Construct Lorentz Hyperboloid Coordinates for LHWR

These helpers turn raw two-component representations or relation matrices into the n x 3 Lorentz hyperboloid coordinate matrix Z used by lhwr() and bw.lhwr(). The coordinates are not required to be longitude and latitude. They may come from geographic coordinates, economic scores, clinical variables, functional or shape summaries, similarity matrices, trade networks, social networks, transportation networks, or graph embeddings.

Usage

check_lhwr_Z(Z, tol = 1e-06)

make_lhwr_Z_features(X, scale = TRUE, tol = 1e-06)

make_lhwr_Z_network(
  A,
  method = c("shortest_path_mds", "spectral"),
  scale = TRUE,
  disconnected = c("stop", "impute"),
  symmetrize = TRUE,
  tol = 1e-06
)

make_lhwr_Z(x, method = c("features", "network"), ...)

check_lorentz_Z(Z, tol = 1e-06)

make_lorentz_Z_features(X, scale = TRUE, tol = 1e-06)

make_lorentz_Z_network(
  A,
  method = c("shortest_path_mds", "spectral"),
  scale = TRUE,
  disconnected = c("stop", "impute"),
  symmetrize = TRUE,
  tol = 1e-06
)

make_lorentz_Z(x, method = c("features", "network"), ...)

Arguments

Z

An n x 3 Lorentz hyperboloid coordinate matrix to validate.

tol

Numerical tolerance for checking the hyperboloid constraint.

X

An n x 2 numeric matrix or data frame containing a two-component representation of the observations.

scale

If TRUE, standardize the two input components before constructing Z. A constant component has undefined sample standard deviation; it is mapped to zero with an explicit warning.

A

An n x n numeric adjacency, similarity, or relation matrix.

method

Construction or network embedding method. In make_lhwr_Z(), use "features" for a two-component feature map or "network" for a relation matrix. In make_lhwr_Z_network(), use "shortest_path_mds" or "spectral".

disconnected

Handling of disconnected graphs for "shortest_path_mds". "stop" reports an error; "impute" replaces infinite graph distances by a large finite distance before MDS.

symmetrize

If TRUE, non-symmetric relation matrices are replaced by (A + t(A)) / 2 with a warning.

x

Input object passed to the generic constructor. Use an n x 2 feature matrix when method = "features", or an n x n relation matrix when method = "network".

...

Arguments passed from make_lhwr_Z() to the selected construction helper.

Value

An n x 3 numeric matrix with columns z0, z1, and z2.

Details

Let x_i denote observation i. Suppose a two-component representation maps each observation to (z_i1, z_i2). LHWR embeds this pair into the upper Lorentz hyperboloid by

$$ z_i = \left(\sqrt{1 + z_{i1}^2 + z_{i2}^2}, z_{i1}, z_{i2}\right). $$

The resulting point satisfies

$$-z_{i0}^2 + z_{i1}^2 + z_{i2}^2 = -1,\quad z_{i0} > 0.$$

For non-geographic data, LHWR should be interpreted as locally weighted regression over a latent hyperbolic geometry. The validity of the model depends on whether the chosen representation captures meaningful neighborhood relations among observations.

Examples

X <- cbind(score1 = c(-1, 0, 1), score2 = c(0.5, 0, -0.5))
Z <- make_lhwr_Z_features(X)
check_lhwr_Z(Z)

A <- matrix(c(
  0, 1, 1, 0,
  1, 0, 1, 1,
  1, 1, 0, 1,
  0, 1, 1, 0
), nrow = 4, byrow = TRUE)
Z_net <- make_lhwr_Z_network(A, method = "spectral")
check_lhwr_Z(Z_net)