Indices of mesh cells for spherical numerical integration

Description

Indices of mesh cells for spherical numerical integration.

Format

A data frame with 2000 rows where each one corresponds to the indices (rows) in the data set points10 that characterizes a cell.

V1

Index of the first cell vertex.

V2

Index of the second cell vertex.

V3

Index of the third cell vertex.

Examples

data(cells10)
## maybe str(cells10) ; plot(cells10) ...

Indices of mesh cells for spherical numerical integration

Description

Indices of mesh cells for spherical numerical integration.

Format

A data frame with 8000 rows where each one corresponds to the indices (rows) in the data set points20 that characterizes a cell.

V1

Index of the first cell vertex.

V2

Index of the second cell vertex.

V3

Index of the third cell vertex.

Indices of mesh cells for spherical numerical integration

Description

Indices of mesh cells for spherical numerical integration.

Format

A data frame with 32000 rows where each one corresponds to the indices (rows) in the data set points40 that characterizes a cell.

V1

Index of the first cell vertex.

V2

Index of the second cell vertex.

V3

Index of the third cell vertex.

Circular smoothing parameter for HDRs estimation

Description

This function provides the specific smoothing parameter for circular HDRs estimation proposed in Saavedra-Nieves and Crujeiras (2021).

Usage

circ.boot.bw(sample, bw = bw.CV(circular(sample),
upper=100), tau = 0.5, B = 50, upper = 1.5 * bw)

Arguments

Details

Saavedra-Nieves and Crujeiras (2021) propose a specific smoothing parameter for HDRs estimation based on the minimization of the Hausdorff distance between the boundaries of the theoretical HDR and the plug-in estimator.

Value

A numeric value corresponding to the selected smoothing parameter.

Author(s)

Paula Saavedra-Nieves and Rosa M. Crujeiras.

References

Saavedra-Nieves, P. and Crujeiras, R. M. (2021). Nonparametric estimation of directional highest density regions. Advances in Data Analysis and Classification, 1-36.

Examples

# HDR selector from a sample of size 500 of model 5 in NPCirc
library(NPCirc)
set.seed(1)
sample<- rcircmix(500, model=5)
circ.boot.bw(sample,tau=0.4,B=2)

Bootstrap samples from a circular kernel estimator

Description

This function generates bootstrap samples from a circular kernel estimator.

Usage

circ.boot.sample(sample, n, bw)

Arguments

Value

A numeric vector containing a bootstrap sample of angles in radians of size n.

Author(s)

Paula Saavedra-Nieves and Rosa M. Crujeiras.

Euclidean and Hausdorff distances between two sets of points on the unit circle

Description

This function determines the Euclidean and Hausdorff distances between two sets of points on the unit circle.

Usage

circ.distances(x, y)

Arguments

Details

If x and y corresponds to two HDRs boundaries, this function returns the Euclidean and Hausdorff distances between the HDRs frontiers, but the function computes the Euclidean and Hausdorff distance for two sets of points on the circle, no matter their nature. See Saavedra-Nieves and Crujeiras (2021) for more details on these two distances.

Value

A list with two components:

Author(s)

Paula Saavedra-Nieves and Rosa M. Crujeiras.

References

Saavedra-Nieves, P. and Crujeiras, R. M. (2021). Nonparametric estimation of directional highest density regions. Advances in Data Analysis and Classification, 1-36.

Examples

# Distances between boundaries of two plug-in HDR estimators for orientations of saltator specie
data(sandhoppers)
attach(sandhoppers)
#Orientations in October
saltatorO<-angle[(species=="salt")&(time=="afternoon")&(sex=="M")&(month=="October")]
hdr1<-circ.plugin.hdr(sample=saltatorO,tau=0.8,plot.hdrconf=FALSE)$hdr
#Orientations in April
saltatorA<-angle[(species=="salt")&(time=="afternoon")&(sex=="M")&(month=="April")]
hdr2<-circ.plugin.hdr(sample=saltatorA,tau=0.8,plot.hdrconf=FALSE)$hdr
circ.distances(hdr1,hdr2)

Computation of HDRs for a circular density and of general level sets for circular real-valued functions

Description

This function computes HDRs for a circular density and general level sets for real-valued functions defined on the unit circle.

