CircNNTSR (version 2.0)

CircNNTSR-package: CircNNTSR: An R package for the statistical analysis of circular data using nonnegative trigonometric sums (NNTS) models

Description

A collection of utilities for the statistical analysis of circular and spherical data using nonnegative trigonometric sums (NNTS) models

Details

ll{ Package: CircNNTSR Type: Package Version: 2.0 Date: 2011-12-27 License: GLP (>=2) LazyLoad: yes } Fernandez-Duran, J.J. (2004) proposed a new family of distributions for circular random variables based on nonnegative trigonometric sums. This package provides functions for working with circular distributions based on nonnegative trigonometric sums, including functions for the estimation of parameters and plotting the densities. The distribution function in this package is a circular distribution based on nonnegative trigonometric sums (Fernandez-Duran, 2004). Fejer (1915) expressed a univariate nonnegative trigonometric (Fourier) sum (series), for a variable $\theta$, as the squared modulus of a sum of complex numbers, i.e., $$\left\|\sum_{k=0}^M c_ke^{ik\theta}\right\|^2\;\;\; (1)$$ where $i=\sqrt{-1}$. From this result, the parameters $(a_k,b_k)$ for $k=1,\ldots, M$ of the trigonometric sum of order $M$,$T(\theta)$, $$T(\theta)=a_0 + \sum_{k=1}^M(a_kcos(k\theta) + b_ksin(k\theta))$$ are expressed in terms of the complex parameters in equation (1) , $c_k$, for $k=0,\ldots, M$, as $a_k - ib_k= 2\sum_{\nu=0}^{n-k}c_{\nu + k}\overline{c}_{\nu}$. The additional constraint, $\sum_{k=0}^n\left\|c_k\right\|^2=\frac{1}{2\pi}=a_0$, is imposed to make the trigonometric sum to integrate 1. Thus, $c_0$ must be real and positive and there are 2*M free parameters. Then, the probability density function for a circular (angular) random variable is defined as (Fernandez-Duran, 2004) $$f(\theta; \underline{a},\underline{b},M)=\frac{1}{2\pi} + \frac{1}{\pi}\ \sum_{k=1}^M(a_kcos(k\theta) + b_ksin(k\theta)).$$ Note that Equation (1) can also be expressed as a double sum as $$\sum_{k=0}^{M}\sum_{m=0}^{M}c_k\bar{c}_me^{i(k-m)\theta}$$ The $\underline{c}$ parameters can also be expressed in polar coodinates as $c_k=\rho_k e^{i\phi_k}$ for $\rho_k \geq 0$ and $\phi_k \in [0,2\pi)$; where $\rho_k$ is the modulus of $c_k$ and $\phi_k$ is the argument of $c_k$ for $k=1,\ldots,M$. Many functions of the packages use as parameters the squared moduli and the arguments of $c_k$, $\rho_k^2$ and $\phi_k$, for $k=1,\ldots,M$. We refer to the parameter $M$ as the number of components in the NNTS.

References

Fernandez-Duran, J.J. (2004). Circular Distributions Based on Nonnegative Trigonometric Sums, Biometrics, 60(2), 499-503. Fernandez-Duran, J.J., Gregorio-Dominguez, M.M. (2009) Symmetric Circular Distributions Based on Nonnegative Trigonometric Sums. Working Paper, DE-C09.12, Department of Statistics, ITAM, Mexico

Examples

a<-c(runif(10,3*pi/2,2*pi-0.00000001),runif(10,pi/2,pi-0.00000001))
#Estimation of the NNTSdensity with 2 components for data and 1000 iterations
est<-nntsmanifoldnewtonestimation(a,2,1000)
#plot the estimated density
nntsplot(est$cestimates[,2],2)

data(Turtles_radians)
#Empirical analysis of data
Turtles_hist<-hist(Turtles_radians,breaks=10,freq=FALSE)
#Estimation of the NNTS density with 3 componentes for data
est<-nntsmanifoldnewtonestimation(Turtles_radians,3)
est
#plot the estimated density
nntsplot(est$cestimates[,2],3)
#add the histogram to the estimated density plot
plot(Turtles_hist, freq=FALSE, add=TRUE)

b<-c(runif(10,3*pi/2,2*pi-0.00000001),runif(10,pi/2,pi-0.00000001))
estS<-nntsestimationSymmetric(2,b)
nntsplotSymmetric(estS$coef,2)

M<-c(2,3)
R<-length(M)
data(Nest)
data<-Nest
est<-mnntsmanifoldnewtonestimation(data,M,R,1000)
est
cest<-est$cestimates
mnntsplot(cest, M)

data(Datab6fisher_ready)
data<-Datab6fisher_ready
M<-c(4,4)
cpars<-rnorm(prod(M+1))+rnorm(prod(M+1))*complex(real=0,imaginary=1)
cpars[1]<-Re(cpars[1])
cpars<- cpars/sqrt(sum(Mod(cpars)^2))
snntsdensity(data, cpars, M)
snntsloglik(data, cpars, M)

data(Datab6fisher_ready)
data<-Datab6fisher_ready
M<-c(1,2)
cest<-snntsmanifoldnewtonestimation(data, M)
lat<-snntsmarginallatitude(seq(0,pi,.1),cest$cestimates[,3],M)
plot(seq(0,pi,.1),lat,type="l")