generalized_wishart: Generalized Wishart distance regression

In nspope/radish: Fast gradient-based optimization of resistance surfaces

Description

A function of class measurement_model that calculates likelihood, gradient, hessian, and partial derivatives of nuisance parameters and the Laplacian generalized inverse, using the generalized Wishart model described in McCullagh (2009), Peterson et al (2019).

Usage

generalized_wishart(
  E,
  S,
  phi,
  nu,
  gradient = TRUE,
  hessian = TRUE,
  partial = TRUE,
  nonnegative = TRUE,
  validate = FALSE
)

Arguments

E	A submatrix of the generalized inverse of the graph Laplacian (e.g. a covariance matrix)
S	A matrix of observed genetic distances
phi	Nuisance parameters (see details)
nu	Number of genetic markers
gradient	Compute gradient of negative loglikelihood with regard to phi?
hessian	Compute Hessian matrix of negative loglikelihood with regard to phi?
partial	Compute second partial derivatives of negative loglikelihood with regard to phi, E, S?
nonnegative	Unused
validate	Numerical validation via package numDeriv (very slow, use for debugging small examples)

Details

The nuisance parameters are the scaling of the generalized inverse of the graph Laplacian ("tau"; can be zero) and a log scalar multiple of the identity matrix that is added to the generalized inverse ("sigma").

TODO: formula

Value

A list containing:

covariance	rows/columns of the generalized inverse of the graph Laplacian for a subset of target vertices
objective	(if objective) the negative loglikelihood
fitted	((if objective) a matrix of expected genetic distances among target vertices
boundary	(if objective) is the MLE on the boundary (e.g. no genetic structure)?
gradient	(if gradient) gradient of negative loglikelihood with respect to phi
hessian	(if hessian) Hessian matrix of the negative loglikelihood with respect to phi
gradient_E	(if partial) gradient with respect to the generalized inverse of the graph Laplacian
partial_E	(if partial) Jacobian of gradient_E with respect to phi
partial_S	(if partial) Jacobian of gradient with respect to S
jacobian_E	(if partial) a function used for reverse algorithmic differentiation
jacobian_S	(if partial) a function used for reverse algorithmic differentiation

References

McCullagh P. 2009. Marginal likelihood for distance matrices. Statistica Sinica 19 Peterson et al. 2019. TODO

See Also

radish_measurement_model

Examples

library(raster)

data(melip)

covariates <- raster::stack(list(altitude=melip.altitude, forestcover=melip.forestcover))
surface <- conductance_surface(covariates, melip.coords, directions = 8)

# inverse of graph Laplacian at null model (IBD) 
laplacian_inv <- radish_distance(theta = matrix(0, 1, 2), 
                                 formula = ~forestcover + altitude,
                                 data = surface,
                                 radish::loglinear_conductance, 
                                 covariance = TRUE)$covariance[,,1]

generalized_wishart(laplacian_inv, melip.Fst, nu = 1000, phi = c(0.1, -0.1))

nspope/radish documentation built on July 12, 2020, 11:50 a.m.