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 sum (NNTS) models

Details

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 estimating the 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 one. 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 coordinates 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.

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

Maintainer: Maria Mercedes Gregorio Dominguez <mercedes@itam.mx>

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. (2010). A Likelihood-Ratio Test for Homogeneity in Circular Data. Journal of Biometrics & Biostatistics, 1(3), 107. doi:10.4172/2155-6180.1000107

Fernandez-Duran, J.J., Gregorio-Dominguez, M.M. (2010). Maximum Likelihood Estimation of Nonnegative Trigonometric Sums Models Using a Newton-Like Algorithm on Manifolds. Electronic Journal of Statistics, 4, 1402-1410. doi:10.1214/10-ejs587

Fernandez-Duran, J.J., Gregorio-Dominguez, M.M. (2014). Distributions for Spherical Data Based on Nonnegative Trigonometric Sums. Statistical Papers, 55(4), 983-1000. doi:10.1007/s00362-013-0547-5

Fernandez-Duran, J.J., Gregorio-Dominguez, M.M. (2014). Modeling Angles in Proteins and Circular Genomes Using Multivariate Angular Distributions Based on Nonnegative Trigonometric Sums. Statistical Applications in Genetics and Molecular Biology, 13(1), 1-18. doi:10.1515/sagmb-2012-0012

Fernandez-Duran, J.J., Gregorio-Dominguez, M.M. (2014). Testing for Seasonality Using Circular Distributions Based on Nonnegative Trigonometric Sums as Alternative Hypotheses. Statistical Methods in Medical Research, 23(3), 279-292. doi:10.1177/0962280211411531.

Juan Jose Fernandez-Duran, Maria Mercedes Gregorio-Dominguez (2016). CircNNTSR: An R Package for the Statistical Analysis of Circular, Multivariate Circular, and Spherical Data Using Nonnegative Trigonometric Sums. Journal of Statistical Software, 70(6), 1-19. doi:10.18637/jss.v070.i06

Examples

set.seed(200)
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,iter=100)
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,100)
est
cest<-est$cestimates
mnntsplot(cest, M)

Movements of ants

Description

Directions chosen by 100 ants in response to an evenly illuminated black target.

Usage

data(Ants)

Format

Directions chosen by 100 ants in degrees

Source

Randomly selected values by Fisher (1993) from Jander (1957)

References

N.I. Fisher (1993) Statistical analysis of circular data. Cambridge University Press.

Movements of ants

Description

Direction chosen by 100 ants in response to an evenly illuminated black target.

Usage

data(Ants_radians)

Format

Directions chosen by 100 ants in radians

Source

Randomly selected values by Fisher (1993) from Jander (1957)

References

N.I. Fisher (1993) Statistical analysis of circular data. Cambridge University Press.

Spherical Data on Magnetic Remanence

Description

Measurements of magnetic remanence from 52 specimens of red beds from the Bowen Basin, Queensland.

Usage

data(DataB5FisherSpherical)

Format

Declination -inclination in degrees

Source

P.W. Schmidt

References

Fisher N.I., Lewis T. and Embleton B.J.J. (1987) Statistical Analysis of Spherical Data. Cambridge University Press, Cambridge. Data B.5.

Examples

data(DataB5FisherSpherical)

Uniform Bivariate Circular Data

Description

200 realizations of a uniform distribution on the torus

Usage

data(DataUniformBivariate200obs)

Format

Angles in radians

Database B3 from Fisher

Description

Database B3 from Fisher et al. (1987)

Usage

data(Datab3fisher)

Format

Datab3fisher

Details

The dataset Datab3fisher consists of 148 observations of the arrival directions of low-mu showers of cosmic rays (Toyoda et al., 1965; see Fisher et al., 1987, pp. 280-281). The observations are measured in declination and right ascension coordinates.

Source

Fisher, et al. (1987)

References

Toyoda, Y., Suga, K., Murakami, K., Hasegawa, H., Shibata, S., Domingo, V., Escobar, I., Kamata, K., Bradt, H., Clark, G. and La Pointe, M. (1965). Studies of Primary Cosmic Rays in the Energy Region 10^{14} eV to 10^{17} eV (Bolivian Air Shower Joint Experiment), Proceedings of the International Conference on Cosmic Rays, vol. 2, London, September, 1965, 708–711. London: The Institute of Physics and the Physical Society.

Fisher, N.I., Lewis, T. and Embleton, B.J.J. (1987). Statistical Analysis of Spherical Data, Cambridge U.K.: Cambridge University Press.

Data transformed from Datab3fisher

Description

Data transformed from Datab3fisher

Usage

data(Datab3fisher_ready)

Format

Datab3fisher_ready

Details

datab3fisher[,2] <- 90 + datab3fisher[,2]; datab3fisher_ready <- datab3fisher*(pi/180)

Database B6 from Fisher et al. (1987)

Description

datab6fisher

Usage

data(Datab6fisher)

Format

The coordinates are declination and inclination measured in degrees

Details

The data-set Datab6fisher contains 107 measurements of magnetic remanence in samples of Precambrian volcanics collected in Northwest Australia. (Schmidt and Embleton, 1985; see Fisher et al., 1987, pp. 285).

