How to Finding Principle Axis and Determining the Specific Shape?

I have the scattered data as shown in below:

I want to determine the principle axis of these scattered data,

and then I want to determine the specific rectangular in which the error(?) is minimized.

In other words, the sum of distances between the outer line of square and each point is minimum.

It seems that the matlab comand 'pca' partically supports it, but I don't know how to use it effectively.

How to realize this idea? Is there other useful matlab command? I eagerly await reply.

Answer 1

[Matt J]

The principle axes you can find with svd(),

XY=[X(:),Y(:)]; %the scattered data
mu=mean(XY);
P=svd(XY-mu,0); %P contains the principle axes

Once you've done the pca, you can rotate/translate your scattered (X,Y) data so that the principle axes are aligned with the coordinate axes and so that the data is centered at the origin.

 xy=P.'*(XY-mu); %rotated data

Then, the problem boils down to estimating the dimensions a and b of a non-rotated rectangle. The latter, you might be able to do with fminsearch:

xp=abs(xy(:,1)); yp=abs(xy(:,2));

ab=fminsearch(@(ab) objective(ab,xp,yp), [max(xp),max(yp)] );

    function fval = objective(ab,x, y)
    
         a=ab(1)/2; b=ab(2)/2; %half length and width of rectangle 
         
         %%% Separate points into 4 regions (positive quadrant only)

           in=   x<a & y<b;  %inside rectangle
         out1=  x>=a & y<b;
         out2=  x>=a & y>=b;
         out3=  x<a  & y>=b;
     
         %%% Distances of inner points to boundary 
         
         xin=x(in); y=y(in); %points inside
         Din=sum( min([abs(xin-a), abs(xin+a), abs(yin-b), abs(yin+b)],[],2) ); 
         
         %%% Distances of outer points to boundary
         Dout1 = sum(x(out1)-a);
         Dout2 = sum(vecnorm([x(out2),y(out2)] -[a,b],2,2));
         Dout3 = sum(y(out3)-b);
        
         fval=Din+Dout1+Dout2+Dout3;
     
    end

[Changwoo Lee]

Thank you for your kindness