Calculate pairwise distances of xyz values within a matrix using matlab - matlab

Question

I am fairly new to Matlab and have been trying to solve this issue for some weeks.
I have several large matrices, which vary in their amount of rows, although all have 69 columns. Within these matrices are 23 xyz values, e.g. column 1,2,3 are the x,y,z values for point 1, column 4,5,6 the x,y,z values for point 2 and so on, up to column 69.
There are over 1000 rows for each matrix, with each row representing a point in time.
I am trying to use a for loop to calculate the difference between row 1 and 2, then row 2 and 3, then 3 and 4 etc, using pythagoras theorem.
Essentially, the end output should be a N*23 matrix of the distance values but I'm getting stuck on generating the for loop.
I've been trying to solve this using a smaller 4*3 array for the meantime, (n is the matrix) with no luck.
for i = 1:row
for j = 1:col
pythag(i,j) = sqrt((n(i,1)-n(j,1))^2 +((n(i,2)-n(j,2))^2 +((n(i,3)-n(j,3))^2)));
end
end
Any help would be appreciated.

Answer 1

You can actually solve the problem in a single line of code.
The end output should be (N-1)*23 and not N*23 matrix.
In your loop, the column index in n should be advanced by 3 for each advancement of j.
You also need to subtract row i+1 from row i.
I renamed your matrix n with A.
Here is a sample solution using a for loop (please read the comments):
%I have a large number of N*69 matrices
%Within these matrices are 23 xyz values, e.g. column 1,2,3 are the x,y,z values for point 1, column 4,5,6 the x,y,z values for point 2 and so on, up to column 69.
%One row:
%A = [x1, y1, z1, x2, y2, z2, x3, y3, z3, x4, y4, z4]; %For example take 4 xyz values (instead of 23)
%There are over 1000 rows for each matrix,
%Example include 4 rows:
% A = [x11, y11, z11, x12, y12, z12, x13, y13, z13, x14, y14, z14;...
% x21, y21, z21, x22, y22, z22, x23, y23, z23, x24, y24, z24;...
% x31, y31, z31, x32, y32, z32, x33, y33, z33, x34, y34, z34;...
% x41, y41, z41, x42, y42, z42, x43, y43, z43, x44, y44, z44];
%Initialize A with arbitrary values (I decided to name the matrix A instead of n).
A = [magic(4) magic(4) magic(4)];
% A = [16 2 3 13 16 2 3 13 16 2 3 13
% 5 11 10 8 5 11 10 8 5 11 10 8
% 9 7 6 12 9 7 6 12 9 7 6 12
% 4 14 15 1 4 14 15 1 4 14 15 1];
%Calculate the difference between row 1 and 2, then row 2 and 3, then 3 and 4
%Essentially, the end output should be a N*23 matrix. Wrong!!! it should be (N-1)*23 matrix
%Assume N is the number of rows (over 1000 rows).
N = size(A, 1);
%Size of pythag matrix is (N-1)x(69/3) and in the example size is 3x4
row = N-1;
col = size(A,2)/3;
%Initialize output matrix with zeros.
pythag = zeros(row, col);
%Solving using a nested for loops:
for i = 1:row
%k, k+1, k+2 are the x,y,z, columns indeces in matrix A
k = 1;
for j = 1:col
%pythag(i,j) = sqrt((n(i,1)-n(j,1))^2 +((n(i,2)-n(j,2))^2 +((n(i,3)-n(j,3))^2)));
%We want the to distance from row i+1 to row i (in matrix A).
pythag(i, j) = sqrt((A(i+1, k)-A(i, k))^2 +((A(i+1, k+1)-A(i, k+1))^2 +((A(i+1, k+2)-A(i, k+2))^2)));
% | | |
% x y z
%The column index in A must skip by 3 for each increment of j
k = k + 3;
end
end
More elegant: replacing the sqrt(x^2 + y^2 + z^2) with norm([x,y,z]):
for i = 1:row
%k, k+1, k+2 are the x,y,z, columns indices in matrix A
k = 1;
for j = 1:col
%We want the to distance from row i+1 to row i (in matrix A).
pythag(i, j) = norm([A(i+1, k) - A(i, k), A(i+1, k+1) - A(i, k+1), A(i+1, k+2) - A(i, k+2)]);
%The column index in A must skip by 3 for each increment of j
k = k + 3;
end
end
Vectorizing the inner loop (use vecnorm instead of norm):
for i = 1:row
pythag(i, 1:col) = vecnorm([A(i+1, 1:3:end) - A(i, 1:3:end); A(i+1, 2:3:end) - A(i, 2:3:end); A(i+1, 3:3:end) - A(i, 3:3:end)]);
end
Vectorizing the outer loop (too confusing with vecnorm - back to Pythagoras):
pythag = sqrt((A(2:end, 1:3:end)-A(1:end-1, 1:3:end)).^2 +((A(2:end, 2:3:end)-A(1:end-1, 2:3:end)).^2 +((A(2:end, 3:3:end)-A(1:end-1, 3:3:end)).^2)));

Answer 2

Make use of the full power of Matlab using vector-wise calculations: you can omit the inner loop and calculate the distance immediately for a vector information (note that you need to provide the direction of the sum()).
Matlab stores all data in the RAM consistency, i.e. it will sometimes have to copy the vector to a different location in the RAM as it grows. Therefore, allocate the full space right at the start e.g. with NaN().
% allocate memory
pythag = NaN(row,col);
for i = 1:row
pythag(i,:) = sqrt(sum( (n(i,1:3)-n(:,1:3)).^2,2 ));
end
If you provide a minimal working example, we can check your full code and walk you through the problem.