Mean and 3-sigma in Lognormal plots

Hi,

I have a lognormal distribution in which I want to mark clearly the mean and 3 sigma ranges. I amusing the below code snippet but it doesn't seem to work.

pd = fitdist(A(i,:)','LogNormal');
y = pdf(pd,A(i,:)');
figure(22); plot(A(i,:)',y,'LineStyle','None','Marker','x');
lowBound = pd.mu - 3 * pd.sigma;
highBound = pd.mu + 3 * pd.sigma;
yl = ylim;
x1 = xlim;
line([lowBound, lowBound], yl, 'Color', [0, .6, 0], 'LineWidth', 1);
line([highBound, highBound], yl, 'Color', [0, .6, 0], 'LineWidth', 1);
line([pd.mu, pd.mu], yl, 'Color', [0, .6, 0], 'LineWidth', 1);
grid on;
set(gcf, 'Toolbar', 'none', 'Menu', 'none');
% Give a name to the title bar.
set(gcf, 'Name', 'Line segmentation', 'NumberTitle', 'Off')
hold on;

I have attached the lognormal plot in which I would like to view the mean and sigma. I have also attached a sample image of how it should look (from a different simulator). Thanks!

Answer 1

[John BG]

Hi Kash022

1.

The attached function lognormal_pdf_123sigma_locations.m calculates the non symmetric locations of +-sigma1 +-sigma2 +-sigma3, along with the mean value and also returns the interpolated curve of the pdf to achieve 1 decimal accuracy, on the pdf, not the data.

    mu0: mean
    sm1: sigma 1 minus
    sp1: sigma 1 plus
    sm2: sigma 2 minus 
    sp2: sigma2 plus
    sm3: sigma 3 minus
    sp3: sigma 3 plus

call example, replace b with data:

b = betarnd(3,10,100,1);  % replace betarand() with data
[mu0,sm1,sp1,sm2,sp2,sm3,sp3,y2,ny2]=lognormal_pdf_123sigma_locations(b)

2.

lognormal_pdf_123sigma_locations.m also returns a couple figures showing data, pdf approximation, and the location of the requested points:

.

.

.

3.

Copy of the function

function [mu0,sm1,sp1,sm2,sp2,sm3,sp3,y2,ny2]=lognormal_pdf_123sigma_locations(b)
% mu0: mean
% sm1: sigma 1 minus
% sp1: sigma 1 plus
% sm2: sigma 2 minus 
% sp2: sigma2 plus
% sm3: sigma 3 minus
% sp3: sigma 3 plus
% author: John Bofarull Guix, [e-mail]

% b = betarnd(3,10,100,1);  % simulated data, replace betarand() with real data

figure(1);
% h1=histfit(b,20,'lognormal') % test
h1=histfit(b,20,'lognormal')
b_histogram=h1(1);grid on;
b_pdf_lognormal_fit=h1(2)

y=b_pdf_lognormal_fit.YData;
sumy=sum(y);y=y/sumy;
ny=b_pdf_lognormal_fit.XData;
N=100;
y2=interp(y,N);y2=y2/sum(y2);
ny2=[1:1:numel(y)]*N;

pc_target=50;                                      %  mu: distribution mean  value
n=2
pc=sum(y2([1:n])) *100
while pc<pc_target
    n=n+1;pc=sum(y2([1:n])) *100;
end
mu=n;  
figure(2);plot(y2);hold all
figure(2);plot(mu,y2(n),'bo');grid on

pc_target=34; [s1_min,s1_plus]=go_get_it(pc_target,mu,y2);               % +- sigma
figure(2);plot(s1_min,y2(s1_min),'ro');
figure(2);plot(s1_plus,y2(s1_plus),'ro');

pc_target=47.5; [s2_min,s2_plus]=go_get_it(pc_target,mu,y2);          %  +-2*sigma
figure(2);plot(s2_min,y2(s2_min),'ro')
figure(2);plot(s2_plus,y2(s2_plus),'ro');

pc_target=49.7;[s3_min,s3_plus]=go_get_it(pc_target,mu,y2);            %  +-3*sigma
figure(2);plot(s3_min,y2(s3_min),'ro');
figure(2);plot(s3_plus,y2(s3_plus),'ro');

mu0=mu;
sm1=s1_min;
sp1=s1_plus;
sm2=s2_min;
sp2=s2_plus;
sm3=s3_min;
sp3=s3_plus;

function [sdown,sup]=go_get_it(pc_target,mu,y2)
% support function finds t1 t2 such sum(y([t1:mu]))=sum(y([mu:t2]))

n=1;pc=sum(y2([mu-n:mu])) *100;  % search - s
while pc<pc_target
        n=n+1;pc=sum(y2([mu-n:mu])) *100;
end
sdown=mu-n;
% figure(h);
% plot(h,sdown,y2(sdown),'ro');hold all;

n=1;pc=sum(y2([mu:mu+n])) *100;   % search + s
while pc<pc_target
        n=n+1;pc=sum(y2([mu:mu+n])) *100 ;
end
sup=n+mu;
% figure(2);
% plot(h,sup,y2(sup),'ro');

end

end

4.

checks

      sum(y2([mu0:sp1]))
      sum(y2([sm1:mu0]))

      sum(y2([mu0:sp2]))
      sum(y2([sm2:mu0]))

      sum(y2([mu0:sp3]))
      sum(y2([sm3:mu0]))

    % upper tail

      sum(y2([sp3:end]))

    % lower tail

      sum(y2([1:sm3]))

     =
       0.340091218398777
     =
       0.340121503890558
     =
       0.475010605933624
     =
       0.475120352505811
     =
       0.497000712621476
     =
       0.497028227492323
     =
       0.003110773668274
  ans =
     0.003254677216922

5.

lognormal pdf is normal pdf skewed righ, the sigma1,2,3 locations are no longer symmetric.

.

.

Kash022

if you find this answer useful would you please be so kind to mark my answer as Accepted Answer?

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please click on the thumbs-up vote link,

thanks in advance

John BG

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