### Highlights from Random Vectors with Fixed Sum

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# Random Vectors with Fixed Sum

### Roger Stafford (view profile)

19 Jan 2006 (Updated )

Randomly and uniformly generates vectors with a specified sum and values in a specified interval.

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Description

This generates m random n-element column vectors of values, [x1;x2;...;xn], each with a fixed sum, s, and subject to a restriction a<=xi<=b. The vectors are randomly and uniformly distributed in the n-1 dimensional space of solutions. This is accomplished by decomposing that space into a number of different types of simplexes (the many-dimensional generalizations of line segments, triangles, and tetrahedra.) The 'rand' function is used to distribute vectors within each simplex uniformly, and further calls on 'rand' serve to select different types of simplexes with probabilities proportional to their respective n-1 dimensional volumes. This algorithm does not perform any rejection of solutions - all are generated so as to already fit within the prescribed hypercube.

MATLAB release MATLAB 5.2 (R10)

Dear John, Thank you for your comment on my opinion. However my opinion is still the same with Juan. If you want to generate a random vectors with fixed sum, each vector that randomly come out of the algorithm has different probability of being chosen. And yes, this file seems to work properly as long as you don't care to examine the probability of each vector coming out of the algorithm. Thank you Juan. You got my point. This problem cannot be solved without using estimations, by the way. This problem can only be solved by discretization of the domain and finding all possible solutions and define a single value for each one and then if you pick one by chance its probability is the same with any other solution (vector). However, it is a stupid solution (Time and Cost). Again, it depends on your purpose. If you just want to produce some vectors with fixed summation this algorithm is a good choice for you. But if you want to use it in an optimization algorithm it is waste of time. For there are some solutions that with the confidence interval of 99.99 % you cannot produce that solution with the algorithm. In other words, if you try infinite number of times to produce them it will happen once. This is a simple and obvious probability problem. See what Juan wrote for you.

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04 Aug 2015 juan

### juan (view profile)

I think this script is not working good, the histograms of the generated variables are nos uniform, you can easily see that with large m. on the other side, if the sum is very extreme (for example large sum) the distribution is even worst.
On the other side, i think this problem has no solution, based on a very simple calculus:
let x1 x2 and x3 be 3 random variables with U[0 1] distribution subjected to the constraint: x1+x2+x3=1, then:
E(x1+x2+x3)=E(x1)+E(x2)+E(x3)=E(1)=1
so, as x1, x2 and x3 are identically and independent, the conclution is that E(x1)=E(x2)=E(x3)=1/3.

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03 Aug 2015 Guillaume

### Guillaume (view profile)

Great work!

I am trying to adapt this code so that it generates a vector with random values from the beta distribution.

It should, of course, remain possible to specify sum of the generated vector in advance.

Any suggestions? This is difficult for me.

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17 Apr 2015 J.E. Wong

### J.E. Wong (view profile)

may i ask what if i want to produce random numbers with fixed sum following a Gaussian distribution but without limit on the intervals?
i have tried to put a=-inf and b=inf but it ends up with NaN. (may be becoz s = (s-n*a)/(b-a) produces inf/inf?). is the rand generated from this algorithm following Gaussian too? many thx!

24 Dec 2014 John D'Errico

### John D'Errico (view profile)

I think Mohammad does not understand what this code does. In fact, it does work properly, and it does that job quite well and efficiently.

Besides, one should never just say simply that something does not work. Instead, show what you tried, and explain why you think it did not work. Then others can see either what you misunderstand about the code, or they can see why there may be a problem in the code. In this case, I happen to know the code does work as designed.

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This Function does not work properly. I tried this file several times and I got histogram for the values. The probabilities of the random values are not the same. It may cause serious problem when you are trying to run an optimization problem. However, It is a good job!

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05 Apr 2014 vachelard

### vachelard (view profile)

Hi
I try to use the function, but I have a problem when I'm driving.
I have a matrix A (193.1) I would like to create a matrix (193.3) whose sum equals one line to my matrix A.
I tried the following code but I meet an error:

Rhedgefund = Y (:, 1);

cols_to_generate = 3;

for K = 1: length (Y)

Neva (K, :) = randfixedsum (1 cols_to_generate, Y (K), -0.15, 0.15);
end

He told me:

Index EXCEEDS matrix dimensions.

Error in randfixedsum (line 95)

x = (b-a) * x (p + repmat ([0: n: n * (m-1)] n, 1)) + a; % Switches & rescale x

95 x = (b-a) * x (p + repmat ([0: n: n * (m-1)] n, 1)) + a; % Switches & rescale x

I do not know how to solve problem

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10 Apr 2013 shatakshi sharma

### shatakshi sharma (view profile)

i am trying to generate 6 random nmbrs within given range and sum:
xmin=[10 10 40 35 130 125];
xmax=[125 150 250 210 325 315];
Pg=randfixedsum(1,6,200,xmin, xmax);
it is giving following error:
?? Error using ==> minus
Matrix dimensions must agree.

Error in ==> randfixedsum at 56
s1 = s - (k:-1:k-n+1); % s1 & s2 will never be negative

Error in ==> busdatas at 47
Qg=randfixedsum(30,1,total(8),xmin, xmax);

can sm1 tell wats wrong..i cnt figure it out..

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08 Jun 2012 Olga Petrik

### Olga Petrik (view profile)

24 Feb 2012 Matlab2010

### Matlab2010 (view profile)

excellent! well done

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24 Feb 2012 Matlab2010

### Matlab2010 (view profile)

excellent! well done

13 Oct 2011 Christophe Lauwerys

### Christophe Lauwerys (view profile)

Nice. I'm trying to generate random data within a simplex defined by linear inequality constraints.

Lets say I already have the N vertices of the simplex defined by the inequalities. Is it then correct to first generate a random sample in the interval [0,1] with a sum equal to 1, and then take the inner product of this sample with the vector of vertices?

Something along the lines of:

X = rand(6,2);
k = convhull(X);
plot(X(k,1),X(k,2),'b'), hold on
nv = numel(k)-1; % Nmuber of vertices
X = X(k(1:end-1),:); % Remove repeated first vertex
L = randfixedsum(size(X,1),1000,1,0,1);
Y = L'*X;
plot(Y(:,1),Y(:,2),'r.'), hold off

Maybe I shouldn't trust my vision on this, but the samples don't really look uniformly spread within the simplex. For some reason they only seem to do for a triangle.

Any thoughts?

Thanks

Christophe

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27 Feb 2011 Bruno Luong

### Bruno Luong (view profile)

12 Jan 2011 Mate 2u

### Mate 2u (view profile)

Hi when i try to use the function on a very large a array it gives me the following error...

??? Maximum variable size allowed by the program is exceeded.

Error in ==> randfixedsum at 58
w = zeros(n,n+1); w(1,2) = realmax; % Scale for full 'double' range

Respect....

02 Apr 2010 Florian

### Florian (view profile)

Exactly what I was looking for!!! Many thanks for the great work!!!

03 Apr 2007 Per-Anders Ekström

Excellent!

01 Sep 2006 Mike Edwards

very useful! beautiful code!

30 Jan 2006 John D'Errico

This took a bit of work to verify uniformity in a slice of an n-dimensional hypercube. I'm now confident that Roger has done what he claimed, having checked samplings in several different dimensions, as well as having thought through the process he used to generate the sampling.