.

I know that (a) is fitting the data more closely (it has a higher risk of overfitting), but I am wondering if there is a name or denomination for the actual phenomenon of having a model that takes on the actual values of the observed data instead of "averaging" through it.

a) is interpolation, b) is regression. See https://stats.stackexchange.com/questions/33659/how-is-interpolation-related-to-the-concept-of-regression — Sergio, Jul 26 '20 at 17:25
Ahh. I think that's the answer I was looking for @Sergio. So it's not correct to say that model (b) is interpolating the data? — Josh, Jul 26 '20 at 17:28
It depends on what you mean by "interpolating." Many people do insist that a true interpolator predict the observed value at each point. (In spatial stats they call this "honoring" the data.) In full generality (b) might include more than regression (for instance, if the red curve is not a function then this would not be a regression setting); a generic term is that (b) "smooths" the data. — whuber, Jul 26 '20 at 19:07

Types of data interpolation

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