Source

Fisher, et al. (1987)

References

Schmidt, P.W. and Embleton, B.J.J. (1985). Pre-folding and overprint magnetic signatures in Precambrian (~2.9-2.7ga) igneous rocks from the Pilbara Craton and Hamersley Basin, N.W. Australia, Journal of Geophysical Research, 90 (B4), 2967–2984.

Fisher, N.I., Lewis, T. and Embleton, B.J.J. (1987). Statistical Analysis of Spherical Data, Cambridge U.K.: Cambridge University Press.

Data transformed from datab6fisher

Description

Data transformed from datab6fisher

Usage

data(Datab6fisher_ready)

Format

Datab6fisher_ready

Details

dataaux <- datab6fisher; datab6fisher[,1] <- dataaux[,2]; datab6fisher[,2] <- dataaux[,1]; datab6fisher[,1] <- 360 - datab6fisher[,1]; datab6fisher[,2] <- 90 + datab6fisher[,2]; datab6fisher_ready <- datab6fisher*(pi/180)

Date of Occurrence of Earthquakes

Description

The time of occurrence of earthquakes of intensity greater than 6.0^o Richter with an epicenter occurring in the coast of the Pacific Ocean in Mexico from 1920 to 2002. There is a total of 241 observations.

Usage

data(EarthquakesPacificMexicogt6)

Format

Time in years. All observations in the interval (0,1]

Source

Mexican Database of Strong Earthquakes. CENAPRED.

Date of Occurrence of Earthquakes 2

Description

The time of occurrence of earthquakes of intensity greater than 7.0^o Richter with an epicenter occurring in the coast of the Pacific Ocean in Mexico from 1920 to 2002. There are a total of 76 observations.

Usage

data(EarthquakesPacificMexicogt7)

Format

Time in years. All observations in the interval (0,1]

Source

Mexican Database of Strong Earthquakes. CENAPRED.

Homicides in Mexico during 2005

Description

Monthly number of homicides in Mexico during 2005

Usage

data(HomicidesMexico2005)

Format

Integer values

Source

INEGI (Mexican National Statistical Agency) www.inegi.gob.mx

Hurricanes in Mexico from 1951 to 1970

Description

The time of occurrence (starting times) of hurricanes in the Gulf of Mexico for the 1951-1970 period. There are a total of 196 observations.

Usage

data(HurricanesGulfofMexico1951to1970)

Format

Time in years. All observations in the interval (0,1]

Source

http://weather.unisys.com/hurricane/atlantic/1978/index.html

Hurricanes in Mexico from 1971 to 2008

Description

The time of occurrence (starting times) of hurricanes in the Gulf of Mexico for the 1971-2008 period. There are a total of 417 observations

Usage

data(HurricanesGulfofMexico1971to2008)

Format

Time in years. All observations in the interval (0,1]

Source

http://weather.unisys.com/hurricane/atlantic/1978/index.html

Nest orientations and creek directions

Description

Orientation of nests of 50 noisy scrub birds (theta) along the bank of a creek bed, together with the corresponding directions (phi) of creek flow at the nearest point to the nest.

Usage

data(Nest)

Format

Orientation of 50 nests (vectors)

Source

Data supplied by Dr. Graham Smith

References

N.I. Fisher (1993) Statistical analysis of circular data. Cambridge University Press.

Dihedral angles in protein

Description

Dataset of the dihedral angles in a protein between three consecutive Alanine (Ala) amino acids. This dataset was constructed from the recommended July 2003 list of proteins via the algorithm in Hobohm et al. (1992). This algorithm selects a representative sample of proteins from the vast Protein Data Bank (PDB, Berman et al., 2000). The dataset contains 233 pairs of dihedral angles.

Usage

data(ProteinsAAA)

Format

Two columns of angles in radians

Source

Protein Data Bank (PDB)

References

Hobohm, U. and Scharf, M. and Schneider, R. and Sander, C. (1992) Selection of a Representative Set of Structures from the Brookhaven Protein Data Bank, Protein Science, 1, 409-417. Berman, H. M. and Westbrook, J. and Feng, Z. and Gilliand, G. and Bhat, T. N. and Weissing, H. and Shyndialov, I. N. and Bourne, P. E. (2000) The Protein Data Bank, Nucleic Acids Research, 28, 235-242.

Suicides in Mexico during 2005

Description

Monthly number of suicides in Mexico during 2005

Usage

data(SuicidesMexico2005)

Format

Integer values

Source

INEGI (Mexican National Statistical Agency) www.inegi.gob.mx

Movements of turtles

Description

Data measurement of the directions taken by 76 turtles after treatment

Usage

data(Turtles)

Format

Directions of turtles in degrees

Source

Stephens (1969) Techniques for directional data. Technical Report 150. Dept. of Statistics, Stanford University. Stanford, CA.

References

N.I. Fisher (1993) Statistical analysis of circular data. Cambridge University Press.

