Lognormal Parameter Estimation using the mode value and frequency.

If we have to estimate the mean (mu) and standard deviation (sigma) of the lognormal distribution, and we only have the mode available, here's what we can do. I'll show it with an example along with the Matlab code.

Let's say that out mode is occurs at x = 1.5 and its frequency is 10%. Since there's two unkowns, we need two equations. Here's the equation of the lognormal PDF mode:

$\normal \mathrm{Mode} = e^{\mu-\sigma^2}$

and here the frequency equation (which happens to be the PDF expression), i.e.:

$\normal \mathrm{Frequency}=\frac{1}{x\sqrt{2\pi\sigma^2}}\, e^{-\frac{\left(\ln x-\mu\right)^2}{2\sigma^2}}$

So, first step is to equal the mode expression to our given value and take logs

$\normal e^{\mu-\sigma^2} = 1.5 \\ \mu - \sigma^2 = \ln(1.5)\\ \mu = \ln(1.5) + \sigma^2$

Next, substitute in the frequency equation the value of the mode, (i.e. x = 1.5) and the expression of mu as a function of \sigma that we just found in the previous step

$\normal \frac{1}{{\color{Blue} 1.5}\sqrt{2\pi\sigma^2}}\,%20e^{-\frac{\left(\ln%20{\color{Blue} 1.5}-{\color{Red} (\ln(1.5) + \sigma^2)}\right)^2}{2\sigma^2}}$

Taking logs and simplifying we get:

$\normal -\frac{\sigma^2}{2} - \ln(1.5\sqrt{2\pi}\sigma) = \ln(0.10)$

We can use Matlab to find out the value of \sigma as:

>> solve('-(s^2)^2/(2*s^2)-log(1.5*s*sqrt(2*3.14))=-2.3026')
>> 1.237330637937465265170425634383

We're now good to go with:

$\normal \sigma = {\color{Red} 1.2373}\\ \mu = \ln(1.5) + \sigma^2 = {\color{Red} 1.9365}$

All the math is ready now. Let's plot the function with Matlab:

>> x = (.01:20/1000:100)';
>> sig = 1.2373;
>> mu = log(1.5)+sig^2;
>> y = lognpdf(x,mu,sig);
>> semilogx(x,y)

Which is a nice lognormal distribution with mode at x = 1.5 and frequency 10%.

Finally, we can generalize the above equation as:

$\normal -\frac{\sigma^2}{2} - \ln(m_l\sqrt{2\pi}\sigma) = \ln(m_f)$

where m_l is the mode location and m_f is the mode frequency (from 0 to 1). Once we solve the above equation for sigma, we can find the mean with the following:

$\normal \mu = \ln(m_l) + \sigma^2$

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Lognormal Parameter Estimation using the mode value and frequency.

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