Main diagonal operations problem
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Giuseppe Pintori
el 13 de Sept. de 2019
Editada: Bruno Luong
el 14 de Sept. de 2019
Hi guys, I need your help.
I want to create a matrix(4,4) in which the main diagonal have values between 0.3 and 1 and the other cells assume values such as to have a horizontal sum equal to 1.
By now I'm using the following code but the only result is to have a main diagonal composed by the same numbers:
x = eye(4)
x(1,1) = 1+(0.3-1)*rand(1,1)
x(2,2) = x(1,1)
x(3,3) = x(1,1)
x(4,4) = x(1,1)
Any suggestion?
PS : I've tried even with diag
2 comentarios
Stephen23
el 13 de Sept. de 2019
Editada: Stephen23
el 13 de Sept. de 2019
"...the other cells assume values such as to have a horizontal sum equal to 1"
Are there any other requirements on the other elements? Positive, negative, fractional values, integer, >1, >100, >1e100 ... what values are allowed?
What is the "horizontal sum": do you mean to sum along the 2nd dimension (i.e. along each row) ?
Respuesta aceptada
Stephen23
el 13 de Sept. de 2019
Editada: Stephen23
el 13 de Sept. de 2019
>> N = 4; % matrix size
>> M = nan(N,N); % preallocate
>> V = 0.3+(1-0.3)*rand(1,1) % diagonal value
V =
0.47505
>> M(~eye(N)) = randfixedsum(N-1,N,1-V,0,1); % other values
>> M = M.'; % transpose
>> M(1:N+1:end) = V % assign diagonal value
M =
0.47505 0.40657 0.0087969 0.10958
0.12917 0.47505 0.21287 0.1829
0.35794 0.15519 0.47505 0.011825
0.41335 0.032696 0.078907 0.47505
Checking the sum of each row:
>> sum(M,2)
ans =
1
1
1
1
and diagonal:
>> diag(M)
ans =
0.47505
0.47505
0.47505
0.47505
6 comentarios
John D'Errico
el 13 de Sept. de 2019
Editada: John D'Errico
el 13 de Sept. de 2019
Yes. I had come to that conclusion after some thought too. But there still seems to be the question of what is the true goal, since only Giuseppe can know that. I think Stephen's solution comes closer than mine.
On the other hand, HAD I generated the diagonal elements using a better distribution than uniform, then my solution would be an avenue to not needing to use a rejection while loop at all.
Bruno Luong
el 13 de Sept. de 2019
Editada: Bruno Luong
el 14 de Sept. de 2019
Oh I see your point. I think you are right by forcing the right PDF to generate D, it's equivalent to rejection method. This I'm pretty sure it's right because of the special property of simplex and barycentric coordinates.
So the right way (I simplify the procedure to a single row without loosing the generality) is
N = 4;
dmin = 0.3;
dmax = 1;
d = dmax-(dmax-dmin)*rand().^(1/(N-1)); % rather than dmin+(dmax-dmin)*rand;
v = (1-d) * randfixedsum(N-1,1,1,0,1);
then insert d to v....
Más respuestas (3)
John D'Errico
el 13 de Sept. de 2019
Editada: John D'Errico
el 13 de Sept. de 2019
Easy enough, it seems. First, determine the diagonal elements.
x = diag(rand(1,4)*.7 + .3);
Next, you need to choose the other row elements randomly so the sum will be 1. But that sum will now depend on the diagonal element you just chose. Stilll simple, as long as you use randfixedsum, by Roger Stafford, found on the file exchange.
for i = 1:4
x(i,setdiff(1:4,i)) = randfixedsum(3,1,1 - x(i,i),0,1)';
end
Did it work? Of course.
x
x =
0.83586 0.075979 0.057706 0.030454
0.012356 0.85664 0.11425 0.016757
0.13748 0.21163 0.43081 0.22009
0.15838 0.037488 0.16129 0.64284
>> sum(x,2)
ans =
1
1
1
1
Find randfixedsum here:
0 comentarios
Bruno Luong
el 13 de Sept. de 2019
Editada: Bruno Luong
el 13 de Sept. de 2019
Here is a method that has two advantages:
- without the need of Roger's FEX randfixedsum
- Produce matrix with rigourous uniform conditional probability
N = 4; % matrix size
% diagonal lo/up bounds
dmin = 0.3;
dmax = 1;
% random (common) diagonal value
d = dmax-(dmax-dmin)*rand().^(1/(N-1)); % Edit see comment above, equiv to rejection method
% d = dmin+(dmax-dmin)*rand;
% Generate N random vectors of length N-1 required sum == (1-d)
V = -log(rand(N-1,N)); % Marsaglia's [1961] method
V = V .* ((1-d)./sum(V,1));
% Arrange in the final matrix
A = zeros(N);
isdiag = sparse(1:N,1:N,true);
A(isdiag) = d;
A(~isdiag) = V(:);
A = A.';
% Check result
disp(A)
sum(A,2)
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