Usage

circ.hdr(f,tau=NULL,level=NULL,plot.hdr=TRUE,col=NULL,
         lty=NULL,shrink=NULL,cex=NULL,pch=NULL)

Arguments

Details

A detailed definition of HDRs for circular and spherical densities is given in Saavedra-Nieves and Crujeiras (2021). Trapezoidal rule is used to compute the threshold of HDR when tau is provided.

Value

If tau is provided, a list with the next components:

If level is provided, a list with the next components:

Author(s)

Paula Saavedra-Nieves and Rosa M. Crujeiras.

References

Saavedra-Nieves, P. and Crujeiras, R. M. (2021). Nonparametric estimation of directional highest density regions. Advances in Data Analysis and Classification, 1-36.

Examples

# HDRs of model 11 in library NPCirc
library(NPCirc)
f1<-function(x){return(dcircmix(x,11))}
circ.hdr(f1,tau=0.2,shrink=1.5)
circ.hdr(f1,tau=0.8,shrink=1.5)

# Plug-in level set estimation for regression
# with circular (x) - linear (y) data by using
# the Nadaraya-Watson estimator
f2<-function(t){
  set.seed(1012)
  n <- 100
  x <- runif(n, 0, 2*pi)
  y <- sin(x)+0.5*rnorm(n)
  return(kern.reg.circ.lin(circular(x),y,t,bw=10,method="NW")$y)
}
circ.hdr(f2,level=.5,plot.hdr=FALSE)

Circular plug-in estimation of HDRs and confidence regions

Description

This function computes the circular plug-in estimator of HDRs and confidence regions in Saavedra-Nieves and Crujeiras (2021).

Usage

circ.plugin.hdr(sample,bw=bw.CV(circular(sample),upper=100),tau=NULL,
                tau.method="quantile",level=NULL,conf=.95,plot.hdr=TRUE,
                plot.hdrconf=TRUE,boot=FALSE,k=3,col=NULL,lty=NULL,shrink=NULL,
                lwd=NULL,pch=NULL,cex=NULL)

Arguments

Details

A detailed definition of plug-in estimators for directional HDRs is given in Saavedra-Nieves and Crujeiras (2021). The density quantile algorithm proposed in Hyndman (1996) or the numerical integration method of trapezoidal rule can be used to compute the threshold of HDR. The confidence region for the estimated HDR is calculated also following Hyndman (1996).

Value

If tau is provided, a list with the next components:

If level is provided, a list with the next components:

Author(s)

Paula Saavedra-Nieves and Rosa M. Crujeiras.

References

Hyndman, R.J. (1996). Computing and graphing highest density regions, The American Statistician, 50, 120-126.
Oliveira, M., Crujeiras. R.M. and Rodríguez-Casal, A. (2014). NPCirc: An R Package for Nonparametric Circular Methods, Journal of Statistical Software, 61, 1-26.
Saavedra-Nieves, P. and Crujeiras, R. M. (2021). Nonparametric estimation of directional highest density regions. Advances in Data Analysis and Classification, 1-36.

Examples

# Plug-in HDR for orientations of saltator specie in April and October
data(sandhoppers)
attach(sandhoppers)
#Orientations in October
saltatorO<-angle[(species=="salt")&(time=="afternoon")&(sex=="M")&(month=="October")]
circ.plugin.hdr(sample=saltatorO,tau=0.8,plot.hdrconf=FALSE)
#Orientations in April
saltatorA<-angle[(species=="salt")&(time=="afternoon")&(sex=="M")&(month=="April")]
circ.plugin.hdr(sample=saltatorA,tau=0.8,plot.hdrconf=FALSE)
#HDR confidence bands for model 5 in NPCirc package
library(NPCirc)
set.seed(1)
sample<- rcircmix(500, model=5)
circ.plugin.hdr(sample,bw=bw.CV(circular(sample),upper=100),tau=0.6)

Circular scatterplot for plug-in HDRs

Description

This function produces a circular scatterplot with points coloured according to the HDRs in which they fall.

Usage

circ.scatterplot(sample,tau=c(0.25,0.5,.75),
bw=bw.CV(circular(sample),upper=100),tau.method="quantile",
plot.density=TRUE,col=NULL,shrink=NULL,cex=NULL,lty=NULL)

Arguments

Details

A detailed definition of directional HDRs and of their plug-in estimators is given in Saavedra-Nieves and Crujeiras (2021).
Package NPCirc is used to estimate the circular density using the classical kernel density estimator. See Oliveira et al. (2014) for more details.
Moreover, the density quantile algorithm proposed in Hyndman (1996) or the trapezoidal rule can be used to compute the threshold of HDR.
The scatterplot is created colouring the sample points according to which HDR they fall.

