У меня есть следующий код, который я хочу оценить параметры пользовательского распределения. Для более подробной информации о распространении . Затем, используя оценочные параметры, я хочу посмотреть, напоминает ли предполагаемый PDF распределение заданных данных (предполагается, что оно соответствует распределению заданных данных).
[РЕДАКТИРОВАТЬ]: «x» теперь содержит образец данных, а не PDF
Основной код:
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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0.0350000000000000 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')
Функции:
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
Однако мне не удалось получить желаемый PDF. Как видно на рисунке, предполагаемый PDF (оранжевая линия) не отслеживает гистограмму 'x' (синие столбцы).
[ОБНОВЛЕНИЕ] Нормализованная гистограмма
Обратите внимание, что я изменил начальные значения параметров. Но это очень много времени. Я также увеличил количество итераций и минимизировал допуск, но пока не повезло. Есть ли лучший способ оценить параметры, отличные от mle?
Буду признателен за любую помощь.
Заранее спасибо.