Movements of turtles

Description

Data measurement of the directions taken by 76 turtles after treatment

Usage

data(Turtles_radians)

Format

Directions of turtles in radians

Source

Stephens (1969) Techniques for directional data. Technical Report 150. Dept. of Statistics, Stanford University. Stanford, CA.

References

N.I. Fisher (1993) Statistical analysis of circular data. Cambridge University Press.

Wind directions

Description

Wind directions registered at the monitoring stations of San Agustin located in the north, Pedregal in the southwest, and Hangares in the southeast of the Mexico Central Valley's at 14:00 on days between January 1, 1993 and February 29, 2000. There are a total of 1,682 observations

Usage

data(WindDirectionsTrivariate)

Format

Three columns of angles in radians

Source

Mexico Central Valleys pollution monitoring network. RAMA SIMAT (Red Automatica de Monitoreo Ambiental)

MNNTS density function

Description

Density function for the MNNTS model

Usage

mnntsdensity(data, cpars = 1/sqrt(2 * pi), M = 0, R=1)

Arguments

Value

The function returns the density function evaluated at each row in data

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

References

Juan Jose Fernandez-Duran, Maria Mercedes Gregorio-Dominguez (2016). CircNNTSR: An R Package for the Statistical Analysis of Circular, Multivariate Circular, and Spherical Data Using Nonnegative Trigonometric Sums. Journal of Statistical Software, 70(6), 1-19. doi:10.18637/jss.v070.i06

Examples

	set.seed(200)
M<-c(2,3)
R<-length(M)
data<-c(0,pi,pi/2,pi,pi,3*pi/2,pi,2*pi,2*pi,pi)
data<-matrix(data,ncol=2,byrow=TRUE)
data
ccoef<-mnntsrandominitial(M,R)
mnntsdensity(data,ccoef,M,R)

M<-c(2,3)
R<-length(M)
data(Nest)
data<-Nest
est<-mnntsmanifoldnewtonestimation(data,M,R,100)
est
ccoef<-est$cestimates[,3]
mnntsdensity(data,ccoef,M,R)

MNNTS density function (the particular case has two dimensions)

Description

Density function for the MNNTS model (the particular case has two dimensions). This is an internal function used by mnntsplot

Usage

mnntsdensity2d(x,y, cpars,M)

Arguments

Value

The function returns the density function evaluated at each row in data

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

References

Fernandez-Duran, J.J. and Gregorio-Dominguez, M.M. (2009) Multivariate Angular Distributions Based on Multiple Nonnegative Trigonometric Sums, Working Paper, Statistics Department, ITAM, DE-C09.1

MNNTS log-likelihood function

Description

Computes the log-likelihood function with MNNTS density for data

Usage

mnntsloglik(data, cpars = 1/sqrt(2 * pi), M = 0, R = 1)

Arguments

Value

The function returns the value of the log-likelihood function for the data.

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

References

Fernandez-Duran, J.J. and Gregorio-Dominguez, M.M. (2009) Multivariate Angular Distributions Based on Multiple Nonnegative Trigonometric Sums, Working Paper, Statistics Department, ITAM, DE-C09.1

Examples

M<-c(2,3)
R<-length(M)
data<-c(0,pi,pi/2,pi,pi,3*pi/2,pi,2*pi,2*pi,pi)
data<-matrix(data,ncol=2,byrow=TRUE)
data
ccoef<-mnntsrandominitial(M,R)
mnntsdensity(data,ccoef,M,R)
mnntsloglik(data,ccoef,M,R)

Parameter estimation for the MNNTS distributions

Description

Computes the maximum likelihood estimates of the MNNTS parameters using a Newton algorithm on the hypersphere

Usage

mnntsmanifoldnewtonestimation(data,M=0,R=1,iter=1000,initialpoint=FALSE,cinitial)

Arguments

Value

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

References

Fernandez-Duran, J.J. and Gregorio-Dominguez, M.M. (2009) Multivariate Angular Distributions Based on Multiple Nonnegative Trigonometric Sums, Working Paper, Statistics Department, ITAM, DE-C09.1

Fernandez-Duran, J.J., Gregorio-Dominguez, M.M. (2014). Modeling Angles in Proteins and Circular Genomes Using Multivariate Angular Distributions Based on Nonnegative Trigonometric Sums. Statistical Applications in Genetics and Molecular Biology, 13(1), 1-18. doi:10.1515/sagmb-2012-0012

Examples

	set.seed(200)
M<-c(2,3)
R<-length(M)
data(Nest)
data<-Nest*(pi/180)
est<-mnntsmanifoldnewtonestimation(data,M,R,100)
est

Marginal density function of the MNNTS model

Description

Marginal density function for one dimension of the MNNTS model evaluated at a point

Usage

mnntsmarginal(cestimatesarray, M, component, theta)

Arguments

Value

The function returns the density function evaluated at theta

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

References

Fernandez-Duran, J.J. and Gregorio-Dominguez, M.M. (2009) Multivariate Angular Distributions Based on Multiple Nonnegative Trigonometric Sums, Working Paper, Statistics Department, ITAM, DE-C09.1

Fernandez-Duran, J.J., Gregorio-Dominguez, M.M. (2014). Modeling Angles in Proteins and Circular Genomes Using Multivariate Angular Distributions Based on Nonnegative Trigonometric Sums. Statistical Applications in Genetics and Molecular Biology, 13(1), 1-18. doi:10.1515/sagmb-2012-0012

Examples

	set.seed(200)
M<-c(2,3)
R<-length(M)
data(Nest)
data<-Nest
est<-mnntsmanifoldnewtonestimation(data,M,R,100)
est
cest<-est$cestimates
mnntsmarginal(cest,M,1,pi)

Plots an MNNTS bivariate density

Description

Plots the MNNTS bivariate density function

Usage

mnntsplot(cestimates, M, ...)

