Solving Normal Equations 3-Parameter Weibull Distribution

Hello, I am new to Matlab and would like to know how to input and solve the following Normal equations

[Torsten]

What is the function F from which you derived the three equations as dF/dm=0, dF/dc=0 and dF/dg=0 ?

Answer 1

[Saarthak Gupta]

Hi Danny,

Looks like you are trying to solve a system of nonlinear equations in the variables “m”, “c”, “g”.

This qualifies as a typical root-finding problem. MATLAB offers the “fsolve” function as part of the Optimization Toolbox for this purpose.

Refer to the following code:

% Define x and w vectors, to be used in paramfunc
x = randi(10,10,1);
w = randi(10,10,1);

% Parameterize paramfunc using anonymous function since the problem is defined in terms of p, and
% not x and w
fun = @(p)paramfunc(p,x,w);

p0 = [1,-0.0001,-10];
options = optimoptions('fsolve','Display','iter');
[p,fval] = fsolve(fun,p0,options);

                                             Norm of      First-order   Trust-region
 Iteration  Func-count     ||f(x)||^2           step       optimality         radius
     0          4             19177.1                        1.13e+04              1
     1          8             2664.67              1         4.21e+03              1
     2         12              20.189        1.84879              338            2.5
     3         13              20.189        4.62196              338           4.62
     4         17            0.996424        1.15549             51.6           1.16
     5         18            0.996424        2.88873             51.6           2.89
     6         22            0.559938       0.722182             18.2          0.722
     7         26            0.548133       0.722182             18.2          0.722
     8         30            0.526196       0.722182             19.8          0.722
     9         34            0.477128       0.722182             21.7          0.722
    10         38            0.396303       0.722182             23.8          0.722
    11         42            0.306712       0.722182               26          0.722
    12         46           0.0997196       0.525304             15.5          0.722
    13         50         1.61082e-05      0.0711707            0.183          0.722
    14         54         8.89285e-11     0.00299213         0.000458          0.722
    15         58         2.67257e-22    3.94812e-06         7.88e-10          0.722

Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.

% To verify if the solution is reliable, calculate the residual (sum of squares of fval) to see how close it is to zero.
% A small residual confirms that p is a solution.
sum(sum(fval.*fval))

ans = 2.6726e-22

% System of nonlinear equations to solve
function F = paramfunc(p,x,w)
m = p(1);
c = p(2);
g = p(3);
F = [sum(m.*log(x-g).^2 + log(x-g).*(c-w))
    sum(c-w+m.*log(x-g))
    sum((log(exp(1))./(x-g)).*(w-m.*log(x-g))-c)];
end

“fsolve” uses Levenberg-Marquardt and trust-region algorithms to find roots of the system of equations. Use one of these methods if the system may not have a zero. The algorithm still returns a point where the residual is small.

By default, “fsolve” chooses the trust-region dogleg algorithm, and its success in finding a solution will depend on the values in x, w, and the initial guess p0.

Refer to the following MATLAB documentation for further reference:

• fsolve: https://www.mathworks.com/help/optim/ug/fsolve.html
• Equation Solving Algorithms: https://www.mathworks.com/help/optim/ug/equation-solving-algorithms.html