Transform a set of rectangles into canonical rectangles
This function transforms a set of rectangles R1,..,Rn into canonical rectangles R1',..,Rn' with the following properties: (1) R1',..Rn' have the same intersection structure as the original rectangles, i.e., Ri' and Rj' intersect if and only if Ri and Rj intersect. (2) All x-coordinates of R1',..,Rn' are distinct, and take values in 1,..,2n. (3) All y-coordinates of R1',..,Rn' are distinct, and take values in 1,..,2n.
The function real2canon performs the inverse operation of the function
canon2real.
real2canon(R,B=c(0,1))
R |
A nx4 real matrix of rectangles. Each row corresponds to a rectangle, represented as (x1,x2,y1,y2). The point (x1,y1) is the lower left corner of the rectangle and the point (x2,y2) is the upper right corner of the rectangle. |
B |
This describes the boundaries of the rectangles (0=open or 1=closed). It can be specified in three ways: (1) A nx4 matrix containing 0's and 1's. Each row corresponds to a rectangle, and is denoted as (cx1, cx2, cy1, cy2). Here cx1 denotes the boundary type of x1, cx2 denotes the boundary type of x2, etc. (2) A vector (cx1, cx2, cy1, cy2) containing 0's and 1's. This representation can be used if all rectangles have the same type of boundaries. (3) A vector (c1, c2) containing 0's and 1's. This representation can be used if all x and y intervals have the same type of boundaries. c1 denotes the boundary type of x1 and y1, and c2 denotes the boundary type of x2 and y2. The default value is c(0,1). |
The functions real2canon and canon2real are carried out
automatically in C-code as part of the functions reduc and computeMLE.
We chose to make the functions available separately as well, in order to illustrate our
algorithm for computing the MLE.
As a first step in the computation of the MLE, we
transform rectangles into canonical rectangles, using real2canon).
This is useful for two reasons. Firstly, it forces us in the
very beginning to deal with possible ties and with the fact whether endpoints are
open or closed. As a consequence, we do not have to account for ties and open or
closed endpoints in the actual computation of the MLE. Secondly, it is convenient to work
with the integer coordinates of the canonical rectangles in the computation of the MLE.
After all computations are done, we transform the canonical rectangles back to their original
coordinates, using canon2real. For more details, see Maathuis (2005, Section 2.1).
A nx4 matrix of canonical observation rectangles. Each row (x1,x2,y1,y2) represents a rectangle.
Marloes Maathuis: maathuis@stat.math.ethz.ch
M.H. Maathuis (2005). Reduction algorithm for the NPMLE for the distribution function of bivariate interval censored data. Journal of Computational and Graphical Statistics 14 252–262.
# An example with 3 arbitrarily chosen observation rectangles
R <- rbind(c(3.5, 4.2, 3.3, 9.1), # first rectangle
c(4.2, 4.9, 3, 4.5), # second rectangle
c(3.8, 5.1, 8.1, 9.5)) # third rectangle
# Plot the rectangles
par(mfrow=c(2,2))
plotRects(R, lwd=2, main="Original rectangles")
# Transform rectangles to canonical rectangles. Since the
# boundaries of R1 and R2 coincide, it matters which boundaries
# we define to be open or closed.
# With boundary structure c(0,1), R1 and R2 do *not* overlap:
res1 <- real2canon(R, c(0,1))
plotRects(res1, grid=TRUE, lwd=2, main="Canonical rectangles
boundary c(0,1)")
# But with boundary structure c(1,1), R1 and R2 *do* overlap:
res2<-real2canon(R, c(1,1))
plotRects(res2, grid=TRUE, lwd=2, main="Canonical rectangles
boundary c(1,1)")Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.