Arguments

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

References

Fernandez-Duran, J.J. and Gregorio-Dominguez, M.M. (2009) Multivariate Angular Distributions Based on Multiple Nonnegative Trigonometric Sums, Working Paper, Statistics Department, ITAM, DE-C09.1

Fernandez-Duran, J.J., Gregorio-Dominguez, M.M. (2014). Modeling Angles in Proteins and Circular Genomes Using Multivariate Angular Distributions Based on Nonnegative Trigonometric Sums. Statistical Applications in Genetics and Molecular Biology, 13(1), 1-18. doi:10.1515/sagmb-2012-0012

Examples

	set.seed(200)
M<-c(2,3)
R<-length(M)
data(Nest)
data<-Nest
est<-mnntsmanifoldnewtonestimation(data,M,R,100)
est
cest<-est$cestimates
mnntsplot(cest, M)

Plots an MNNTS marginal density

Description

Plots the MNNTS marginal density function

Usage

mnntsplotmarginal(cestimates, M, component, ...)

Arguments

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

References

Fernandez-Duran, J.J. and Gregorio-Dominguez, M.M. (2009) Multivariate Angular Distributions Based on Multiple Nonnegative Trigonometric Sums, Working Paper, Statistics Department, ITAM, DE-C09.1

Fernandez-Duran, J.J., Gregorio-Dominguez, M.M. (2014). Modeling Angles in Proteins and Circular Genomes Using Multivariate Angular Distributions Based on Nonnegative Trigonometric Sums. Statistical Applications in Genetics and Molecular Biology, 13(1), 1-18. doi:10.1515/sagmb-2012-0012

Examples

	set.seed(200)
M<-c(2,3)
R<-length(M)
data(Nest)
data<-Nest
est<-mnntsmanifoldnewtonestimation(data,M,R,100)
est
cest<-est$cestimates
mnntsplotmarginal(cest, M, 1)
mnntsplotmarginal(cest, M, 2)

Plots an MNNTS bivariate density together with the marginals

Description

Plots the MNNTS bivariate density function together with the marginals

Usage

mnntsplotwithmarginals(cestimates, M, ...)

Arguments

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

References

Fernandez-Duran, J.J. and Gregorio-Dominguez, M.M. (2009) Multivariate Angular Distributions Based on Multiple Nonnegative Trigonometric Sums, Working Paper, Statistics Department, ITAM, DE-C09.1

Fernandez-Duran, J.J., Gregorio-Dominguez, M.M. (2014). Modeling Angles in Proteins and Circular Genomes Using Multivariate Angular Distributions Based on Nonnegative Trigonometric Sums. Statistical Applications in Genetics and Molecular Biology, 13(1), 1-18. doi:10.1515/sagmb-2012-0012

Examples

	set.seed(200)
M<-c(2,3)
R<-length(M)
data(Nest)
data<-Nest
est<-mnntsmanifoldnewtonestimation(data,M,R,100)
est
cest<-est$cestimates
mnntsplotwithmarginals(cest, M)

Initial random point

Description

This function generates a random point on the surface of the prod(M+1)-dimensional unit hypersphere

Usage

mnntsrandominitial(M = 1, R = 1)

Arguments

Value

Returns a valid initial point for estimation functions

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

References

Fernandez-Duran, J.J. and Gregorio-Dominguez, M.M. (2009) Multivariate Angular Distributions Based on Multiple Nonnegative Trigonometric Sums, Working Paper, Statistics Department, ITAM, DE-C09.1

Examples

M<-c(2,3)
R<-length(M)
mnntsrandominitial(M,R)

MNNTS density simulation function

Description

Simulation for the density function for the MNNTS model

Usage

mnntssimulation(nsim=1, cpars = 1/(2 * pi), M = c(0,0), R=2)

Arguments

Value

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

References

Fernandez-Duran, J.J. and Gregorio-Dominguez, M.M. (2009) Multivariate Angular Distributions Based on Multiple Nonnegative Trigonometric Sums, Working Paper, Statistics Department, ITAM, DE-C09.1

Examples

M<-c(2,3)
R<-length(M)
ccoef<-mnntsrandominitial(M,R)
data<-mnntssimulation(10,ccoef,M,R)
data

Density function with AB coefficients

Description

Density function expressed in terms of the ab parameters at theta

Usage

nntsABDensity(theta, cpars = 1/sqrt(2 * pi), M = 0)

