Bingham {scorematchingad}

Score Matching Estimators for the Bingham Distribution

Description

Score matching estimators for the Bingham distribution's parameter matrix. Two methods are available: a full score matching method that estimates the parameter matrix directly and a hybrid method by Mardia et al. (2016) that uses score matching to estimate just the eigenvalues of the parameter matrix.

Usage

Bingham(Y, A = NULL, w = rep(1, nrow(Y)), method = "Mardia")

Arguments

Y

A matrix of multivariate observations in Cartesian coordinates. Each row is a multivariate measurement (i.e. each row corresponds to an individual).

A

For full score matching only: if supplied, then NA elements of A are estimated and the other elements are fixed. For identifiability the final element of diag(A) must be NA.

w

An optional vector of weights for each measurement in Y

method

Either "Mardia" or "hybrid" for the hybrid score matching estimator from Mardia et al. (2016) or "smfull" for the full score matching estimator.

Details

The Bingham distribution has a density proportional to

\exp(z^T A z),

where A is a symmetric matrix and the trace (sum of the diagonals) of A is zero for identifiability (p181, Mardia and Jupp 2000).

The full score matching method estimates all elements of A directly except the final element of the diagonal, which is calculated from the sum of the other diagonal elements to ensure that the trace of A is zero.

The method by Mardia et al. (2016) first calculates the maximum-likelihood estimate of the eigenvectors G of A. The observations Y are then standardised to YG. This standardisation corresponds to diagonalising A where the eigenvalues of A become the diagonal elements of the new A. The diagonal elements of the new A are then estimated using score matching, with the final diagonal element calculated from the sum of the other elements. See Mardia et al. (2016) for details.

Value

A list of est, SE and info.

• est contains the estimated matrix A and a vector form, paramvec, of A (ordered according to c(diag(A)[1:(p-1)], A[upper.tri(A)]) ). For the Mardia method, the estimated eigenvalues of A (named evals) and eigenvectors of A (named G) are also returned.

• SE contains estimates of the standard errors if computed. See cppad_closed().

• info contains a variety of information about the model fitting procedure and results.

References

Mardia KV, Jupp PE (2000). Directional Statistics, Probability and Statistics. Wiley, Great Britain. ISBN 0-471-95333-4.

Mardia KV, Kent JT, Laha AK (2016). “Score matching estimators for directional distributions.” doi:10.48550/arXiv.1604.08470.

See Also

Other directional model estimators: FB(), vMF_robust(), vMF()

Examples

p <- 4
A <- rsymmetricmatrix(p)
A[p,p] <- -sum(diag(A)[1:(p-1)]) #to satisfy the trace = 0 constraint
if (requireNamespace("simdd")){
  Y <- simdd::rBingham(100, A)
  Bingham(Y, method = "Mardia")
}