vMF {scorematchingad}

Score Matching Estimator for the von-Mises Fisher Distribution

Description

In general the normalising constant in von Mises Fisher distributions is hard to compute, so Mardia et al. (2016) suggested a hybrid method that uses maximum likelihood to estimate the mean direction and score matching for the concentration. We can also estimate all parameters using score matching (smfull method), although this estimator is likely to be less efficient than the hybrid estimator. On the circle the hybrid estimators were often nearly as efficient as maximum likelihood estimators (Mardia et al. 2016). For maximum likelihood estimators of the von Mises Fisher distribution, which all use approximations of the normalising constant, consider movMF::movMF().

Usage

vMF(Y, paramvec = NULL, method = "Mardia", w = rep(1, nrow(Y)))

Arguments

Y

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

paramvec

smfull method only: Optional. A vector of same length as the dimension, representing the elements of the \kappa \mu vector.

method

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

w

An optional vector of weights for each measurement in Y

Details

The full score matching estimator (method = "smfull") estimates \kappa \mu. The hybrid estimator (method = "Mardia") estimates \kappa and \mu separately. Both use cppad_closed() for score matching estimation.

Value

A list of est, SE and info.

• est contains the estimates in vector form, paramvec, and with user friendly names k and m.

• 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.

von Mises Fisher Model

The von Mises Fisher density is proportional to

\exp(\kappa \mu^T z),

where z is on a unit sphere, \kappa is termed the concentration, and \mu is the mean direction unit vector. The effect of the \mu and \kappa can be decoupled in a sense (p169, Mardia and Jupp 2000), allowing for estimating \mu and \kappa separately.

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: Bingham(), FB(), vMF_robust()

Examples

if (requireNamespace("movMF")){
  Y <- movMF::rmovMF(1000, 100 * c(1, 1) / sqrt(2))
  movMF::movMF(Y, 1) #maximum likelihood estimate
} else {
  Y <- matrix(rnorm(1000 * 2, sd = 0.01), ncol = 2)
  Y <- Y / sqrt(rowSums(Y^2))
}
vMF(Y, method = "smfull")
vMF(Y, method = "Mardia")
vMF(Y, method = "hybrid")