# Should an interpolation coincide the original function on the given data points?

Suppose having a model $f(x)=y$ where $f$ is unkown. Moreover, suppose you have some data points for this model i.e. $(x_1,y_1), (x_2,y_2), \dots , (x_n,y_n)$. If one can find an approximate of $f$ called $\tilde{f}$ using the given data points.

When such aproximation is called interpolation? should the approximation vanish on the given data points in order to be considered as an interpolation ( i.e. $\tilde{f}(x_i)=y_i$ for all $i$) ? Thank you in advance.

• What do you mean by "approximate a model by an explicit expression,using given data points" and "vanishing" approximation? What do you mean by approximating a model?
– Tim
Feb 8, 2016 at 15:36
• I edited my question, please check it @Tim Feb 8, 2016 at 15:41
• Interpolation takes certain data points as given and could not do anything otherwise. Any estimation method that doesn't reproduce those points, or leave them as given, is not an interpolation method, as least as I have ever met the term. Feb 8, 2016 at 15:47
• – user83346
Feb 8, 2016 at 16:07
• @Nick Your sense of "interpolation" may be unduly restrictive, because it would exclude very popular techniques such as various forms of splines, kriging with a nugget, loess (and other smoothers), and even linear regression. Many people distinguish interpolators that "honor the data" from those that do not, but they do not declare the latter techniques to be non-interpolators.
– whuber
Feb 9, 2016 at 3:26

Interpolation usually means $\tilde{f}(x_i)=y_i$ for all $i$.

There are other methods where we don't require $\tilde{f}(x_i)=y_i$ for all $i$, but these methods are not usually referred to as interpolation.

• Thank you for your aswer, but would you please add on what evidence did you base you answer ? a definition , reference ...; Feb 8, 2016 at 16:06
• I just knew this, but here's one source: mathworld.wolfram.com/Interpolation.html Feb 8, 2016 at 16:14
• As I clarify in a comment to the question, this is a mathematical answer to a statistical question. As such I think it's misleading in this context.
– whuber
Feb 9, 2016 at 14:34
• @whuber I have said in my answer that there are other methods, but that these methods are not referred to as interpolation. Feb 9, 2016 at 14:51
• Alas, they are referred to as interpolation. See the literature on time series, spatial analysis, or stochastic processes, inter alia. Some people, but definitely not all, call this "inexact interpolation". One well recognized authority, Noel Cressie, actually equates the words "smoothing" and "interpolation." He sees "smoothing (or interpolation) , filtering, and prediction" as all being the same thing, which he calls "spatial prediction" (Statistics for Spatial Data, 1st Ed., pp 105-6).
– whuber
Feb 9, 2016 at 15:18