| it {simEd} | R Documentation |
Visualization of Random Variate Generation for the Student T Distribution
Description
Generates random variates from the Student T distribution by inversion. Optionally graphs the population cumulative distribution function and associated random variates, the population probability density function and a histogram of the random variates, and the empirical cumulative distribution function versus the population cumulative distribution function.
Usage
it(
u = runif(1),
df,
ncp,
minPlotQuantile = 0.01,
maxPlotQuantile = 0.99,
plot = TRUE,
showCDF = TRUE,
showPDF = TRUE,
showECDF = TRUE,
show = NULL,
maxInvPlotted = 50,
plotDelay = 0,
sampleColor = "red3",
populationColor = "grey",
showTitle = TRUE,
respectLayout = FALSE,
restorePar = TRUE,
...
)
Arguments
u |
vector of uniform(0,1) random numbers, or NULL to show population figures only |
df |
Degrees of freedom > 0 |
ncp |
Non-centrality parameter delta (default NULL) |
minPlotQuantile |
minimum quantile to plot |
maxPlotQuantile |
maximum quantile to plot |
plot |
logical; if |
showCDF |
logical; if |
showPDF |
logical; if |
showECDF |
logical; if |
show |
octal number (0-7) indicating plots to display; 4: CDF, 2: PDF, 1: ECDF; sum for desired combination |
maxInvPlotted |
number of inversions to plot across CDF before switching to plotting quantiles only |
plotDelay |
delay in seconds between CDF plots |
sampleColor |
Color used to display random sample from distribution |
populationColor |
Color used to display population |
showTitle |
logical; if |
respectLayout |
logical; if |
restorePar |
logical; if |
... |
Possible additional arguments. Currently, additional arguments not considered. |
Details
Generates random variates from the Student T 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 t-distribution with df = v degrees of freedom has density
f(x) = \frac{\Gamma((v+1)/2)}{\sqrt{v\pi} \
\Gamma(v/2)} \ (1 + x^2/v)^{-(v+1)/2}
for all real x. It has mean 0 (for v > 1) and variance
v/(v-2) (for v > 2).
The general non-central t with parameters (\nu, \delta) =
(df, ncp) is defined as the distribution of
T_{\nu}(\delta) := (U + \delta) \ / \ \sqrt{(V/\nu)}
where U and V are
independent random variables, U \sim \mathcal{N}(0,1)
and V \sim \chi^2(\nu).
The algorithm for generating random variates from the Student t 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 Student t random variate is plotted.
This is not a particularly fast variate generation algorithm because it uses
the base R qt 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, andshowECDF, 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.
Value
A vector of Student T random variates
Author(s)
Barry Lawson (blawson@bates.edu),
Larry Leemis (leemis@math.wm.edu),
Vadim Kudlay (vkudlay@nvidia.com)
See Also
Examples
it(0.5, df = 5, ncp = 10)
set.seed(8675309)
it(runif(10), 4, showPDF = TRUE)
set.seed(8675309)
it(runif(10), 4, showECDF = TRUE)
set.seed(8675309)
it(runif(10), 4, showPDF = TRUE, showECDF = TRUE, sampleColor = "blue3")
set.seed(8675309)
it(runif(10), 4, showPDF = TRUE, showCDF = FALSE)
it(runif(100), 4, showPDF = TRUE, minPlotQuantile = 0.02, maxPlotQuantile = 0.98)
# plot the PDF and CDF without any variates
it(NULL, 4, showPDF = TRUE, showCDF = TRUE)
# plot CDF with inversion and PDF using show
it(runif(10), 4, show = c(1,1,0))
it(runif(10), 4, show = 6)
# plot CDF with inversion and ECDF using show, using vunif
it(vunif(10), 4, show = c(1,0,1))
it(vunif(10), 4, show = 5)
# plot CDF with inversion, PDF, and ECDF using show
it(vunif(10), 4, show = c(1,1,1))
it(vunif(10), 4, show = 7)
# plot three different CDF+PDF+ECDF horizontal displays,
# with title only on the first display
oldpar <- par(no.readonly = TRUE)
par(mfrow = c(3,3)) # 3 rows, 3 cols, filling rows before columns
set.seed(8675309)
it(runif(20), 4, show = 7, respectLayout = TRUE, restorePar = FALSE)
it(runif(20), 4, show = 7, respectLayout = TRUE, restorePar = FALSE, showTitle = FALSE)
it(runif(20), 4, show = 7, respectLayout = TRUE, restorePar = TRUE, showTitle = FALSE)
par(oldpar)
# display animation of all components
it(runif(10), 4, show = 7, plotDelay = 0.1)
# display animation of CDF and PDF components only
it(runif(10), 4, show = 5, plotDelay = 0.1)
if (interactive()) {
# interactive -- pause at each stage of inversion
it(runif(10), 4, show = 7, plotDelay = -1)
}