# Distance between data and curve

I have two datasets:

dataset1 = {{3514147200, 5.83}, {3514233600, 7.48}, {3514320000, 7.86}, {3514406400, 6.74}}
dataset2 = {{2407708800, 131.3}, {2407795200, 131.7}, {2407881600, 130.9}, {2407968000, 131}}


By down sampling to get the datasets to be the same "length", I know that there is a correlation of approximately 0.95 between the two datasets.

I have created a very large polynomial from dataset2 to approximate dataset1.

polynomial[x_] := -1.19937*10^11 - 16.2653 x - 9.10704*10^-10 x^2 + 4.74474*10^-19 x^3 + ...


When polynomial and dataset1 are plotted together, I can see that the polynomial is close to the data. I am looking for a way to measure the "distance" between the dataset and the polynomial. I tried using KolmogorovSmirnovTest but it wasn't much help, any ideas about how to do this?

• Can you clarify what kind of data you have? "Dataset" in itself is not descriptive. Commented Jul 14, 2014 at 16:39
• Another thing you might want to clarify is what you mean by "distance". If you are asking about what is a good distance measure, it probably depends on the applications you have in mind, and it's not a question about Mathematica. stats.stackexchange.com is likely a better place to ask. If you already know what distance measure you need, then asking about how to compute it is appropriate here. Commented Jul 14, 2014 at 16:47
• So you have $(x_i,y_i)$ pairs that could be considered to approximate a function. If the domain of the two curves is the same, you could build an interpolation function from them (Interpolation) and NIntegrate the square of their difference. Commented Jul 14, 2014 at 16:52
• just fyi a high order polynomial is usually a terrible way to approximate data. You will tend to hit the points well and deviate wildly in between. Commented Jul 14, 2014 at 18:06
• Have you considered functions like Fit and NonlinearModelFit? Commented Jul 14, 2014 at 19:37

dataset1 = Table[{x1[n], y1[n]}, {n, 5}];

poly2[x_] = a*x^2 + b*x + c;

rms1 = Norm[(poly2 /@ dataset1[[All, 1]]) - dataset1[[All, 2]]]/
Sqrt[Length[dataset1]];

rms2 = RootMeanSquare[(poly2 /@ dataset1[[All, 1]]) -
dataset1[[All, 2]]];

(rms1 // ComplexExpand) == rms2


True