# npf: Nonparametric Position Formulation In edmcr: Tools to Complete Euclidean Distance Matrices

## Description

npf returns a completed Euclidean Distance Matrix D, with dimension d, from a partial Euclidean Distance Matrix using the methods of Fang & O'Leary (2012)

## Usage

 1 2 npf(D, A = NA, d, dmax = (nrow(D) - 1), decreaseDim = 1, stretch = NULL, method = "Linear", toler = 1e-08) 

## Arguments

 D An nxn partial-distance matrix to be completed. D must satisfy a list of conditions (see details), with unkown entries set to NA. A a weight matrix, with h_{ij} = 0 implying a_{ij} is unknown. Generally, if a_{ij} is known, h_{ij} = 1, although any non-negative weight is allowed. d the dimension of the resulting completion dmax the maximum dimension to consider during dimension relaxation decreaseDim during dimension reduction, the number of dimensions to decrease each step stretch should the distance matrix be multiplied by a scalar constant? If no, stretch = NULL, otherwise stretch is a positive scalar method The method used for dimension reduction, one of "Linear" or "NLP". toler convergence tolerance for the algorithm

## Details

This is an implementation of the Nonconvex Position Formulation (npf) for Euclidean Distance Matrix Completion, as proposed in 'Euclidean Distance Matrix Completion Problems' (Fang & O'Leary, 2012).

The method seeks to minimize the following:

||A \cdot (D - K(XX'))||_{F}^{2}

where the function K() is that described in gram2edm, and the norm is Frobenius. Minimization is over X, the nxp matrix of node locations.

The matrix D is a partial-distance matrix, meaning some of its entries are unknown. It must satisfy the following conditions in order to be completed:

• diag(D) = 0

• If a_{ij} is known, a_{ji} = a_{ij}

• If a_{ij} is unknown, so is a_{ji}

• The graph of D must be connected. If D can be decomposed into two (or more) subgraphs, then the completion of D can be decomposed into two (or more) independent completion problems.

## Value

 D an nxn matrix of the completed Euclidean distances optval the minimum value achieved of the target function during minimization

gram2edm
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 D <- matrix(c(0,3,4,3,4,3, 3,0,1,NA,5,NA, 4,1,0,5,NA,5, 3,NA,5,0,1,NA, 4,5,NA,1,0,5, 3,NA,5,NA,5,0),byrow=TRUE, nrow=6) A <- matrix(c(1,1,1,1,1,1, 1,1,1,0,1,0, 1,1,1,1,0,1, 1,0,1,1,1,0, 1,1,0,1,1,1, 1,0,1,0,1,1),byrow=TRUE, nrow=6) edmc(D, method="npf", d=3, dmax=5)