determine the normal to a plane using PCA

I have measured data that should represent a plane. I use princomp to remove tilt to the data in Z. I am interested in knowing the angle of tilt from Z to the normal of the original data. It seems like you should be able to use the coeffs calculated from princomp to get the normal vector which you can use to calculate the angle. Is this true? Alternatively, is there a way to calculate the normal vector of a data set?

Answer 1

[Aditya]

Hi Victor,

Yes, you can use the principal component analysis (PCA) coefficients to determine the normal vector of your data plane. The princomp function (or pca in more recent versions of MATLAB) provides you with principal component coefficients, which are essentially the directions of maximum variance in your data. For a plane, the normal vector can be derived from these coefficients.

Here is the documentation link for PCA:

https://www.mathworks.com/help/stats/pca.html

Here is an example code for the same:

% Example data
data = [x, y, z]; % Replace with your actual data

% Perform PCA
[coeff, ~, ~, ~, explained] = pca(data);

% The normal vector to the plane is the principal component with the smallest variance
% Since the components are sorted by explained variance, this is the last column in coeff
normal_vector = coeff(:, end);

% Z-axis unit vector
z_axis = [0; 0; 1];

% Calculate the angle between the normal vector and the Z-axis
cos_theta = dot(normal_vector, z_axis) / (norm(normal_vector) * norm(z_axis));
angle_rad = acos(cos_theta); % Angle in radians
angle_deg = rad2deg(angle_rad); % Angle in degrees

disp(['Angle of tilt: ', num2str(angle_deg), ' degrees']);

Hope this helps you resolve your query.