R: Compute Sum of q-th Powers of Distances Between a Point...

sumppdistnet {ttbary}

R Documentation

Compute Sum of q-th Powers of Distances Between a Point Pattern and a Collection of Point Patterns on a Network

Description

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.

Usage

sumppdistnet(
  dmat,
  zeta,
  ppmatrix,
  penalty = 1,
  type = c("tt", "rtt", "TT", "RTT"),
  p = 1,
  q = 1
)

Arguments

`dmat`	the distance matrix of a network containing all shortest-path distances between its vertices.
`zeta`	a vector specifying the vertex-indices of zeta.
`ppmatrix`	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.
`penalty`	a positive number. The penalty for adding/deleting points.
`type`	either `"tt"`/`"TT"` for the transport-transform metric or `"rtt"`/`"RTT"` for the relative transport-transform metric.
`p`	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.
`q`	a number `>0`.

Details

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.

Value

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.

Author(s)

Raoul Müller raoul.mueller@uni-goettingen.de
Dominic Schuhmacher schuhmacher@math.uni-goettingen.de

Examples

  # See examples for kmeansbarynet

[Package ttbary version 0.3-1 Index]