使用 fminsearch 进行参数估计

Question

我试图找到高斯分布的对数最大似然估计，以便估计参数。我知道 Matlab 有一个内置函数，它通过拟合高斯分布来执行此操作，但我需要使用 logMLE 来执行此操作，以便稍后将此方法扩展到其他分布。所以这里是 gaussian dist 的对数似然函数： Gaussian Log MLE

我使用这段代码通过 fminsearch 估计一组变量 (r) 的参数。但我的搜索没有涵盖，我也不完全明白问题出在哪里：

clear
clc
close all
%make random numbers with gaussian dist
r=[2.39587291079469
1.57478022109723
-0.442284350603745
4.39661178526569
7.94034385633171
7.52208574723178
5.80673144943155
-3.11338531920164
6.64267230284774
-2.02996003947964];
% mu=2 sigma=3

%introduce f
f=@(x,r)-(sum((-0.5.*log(2*3.14.*(x(2))))-(((r-(x(2))).^2)./(2.*(x(1))))))
fun = @(x)f(x,r);

% starting point
x0 = [0,0];
 [y,fval,exitflag,output] = fminsearch(fun,x0)


f = 
    @(x,r)-(sum((-0.5.*log(2*3.14.*(x(2))))-(((r-(x(2))).^2)./(2.*(x(1))))))


Exiting: Maximum number of function evaluations has been exceeded
         - increase MaxFunEvals option.
         Current function value: 477814.233176 
y = 1×2    
1.0e+-3 *

    0.2501   -0.0000

fval = 4.7781e+05 + 1.5708e+01i
exitflag = 0
output = 
    iterations: 183
     funcCount: 400
     algorithm: 'Nelder-Mead simplex direct search'
       message: 'Exiting: Maximum number of function evaluations has been exceeded↵         - increase MaxFunEvals option.↵         Current function value: 477814.233176 ↵'

Answer 1

Rewrite f as follows:

function y = g(x, r)

     n = length(r);

     log_part = 0.5.*n.*log(x(2).^2);

     sum_part = ((sum(r-x(1))).^2)./(2.*x(2).^2);

     y = log_part + sum_part;

 end

Use fmincon instead of fminsearch because standard deviation is always a positif number.

Set standard deviation lower bound to zero 0

整个代码如下：

%make random numbers with gaussian dist
r=[2.39587291079469
1.57478022109723
-0.442284350603745
4.39661178526569
7.94034385633171
7.52208574723178
5.80673144943155
-3.11338531920164
6.64267230284774
-2.02996003947964];
% mu=2 sigma=3

fun = @(x)g(x, r);
% starting point
x0 = [0,0];

% borns 
lb = [-inf, 0];
ub = [inf, inf];
[y, fval] = fmincon(fun,x0,[],[],[],[],lb,ub, []);
function y = g(x, r)

     n = length(r);

     log_part = 0.5.*n.*log(x(2).^2);

     sum_part = ((sum(r-x(1))).^2)./(2.*x(2).^2);

     y = log_part + sum_part;
end

解决方案

y = [3.0693    0.0000]

为了更好的估计直接使用mle()

代码非常简单：

y = mle(r,'distribution','normal')

解决方案

y = [3.0693    3.8056]

使用 fminsearch 进行参数估计

Using fminsearch for parameter estimation

matlab

fminsearch

mle

Rewrite f as follows: