CircNNTSRaxial (version 0.1.0)

axialnntsmanifoldnewtonestimationgradientstopsymmetric: Parameter estimation for axial symmetric NNTS distributions with gradient stop

Description

Computes the maximum likelihood estimates of the parameters of an axial symmetric NNTS distribution, 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

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

Value

A list with 13 elements:

cestimatessym

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

mu

Estimate of the angle of symmetry of the NNTS symmetric axial model

logliksym

Optimum log-likelihood value for the NNTS symmetric axial model

AICsym

Value of Akaike's Information Criterion for the NNTS symmetric axial model

BICsym

Value of Bayesian Information Criterion for the NNTS symmetric axial model

gradnormerrorsym

Gradient error after the last iteration for the estimation of the parameters of the NNTS symmetric axial model

cestimatesnonsym

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

logliknonsym

Optimum log-likelihood value for the general (non-symmetric) NNTS axial model

AICnonsym

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

BICnonsym

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

gradnormerrornonsym

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

loglikratioforsym

Value of the likelihood ratio test statistic for symmetry

loglikratioforsympvalue

Value of the asymptotic chi squared p-value of the likelihood ratio test statistic for symmetry

Arguments

data

Vector of axial angles in radians

M

Number of components in the NNTS axial model

iter

Number of iterations

gradientstop

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

pevalmu

Number of equidistant points in the interval 0 to pi to search for the maxima of the angle of symmetry

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.

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