Arguments

Value

Returns the density function in terms of the ab coefficients evaluated at theta

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

Examples

ccoef<-nntsrandominitial(3)
nntsABDensity(1,ccoef,3)
nntsABDensity(1+2*pi,ccoef,3)

Density function with AB coefficients

Description

Density function expressed in terms of the ab parameters at theta

Usage

nntsABDensitySymmetric(cpars = c(0, 0), M = 0, theta)

Arguments

Value

Returns the density function in terms of the ab coefficients evaluated at theta

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

AB coefficients

Description

This function transforms the complex parameters c to the parameters ab for a reparameterization of the density function

Usage

nntsABcoefficients(cpars = 1/sqrt(2 * pi), M = 0)

Arguments

Value

The function returns the parameters ab associated with the parameters cpars and returns a vector of real numbers of size 2*M, where the first M elements are the a_k, k=1,...,M, and the next M elements are the b_k, k=1,...,M

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

Examples

#random generation of c parameters
ccoef<-nntsrandominitial(3)
ccoef
ab<-nntsABcoefficients(ccoef,3)
ab

AB coefficients

Description

This function transforms the complex parameters c to the parameters ab for a reparameterization of the density function

Usage

nntsABcoefficientsSymmetric(cpars = c(0,0), M = 0)

Arguments

Value

The function returns the parameters ab associated with the pararameters cpars

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

NNTS density function for a variable defined in the interval [0,1)

Description

Computes the density function at theta for a variable defined in the interval [0,1))

Usage

nntsDensityInterval0to1(S, cpars = 1/sqrt(2 * pi), M = 0)

Arguments

Details

This function computes the density function of a variable S (S in the interval [0,1)). If theta is defined in radians (theta in the interval [0,2*pi)), the relation between S and theta is theta=2*pi*S.

Value

Value of the density function at each component of S

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

Examples

ccoef<-nntsrandominitial(3)
nntsDensityInterval0to1(c(.8,1.8),ccoef,3)

NNTS Distribution function

Description

Cumulative distribution function in terms of the c parameters at theta, measured in radians [0,2*pi).

Usage

nntsDistribution(theta, cpars = 1/sqrt(2 * pi), M = 0)

Arguments

Value

The function returns the value of the distribution function evaluated at each component of theta

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

Examples

ccoef<-nntsrandominitial(3)
nntsDistribution(c(0,pi/2,pi,2*pi-0.00000001,2*pi),ccoef,3)

NNTS distribution function for the incidence data defined in the interval [0,1)

Description

Computes the distribution function at theta for the incidence data (number of observed values in certain intervals defined in the interval [0,1))

Usage

nntsDistributioninterval0to1(theta, cpars = 1/sqrt(2 * pi), M = 0)

Arguments

Value

The function returns the value of the distribution function at theta

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

Examples

cpars<-nntsrandominitial(2)
nntsDistributioninterval0to1(pi, cpars, 2)

NNTS distribution function for data defined in the interval [0,2*pi)

Description

Computes the distribution function for the data at theta

Usage

nntsDistributioninterval0to2pi(theta, cpars = 1/sqrt(2 * pi), M = 0)

Arguments

Value

The function returns the value of the distribution function at theta

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

Examples

cpars<-nntsrandominitial(3)
nntsDistributioninterval0to2pi(0, cpars, 3)
nntsDistributioninterval0to2pi(pi, cpars, 3)
nntsDistributioninterval0to2pi(2*pi-0.00000001, cpars, 3)
nntsDistributioninterval0to2pi(2*pi, cpars, 3)
nntsDistributioninterval0to2pi(3*pi, cpars, 3) 

Symmetric NNTS density function

Description

Density function for the Symmetric NNTS

Usage

nntsSymmetricDensity(cpars = c(0, 0), M = 0, theta)

Arguments

Value

The function returns the density function evaluated at theta

Note

The default values provide the uniform circular density

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

References

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

NNTS density function

Description

Density function for the NNTS model

Usage

nntsdensity(data, cpars = 1/sqrt(2 * pi), M = 0)

Arguments

Value

The function returns the density function evaluated at each point in data

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

References

Fernandez-Duran, J.J. (2004). Circular Distributions Based on Nonnegative Trigonometric Sums, Biometrics, 60(2), 499-503.

Juan Jose Fernandez-Duran, Maria Mercedes Gregorio-Dominguez (2016). CircNNTSR: An R Package for the Statistical Analysis of Circular, Multivariate Circular, and Spherical Data Using Nonnegative Trigonometric Sums. Journal of Statistical Software, 70(6), 1-19. doi:10.18637/jss.v070.i06

Examples

ccoef<-nntsrandominitial(3)
nntsdensity(1,ccoef,3)
nntsdensity(1+pi,ccoef,3)
nntsdensity(c(1,1+pi),ccoef,3)

NNTS Symmetric Coefficient estimation

Description

Computes the maximum likelihood estimates of the symmetric NNTS parameters

Usage

nntsestimationSymmetric(M = 0, data, maxit = 500)

Arguments

Value

Note

For the maximization of the loglikelihood function the function constrOptim from the package stats is used

Author(s)

Juan Jose Fernandez-Duran y Maria Mercedes Gregorio-Dominguez

References

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

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)

NNTS log-likelihood function

Description

Computes the log-likelihood function with NNTS density for data

Usage

nntsloglik(data, cpars = 1/sqrt(2 * pi), M = 0)

Arguments

Value

The function returns the value of the log-likelihood function for the data

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

References

Fernandez-Duran, J.J. (2004). Circular Distributions Based on Nonnegative Trigonometric Sums, Biometrics, 60(2), 499-503.

