Based on the shortest-path metric in a network, determine the TT-p-distances (or RTT-p-distances)
between a single point pattern zeta and a collection of point patterns. Then
compute the sum of \(q\)-th powers of these distances. The point patterns are
specified by vectors of indices referring to the vertices in the network.
sumppdistnet(
dmat,
zeta,
ppmatrix,
penalty = 1,
type = c("tt", "rtt", "TT", "RTT"),
p = 1,
q = 1
)A nonnegative number, the q-th order sum of the TT-p- or RTT-p-distances
between the patterns represented by zeta and ppmatrix. This number has an attribute
distances that contains the individual distances.
the distance matrix of a network containing all shortest-path distances between its vertices.
a vector specifying the vertex-indices of zeta.
a matrix specifying in its columns the vertex-indices of the point patterns in the collection. A virtual index that is one greater than the maximum vertex-index in the network can be used to fill up columns so that they all have the same length.
a positive number. The penalty for adding/deleting points.
either "tt"/"TT" for the transport-transform metric
or "rtt"/"RTT" for the relative transport-transform metric.
a number \(>0\). Matchings between zeta and the patterns in
ppmatrix are chosen such that the p-th order sums (\(\ell_p\)-norms)
of the shortest-path distances are minimized.
a number \(>0\).
Raoul Müller raoul.mueller@uni-goettingen.de
Dominic Schuhmacher schuhmacher@math.uni-goettingen.de
The main purpose of this function is to evaluate the relative performance of approximate \(q\)-th order barycenters of point patterns. A true \(q\)-th order barycenter of the point patterns \(\xi_1,\ldots,\xi_k\) with respect to the TT-p metric \(\tau_p\) minimizes $$\sum_{j=1}^k \tau_p(\xi_j, \zeta)^q$$ in \(\zeta\).
The most common choices are p = q = 1 and p = q = 2. Other
choices have not been tested.
kmeansbarynet, sumppdist
# See examples for kmeansbarynet
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