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

In ttbary: Barycenter Methods for Spatial Point Patterns

Compute Sum of q-th Powers of Distances Between a Point Pattern and a List of Point Patterns

Description

Determine the Euclidean distance based TT-p-distances (or RTT-p-distances) between a single point pattern zeta and each point pattern in a list pplist. Then compute the sum of q-th powers of these distances.

Usage

sumppdist(
  zeta,
  pplist,
  penalty = 1,
  type = c("tt", "rtt", "TT", "RTT"),
  p = 1,
  q = 1
)

Arguments

zeta	an object of class ppp.
pplist	an object of class ppplist or an object that can be coerced to this class, such as a list of ppp objects.
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 pplist are chosen such that the p-th order sums (l_p-norms) of the Euclidean 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, ..., 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 zeta and each pattern in pplist. This number has an attribute distances that contains the individual distances.

Author(s)

Dominic Schuhmacher [e-mail]

See Also

ppdist for computation of TT-p- and RTT-p-metrics,
kmeansbary for finding a local minimum of the above sum for p = q = 2

Examples

  # See the examples for kmeansbary
  

ttbary documentation built on Nov. 16, 2022, 5:15 p.m.