Plane Fitting a 3D Scatter Plot

I have a 3D scatter plot that I want to have a planar fit. How do you do this?

x = [14.9 1.7 0.0 10.9 0.0];
y = [11.3 9.1 23.7 12.8 2.9];
z = [5.32787E-17 2.93234E-16 2.09997E-16 5.45E-17 4.55E-16];
scatter3(x,y,z)

Answer 1

[Star Strider]

A simple linear regression is straightforward:

x = [14.9 1.7 0.0 10.9 0.0];
y = [11.3 9.1 23.7 12.8 2.9];
z = [5.32787E-17 2.93234E-16 2.09997E-16 5.45E-17 4.55E-16];

B = [x(:) y(:) ones(size(x(:)))] \ z(:);

xv = linspace(min(x), max(x), 10)';
yv = linspace(min(y), max(y), 10)';
[X,Y] = meshgrid(xv, yv);
Z = reshape([X(:), Y(:), ones(size(X(:)))] * B, numel(xv), []);

scatter3(x,y,z, 'filled')
hold on
mesh(X, Y, Z, 'FaceAlpha', 0.5)
hold off
view(-120, 35)
title(sprintf('Z = %+.3E\\cdotX %+.3E\\cdotY %+.3E', B)) 

producing:

[Sarah]

Two Questions:

1. Is there a way to visually indicate the distance or "error" between the measured points and the fit plane?
2. How would you calculate and display the coefficient of determination (R^2) on the graph?

[Star Strider]

Try this:

x = [14.9 1.7 0.0 10.9 0.0];
y = [11.3 9.1 23.7 12.8 2.9];
z = [5.32787E-17 2.93234E-16 2.09997E-16 5.45E-17 4.55E-16];

B = [x(:) y(:) ones(size(x(:)))] \ z(:);

xv = linspace(min(x), max(x), 50)';
yv = linspace(min(y), max(y), 50)';
[X,Y] = meshgrid(xv, yv);
Z = reshape([X(:), Y(:), ones(size(X(:)))] * B, numel(xv), []);
Ze = [x(:) y(:) ones(size(x(:)))] * B;                                      % Calculate Surface At Each Data Triplet
Err = Ze - z(:);                                                            % Calculate Errors
SStot = sum((z - mean(z)).^2);                                              % Total Sum-Of-Squares
SSres = sum(Err.^2);                                                        % Residual Sum-Of-Squares
Rsq = 1 - SSres/SStot;                                                      % Coefficient Of Determination

figure
surf(X, Y, Z, 'FaceAlpha', 0.5, 'EdgeColor','none')
hold on
plot3([x; x], [y; y], [z(:) Ze]', '-r', 'LineWidth',1)                      % Plot Errors (Red Lines From Surface To Data)
scatter3(x,y,z, 'filled')
hold off
view(-120, 35)
xlabel('X')
ylabel('Y')
zlabel('Z')
% title(sprintf('Z = %+.3E\\cdotX %+.3E\\cdotY %+.3E', B)) 
title('Z = -2.0998\times10^{-17}X - 1.135\times10^{-17}Y + 4.644\times10^{-16}')
text(10, 20, 3.5E-16, sprintf('R^2 = %0.3f', Rsq))

I tweaked my previous code slightly to incorporate the additions you requested. I also manually entered the title string, with the same data as previously. The first one still works, this one is easier to read.

Experiment to get the result you want.

The updated plot:

[Sarah]

Wonderful!!

[Star Strider]

As always, my pleasure!

[Damla Yüksel]

Hello, in the following line, matlab returns an error:

"Error using plot3

Vectors must be the same length.

Error in (line 33)

plot3([x; x], [y; y], [z(:) Ze]', '-r','LineWidth',1) % Plot Errors (Red Lines From Surface To Data)"

What should I do for this error?

[Star Strider]

You should make the vectors the same length!

It is best for you to post this as a new Question, supplying ‘x’, ‘y’, ‘z’ and ‘Ze’.

Your Comment does not relate to the substance of this thread, so I will delete it (and my Comment here) in a few yours.

Answer 2

[KSSV]

x = [14.9 1.7 0.0 10.9 0.0];
y = [11.3 9.1 23.7 12.8 2.9];
z = [5.32787E-17 2.93234E-16 2.09997E-16 5.45E-17 4.55E-16];

nx = 10 ; ny = 10 ;
xi = linspace(min(x),max(x),nx) ; 
yi = linspace(min(y),max(y),ny) ; 

[X,Y] = meshgrid(xi,yi) ;

F = scatteredInterpolant(x',y',z') ; 
Z = F(X,Y) ;

scatter3(x,y,z)
hold on
surf(X,Y,Z)