Generates random variates from the Log-Normal distribution, and optionally,
illustrates
the use of the inverse-CDF technique,
the effect of random sampling variability in relation to the PDF and CDF.
When all of the graphics are requested,
the first graph illustrates the use of the inverse-CDF technique by
graphing the population CDF and the transformation of the random numbers
to random variates,
the second graph illustrates the effect of random sampling variability
by graphing the population PDF and the histogram associated with the
random variates, and
the third graph illustrates effect of random sampling variability by
graphing the population CDF and the empirical CDF associated with the
random variates.
All aspects of the random variate generation algorithm are output in red by
default, which can be changed by specifying sampleColor.
All aspects of the population distribution are output in gray by default,
which can be changed by specifying populationColor.
The log-normal distribution has density
\deqn{f(x) = \frac{1}{\sqrt{2 \pi} \sigma x}
e^{-(\log{x} - \mu)^2 / (2 \sigma^2)} }{
f(x) = 1/(\sqrt(2 \pi) \sigma x) e^-((log x - \mu)^2 / (2 \sigma^2))}
where \(\mu\) and \(\sigma\) are the mean and standard deviation
of the logarithm.
The mean is
\(E(X) = \exp(\mu + 1/2 \sigma^2)\),
the median is
\(med(X) = \exp(\mu)\),
and the variance is
\(Var(X) = \exp(2\times \mu +\sigma^2)\times (\exp(\sigma^2)-1)\)
and hence the coefficient of variation is
\(sqrt(\exp(\sigma^2)-1)\)
which is approximately \(\sigma\) when small
(e.g., \(\sigma < 1/2\)).
The algorithm for generating random variates from the log-normal distribution is
synchronized (one random variate for each random number) and monotone in u.
This means that the variates generated here might be useful in some variance
reduction techniques used in Monte Carlo and discrete-event simulation.
Values from the u vector are plotted in the cdf plot along the vertical axis
as colored dots. A horizontal, dashed, colored line extends from the dot to
the population cdf. At the intersection, a vertical, dashed colored line
extends downward to the horizontal axis, where a second colored dot, denoting
the associated log-normal random variate is plotted.
This is not a particularly fast variate generation algorithm because it uses
the base R qlnorm function to invert the values contained in u.
All of the elements of the u vector must be between 0 and 1.
Alternatively, u can be NULL in which case plot(s) of the
theoretical PDF and cdf are displayed according to plotting parameter
values (defaulting to display of both the PDF and cdf).
The show parameter can be used as a shortcut way to denote plots to
display. The argument to show can be either:
a binary vector of length three, where the entries from left to right
correspond to showCDF, showPDF, and showECDF,
respectively. For each entry, a 1 indicates the plot should be
displayed, and a 0 indicates the plot should be suppressed.
an integer in [0,7] interpreted similar to the Unix chmod command. That
is, the integer's binary representation can be transformed into a
length-three vector discussed above (e.g., 6 corresponds to c(1,1,0)).
See examples.
Any valid value for show takes precedence over existing individual
values for showCDF, showPDF, and showECDF.
If respectLayout is TRUE, the function respects existing
settings for device layout. Note, however, that if the number of plots
requested (either via show or via showCDF, showPMF, and
showECDF) exceeds the number of plots available in the current layout
(as determined by prod(par("mfrow"))), the function will display all
requested plots but will also display a warning message indicating that the
current layout does not permit simultaneous viewing of all requested plots.
The most recent plot with this attribute can be further annotated after the call.
If respectLayout is FALSE, any existing user settings for device
layout are ignored. That is, the function uses par to explicitly set
mfrow sufficient to show all requested plots stacked vertically to
align their horizontal axes, and then resets row, column, and margin settings
to their prior state on exit.
The minPlotQuantile and maxPlotQuantile arguments are present in
order to compress the plots horizontally. The random variates generated are
not impacted by these two arguments. Vertical, dotted, black lines are
plotted at the associated quantiles on the plots.
plotDelay can be used to slow down or halt the variate generation for
classroom explanation.
In the plot associated with the PDF, the maximum plotting height is
associated with 125\
that extends above this limit will have three dots appearing above it.