# matchingdist

##### Distance for a Point Pattern Matching

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.
- q
- 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.

##### 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"`

.

##### Value

- Numeric value of the distance associated with the matching.

##### See Also

##### Examples

```
# 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)
```

*Documentation reproduced from package spatstat, version 1.31-2, License: GPL (>= 2)*