# How can I determine the equation of the best-fit line, plane, or N-D surface using MATLAB?

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MathWorks Support Team on 10 Sep 2012
I have the coordinates of points on a line, plane, or higher dimensional surface, and I would like to know how I can fit these to a line, plane or surface, respectively, using MATLAB.

### Accepted Answer

MathWorks Support Team on 17 Feb 2021
Edited: MathWorks Support Team on 17 Feb 2021
Fitting surfaces to data can be performed both interactively and programmatically using Curve Fitting Toolbox 2.0 (R2009a). For more information on surface fitting in Curve Fitting Toolbox 2.0 (R2009a), refer the following online documentation:
For previous versions of MATLAB, try the following workarounds:
To fit a line, plane, or higher-dimensional surface to a set of data using MATLAB, use the backslash operator. The first example listed below shows how to determine the best-fit line for two-dimensional data; the second demonstrates how to fit a plane to three-dimensional data.
Example 1: Fitting a best-fit line to data
% Fitting a best-fit line to data, both noisy and non-noisy
x = rand(1,10);
n = rand(size(x)); % Noise
y = 2*x + 3; % x and y satisfy y = 2*x + 3
yn = y + n; % x and yn roughly satisfy yn = 2*x + 3 due to the noise
% Determine coefficients for non-noisy line y=m1*x+b1
Xcolv = x(:); % Make X a column vector
Ycolv = y(:); % Make Y a column vector
Const = ones(size(Xcolv)); % Vector of ones for constant term
Coeffs = [Xcolv Const]\Ycolv; % Find the coefficients
m1 = Coeffs(1);
b1 = Coeffs(2);
% To fit another function to this data, simply change the first
% matrix on the line defining Coeffs
% For example, this code would fit a quadratic
% y = Coeffs(1)*x^2+Coeffs(2)*x+Coeffs(3)
% Coeffs = [Xcolv.^2 Xcolv Const]\Ycolv;
% Note the .^ before the exponent of the first term
% Plot the original points and the fitted curve
figure
plot(x,y,'ro')
hold on
x2 = 0:0.01:1;
y2 = m1*x2+b1; % Evaluate fitted curve at many points
plot(x2, y2, 'g-')
title(sprintf('Non-noisy data: y=%f*x+%f',m1,b1))
% Determine coefficients for noisy line yn=m2*x+b2
Xcolv = x(:); % Make X a column vector
Yncolv = yn(:); % Make Yn a column vector
Const = ones(size(Xcolv)); % Vector of ones for constant term
NoisyCoeffs = [Xcolv Const]\Yncolv; % Find the coefficients
m2 = NoisyCoeffs(1);
b2 = NoisyCoeffs(2);
% Plot the original points and the fitted curve
figure
plot(x,yn,'ro')
hold on
x2 = 0:0.01:1;
yn2 = m2*x2+b2;
plot(x2, yn2, 'g-')
title(sprintf('Noisy data: y=%f*x+%f',m2,b2))
Example 2: Fitting a plane to data
x = rand(1,10);
y = rand(1,10);
z = (3-2*x-5*y)/4; % Equation of the plane containing
% (x,y,z) points is 2*x+5*y+4*z=3
Xcolv = x(:); % Make X a column vector
Ycolv = y(:); % Make Y a column vector
Zcolv = z(:); % Make Z a column vector
Const = ones(size(Xcolv)); % Vector of ones for constant term
Coefficients = [Xcolv Ycolv Const]\Zcolv; % Find the coefficients
XCoeff = Coefficients(1); % X coefficient
YCoeff = Coefficients(2); % X coefficient
CCoeff = Coefficients(3); % constant term
% Using the above variables, z = XCoeff * x + YCoeff * y + CCoeff
L=plot3(x,y,z,'ro'); % Plot the original data points
set(L,'Markersize',2*get(L,'Markersize')) % Making the circle markers larger
set(L,'Markerfacecolor','r') % Filling in the markers
hold on
[xx, yy]=meshgrid(0:0.1:1,0:0.1:1); % Generating a regular grid for plotting
zz = XCoeff * xx + YCoeff * yy + CCoeff;
surf(xx,yy,zz) % Plotting the surface
title(sprintf('Plotting plane z=(%f)*x+(%f)*y+(%f)',XCoeff, YCoeff, CCoeff))
% By rotating the surface, you can see that the points lie on the plane
% Also, if you multiply both sides of the equation in the title by 4,
% you get the equation in the comment on the third line of this example
The syntax used in these two examples generalizes to higher dimensions and higher order polynomials, by assembling the data matrices and using the backslash operator as shown on the lines labelled "Find the coefficients". This is successful as long as the data matrices are linear in powers of the independent variables.
For information on N-dimensional curve fitting tools, see the demos in the documentation for the following products:
Curve Fitting Toolbox:
http://www.mathworks.com/help/toolbox/curvefit/curvefit_product_page.html
Statistics Toolbox:
http://www.mathworks.com/access/helpdesk/help/toolbox/stats/
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