问题
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).
[EDIT]: 'x' now holds a sample of data and not a PDF
The main code is:
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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0.0270000000000000 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')
The functions are:
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
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] Normalized histogram
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?
Any help would be appreciated.
Thanks in advance.
回答1:
Noting this from the help
If the 'censoring' name/value pair is not present, you may omit the 'cdf' name/value pair.
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
yields the figure
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);
should have been
second_integral = integral(@(x) eval_K(x,a,b,c),0,incomplete_gamma./n_gamma);
来源：https://stackoverflow.com/questions/56625339/estimating-the-parameters-of-a-custom-distribution-using-mle