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我有以下代码,我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
主要代码为:
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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0.0360000000000000 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 0.0650000000000000 0.0360000000000000 0.0650000000000000 0.0860000000000000 0.0290000000000000 0.0800000000000000 0.0430000000000000 0.0360000000000000 0.0490000000000000 0.0580000000000000 0.0310000000000000 0.0300000000000000 0.0330000000000000 0.0390000000000000 0.0330000000000000 0.0620000000000000 0.0330000000000000 0.0940000000000000 0.0270000000000000 0.0410000000000000 0.0570000000000000 0.0540000000000000 0.0390000000000000 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')
函数是:
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,我一直没有成功获得想要的PDF。正如您在图中看到的,估计的 PDF(橙色线)没有追踪 'x'(蓝色条)的直方图。
[更新] 归一化直方图
请注意,我改变了参数的初始值。但这是非常耗时的。我还增加了迭代次数并最小化了公差,但还没有成功。有没有比 mle
更好的方法来估计参数?
如有任何帮助,我们将不胜感激。
提前致谢。
最佳答案
从帮助中注意到这一点
If the 'censoring' name/value pair is not present, you may omit the 'cdf' name/value pair.
为我们提供了第一个调试建议。因此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
产量图
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);
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