CircNNTSRaxial (version 0.1.0)

axialnntsmanifoldnewtonestimationgradientstopknownmusymmetric: Parameter estimation for axial symmetric NNTS distributions with known location angle gradient stop

Description

Computes the maximum likelihood estimates of the parameters of an axial symmetric NNTS distribution with known location angle, using a Newton algorithm on the hypersphere and considering a maximum number of iterations determined by a constraint in terms of the norm of the gradient

Usage

axialnntsmanifoldnewtonestimationgradientstopknownmusymmetric(data, muknown=0, M = 0, 
iter = 1000, initialpoint = FALSE, cinitial,gradientstop=1e-10)

Value

A list with 13 elements:

cestimatesmuknown

Matrix of (M+1)x2. The first column is the parameter numbers, and the second column is the c parameter's estimators of the symmetric NNTS axial model with known location angle

muknown

Known value of the location angle of the symmetric NNTS axial model

loglikmuknown

Optimum log-likelihood value for the symmetric NNTS axial model with known location angle

AICmuknown

Value of Akaike's Information Criterion for the symmetric NNTS axial model with known location angle

BICmuknown

Value of Bayesian Information Criterion for the symmetric NNTS axial model with known location angle

gradnormerrormuknown

Gradient error after the last iteration for the estimation of the parameters of the symmetric NNTS axial model with known location angle

cestimatesmuunknown

Matrix of (M+1)x2. The first column is the parameter numbers, and the second column is the c parameter's estimators of the general (non-symmetric) NNTS axial model with unknown location angle

loglikmuunknown

Optimum log-likelihood value for the general (non-symmetric) NNTS axial model with unknown location angle

AICmuunknown

Value of Akaike's Information Criterion for the general (non-symmetric) NNTS axial model with unknown location angle

BICmuunknown

Value of Bayesian Information Criterion for the general (non-symmetric) NNTS axial model with unknown location angle

gradnormerrormuunknown

Gradient error after the last iteration for the estimation of the parameters of the general (non-symmetric) NNTS axial model with unknown location angle

loglikratioformuknown

Value of the likelihood ratio test statistic for known location angle

loglikratioformuknownpvalue

Value of the asymptotic chi squared p-value of the likelihood ratio test statistic for known location angle

Arguments

data

Vector of axial angles in radians

muknown

Value of the known location angle

M

Number of components in the NNTS axial model

iter

Number of iterations

initialpoint

TRUE if an initial point for the optimization algorithm for the axial NNTS density will be used

cinitial

Vector of size M+1. The first element is real and the next M elements are complex (values for $c_0$ and $c_1, ...,c_M$). The sum of the squared moduli of the parameters must be equal to 1/pi. This is the vector of parameters for the general (asymmetric) NNTS axial density

gradientstop

The minimum value of the norm of the gradient to stop the Newton algorithm on the hypersphere

Author

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

References

Fernandez-Duran, J.J. and Gregorio-Dominguez, M.M. (2025). Multimodal distributions for circular axial data. arXiv:2504.04681 [stat.ME] (available at https://arxiv.org/abs/2504.04681)

Fernández-Durán, J.J., Gregorio-Domínguez, M.M. (2025). Multimodal Symmetric Circular Distributions Based on Nonnegative Trigonometric Sums and a Likelihood Ratio Test for Reflective Symmetry, arXiv:2412.19501 [stat.ME] (available at https://arxiv.org/abs/2412.19501)

Examples

data(Datab2fisher)
feldsparsangles<-Datab2fisher
feldsparsangles<-feldsparsangles$orientations*(pi/180)
resfeldsparknownanglesymmetric<-axialnntsmanifoldnewtonestimationgradientstopknownmusymmetric(
data=feldsparsangles, muknown=pi/3, M = 3, iter =1000, gradientstop=1e-10)
resfeldsparknownanglesymmetric
hist(feldsparsangles,breaks=seq(0,pi,pi/7),xlab="Orientations (radians)",freq=FALSE,
ylab="",main="",ylim=c(0,.8),axes=FALSE)
axialnntsplot(resfeldsparknownanglesymmetric$cestimatesmuunknown[,2],3,add=TRUE)
axialnntsplot(resfeldsparknownanglesymmetric$cestimatesmuknown[,2],3,add=TRUE,lty=2)
axis(1,at=c(0,pi/2,pi),labels=c("0",expression(pi/2),expression(pi)),las=1)
axis(2)