CircNNTSRSymmetric (version 0.1.0)

nntsmanifoldnewtonestimationsymmetry: Parameter estimation for NNTS symmetric distributions

Description

Computes the maximum likelihood estimates of the NNTS parameters of an NNTS symmetric distribution, using a Newton algorithm on the hypersphere

Usage

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

Value

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 model

mu

Estimate of the angle of symmetry of the NNTS symmetric model

logliksym

Optimum log-likelihood value for the NNTS symmetric model

AICsym

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

BICsym

Value of Bayesian Information Criterion for the NNTS symmetric model

gradnormerrorsym

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

logliknonsym

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

AICnonsym

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

BICnonsym

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

gradnormerrornonsym

Gradient error after the last iteration for the estimation of the parameters of the general (non-symmetric) NNTS 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 angles in radians

M

Number of components in the NNTS symmetric density

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 2pi to search for the maxima of the angle of symmetry

initialpoint

TRUE if an initial point for the optimization algorithm for the general (asymmetric) 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/(2*pi). This is the vector of parameters for the general (asymmetric) NNTS density

Author

Juan Jose Fernandez-Duran y Maria Mercedes Gregorio-Dominguez

References

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(Turtles_radians)
resturtlessymm<-nntsmanifoldnewtonestimationsymmetry(data=Turtles_radians, M = 2, iter =1000,
gradientstop=1e-10,pevalmu=1000)
resturtlessymm
hist(Turtles_radians,breaks=seq(0,2*pi,2*pi/13),xlab="Direction (radians)",freq=FALSE,
ylab="",main="",ylim=c(0,.8),axes=FALSE)
nntsplot(resturtlessymm$cestimatessym[,2],2,add=TRUE)
nntsplot(resturtlessymm$cestimatesnonsym[,2],2,add=TRUE,lty=2)
axis(1,at=c(0,pi/2,pi,6*(pi/4),2*pi),labels=c("0",expression(pi/2),expression(pi),
expression(3*pi/2),expression(2*pi)),las=1)
axis(2)

data(Ants_radians)
resantssymm<-nntsmanifoldnewtonestimationsymmetry(data=Ants_radians, M = 4, iter =1000,
gradientstop=1e-10,pevalmu=1000)
resantssymm
hist(Ants_radians,breaks=seq(0,2*pi,2*pi/13),xlab="Direction (radians)",freq=FALSE,
ylab="",main="",ylim=c(0,.8),axes=FALSE)
nntsplot(resantssymm$cestimatessym[,2],4,add=TRUE)
nntsplot(resantssymm$cestimatesnonsym[,2],4,add=TRUE,lty=2)
axis(1,at=c(0,pi/2,pi,6*(pi/4),2*pi),labels=c("0",expression(pi/2),expression(pi),
expression(3*pi/2),expression(2*pi)),las=1)
axis(2)