reduc: Determine areas of possible mass support of the MLE

Description

The MLE for censored data can only assign mass to a finite set distinct regions, called maximal intersections. The function reduc computes these areas, using the height map algorithm described in Maathuis (2005). In addition to the maximal intersections, the function can also output the height map and the clique matrix.

Usage

reduc(R,B=c(0,1),hm=FALSE,cm=FALSE)

Arguments

A nx4 matrix containing the observation 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

This describes the boundaries of the 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). Here c

Logical, indicating if the heightmap must be outputted. The default value is FALSE. The height map is a (2n+1)x(2n+1) matrix, where n is the number of observation rectangles. Its values give the number of rectangles that ov

Logical, indicating if the clique matrix must be outputted. The default value is FALSE. The clique matrix is a mxn matrix, where m is the number of maximal intersections and n is the number of observation rectangles. The (i,

Value

A list containing the following elements:
rectsA mx4 matrix of maximal intersections. Each row (x1,x2,y1,y2) represents a maximal intersection, i.e., an area where the MLE can possibly assign mass.
boundsThis describes the boundaries of rects. It is given in the same format as B.
hm(Optional) A (2n)x(2n) matrix containing the height map. Its values represent the number of rectangles that overlap at any given point.
cm(Optional) A mxn matrix containing the clique matrix. The (i,j)th element of the matrix is 1 if the ith maximal intersection is contained in the jth observation rectangle, and it is 0 otherwise.

concept

nonparametric maximum likelihood estimator
censored data
parameter reduction

Details

The computation of the MLE for censored data can be split into two steps: a reduction step and an optimization step. In the reduction step, the areas of possible mass support are computed. Next, in the optimization step, it is determined how much probability mass should be assigned to each of these areas. The function reduc can be used for the reduction step. It is carried out automatically as part of the function computeMLE. The time and space complexity of the function reduc depend on the parameters 'hm' and 'cm'. If hm=FALSE and cm=FALSE, then the algorithm is $O(n^2)$ in time and $O(n)$ in memory space. If hm=TRUE, then the algorithm is $O(n^2)$ in both time and space. If cm=TRUE, then the algorithm is $O(n^3)$ in time and space. The function reduc uses the height map algorithm of Maathuis (2005). It first converts the observation rectangles to canonical rectangles. Next, it computes the local maxima of the height map, and then it convert these back to the original coordinates. This process can be mimicked by hand as follows: (1) use real2canon to convert the rectangles to canonical rectangles; (2) use reduc to find the canonical maximal intersections (local maxima of the height map of the canonical rectangles); (3) use canon2real to convert the canonical maximal intersections back to the original coordinates.

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.

Examples

Run this code

# Load example data:
data(ex)
par(mfrow=c(1,1))

# Plot the observation rectangles
plotRects(ex,main="Example")

# Perform the reduction step
res<-reduc(ex, hm=TRUE, cm=TRUE)

# Shade the maximal intersections
plotRects(res$rects, density=15, add=TRUE, border=NA)

# Plot the height map, together with the observation 
# rectangles (in black) and the maximal intersections (shaded)
plotHM(res$hm, ex)
plotRects(ex, add=TRUE, border="black")
plotRects(res$rects, add=TRUE, border=NA, density=15)

# Print the clique matrix 
res$cm

# Make a plot of the clique matrix (useful for large data sets)
plotCM(res$cm)

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