使用 mle() 估计自定义分布的参数
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【中文标题】使用 mle() 估计自定义分布的参数【英文标题】：Estimating the parameters of a custom distribution using mle() 【发布时间】：2019-06-17 05:46:56 【问题描述】：我有以下代码，希望估计自定义分布的参数。 For more details on the distribution。然后使用估计的参数，我想看看估计的 PDF 是否类似于给定数据的分布（它应该与给定数据的分布相匹配）。
[编辑]：“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 0.0370000000000000 0.0350000000000000 0.0280000000000000 0.0340000000000000 0.0400000000000000 0.0370000000000000 0.0570000000000000 0.0610000000000000 0.0440000000000000 0.0680000000000000 0.0470000000000000 0.0400000000000000 0.0270000000000000 0.0470000000000000 0.0350000000000000 0.0290000000000000 0.0280000000000000 0.0300000000000000 0.0770000000000000 0.0310000000000000 0.0560000000000000 0.0640000000000000 0.0420000000000000 0.0340000000000000 0.0290000000000000 0.0520000000000000 0.0260000000000000 0.0460000000000000 0.0380000000000000 0.0720000000000000 0.0750000000000000 0.0510000000000000 0.0560000000000000 0.0440000000000000 0.0390000000000000 0.0470000000000000 0.0450000000000000 0.0400000000000000 0.0880000000000000 0.0300000000000000 0.0440000000000000 0.0610000000000000 0.0370000000000000 0.0350000000000000 0.0490000000000000 0.0510000000000000 0.0820000000000000 0.0390000000000000 0.0440000000000000 0.0600000000000000 0.0310000000000000 0.0720000000000000 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0.0270000000000000 0.0590000000000000 0.0320000000000000 0.0390000000000000 0.0400000000000000 0.0720000000000000 0.0480000000000000 0.0480000000000000 0.0560000000000000 0.0730000000000000 0.0410000000000000 0.0520000000000000 0.0840000000000000 0.0590000000000000 0.0690000000000000 0.0330000000000000 0.0400000000000000 0.0320000000000000 0.0320000000000000 0.0310000000000000 0.0520000000000000 0.0760000000000000 0.0420000000000000 0.0370000000000000 0.0360000000000000 0.0780000000000000 0.0590000000000000 0.0390000000000000 0.0590000000000000 0.0880000000000000 0.0410000000000000 0.0640000000000000 0.0350000000000000 0.0350000000000000 0.0530000000000000 0.0490000000000000 0.0330000000000000 0.0640000000000000 0.0320000000000000 0.0880000000000000 0.0310000000000000 0.0980000000000000 0.0380000000000000 0.0270000000000000 0.0690000000000000 0.0530000000000000 0.0610000000000000 0.0380000000000000 0.0470000000000000 0.0620000000000000 0.0400000000000000 0.0400000000000000 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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')
功能有：
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。如图所示，估计的 PDF（橙色线）不跟踪“x”（蓝色条）的直方图。
[更新]归一化直方图
请注意，我已经改变了参数的初始值。但这非常耗时。我还增加了迭代次数并最小化了容差，但还没有运气。除了mle之外，还有更好的方法来估计参数吗？
任何帮助将不胜感激。
提前致谢。
【问题讨论】：
你得到了什么输出，它与你想要的有什么不同？ MLE 需要样本数据。看起来你给它一个PDF？如果您想适应 PDF 而不是样本，那么 MLE 是错误的工具。您正在寻找的是曲线拟合而不是分布拟合。 @John 你是对的。我提供的是 PDF 而不是样本。我会改变它，看看是否提供样品。非常感谢您指出这一点。 @John 我用示例数据更新了“x”。我运行了代码，但仍然无法得到想要的结果。 你现在得到了什么结果？您是否通过比较 PDF 和直方图来评估成功？如果是这样，它们是否以相同的方式归一化？ @John 该图显示了我现在得到的结果。是的，我将 PDF 与直方图进行比较以评估成功。我相信它们以同样的方式被规范化。 【参考方案1】：从帮助中注意到这一点
如果“审查”名称/值对不是 现在，您可以省略 'cdf' 名称/值对。
为我们提供了第一个调试建议。所以从输入列表中删除审查部分和 CDF 并运行
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
生成图形
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
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);
【讨论】：
这太棒了，尼基，我喜欢你分解解释的方式。现在可以了。非常感谢你。我还发现在PDF函数中denominator = mytheta.* n_gamma.^(a+b-1).* (n_gamma-incomplete_gamma.^mybeta).^(1-b);应该是denominator = mytheta.* n_gamma.^(a+b-1).* (n_gamma-incomplete_gamma).^(1-b);。以上是关于使用 mle() 估计自定义分布的参数的主要内容，如果未能解决你的问题，请参考以下文章
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