Plot 3d surface or ploygon in R based on specific combinations of 3 variables

Question

I'm trying to make a 3D scatterplot with boudaries or zones based on combinations of 3 variables that return certain values. The variables each range between 0:1 , and combine to make an index that ranges from -1:1 as follows:

f(x,y,z) = (x*y)-z

I'd like to create a visual representation that will highlight all combinations of variables that return a certain index value. As an example, I can easily show those variables where index > 0 using scatterplot3d ( rgl would also work):

# Create imaginary dataset of 50 observations for each variable
x<-runif(50,0,1)
y<-runif(50,0,1)
z<-runif(50,0,1)

# Create subset where f(x,y,z) > 0
x1<-y1<-z1<-1
for (i in 1:length(x)){ if ((x[i]*y[i])-z[i] > 0) {
x1<-rbind(x1, x[i])
y1<-rbind(y1, y[i])
z1<-rbind(z1, z[i])}
}

s3d<-scatterplot3d(x,y,z) # Plot entire dataset
s3d$points3d(x1,y1,z1,pch=19, col="red") # Highlight subset where f(x,y,z) > 0

This gives me the following graph:

It seems fairly intuitive that there should be an easy way to plot either the surface (extending from top/right/back to bottom/left/front) separating the subset from the rest of the data, or else a volume/3D area within which these plots lie. Eg my first instinct was to use something like surface3d , persp3d or planes3d . However, all attempts so far have only yielded error messages. Most solutions seem to use some form of z<-lm(y~x) but I obviously need something like q<-func((x*y)-z) for all values of x , y and z that yield q > 0 .

I know I could calculate extreme points and use them as vertices for a 3D polygon, but that seems too "manual". It feels like I'm overlooking something fairly simple and obvious. I've looked at many similar questions on Stack but can't seem to find one that fits my particular problem. If I've missed any and this question has been answered already, please do point me in the right direction!

Answer 1

Here is a suggestion for an interactive 3D plot that is based on an example from the "R Graphics Cookbook" by Winston Chang.

set.seed(4321)
library(rgl)
interleave <- function(v1,v2) as.vector(rbind(v1,v2))
x <- runif(50)
y <- runif(50)
z <- runif(50)
plot3d(x, y, z, type="s", size=0.6, col=(2+(x*y<z)))
x0 <- y0 <- seq(0, 1, 0.1)
surface3d(x0, y0, outer(x0, y0), alpha=0.4) #plot the surface f(x,y)=x*y
x1 <- x[x * y > z] #select subset that is below the separating surface
y1 <- y[x * y > z]
z1 <- z[x * y > z]
segments3d(interleave(x1, x1), #highlight the distance of the points below the surface
           interleave(y1, y1),
           interleave(x1 * y1, z1), col="red", alpha=0.4)

If you don't like the red lines and only want the surface and the colored points, this will be sufficient:

plot3d(x,y,z,type="s",size=0.6,col=(2+(x*y<z)))
x0 <- y0 <- seq(0,1,0.1)
surface3d(x0,y0,outer(x0,y0),alpha=0.4)

Does this representation provide the information that you wanted to highlight?

Answer 2

The first thought was to see if the existing functions within scatterplot3d could handle the problem but I think not:

 my.lm <- lm(z ~ I(x) * I(y)+0)
 s3d$plane3d(my.lm, lty.box = "solid", col="red")

pkg:scatterplot3d doesn't really have a surface3d function so you will need to choose a package that provides that capability; say 'rgl', 'lattice', or 'plot3d'. Any of them should provide the needed facilities.