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
axialnntsmanifoldnewtonestimationgradientstopknownmusymmetric(data, muknown=0, M = 0,
iter = 1000, initialpoint = FALSE, cinitial,gradientstop=1e-10)
A list with 13 elements:
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
Known value of the location angle of the symmetric NNTS axial model
Optimum log-likelihood value for the symmetric NNTS axial model with known location angle
Value of Akaike's Information Criterion for the symmetric NNTS axial model with known location angle
Value of Bayesian Information Criterion for the symmetric NNTS axial model with known location angle
Gradient error after the last iteration for the estimation of the parameters of the symmetric NNTS axial model with known location angle
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
Optimum log-likelihood value for the general (non-symmetric) NNTS axial model with unknown location angle
Value of Akaike's Information Criterion for the general (non-symmetric) NNTS axial model with unknown location angle
Value of Bayesian Information Criterion for the general (non-symmetric) NNTS axial model with unknown location angle
Gradient error after the last iteration for the estimation of the parameters of the general (non-symmetric) NNTS axial model with unknown location angle
Value of the likelihood ratio test statistic for known location angle
Value of the asymptotic chi squared p-value of the likelihood ratio test statistic for known location angle
Vector of axial angles in radians
Value of the known location angle
Number of components in the NNTS axial model
Number of iterations
TRUE if an initial point for the optimization algorithm for the axial NNTS density will be used
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
The minimum value of the norm of the gradient to stop the Newton algorithm on the hypersphere
Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez
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)
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)
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