This should be an optimization problem with nonlinear constraints, and the problem should be able to be solved using fmincon (See Documentation). Before using fmincon, two functions need to be defined. One is the objective function, one is the nonlinear constraint function.
Let us denote the nonlinear function as foo(x,a,b,c,d), which is the mixture Beta distribution as you mentioned previously. Then, the objective function can be defined as
function err = myObj(p,x,y)
a = p(1);
b = p(2);
c = p(3);
d = p(4);
err = 0;
for i = 1:length(x)
err = (foo(x(i) , a, b, c, d) - y(i))^2;
end
end
Save the above two functions as two separate MATLAB files of which the filenames are the same as the function names.
Now, the nonlinear constraint also needs to be defined. Since it is required that f(x,a,b,c,d)>=0 for all x, a way to defined the nonlinear constraint is
function [nl,nl2] = nlcon(p,x)
a = p(1);
b = p(2);
c = p(3);
d = p(4);
nl = [];
for i = 1:length(x)
nl = [nl; -foo(x(i), a,b,c,d)];
end
nl2= [];
end
Notice that I put a negative sign before foo. This is because fmincon requires the nonlinear constraint function to be smaller or equal to zero.
Now, in MATLAB command line or a script file, you can try the following steps
Step 1) Give values to x and y, such as
>> x =[10:52]; y = [0.006, 0.017, 0.026, ,...
Step 2) Specify the upper and lower bounds
>> lb = [10; 10; -inf; -inf];
>> ub = [52; 52; inf; inf];
Step 3) Specify the initial condition
Step 4) Solve the optimization problem using fmincon
>> xx = fmincon(@(p)myObj(p,x,y),p0,[],[],[],[],lb,ub,@(p)nlcon(p,x))
Note that in the above code I use anonymous functions (See Documentation) to pass x and y. This is because fmincon requires the objective function and the constraint function to only contain the decision variable p as their arguments, while the functions I defined have (p,x,y) or (p,x) as the arguments. To ensure the functions satisfy the requirements of fmincon, I defined to anonymous functions which only depend on p. -Yu
