# 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]

##### 2 Comments

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;

[Y;Y2]

plot(x,y,'y-',X,Y2,'yo',X,Y,'r*')

### Accepted Answer

Jeff Miller
on 22 Nov 2017

Edited: Image Analyst
on 27 Nov 2017

It sounds like you want "orthogonal linear regression".

##### 2 Comments

Image Analyst
on 27 Nov 2017

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