Fitting a log-normal distribution

Hello,

I'm trying to plot a fit to a log-normal distribution. I have the statistics and machine learning toolbox, but I am confused as how to apply the log-normal fit function to this data.

Below is some minimum working code to create a log-normal distribution, but I do not know how to progress further with this fit. The 'lognfit' function requires only a 1 dimensional input vector, not the two input parameters I have (i.e. x and p).

As an alternative, I've also tried fitting using cftool and inputting the log-normal probability distribution function, but unfortunately I did not get a successful fit.

Any advice that you could give would be greatly appreciated.

Lewis

% create x data

x = (0.01:0.02:11); 

% define lognormal distribution parameters

sigma=1;
mu=1;

% create a log normal distribution with these parameters

y = lognpdf(x,sigma,mu);

% plot the resulting distribution

figure, plot(x,y);

Answer 1

[John D'Errico]

Your problem is a not uncommon one for people who don't understand random distributions.

The fact is, you don't have data as samples from a lognormal distribution. You have points taken as values off the lognormal PDF. You cannot use lognfit to fit that data.

Lets see how to do it, in a way that will work. I'll start with a simple example, using a normal.

Here, I'll generate some random samples using randn. The mean should be 2, standard deviation 3. But the estimated values for those parameters will not be exactly those numbers of course since they are estimates, taken from a finite sample.

X = randn(1000,1)*3 + 2;

[MUHAT,SIGMAHAT] = normfit(X)
MUHAT =
           1.9021041197117
SIGMAHAT =
          2.99689596815649

As you can see, it did pretty well. But you need to recognize that normfit (and the cousin, lognfit that you tried to use) work on points that are assumed to follow that distribution, as I did above.

Random samples that are assumed to follow a given distribution are NOT the same thing as points taken off a PDF.

[Lewis MacKenzie]

Thank you very much for your reply. I highlights to me that I am barking up the wrong tree by trying to use functions from the stats toolbox.

I think I will have to go back to the drawing board with my approach on this one.

My ultimate aim is to fit a lognormal curve to some real data. I can do this by taking the log of the x axis and fitting a gaussian to that. However, that's much less satisfying graphically.

[Tom Lane]

Before you stop barking, and in support of what John wrote, here is an example of fitting the lognormal distribution to data:

rng(0)
x = lognrnd(5,.3,100,1);
histfit(x,15,'lognormal')

And here is an example of fitting a model that is of the shape of a lognormal density to data that looks like it has that shape:

xx = linspace(0,500)';
yy = lognpdf(xx,5,.3) + randn(100,1)/10000;
F = @(b,x)lognpdf(x,b(1),b(2));
b = nlinfit(xx,yy,F,[5,.3])
plot(xx,yy,'.', xx,F(b,xx),'r-')

Both use functions from the Statistics and Machine Learning Toolbox.

[John D'Errico]

You CAN use the function lognfit on real data. The problem is that in your test, you did not have real data. You did not really have data at all. i.e., random samples from the lognormal distribution. You had values of the lognormal pdf, taken at a list of points. The two things are very different, though perhaps confusing.