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使用mle()估计自定义分布的参数

matlabdistributionprobability-densitymlefminsearch
7
我有以下代码，希望估计自定义分布的参数。有关分布详细信息，请点击此处。然后使用估计的参数，我想看看估计的概率密度函数是否类似于给定数据的分布（它应该与给定数据的分布匹配）。 [编辑]：'x'现在保存了数据样本，而不是PDF 主要代码如下：
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  0.0390000000000000  0.0480000000000000  0.0680000000000000  0.0330000000000000  0.0510000000000000  0.0430000000000000  0.0270000000000000  0.0330000000000000  0.0590000000000000  0.0380000000000000  0.0270000000000000  0.0600000000000000  0.0310000000000000  0.0520000000000000  0.0350000000000000  0.0640000000000000  0.0570000000000000  0.0520000000000000  0.0330000000000000  0.0480000000000000  0.0530000000000000  0.0380000000000000  0.0320000000000000  0.0340000000000000  0.0380000000000000  0.0470000000000000  0.0950000000000000  0.0510000000000000  0.0280000000000000  0.124000000000000   0.0360000000000000  0.0670000000000000  0.0380000000000000  0.0760000000000000  0.0440000000000000  0.0390000000000000  0.0500000000000000  0.0500000000000000  0.0370000000000000  0.0350000000000000  0.0490000000000000  0.0570000000000000  0.0560000000000000  0.0500000000000000  0.0350000000000000  0.0390000000000000  0.0390000000000000  0.0310000000000000  0.0260000000000000  0.0350000000000000  0.0610000000000000  0.0280000000000000  0.0480000000000000  0.0560000000000000  0.0650000000000000  0.0400000000000000  0.131000000000000   0.0600000000000000  0.0310000000000000  0.0620000000000000  0.0320000000000000  0.0510000000000000  0.0510000000000000  0.0480000000000000  0.0420000000000000  0.0470000000000000  0.0690000000000000  0.0590000000000000  0.0550000000000000  0.0580000000000000  0.0410000000000000  0.0320000000000000  0.0440000000000000  0.0370000000000000  0.0390000000000000  0.0620000000000000  0.0290000000000000  0.0340000000000000  0.0490000000000000  0.0300000000000000  0.0540000000000000  0.0420000000000000  0.0610000000000000  0.159000000000000   0.0520000000000000  0.0450000000000000  0.0670000000000000  0.0590000000000000  0.111000000000000   0.0250000000000000  0.0660000000000000  0.0210000000000000  0.0370000000000000  0.0330000000000000  0.0510000000000000  0.0330000000000000  0.0450000000000000  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0.0270000000000000  0.0830000000000000  0.0440000000000000  0.0680000000000000  0.0500000000000000  0.0480000000000000  0.0520000000000000  0.0510000000000000  0.0290000000000000  0.0360000000000000  0.0290000000000000  0.0390000000000000  0.0290000000000000  0.0600000000000000  0.0310000000000000  0.0440000000000000  0.0410000000000000  0.0350000000000000  0.0510000000000000  0.0660000000000000  0.0450000000000000  0.0520000000000000  0.0530000000000000  0.0260000000000000  0.0690000000000000  0.0590000000000000  0.0760000000000000  0.118000000000000   0.0500000000000000  0.0480000000000000  0.0300000000000000  0.0300000000000000  0.0390000000000000  0.0340000000000000  0.0500000000000000  0.0280000000000000  0.0410000000000000  0.0490000000000000  0.0390000000000000  0.0390000000000000  0.0420000000000000  0.0520000000000000  0.0300000000000000  0.0320000000000000  0.0650000000000000  0.0650000000000000  0.0600000000000000  0.0450000000000000  0.0680000000000000  0.0350000000000000  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0.0440000000000000  0.0290000000000000  0.0280000000000000  0.0350000000000000  0.0840000000000000  0.0660000000000000  0.0410000000000000  0.0300000000000000  0.0440000000000000  0.0450000000000000  0.0470000000000000  0.0620000000000000  0.0420000000000000  0.0300000000000000  0.0330000000000000  0.0320000000000000  0.0440000000000000  0.0700000000000000  0.0340000000000000  0.0420000000000000  0.0480000000000000  0.0360000000000000  0.0590000000000000  0.106000000000000   0.0280000000000000  0.0540000000000000  0.0870000000000000  0.0300000000000000  0.0300000000000000  0.0370000000000000  0.0210000000000000  0.0360000000000000  0.0910000000000000  0.126000000000000   0.0780000000000000  0.0510000000000000  0.0500000000000000  0.0370000000000000  0.0540000000000000  0.0380000000000000  0.0350000000000000  0.0480000000000000  0.0300000000000000  0.0340000000000000  0.133000000000000   0.0330000000000000  0.0340000000000000  0.0480000000000000  0.0590000000000000  0.0460000000000000  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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  0.0310000000000000  0.0740000000000000  0.0280000000000000  0.0310000000000000  0.0330000000000000  0.115000000000000   0.0890000000000000  0.0770000000000000  0.0490000000000000  0.0640000000000000  0.0580000000000000  0.0540000000000000  0.0330000000000000  0.0490000000000000  0.0230000000000000  0.0330000000000000  0.0490000000000000  0.0370000000000000  0.0330000000000000  0.0490000000000000  0.0460000000000000  0.0750000000000000  0.0420000000000000  0.0610000000000000  0.0340000000000000  0.0870000000000000  0.0390000000000000  0.0410000000000000  0.0430000000000000  0.0810000000000000  0.0570000000000000  0.0440000000000000  0.0470000000000000  0.0470000000000000  0.0320000000000000  0.0730000000000000  0.0590000000000000  0.0290000000000000  0.0450000000000000  0.0310000000000000  0.0450000000000000  0.0490000000000000  0.0880000000000000  0.0220000000000000  0.0330000000000000  0.0360000000000000  0.0350000000000000  0.0540000000000000  0.0470000000000000  0.0390000000000000  0.0300000000000000  0.0450000000000000  0.0740000000000000  0.0270000000000000  0.0320000000000000  0.0640000000000000  0.0750000000000000  0.0320000000000000  0.0300000000000000  0.0310000000000000  0.0470000000000000  0.0510000000000000  0.0520000000000000  0.0680000000000000  0.0700000000000000  0.0630000000000000  0.0410000000000000  0.0510000000000000  0.0460000000000000  0.0280000000000000  0.0780000000000000  0.0420000000000000  0.0390000000000000  0.0540000000000000  0.0310000000000000  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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')

