sumppdist {ttbary}

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 (\ell_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,\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 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