Examples

a<-c(runif(10,3*pi/2,2*pi-0.00000001),runif(10,pi/2,pi-0.00000001))
est<-nntsmanifoldnewtonestimation(a,2)
ccoef<-est$cestimates[,2]
nntsloglik(a,ccoef,2)

NNTS log-likelihood function for the incidence data defined in the interval [0,1)

Description

Computes the log-likelihood function for incidence data (number of observed values in certain intervals defined in the interval [0,1))

Usage

nntsloglikInterval0to1(data, cutpoints, cpars = 1/sqrt(2 * pi), M = 0)

Arguments

Value

The function returns the value of the log-likelihood function for data

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

Examples

data<-c(1,2,6,4,1)
cutpoints<-c(0,0.2,0.4,0.6,0.8,0.9999999)
cpars<-nntsrandominitial(1)
nntsloglikInterval0to1(data, cutpoints, cpars, 1)

NNTS log-likelihood function for the incidence data defined in the interval [0,2*pi)

Description

Computes the log-likelihood function for incidence data (number of observed values in certain intervals defined in the interval [0,2*pi))

Usage

nntsloglikInterval0to2pi(data, cutpoints, cpars = 1/sqrt(2 * pi), M = 0)

Arguments

Value

The function returns the value of the log-likelihood function for the data

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

Examples

data<-c(2,3,6,4)
cutpoints<-c(0,pi/2,pi,3*pi/2,2*pi-0.00000001)
est<-nntsmanifoldnewtonestimationinterval0to2pi(data,cutpoints,M=1)
cpars<-est$cestimates[,2]
nntsloglikInterval0to2pi(data,cutpoints,cpars,M=1)

NNTS symmetric log-likelihood function

Description

Computes the log-likelihood function with NNTS symmetric density for the data

Usage

nntsloglikSymmetric(cpars = c(0, 0), M = 0, data)

Arguments

Value

The function returns the value of the log-likelihood function for the data

Note

The default values provide the Uniform circular log-likelihood for the data

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

References

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

nntsloglikSymmetric(c(.01,.02,2),2,t(c(pi,pi/2,2*pi,pi)))

Parameter estimation for NNTS distributions

Description

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

Usage

nntsmanifoldnewtonestimation(data, M=0, iter=1000, initialpoint = FALSE, cinitial)

Arguments

Value

Author(s)

Juan Jose Fernandez-Duran y Maria Mercedes Gregorio-Dominguez

References

Fernandez-Duran, J.J., Gregorio-Dominguez, M.M. (2010). Maximum Likelihood Estimation of Nonnegative Trigonometric Sums Models by Using a Newton-like Algorithm on Manifolds, Working Paper, Department of Statistics, ITAM, DE-C10.8

Examples

set.seed(200)
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 200 iterations
nntsmanifoldnewtonestimation(a,2,iter=200)

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
nntsmanifoldnewtonestimation(Turtles_radians,3,iter=200)

Maximum likelihood estimates of the NNTS parameters

Description

Computes the maximum likelihood estimates of the NNTS parameters, using a Newton algorithm on the hypersphere with the option to specify a minimum value of the norm of the gradient error to stop the algorithm

Usage

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

Arguments

Value

Author(s)

Juan Jose Fernandez-Duran y Maria Mercedes Gregorio-Dominguez

References

Fernandez-Duran, J.J., Gregorio-Dominguez, M.M. (2010). Maximum Likelihood Estimation of Nonnegative Trigonometric Sums Models by Using a Newton-like Algorithm on Manifolds, Working Paper, Department of Statistics, ITAM, DE-C10.8

Examples

set.seed(200)
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 gradientstop at 1e-12
nntsmanifoldnewtonestimationgradientstop(a,2,gradientstop=1e-12)
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 and gradientstop at 1e-12
nntsmanifoldnewtonestimationgradientstop(Turtles_radians,3,gradientstop=1e-12)

Parameter estimation for grouped data defined in [0,1)

Description

Parameter estimation for incidence data (number of observed values in certain intervals defined over [0,1))

Usage

nntsmanifoldnewtonestimationinterval0to1(data, cutpoints, subintervals, M = 0, iter=1000, 
initialpoint = FALSE, cinitial)

Arguments

Value

Author(s)

Juan Jose Fernandez-Duran y Maria Mercedes Gregorio-Dominguez

Examples

data<-c(1,2,4,6,1)
cutpoints<-c(0,0.2,0.4,0.6,0.8,0.999999999)
nntsmanifoldnewtonestimationinterval0to1(data, cutpoints, length(data), 1)

