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3D Coordinates Line of Fit

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Hello, I have an Nx3 matrix which represents sets of coordinates in 3D space. Is there a way to calculate a line of best fit (or any type of regression) to generate an equation for approximating expected data points? Thank you!

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Accepted Answer

Anton Semechko
Anton Semechko on 20 Jun 2018
Hey, David. Here is a demo you may find useful:
function best_fit_3D_line_demo
% Generate sample dataset
% -------------------------------------------------------------------------
% Standard deviation of noise
s=1;
% Random direction in space
r=randn(3,1);
r=r/(norm(r)+eps);
% Random data points along r
N=1E3; % number of point samples
t=(10*s)*(2*rand(N,1)-1);
Xo=bsxfun(@times,t,r');
% Add (isotropic) Gaussian noise to Xo
X=Xo+s*randn(N,3);
% Offset X (relative to the origin) by a random amount
X=bsxfun(@plus,X,5*s*randn(1,3));
% Find line of best fit (in least-squares sense) through X
% -------------------------------------------------------------------------
X_ave=mean(X,1); % mean; line of best fit will pass through this point
dX=bsxfun(@minus,X,X_ave); % residuals
C=(dX'*dX)/(N-1); % variance-covariance matrix of X
[R,D]=svd(C,0); % singular value decomposition of C; C=R*D*R'
% NOTES:
% 1) Direction of best fit line corresponds to R(:,1)
% 2) R(:,1) is the direction of maximum variances of dX
% 3) D(1,1) is the variance of dX after projection on R(:,1)
% 4) Parametric equation of best fit line: L(t)=X_ave+t*R(:,1)', where t is a real number
% 5) Total variance of X = trace(D)
% Coefficient of determineation; R^2 = (explained variance)/(total variance)
D=diag(D);
R2=D(1)/sum(D);
% Visualize X and line of best fit
% -------------------------------------------------------------------------
% End-points of a best-fit line (segment); used for visualization only
x=dX*R(:,1); % project residuals on R(:,1)
x_min=min(x);
x_max=max(x);
dx=x_max-x_min;
Xa=(x_min-0.05*dx)*R(:,1)' + X_ave;
Xb=(x_max+0.05*dx)*R(:,1)' + X_ave;
X_end=[Xa;Xb];
figure('color','w')
axis equal
hold on
plot3(X_end(:,1),X_end(:,2),X_end(:,3),'-r','LineWidth',3) % best fit line
plot3(X(:,1),X(:,2),X(:,3),'.k','MarkerSize',13) % simulated noisy data
set(get(gca,'Title'),'String',sprintf('R^2 = %.3f',R2),'FontSize',25,'FontWeight','normal')
xlabel('X','FontSize',20,'Color','k')
ylabel('Y','FontSize',20,'Color','k')
zlabel('Z','FontSize',20,'Color','k')
view([20 20])
drawnow

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Anton Semechko
Anton Semechko on 21 Jun 2018
No worries. Good luck with the rest of the bird-watching project.
Wes Anderson
Wes Anderson on 3 Dec 2019
Hi,
I have the same issue, but I'd like my regression to go from the axis to those data points, and to get an equation out of it...any idea how to do that?
Thank you
Wes Anderson
Wes Anderson on 3 Dec 2019
*from the origin, not the axis. Sorry.

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R2018a