R help - May 2008 - Is my understanding of rlnorm correct?

rlnorm takes two 'shaping' parameters: meanlog and sdlog.

meanlog would appear from the documentation to be the log of the mean.  
eg if the desired mean is 1 then meanlog=0.

So to generate random values that fit a lognormal distribution I would  
do this:

rlnorm(N , meanlog = log(mean) , sdlog = log(sd))

But when I check the mean I don't get it when sdlog>0. Interestingly I  
found that the median is close to what I want the mean to be.

 > par( mfrow = c(2,2) )
 > set.seed( 1 )
 > x <- 1:100000
 > y <- rlnorm( x , meanlog = log(100) , sdlog = log(2) )
 > plot( x , y )
 > hist( y , breaks = 100)
 > summary( y )
     Min.  1st Qu.   Median     Mean  3rd Qu.     Max.
    4.292   62.360  100.100  127.100  159.300 1989.000

I noticed on wikipedia lognormal page that the median is exp(mu) and  
that the mean is exp(mu + sigma^2/2)

http://en.wikipedia.org/wiki/Log-normal_distribution

So, does this mean that if i want a mean of 100 that the meanlog value  
needs to be log(100) - log(sd)^2/2?

It seems to give me the result I am after.

 > par( mfrow = c(2,2) )
 > set.seed( 1 )
 > x <- 1:100000
 > y <- rlnorm( x , meanlog = log(100) , sdlog = log(2) )
 > plot( x , y )
 > hist( y , breaks = 100)
 > summary( y )
     Min.  1st Qu.   Median     Mean  3rd Qu.     Max.
    4.292   62.360  100.100  127.100  159.300 1989.000
 > set.seed( 1 )
 > y <- rlnorm( x , meanlog = log(100) - log(2)^2/2 , sdlog = log(2) )
 > plot( x , y )
 > hist( y , breaks = 100)
 > summary( y )
     Min.  1st Qu.   Median     Mean  3rd Qu.     Max.
    3.376   49.040   78.690   99.970  125.300 1564.000 
	[[alternative HTML version deleted]]

G'day Phil,

On Sun, 4 May 2008 14:05:09 +1000
phil colbourn <philcolbourn at gmail.com> wrote:
> rlnorm takes two 'shaping' parameters: meanlog and sdlog.
> 
> meanlog would appear from the documentation to be the log of the
> mean. eg if the desired mean is 1 then meanlog=0.
These to parameters are the mean and the sd on the log scale of the
variate, i.e. if you take the logarithm of the produced numbers then
those values will have the given mean and sd.  

If X has an N(mu, sd^2) distribution, then Y=exp(X) has a log-normal
distribution with parameters mu and sd.  

R> set.seed(1)
R> y <- rlnorm(10000, mean=3, sd=2)
R> summary(log(y))
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
 -4.343   1.653   2.968   2.987   4.355  10.620 
R> mean(log(y))
[1] 2.986926
R> sd(log(y))
[1] 2.024713

> I noticed on wikipedia lognormal page that the median is exp(mu) and  
> that the mean is exp(mu + sigma^2/2)
> 
> http://en.wikipedia.org/wiki/Log-normal_distribution
Where mu and sigma are the mean and standard deviation of a normal
variate which is exponentiated to obtain a log normal variate.  And
this holds for the above example (upto sampling variation):

R> mean(y)
[1] 143.1624
R> exp(3+2^2/2)
[1] 148.4132
> So, does this mean that if i want a mean of 100 that the meanlog
> value needs to be log(100) - log(sd)^2/2?
A mean of 100 for the log-normal variate?  In this case any set of mu
and sd for which exp(mu+sd^2/2)=100 (or mu+sd^2/2=log(100)) would do
the trick:

R> mu <- 2
R> sd <- sqrt(2*(log(100)-mu))
R> summary(rlnorm(10000, mean=mu, sd=sd))
     Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
4.010e-04 1.551e+00 7.075e+00 1.006e+02 3.344e+01 3.666e+04 
R> mu <- 4
R> sd <- sqrt(2*(log(100)-mu))
R> summary(rlnorm(10000, mean=mu, sd=sd))
     Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
   0.9965   25.9400   56.0200  101.2000  115.5000 3030.0000 
R> mu <- 1
R> sd <- sqrt(2*(log(100)-mu))
R> summary(rlnorm(10000, mean=mu, sd=sd))
     Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
9.408e-05 4.218e-01 2.797e+00 8.845e+01 1.591e+01 7.538e+04 

Note that given the variation we would expect in the mean in the last
example, the mean is actually "close enough" to the theoretical value
of 100:

R> sqrt((exp(sd^2)-1)*exp(2*mu + sd^2)/10000)
[1] 36.77435

HTH.

Cheers,
	
	Berwin

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