# Very robust regression?

by skyraider
Tags: regression, robust
 P: 3 Hi, I want to model a set of a few dozen points on the x-y plane where y can be anywhere from 0 to 100 and x increases by 1 for each point on the y-axis, ex: (1, 26) (2, 84) (3, 2) etc. . . Is it possible to accurately model such a random array of points with an equation? Someone once suggested using an 'interpolating polynomial in the Lagrange form', but that does not appear to work well with such a random array of points. If it can't be done with a known regression technique, here is my question: Given the points (1, 26) (2, 84) (3, 2) (4, 100) (5, 50), could a function exist - any function of any category - which will hit each point? Thanks.
Mentor
P: 14,242
 Quote by skyraider Given the points (1, 26) (2, 84) (3, 2) (4, 100) (5, 50), could a function exist - any function of any category - which will hit each point? Thanks.
This final question is an easy one: The answer is yes. A fourth-order polynomial will hit each point exactly:
$$-27x^4 + 323\frac1 3x^3-1335x^2+2204\frac2 3x-1140$$

You generally don't want to do that, however. For example, this particular polynomial rapidly goes negative as x goes below 1 or above 5. In other words, it has very little extrapolative capability. You will quickly start to lose even interpolative capability with the exact-fit polynomial as the number of points increases. You want to develop a fit to a less expressive model.

There is no magic one-form-fits-all method. People can still get advanced degrees in statistics, after all.
 P: 1,294 If you tell us what you expect from this "model", we can suggest various methods that are suited to the task.