Value

A scatterplot showing the points coloured according to which HDR they fall. Futhermore, a list where the number of components is equal to the number HDR estimated or, equivalently, to the length of tau vector. Each component contains the sample points in each HDR from the smallest value of tau to the biggest one.

Author(s)

Paula Saavedra-Nieves and Rosa M. Crujeiras.

References

Hyndman, R.J. (1996). Computing and graphing highest density regions, The American Statistician, 50, 120-126.
Oliveira, M., Crujeiras R.M. and Rodríguez-Casal, A. (2014). NPCirc: an R package for nonparametric circular methods. Journal of Statistical Software, 61(9), 1-26. https://www.jstatsoft.org/v61/i09/.
Saavedra-Nieves, P. and Crujeiras, R. M. (2021). Nonparametric estimation of directional highest density regions. Advances in Data Analysis and Classification, 1-36.

Examples

# Scatterplot for orientations of females for saltator specie
data(sandhoppers)
attach(sandhoppers)
saltatorF<-angle[(species=="salt")&(sex=="F")]
circ.scatterplot(saltatorF)
# Scatterplot for sample of size 100 of model 14 in NPCirc
library(NPCirc)
set.seed(1)
sample<- rcircmix(100, model=14)
circ.scatterplot(sample,tau=c(0.2,0.5,0.8))

Density functions for mixtures of spherical von Mises-Fisher

Description

Density functions for nine finite mixtures of spherical von Mises-Fisher allowing different numbers of modes.

Usage

dspheremix(x, model = NULL)

Arguments

Details

These nine spherical models are obtained as mixtures of von Mises distributions where the density f is given by:

f=\sum_{i=1}^I w_i K_{vM}(x;m_i;k_i), w_i\geq 0;\sum_{i=1}^I w_i=1

with K_vM denoting the von Mises-Fisher kernel density; m_i, k_i and w_i the mean, concentration and weight corresponding to each component. More details can be found in Hornik and Grun (2014) and Wood (1994). The combination of means, concentration parameters and the weights of spherical models from Saavedra-Nieves and Crujeiras (2021) are specified below:

S1: (0, 0, 1) (m); 10 (k); 1 (w).

S2: (0, 0, 1), (0, 0, -1) (m); 1, 1 (k); 1/2, 1/2 (w).

S3: (0, 0, 1), (0, 0, -1) (m); 10, 1 (k); 1/2, 1/2 (w).

S4: (0, 0, 1); (0, 1/\sqrt2, 1/\sqrt2) (m); 10, 10 (k); 1/2, 1/2 (w).

S5: (0, 0, 1); (0, 1/\sqrt2, 1/\sqrt2) (m); 10, 10 (k); 2/5, 3/5 (w).

S6: (0, 0, 1); (0, 1/\sqrt2, 1/\sqrt2 ) (m); 10, 5 (k); 1/5, 4/5 (w).

S7: (0, 0, 1), (0, 1, 0), (1, 0, 0) (m); 5, 5, 5 (k); 1/3, 1/3, 1/3 (w).

S8: (0, 0, 1), (0, 1, 0), (1, 0, 0) (m); 5, 5, 5 (k); 2/3, 1/6, 1/6 (w).

S9: (0, 0, 1); (0, 1/\sqrt 2, 1/\sqrt 2), (0, 1, 0) (m); 10, 10, 10 (k); 1/3, 1/3, 1/3 (w).

Value

A numeric vector of density evaluated on x.

Author(s)

Paula Saavedra-Nieves and Rosa M. Crujeiras.

References

Hornik, K. and Grun, B. (2014). movMF: an R package for fitting mixtures of von Mises-Fisher distributions. Journal of Statistical Software, 58(10), 1-31.
Saavedra-Nieves, P. and Crujeiras, R. M. (2021). Nonparametric estimation of directional highest density regions. Advances in Data Analysis and Classification, 1-36.
Wood, A. T. (1994). Simulation of the von Mises Fisher distribution. Communications in Statistics-Simulation and Computation, 23(1), 157-164.

