matchingdist: Distance for a Point Pattern Matching

Description

Computes the distance associated with a matching between two point patterns.

Usage

matchingdist(matching, type = NULL, cutoff = NULL, q = NULL)

Arguments

matching

A point pattern matching (an object of class "pppmatching").

type

A character string giving the type of distance to be computed. One of "spa", "ace" or "mat". See details below.

cutoff

The value $> 0$ at which interpoint distances are cut off.

The order of the average that is applied to the interpoint distances. May be Inf, in which case the maximum of the interpoint distances is taken.

Value

Numeric value of the distance associated with the matching.

Details

Computes the distance specified by type, cutoff, and order for a point matching. If any of these arguments are not provided, the function uses the corresponding elements of matching (if available).

For the type "spa" (subpattern assignment) it is assumed that the points of the point pattern with the smaller cardinality $m$ are matched to a $m$-point subpattern of the point pattern with the larger cardinality $n$ in a 1-1 way. The distance is then given as the q-th order average of the $m$ distances between matched points (minimum of Euclidean distance and cutoff) and $n-m$ "penalty distances" of value cutoff.

For the type "ace" (assignment only if cardinalities equal) the matching is assumed to be 1-1 if the cardinalities of the point patterns are the same, in which case the q-th order average of the matching distances (minimum of Euclidean distance and cutoff) is taken. If the cardinalities are different, the matching may be arbitrary and the distance returned is always equal to cutoff.

For the type mat (mass transfer) it is assumed that each point of the point pattern with the smaller cardinality $m$ has mass $1$, each point of the point pattern with the larger cardinality $n$ has mass $m/n$, and fractions of these masses are matched in such a way that each point contributes exactly its mass. The distance is then given as the q-th order weighted average of all distances (minimum of Euclidean distance and cutoff) of (partially) matched points with weights equal to the fractional masses divided by $m$.

If the cardinalities of the two point patterns are equal, matchingdist(m, type, cutoff, q) yields the same result no matter if type is "spa", "ace" or "mat".

Examples

Run this code

# an optimal matching
  X <- runifpoint(20)
  Y <- runifpoint(20)
  m.opt <- pppdist(X, Y)
  summary(m.opt)
  matchingdist(m.opt)
       # is the same as the distance given by summary(m.opt)
  
  # sequential nearest neighbour matching
  # (go through all points of point pattern X in sequence
  # and match each point with the closest point of Y that is
  # still unmatched)
  am <- matrix(0, 20, 20)
  h <- matrix(c(1:20, rep(0,20)), 20, 2)
  h[1,2] = nncross(X[1],Y)[1,2]
  for (i in 2:20) {
    nn <- nncross(X[i],Y[-h[1:(i-1),2]])[1,2]
    h[i,2] <- ((1:20)[-h[1:(i-1),2]])[nn]
  }
  am[h] <- 1
  m.nn <- pppmatching(X, Y, am)
  matchingdist(m.nn, type="spa", cutoff=1, q=1)
       # is >= the distance obtained for m.opt
       # in most cases strictly >

  par(mfrow=c(1,2))
    plot(m.opt)
    plot(m.nn)
    text(X$x, X$y, 1:20, pos=1, offset=0.3, cex=0.8)