Milhouse - 1 year ago 114
R Question

# Plotting inequalities in r

I have a number of values that come from my data, e.g. x1, x2, x3 etc., and I'm trying to plot shaded regions that are dependent on functionals of these values. These regions are defined by inequalities of the form:

f(x1, x2) <= A <= f(x3, x4)

f(x5, x6) <= A + B <= f(x7, x8)

and I would like to have A on the x-axis, B on the y-axis.

I've tried putting the values in a data frame and playing around with ggplot, but my r skills are lacking and beyond simple line-plots/bar-charts etc. I'm a bit lost, I especially can't figure out the syntax for plotting inequalities as regions.

Thanks

Edit for example:

``````        x1    x2    x3    x4    x5    x6
Plot1   0.2   0.3   0.24  0.14  0.17  0.31
Plot2   0.14  0.35  0.30  0.11  0.21  0.39
``````

with the hope of plotting two separate graphs (one for Plot 1, one for Plot 2) with the regions defined by:

max(x2 + x3, x4 + x5) <= A <= min(1 - x1, 1 - x6)

max(x3, x5) <= A + B <= min(x2 + x3, x5 + x6)

I'm not aware of any build-in R function to do this. You can however make a data.frame with a grid and compute for each gridpoint if the inequalities are satisfied. Then you can plot with geom_tile. A working example:

``````## Equation 1: f1(A,B) <= A < f2(A,B)
## Equation 2: f3(A,B) < A + B < f4(A,B)

library(ggplot2)

grid <- expand.grid( A = seq(-2,2, length.out = 100), B = seq(-1, 1, length.out = 100))

f1 <- function(A, B) B
f2 <- function(A, B) 2*(A^2 + B^2)
f3 <- function(A, B) A*B
f4 <- function(A, B) 2*A+B^3

grid\$inside_eq1 <- (f1(grid\$A, grid\$B) <= grid\$A) & (grid\$A < f2(grid\$A, grid\$B))
grid\$inside_eq2 <- (f3(grid\$A, grid\$B) < grid\$A + grid\$B) & (grid\$A + grid\$B < f4(grid\$A, grid\$B))
grid\$inside <- grid\$inside_eq1 & grid\$inside_eq2

ggplot(grid) +
geom_tile(aes(x = A, y = B, color = inside, fill = inside))
``````

Edit: You asked in the comment how you could draw a border around the region. I think a general soultion is a bit complex, but in your case it's not so hard. Your functions in the inequalities are independent of A and B, they are constant and thus convex. Since the intersection of convex sets is again convex the region in which the inequalities are fulfilled is also convex. This leads to a solution which makes use of the convex hull of the region:

``````## if you want a border for the region (works only if all equations are convex!)

f1 <- function() 0.1
f2 <- function() 1.8
f3 <- function() -1.2
f4 <- function() 0.3

grid\$inside_eq1 <- (f1() <= grid\$A) & (grid\$A < f2())
grid\$inside_eq2 <- (f3() < grid\$A + grid\$B) & (grid\$A + grid\$B < f4())
grid\$inside <- grid\$inside_eq1 & grid\$inside_eq2

hull <- chull(grid\$A[grid\$inside], grid\$B[grid\$inside])
ggplot(grid, aes(x = A, y = B)) +
geom_tile(aes(color = inside, fill = inside)) +
geom_path(data = grid[grid\$inside, ][c(hull, hull[1]), ], size = 2)
``````
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