Hi @Wes Anderson ,
You can utilize the concept of principal component analysis (PCA). The line of best fit can be represented as a vector that points in the direction of maximum variance of the data points. First, create 3-D data points into an Nx3 matrix, where each row corresponds to a point in 3D space.
% Example data points A = [1, 2, 3; 2, 3, 4; 3, 5, 6; 4, 7, 8]; % Nx3 matrix
Then, compute the covariance matrix which will capture how much the dimensions vary together.
C = cov(A); % Compute covariance matrix
Perform eigenvalue decomposition on the covariance matrix to find the principal components.
% V contains eigenvectors, D contains eigenvalues
[V, D] = eig(C);
Identify the principal component, bear in mind that the eigenvector corresponding to the largest eigenvalue indicates the direction of maximum variance, which is the direction of the line of best fit.
[~, i] = max(diag(D)); % Index of the largest eigenvalue
u = V(:, i); % Direction vector of the line of best fit
Formulate the equations by using line_points = t' * u'; % Points along the line where ( t ) is a scalar parameter that allows you to generate points along the line. % Example of generating points along the line
t = linspace(0, 10, 100); % Generate 100 points from 0 to 10
line_points = t' * u'; % Points along the line
Finally, plot the results and visualize the original data points along with the line of best fit.
figure;
scatter3(A(:,1), A(:,2), A(:,3), 'filled'); % Original data points
hold on;
plot3(line_points(:,1), line_points(:,2), line_points(:,3), 'r-', 'LineWidth', 2); %
Line of best fit
xlabel('X-axis');
ylabel('Y-axis');
zlabel('Z-axis');
title('3D Line of Best Fit');
grid on;
hold off;
Please see attached plot.
If you have new data, you can apply the same transformation using the derived direction vector ( u ) to assess its position relative to the established line of best fit. Hope this helps. Please let me know if you have any further questions.