Parameter estimation for grouped data defined in [0,2*pi)

Description

Parameter estimation for incidence data (number of observed values in certain intervals defined over [0,2*pi))

Usage

nntsmanifoldnewtonestimationinterval0to2pi(data, cutpoints, 
subintervals,M = 0, iter=1000, initialpoint = FALSE, cinitial)

Arguments

Value

Author(s)

Juan Jose Fernandez-Duran y Maria Mercedes Gregorio-Dominguez

Examples

data<-c(1,2,6,4)
cutpoints<-c(0,pi/2,pi,3*pi/2,2*pi-0.00000001)
nntsmanifoldnewtonestimationinterval0to2pi(data, cutpoints, length(data),1)

Plots the NNTS density

Description

Plots the NNTS density

Usage

nntsplot(cpars = 1/sqrt(2 * pi), M = 0, ...)

Arguments

Examples

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,iter=200)
est
#plot the histogram
plot(Turtles_hist, freq=FALSE)
#add the estimated density to the histogram
nntsplot(est$cestimates[,2],3,add= TRUE)

Plots an NNTS density for a variable defined in the interval [0,1)

Description

Plots the NNTS density for a variable defined in the interval [0,1)

Usage

nntsplotInterval0to1(cpars = 1/sqrt(2 * pi), M = 0, ...)

Arguments

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

Examples

data<-c(1,2,4,6,2)
cutpoints<-c(0,0.2,0.4,0.6,0.8,0.9999999)
est<-nntsmanifoldnewtonestimationinterval0to1(data,cutpoints,5,1)
cpars<-est$cestimates[,2]
nntsplotInterval0to1(cpars, 1)

Plots a symmetric NNTS density function

Description

Plots the Symmetric NNTS density function

Usage

nntsplotSymmetric(cpars = c(0, 0), M = 0, ...)

Arguments

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

References

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

Initial random point

Description

This function generates a random point on the surface of the (M+1)-dimensional unit hypersphere

Usage

nntsrandominitial(M=1)

Arguments

Value

Returns a valid initial point for the estimation functions

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

Examples

nntsrandominitial(3)
nntsrandominitial(7)

Initial random point

Description

This function generates a random point on the surface of the (M+1)-dimensional unit hypersphere

Usage

nntsrandominitialSymmetric(M)

Arguments

Value

Returns a valid initial point for the estimation functions nntsestimation and nntsestimationSymmetric

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

NNTS density simulation function

Description

Simulation for the density function for the NNTS model

Usage

nntssimulation(nsim=1, cpars = 1/(2 * pi), M = 0)

Arguments

Value

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

Examples

M<-3
ccoef<-nntsrandominitial(M)
data<-nntssimulation(10,ccoef,M)
data

Computes the statistic and critical values of the circular uniformity test

Description

Computes the statistic and critical values at 10%, 5% and 1% of the circular uniformity test based on the NNTS likelihood ratio for M values from 1 to 7 and any sample size.

Usage

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

Arguments

.

Value

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

References

Fernandez-Duran and J. J. and Gregorio-Dominguez and M. M (2022). Sums of Independent Circular Random Variables and Maximum Likelihood Circular Uniformity Tests Based on Nonnegative Trigonometric Sums Distributions, arXiv preprint arXiv:2212.01416

Examples

set.seed(200)
a<-2*pi*runif(50)
#NNTS likelihood ratio circular uniformity test for M=2 and gradientstop at 1e-09
nntsuniformitytestlikelihoodratio(data=a,M=2,gradientstop=1e-09)
data(Turtles_radians)
#NNTS likelihood ratio circular uniformity test for M=5 and gradientstop at 1e-12
nntsuniformitytestlikelihoodratio(data=Turtles_radians,M=5,gradientstop=1e-09)

SNNTS density function for spherical data

Description

Density function for the SNNTS model for spherical data

Usage

snntsdensity(data, cpars = 1, M = c(0,0))

Arguments

Value

The function returns the density function evaluated for each row in the data

Note

The parameters cinitial and cestimates used by this function are the transformed parameters of the SNNTS density function, which lie on the surface of the unit hypersphere

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

References

Juan Jose Fernandez-Duran, Maria Mercedes Gregorio-Dominguez (2016). CircNNTSR: An R Package for the Statistical Analysis of Circular, Multivariate Circular, and Spherical Data Using Nonnegative Trigonometric Sums. Journal of Statistical Software, 70(6), 1-19. doi:10.18637/jss.v070.i06

Examples

data(Datab6fisher_ready)
data<-Datab6fisher_ready
M<-c(2,3)
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)

Plots a SNNTS density for spherical data

Description

Computes the points needed to plot the SNNTS density function for spherical data

Usage

snntsdensityplot(long, lat, cpars = 1, M = c(0,0))

Arguments

Value

The points needed to plot the SNNTS density function

Note

The parameters cpars used by this function are the transformed parameters of the SNNTS density function, which lie on the surface of the unit hypersphere

