How to generate random matlab vector with these constraints

Question

I'm having trouble creating a random vector V in Matlab subject to the following set of constraints: (given parameters N,D, L, and theta)

1. The vector V must be N units long
2. The elements must have an average of theta
3. No 2 successive elements may differ by more than +/-10
4. D == sum(L*cosd(V-theta))

I'm having the most problems with the last one. Any ideas?

Edit
Solutions in other languages or equation form are equally acceptable. Matlab is just a convenient prototyping tool for me, but the final algorithm will be in java.

Edit
From the comments and initial answers I want to add some clarifications and initial thoughts.

I am not seeking a 'truly random' solution from any standard distribution. I want a pseudo randomly generated sequence of values that satisfy the constraints given a parameter set.

The system I'm trying to approximate is a chain of N links of link length L where the end of the chain is D away from the other end in the direction of theta.

My initial insight here is that theta can be removed from consideration until the end, since (2) in essence adds theta to every element of a 0 mean vector V (shifting the mean to theta) and (4) simply removes that mean again. So, if you can find a solution for theta=0, the problem is solved for all theta.

As requested, here is a reasonable range of parameters (not hard constraints, but typical values):
5<N<200
3<D<150
L==1
0 < theta < 360

Answer 1

I would start by creating a "valid" vector. That should be possible - say calculate it for every entry to have the same value.

Once you got that vector I would apply some transformations to "shuffle" it. "Rejection sampling" is the keyword - if the shuffle would violate one of your rules you just don't do it.

As transformations I come up with:

• switch two entries
• modify the value of one entry and modify a second one to keep the 4th condition (Theoretically you could just shuffle two till the condition is fulfilled - but the chance that happens is quite low)

But maybe you can find some more.

Do this reasonable often and you get a "valid" random vector. Theoretically you should be able to get all valid vectors - practically you could try to construct several "start" vectors so it won't take that long.

Answer 2

You don't give us a lot of detail to work with, so I'll assume the following:

• random numbers are to be drawn from [-127+theta +127-theta]
• all random numbers will be drawn from a uniform distribution
• all random numbers will be of type int8

Then, for the first 3 requirements, you can use this:

N = 1e4;
theta = 40;
diffVal = 10;

g = @() randi([intmin('int8')+theta  intmax('int8')-theta], 'int8') + theta;
V = [g(); zeros(N-1,1, 'int8')];
for ii = 2:N
    V(ii) = g();
    while abs(V(ii)-V(ii-1)) >= diffVal
        V(ii) = g();
    end
end

inline the anonymous function for more speed.

Now, the last requirement,

D == sum(L*cos(V-theta))

is a bit of a strange one...cos(V-theta) is a specific way to re-scale the data to the [-1 +1] interval, which the multiplication with L will then scale to [-L +L]. On first sight, you'd expect the sum to average out to 0.

However, the expected value of cos(x) when x is a random variable from a uniform distribution in [0 2*pi] is 2/pi (see here for example). Ignoring for the moment the fact that our limits are different from [0 2*pi], the expected value of sum(L*cos(V-theta)) would simply reduce to the constant value of 2*N*L/pi.

How you can force this to equal some other constant D is beyond me...can you perhaps elaborate on that a bit more?

Answer 3

Here's a way of doing it. It is clear that not all combinations of theta, N, L and D are valid. It is also clear that you're trying to simulate random objects that are quite complex. You will probably have a hard time showing anything useful with respect to these vectors.

The series you're trying to simulate seems similar to the Wiener process. So I started with that, you can start with anything that is random yet reasonable. I then use that as a starting point for an optimization that tries to satisfy 2,3 and 4. The closer your initial value to a valid vector (satisfying all your conditions) the better the convergence.

function series = generate_series(D, L, N,theta)
s(1) = theta;
for i=2:N,
    s(i) = s(i-1) + randn(1,1);
end
f = @(x)objective(x,D,L,N,theta)
q = optimset('Display','iter','TolFun',1e-10,'MaxFunEvals',Inf,'MaxIter',Inf)
[sf,val] = fminunc(f,s,q);
val
series = sf;



function value= objective(s,D,L,N,theta)
a = abs(mean(s)-theta);
b = abs(D-sum(L*cos(s-theta)));
c = 0;
for i=2:N,
    u =abs(s(i)-s(i-1)) ;
    if u>10,
        c = c + u;
    end
end
value = a^2 + b^2+ c^2;

It seems like you're trying to simulate something very complex/strange (a path of a given curvature?), see questions by other commenters. Still you will have to use your domain knowledge to connect D and L with a reasonable mu and sigma for the Wiener to act as initialization.

Answer 4

So based on your new requirements, it seems like what you're actually looking for is an ordered list of random angles, with a maximum change in angle of 10 degrees (which I first convert to radians), such that the distance and direction from start to end and link length and number of links are specified?

Simulate an initial guess. It will not hold with the D and theta constraints (i.e. specified D and specified theta)

angles = zeros(N, 1)

for link = 2:N
    angles (link) = theta(link - 1) + (rand() - 0.5)*(10*pi/180)
end

Use genetic algorithm (or another optimization) to adjust the angles based on the following cost function:

dx = sum(L*cos(angle));
dy = sum(L*sin(angle));

D = sqrt(dx^2 + dy^2);
theta = atan2(dy/dx);

the cost is now just the difference between the vector given by my D and theta above and the vector given by the specified D and theta (i.e. the inputs).

You will still have to enforce the max change of 10 degrees rule, perhaps that should just make the cost function enormous if it is violated? Perhaps there is a cleaner way to specify sequence constraints in optimization algorithms (I don't know how).

I feel like if you can find the right optimization with the right parameters this should be able to simulate your problem.