MLEcens (version 0.1-2)

canon2real: Transform (intersections of) canonical rectangles back to their original coordinates

Description

This function transforms a set of (intersections of) canonical rectangles (see real2canon for a definition) back to their original coordinates. It performs the inverse operation of the function real2canon.

Usage

canon2real(Rcanon, R, B = c(0,1))

Arguments

Rcanon

A mx4 matrix of (intersections of) canonical rectangles that are to be transformed back to their original coordinates. Each row corresponds to a rectangle, represented as (x1,x2,y1,y2). The point (x1,y1) is the lower left corner of the rectangle an

R

A nx4 matrix with the coordinates of the original rectangles. Each row corresponds to a rectangle, represented as (x1,x2,y1,y2).

B

This describes the boundaries of the original rectangles (0=open or 1=closed). It can be specified in three ways: \itemA nx4 matrix containing 0's and 1's. Each row corresponds to a rectangle and is denoted as (cx1, cx2, cy1, cy2), where cx1 de

Value

• A list with the following elements:
• rects
  A mx4 matrix giving the original coordinates of the input rectangles. Each row (x1,x2,y1,y2) represents a rectangle.
• bounds
  This describes the boundaries rects. It is given in the same format as B.

concept

• nonparametric maximum likelihood estimator
• censored data

Details

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).

References

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.

See Also

real2canon

Examples

# 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 into canonical rectangles
res1 <- real2canon(R, c(0,1))   
plotRects(res1, grid=TRUE, lwd=2, main="Canonical rectangles")

# Transform canonical rectangles back to original coordinates   
res2 <- canon2real(res1, R, c(0,1))
plotRects(res2$rects, lwd=2, main="Original rectangles")
   
# Only transform rectangle (2,3)x(4,5), which is the  
#   the intersection of the canonical rectangles R1 and R3. 
#   The result is the intersection of the original rectangles R1 and R3. 
R.1.3 <- matrix(c(2,3,4,5),nrow=1)
res3 <- canon2real(R.1.3, R, c(0,1))
res3$rects
 
# Note that the algorithm keeps track of the boundaries of the rectangles:
B <- rbind(c(1,0,1,0),
           c(1,1,1,1),
           c(0,1,0,1))
res4 <- canon2real(R.1.3, R, B)
res4$bounds