Distance Matrix Computation for Non-Hilbert Data

This function computes and returns the numeric distance matrix computed by using the specified distance measure to compute the distances between the rows of a data matrix.

nhdist(x, method = "geodesic")

Arguments

x: a numeric matrix, data frame or numeric array of dimension $k \times m \times n$ containing $n$ samples in $k \times m$ dimension.
method: the distance measure to be used. This must be one of "geodesic", "compositional", or "riemann". Any unambiguous substring can be given.

Value

$n \times n$ numeric distance matrix

Details

Available distance measures are geodesic, compositional and riemann. Denoting any two sample in the dataset as $x$ and $y$, we give the definition of distance measures as follows.

geodesic:

The shortest route between two points on the Earth's surface, namely, a segment of a great circle. $$\arccos(x^{T}y), \|x\|_{2} = \|y\|_{2} = 1$$

compositional:

First, we apply scale transformation to it, i.e., $(x_{i1}/t, ..., x_{ip}/t_{i}), t_{i} = \sum_{d=1}^{p}{x_{d}}$ . Then, apply the square root transformation to data and calculate the geodesic distance between samples.

riemann:

$k \times m \times n$ array where $k$ = number of landmarks, $m$ = number of dimensions and $n$ = sample size. Detail about riemannian shape distance was given in Kendall, D. G. (1984).

References

Kendall, D. G. (1984). Shape manifolds, Procrustean metrics and complex projective spaces, Bulletin of the London Mathematical Society, 16, 81-121.

Examples

data('bdvmf')
Dmat <- nhdist(bdvmf[['x']], method = "geodesic")

data("ArcticLake")
Dmat <- nhdist(ArcticLake[['x']], method = "compositional")

data("macaques")
Dmat <- nhdist(macaques[["x"]], method = "riemann")

# unambiguous substring also available:
Dmat <- nhdist(macaques[["x"]], method = "rie")