这些功能包括:

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

然而，我没有成功获得所需的PDF文件。如图所示，估计的PDF（橙色线条）未追踪'x'的直方图（蓝色条形）。

[更新] 标准化直方图

enter image description here 请注意，我已经变化了参数的初始值。但这非常耗费时间。我还增加了迭代次数和最小容忍度，但仍然没有运气。是否有比'mle'更好的估计参数的方法？

如果您能提供任何帮助，将不胜感激。

先行致谢。

1
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。看起来你正在输入一个PDF文件？如果你想拟合PDF而不是样本，那么MLE就不是正确的工具。你需要寻找的是曲线拟合而不是分布拟合。 - John
@John 这张图展示了我现在得到的结果。是的，我正在比较PDF和直方图来评估成功率。我相信它们以相同的方式被归一化了。 - nashynash
直方图没有被归一化。概率密度函数可能已经被归一化为总和为1，但直方图的总和是数据点的总数。将直方图计数除以它们的总和，然后看看图形的效果如何。 - nhowe
@nhowe 我已经添加了一个额外的图表，显示了归一化直方图和归一化概率密度函数。结果中没有任何变化。 - nashynash
我认为错误在于您的自定义cdf / pdf，这对我们来说几乎不可能进行调试，因为我们不知道您打算编写哪个函数。因此，为了继续进行，我建议您仔细编写PDF和CDF，然后同样仔细地检查您是否已正确输入它们，如果这样做不起作用，您可以上传您打算编写的函数的图片。另外，剩下的质量在哪里？从直方图中，我可以看到少于3％的概率质量（这可能是错误吗？） - Nicky Mattsson
显示剩余7条评论
1个回答
5

从帮助文档中可以看到：

如果不存在'censoring'名称/值对，则可以省略'cdf'名称/值对。

这为我们提供了第一个调试建议。因此，从输入列表中删除censoring部分和CDF，然后运行程序。

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

产生了这个图形

plot1

提示我们问题可能在CDF函数中。与问题中给出的链接进行比较，我们发现这行代码

second_integral     =  integral(@(x) eval_K(x,a,b,c),0,incomplete_gamma.^mybeta./n_gamma);

应该已经。
second_integral =  integral(@(x) eval_K(x,a,b,c),0,incomplete_gamma./n_gamma);
太棒了，Nicky，我喜欢你分解说明的方式。现在它可以工作了。非常感谢。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);”。 - nashynash
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