本文介绍了使用mle()估算自定义分布的参数的处理方法，对大家解决问题具有一定的参考价值，需要的朋友们下面随着小编来一起学习吧！
问题描述
我有以下代码，希望估算自定义发行版的参数. 有关分发的更多详细信息.然后，使用估计的参数，我想查看估计的PDF是否类似于给定数据的分布(应该与给定数据的分布匹配).
I have the following code that I wish to estimate the parameters of a custom distribution. For more details on the distribution. Then using the estimated parameters I want to see if the estimated PDF resembles the distribution of the given the data (it is supposed to match the distribution of the given data).
:"x"现在包含数据样本，而不是PDF
主要代码是:
x = [0.0320000000000000 0.0280000000000000 0.0280000000000000 0.0270000000000000 0.0320000000000000 0.0320000000000000 0.0480000000000000 0.0890000000000000 0.0500000000000000 0.0620000000000000 0.0480000000000000 0.0300000000000000 0.0520000000000000 0.0460000000000000 0.0540000000000000 0.0520000000000000 0.0510000000000000 0.0310000000000000 0.0330000000000000 0.0330000000000000 0.0380000000000000 0.0850000000000000 0.102000000000000 0.0290000000000000 0.0530000000000000 0.0590000000000000 0.0320000000000000 0.0800000000000000 0.0410000000000000 0.0280000000000000 0.0670000000000000 0.0350000000000000 0.0420000000000000 0.0280000000000000 0.0370000000000000 0.0480000000000000 0.0330000000000000 0.101000000000000 0.0420000000000000 0.0840000000000000 0.0340000000000000 0.0900000000000000 0.0900000000000000 0.0460000000000000 0.0290000000000000 0.0330000000000000 0.0350000000000000 0.0330000000000000 0.0320000000000000 0.0420000000000000 0.0600000000000000 0.0500000000000000 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0.0390000000000000 0.0290000000000000 0.0270000000000000 0.0370000000000000 0.0580000000000000 0.0640000000000000 0.0300000000000000 0.0380000000000000 0.0240000000000000 0.0380000000000000 0.0830000000000000 0.0400000000000000 0.0990000000000000 0.0600000000000000 0.0580000000000000 0.0430000000000000 0.0840000000000000 0.0390000000000000 0.0370000000000000 0.0850000000000000 0.0590000000000000 0.0530000000000000 0.0560000000000000 0.0320000000000000 0.0340000000000000 0.0250000000000000 0.0520000000000000 0.0490000000000000 0.0270000000000000 0.0470000000000000 0.0520000000000000 0.0530000000000000 0.0410000000000000 0.0260000000000000 0.0290000000000000 0.0470000000000000 0.0550000000000000 0.0710000000000000 0.0520000000000000 0.0650000000000000 0.0440000000000000 0.0710000000000000 0.0550000000000000 0.0410000000000000 0.0640000000000000 0.0350000000000000 0.0930000000000000 0.0310000000000000 0.0480000000000000 0.0370000000000000 0.0380000000000000 0.0520000000000000 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0.0350000000000000 0.0370000000000000 0.0630000000000000 0.0760000000000000 0.0830000000000000 0.0360000000000000 0.0590000000000000 0.0430000000000000 0.0790000000000000 0.0330000000000000 0.0520000000000000 0.0530000000000000];
Censored = ones(1,size(x,2));%
custpdf = @eval_custpdf;
custcdf = @eval_custcdf;
options = statset('Display','iter','MaxFunEvals',1000,'MaxIter',1000,...
'FunValCheck','off','TolX',1.0e-10,'TolFun',1.0e-10);
phat = mle(x,'pdf', custpdf,'cdf', custcdf,...
'start',[0.6,0.02,1.01,2,4,-10],...
'lowerbound',[0 0 0 0 0 -inf],...
'upperbound',[inf inf inf inf inf inf],...
'Censoring',Censored,...
'Options',options);;
% Checking how close the estimated PDF and CDF match with those from the data x
figure();
h = histogram(x,'Normalization','probability');hold on
x_times = h.BinEdges(1:end-1) + h.BinWidth/2 ;
y_vals = custpdf(x_times, phat(1), phat(2), phat(3), phat(4), phat(5), phat(6))./...
sum(custpdf(x_times, phat(1), phat(2), phat(3), phat(4), phat(5), phat(6)),'omitnan');
plot(x_times,y_vals,'linewidth',2)
legend('Data','Estimated PDF')
功能是:
function out = eval_custpdf(x,myalpha,mytheta,mybeta,a,b,c)
first_integral = integral(@(x) eval_K(x,a,b,c),0,1).^-1;
theta_t_ratio = (mytheta./x);
incomplete_gamma = igamma(myalpha,theta_t_ratio.^mybeta);
n_gamma = gamma(myalpha);
exponent_term = exp(-theta_t_ratio.^mybeta-(c.*(incomplete_gamma./n_gamma)));
numerator = first_integral.* mybeta.*incomplete_gamma.^(a-1).*...
theta_t_ratio.^(myalpha*mybeta+1).*exponent_term;
denominator = mytheta.* n_gamma.^(a+b-1).* (n_gamma-incomplete_gamma.^mybeta).^(1-b);
out = numerator./denominator;
end
function out = eval_custcdf(x,myalpha,mytheta,mybeta,a,b,c)
out = zeros(size(x));
for i = 1: length(x)
first_integral = integral(@(x) eval_K(x,a,b,c),0,1).^-1;
theta_t_ratio = mytheta./x(i);
incomplete_gamma = igamma(myalpha,theta_t_ratio.^mybeta);
n_gamma = gamma(myalpha);
second_integral = integral(@(x) eval_K(x,a,b,c),0,...
incomplete_gamma.^mybeta./n_gamma);
% second_integral = integral(@(x) eval_K(x,a,b,c),0,2);
out(i) = first_integral*second_integral;
end
end
function out = eval_K(x,a,b,c)
out = x.^(a-1).*(1-x).^(b-1).*exp(-c.*x);
end
但是，我没有成功获得所需的PDF.正如您在图中所看到的，估计的PDF(橙色线)没有追踪到"x"的直方图(蓝色条形).
However, I have not been successfull to obtain the desired PDF. As you can see in the figure, the estimated PDF (orange line) does not trace the histogram of 'x' (blue bars).
[UPDATE]标准化直方图
请注意，我已经更改了参数的初始值.但这非常耗时.我还增加了迭代次数，并最大程度地降低了公差，但还没有运气.除了mle之外，还有没有更好的方法来估算参数?
Note that I have varied the initial values of the parameters. But this is very time consuming. I also increased the number of iterations and minimized the tolerance, but no luck yet. Is there a better way to estimate the paratemters other than mle?
任何帮助将不胜感激.
谢谢.
推荐答案
从帮助中注意到这一点
给我们第一个调试建议.因此，从输入列表中删除检查部分和CDF并运行
gives us a first suggestion for debugging. So removing the censoring part and the CDF from the input list and running
phat = mle(x,'pdf', @eval_custpdf,'start',[0.6,0.02,1.01,2,4,-10]);
phat = mle(x,'pdf', @eval_custpdf,'start',phat); %Restart for better result
产生数字
告诉我们问题可能出在CDF函数中.与问题中给出的链接相比，我们看到这一行
Telling us that the problem might be in the CDF function. Comparing with the link given in the question, we see that this line
second_integral = integral(@(x) eval_K(x,a,b,c),0,incomplete_gamma.^mybeta./n_gamma);
应该是
second_integral = integral(@(x) eval_K(x,a,b,c),0,incomplete_gamma./n_gamma);
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