I have been trying to derivate equation under to find the formula for 3 parameters Weibull distribution as same as picture. Can someone please help me fix it?

syms m sigma_i sigma_0 sigma_th i N

lnL = symsum(ln((m/sigma_0) * ((sigma_i - sigma_th) / sigma_0))^(m-1) * exp(-(((sigma_i - sigma_th) / sigma_0) ^m)), i, 1, N);

Df_m = diff(lnL,m)

Df_sigma_0 = diff(lnL,sigma_0)

Df_sigma_th = diff(lnL,sigma_th)

Answer 1

[Sam Chak]

Is this what you are seeking?

Your equation is too lengthy and contains excessive, unnecessary brackets, as well as mismatched and unclosed brackets.

syms m sigma_i sigma_0 sigma_th i N

%% long equation, too many unnecessary brackets, mismatched brackets, unclosed bracket
% lnL = symsum((m/sigma_0) * ((sigma_i - sigma_th) / sigma_0))^(m-1) * exp(-(((sigma_i - sigma_th) / sigma_0) ^m), i, 1, N);

%% split into two shorter and easy-to-check terms
term1 = ((sigma_i - sigma_th)/sigma_0)^(m-1)

term1 = 

term2 = exp(- ((sigma_i - sigma_th)/sigma_0)^m)

term2 = 

%% function
fcn = (m/sigma_0)*term1*term2

fcn = 

%% sum of function (closed-form expression)
lnL = symsum(fcn, i, 1, N)

lnL = 

●

%% Derivative of lnL w.r.t. m
Df_m = diff(lnL,m)

Df_m = 

%% Derivative of lnL w.r.t. sigma_0
Df_sigma_0 = diff(lnL,sigma_0)

Df_sigma_0 = 

%% Derivative of lnL w.r.t. sigma_th
Df_sigma_th = diff(lnL,sigma_th)

Df_sigma_th = 

[Trong Nhan Tran]

sry i missed ln in euqation. how do i add ln in equation like in photo?

Answer 2

[Torsten]

This code works, but it will not give you the result you want.

syms m sigma_i sigma_0 sigma_th i N
lnL = symsum(log((m/sigma_0) * ((sigma_i - sigma_th) / sigma_0)^(m-1) * exp(-((sigma_i - sigma_th) / sigma_0)^m)), i, 1, N);
Df_m = diff(lnL,m)
Df_sigma_0 = diff(lnL,sigma_0)
Df_sigma_th = diff(lnL,sigma_th)

The problem is that you need to define sigma_i as an array of length N where N is itself a symbolic variable. This is not possible with the symbolic toolbox. You will need to specify a numerical value for N (usually the number of data points you have).

The below code should work:

syms sigma
syms m sigma_0 sigma_th
N = 15;
sigma = sym('sigma',[1,N]);
lnL = sum(log((m/sigma_0) * ((sigma - sigma_th) / sigma_0).^(m-1) .* exp(-((sigma - sigma_th) / sigma_0).^m)))
Df_m = diff(lnL,m)
Df_sigma_0 = diff(lnL,sigma_0)
Df_sigma_th = diff(lnL,sigma_th)

[Trong Nhan Tran]

Thank you for help me. I have question how can I add ln(x) like in photo?

[Torsten]

Thank you for help me. I have question how can I add ln(x) like in photo?

I added it.