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fit a curve with smallest distance in y AND x direction to data points

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I have two sets of measurements x and y that seem to be correlated when plotting them against each other (below a small subsample of my data). I would like to fit a curve (linear, exponential, polynomial ..) through my data, such that it minimizes the distance of my points to the line (in an absolute or mean square way), but not only minimizing the distance on the y axis, but the closest distance in the x,y plane (euclidean distance). Is there a function/way to do that? As I understand it, most curve fitting functions in MATLAB fit the according to the rmse in y-direction only.
x =[1.3049 1.4137 0.2165 0.6538 0.6135 1.0655]
y =[4.0280 4.0865 50.1873 11.8024 7.9184 5.5866]
Roger Stafford
Roger Stafford on 23 Nov 2017
If you use a fifth order polynomial, here's about as close as you can get:
X = [1.3049 1.4137 0.2165 0.6538 0.6135 1.0655];
Y = [4.0280 4.0865 50.1873 11.8024 7.9184 5.5866];
Y2 = 469.165558828655 - 3459.00108010757*X + 9145.4534625995*X.^2 ...
- 11008.7869446613*X.^3 + 6175.46754074994*X.^4 - 1313.19614251125*X.^5;
x = linspace(.2,1.5);
y = 469.165558828655 - 3459.00108010757*x + 9145.4534625995*x.^2 ...
- 11008.7869446613*x.^3 + 6175.46754074994*x.^4 - 1313.19614251125*x.^5;
F S on 27 Nov 2017
Thank you! A fifth order polynomial is likely to hugely over fit the data though. And - more importantly - the way you did it if I'm not mistaken, you were minimizing the differences only in y-direction not in x AND y direction as I would like to (since I have errors on both measurements). Which function were you using for creating the result? Polyfit?

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Accepted Answer

Jeff Miller
Jeff Miller on 22 Nov 2017
Edited: Image Analyst on 27 Nov 2017
It sounds like you want "orthogonal linear regression".

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