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

References

Fernandez-Duran J. J. y Gregorio Dominguez, M. M. (2008) Spherical Distributions Based on Nonnegative Trigonometric Sums, Working Paper, Statistics Department, ITAM, DE-C08.6

Examples

	set.seed(200)
data(Datab6fisher_ready)
data<-Datab6fisher_ready
M<-c(4,4)
cest<-snntsmanifoldnewtonestimation(data, M, iter=150)
cpars<-cest$cestimates[,3]
longitud<-seq(0,360,10)*(pi/180)
latitud<-seq(0,180,5)*(pi/180)
z<-outer(longitud,latitud,FUN="snntsdensityplot",cpars,M)
persp(longitud,latitud,z,theta=45,phi=30)
contour(longitud,latitud,z)
points(data[,1],data[,2])

SNNTS log-likelihood function for spherical data

Description

Computes the log-likelihood function with SNNTS density for spherical data

Usage

snntsloglik(data, cpars = 1, M = c(0,0))

Arguments

Value

The function returns the value of the log-likelihood function for the data

Note

The parameters cpars used by this function are the transformed parameters of the SNNTS density function, which lie on the surface of the unit hypersphere

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

References

Fernandez-Duran J. J. y Gregorio Dominguez, M. M. (2008) Spherical Distributions Based on Nonnegative Trigonometric Sums, Working Paper, Statistics Department, ITAM, DE-C08.6

Examples

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)

Parameter estimation for SNNTS distributions for spherical data

Description

Computes the maximum likelihood estimates of the SNNTS model parameters using a Newton algorithm on the hypersphere

Usage

snntsmanifoldnewtonestimation(data, M = c(0,0), iter = 1000,
initialpoint = FALSE, cinitial)

Arguments

Value

Note

The parameters cinitial and cestimates used by this function are the transformed parameters of the SNNTS density function, which lie on the surface of the unit hypersphere

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

References

Fernandez-Duran J. J. y Gregorio Dominguez, M. M. (2008) Spherical Distributions Based on Nonnegative Trigonometric Sums, Working Paper, Statistics Department, ITAM, DE-C08.6

Examples

set.seed(200)
data(Datab6fisher_ready)
data<-Datab6fisher_ready

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

cest<-snntsmanifoldnewtonestimation(data,c(4,4),100,TRUE,cpar) 
cest
cest<-snntsmanifoldnewtonestimation(data,c(1,2),100)
cest

Marginal density function for latitude of the SNNTS model for spherical data

Description

Marginal density function for latitude of the SNNTS model for spherical data

Usage

snntsmarginallatitude(data, cpars = 1, M = c(0,0))

Arguments

Value

The function returns the SNNTS marginal density function for latitude evaluated at data

Note

The parameters cpars used by this function are the transformed parameters of the SNNTS density function, which lie on the surface of the unit hypersphere

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

References

Fernandez-Duran J. J. y Gregorio Dominguez, M. M. (2008) Spherical Distributions Based on Nonnegative Trigonometric Sums, Working Paper, Statistics Department, ITAM, DE-C08.6

Examples

	set.seed(200)
data(Datab6fisher_ready)
data<-Datab6fisher_ready
M<-c(1,2)
cest<-snntsmanifoldnewtonestimation(data, M,iter=150)
lat<-snntsmarginallatitude(seq(0,pi,.1),cest$cestimates[,3],M)
plot(seq(0,pi,.1),lat,type="l")

Marginal density function for the longitude of the SNNTS model for spherical data

Description

Marginal density function for the longitude of the SNNTS model for spherical data

Usage

snntsmarginallongitude(data, cpars = 1, M = c(0,0))

Arguments

Value

The function returns the density function evaluated for the data

Note

The parameters cpars used by this function are the transformed parameters of the SNNTS density function, which lie on the surface of the unit hypersphere

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

References

Fernandez-Duran J. J. y Gregorio Dominguez, M. M. (2008) Spherical Distributions Based on Nonnegative Trigonometric Sums, Working Paper, Statistics Department, ITAM, DE-C08.6

Examples

	set.seed(200)
data(Datab6fisher_ready)
data<-Datab6fisher_ready
M<-c(1,2)
cest<-snntsmanifoldnewtonestimation(data, M,iter=150)
long<-snntsmarginallongitude(seq(0,2*pi,.1),cest$cestimates[,3],M)
plot(seq(0,2*pi,.1),long,type="l")

SNNTS density simulation function

Description

Simulation for the density function for the SNNTS model

Usage

snntssimulation(nsim=1, cpars =(1/(2*pi))^2, M = c(0,0))

Arguments

Value

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez

Examples

M<-c(2,3)
R<-length(M)
ccoef<-mnntsrandominitial(M,R)
data<-mnntssimulation(10,ccoef,M,R)
data

Auxiliary function for plotting

Description

Auxiliary function for plotting

Usage

trans3d(x,y,z,pmat)

Arguments

Value

The function returns the density function evaluated for each row in the data

Author(s)

Juan Jose Fernandez-Duran and Maria Mercedes Gregorio-Dominguez