Examples

# Density function evaluation from model S1
data <- rbind(c(1,0,0),c(0,1,0),c(0,0,1))
dspheremix(data, model=1)

Earthquakes on Earth between October 2004 and April 2020

Description

Geographical coordinates (latitude and longitude) of earthquakes of magnitude greater than or equal to 2.5 degrees.

Usage

data("earthquakes")

Format

A data frame with 272 observations on the following 2 variables:

Latitude

A numeric vector containing the latitude coordinates.

Longitude

A numeric vector containing the longitude coordinates.

Details

To map this dataset on the unit sphere, function euclid in package Directional can be used.

Source

European-Mediterranean Seismological Centre, https://www.emsc-csem.org.

Examples

if (requireNamespace("ggplot2", quietly = TRUE)) {
library(ggplot2)
}
if (requireNamespace("maps", quietly = TRUE)) {
library(maps)
}
if (requireNamespace("mapproj", quietly = TRUE)) {
library(mapproj)
}
data(earthquakes)
world <- map_data("world")
g.earthquakes <- ggplot() +
  geom_map(data = world, map = world,
           mapping = aes(map_id = region),
           color = "grey90", fill = "grey80") +
  geom_point(data = earthquakes,
            mapping = aes(x = Longitude, y = Latitude),
            color = "red",alpha=.2,size=.75,stroke=0) +
    scale_y_continuous(breaks = NULL, limits = c(-90, 90)) +
  scale_x_continuous(breaks = NULL, limits = c(-180, 180)) +
  coord_map("mercator")
g.earthquakes

Calculation of the boundaries of HDRs in the circular setting

Description

This function calculates the boundaries of HDRs in the circular setting.

Usage

find.circ.hdr(x, f, level)

Arguments

Value

A numeric vector of angles in radians corresponding to the frontiers of the HDR.

Author(s)

Paula Saavedra-Nieves and Rosa M. Crujeiras.

Checking if a value is integer

Description

This function checks if a value is integer.

Usage

is.wholenumber(x, tol = .Machine$double.eps^0.5)

Arguments

Value

Logical value: TRUE if x is integer and FALSE if x is not integer.

Author(s)

Paula Saavedra-Nieves and Rosa M. Crujeiras.

Coordinates of mesh vertices for spherical numerical integration

Description

Coordinates of mesh vertices for spherical numerical integration.

Format

A data frame with 1002 rows where each one corresponds to the coordinates of vertices that characterize a cell.

V1

Coordinate of the first cell vertex.

V2

Coordinate of the second cell vertex.

V3

Coordinate of the third cell vertex.

Coordinates of mesh vertices for spherical numerical integration

Description

Coordinates of mesh vertices for spherical numerical integration.

Format

A data frame with 4002 rows where each one corresponds to the coordinates of vertices that characterize a cell.

V1

Coordinate of the first cell vertex.

V2

Coordinate of the second cell vertex.

V3

Coordinate of the third cell vertex.

Coordinates of mesh vertices for spherical numerical integration

Description

Coordinates of mesh vertices for spherical numerical integration.

Format

A data frame with 16002 rows where each one corresponds to the coordinates of vertices that characterize a cell.

V1

Coordinate of the first cell vertex.

V2

Coordinate of the second cell vertex.

V3

Coordinate of the third cell vertex.

Random generation functions for mixtures of spherical von Mises-Fisher

Description

Random generation functions for nine finite mixtures of spherical von Mises-Fisher allowing different numbers of modes.

Usage

rspheremix(n, model = NULL)

Arguments

Details

These nine spherical models are obtained as mixtures of von Mises distributions where the density f is given by:

f=\sum_{i=1}^I w_i K_{vM}(x;m_i;k_i), w_i\geq 0;\sum_{i=1}^I w_i=1

with K_vM denoting the von Mises-Fisher kernel density; m_i, k_i and w_i the mean, concentration and weight corresponding to each component. More details can be found in Hornik and Grun (2014) and Wood (1994). The combination of means, concentration parameters and the weights of spherical models from Saavedra-Nieves and Crujeiras (2021) are specified below:

S1: (0, 0, 1) (m); 10 (k); 1 (w).

S2: (0, 0, 1), (0, 0, -1) (m); 1, 1 (k); 1/2, 1/2 (w).

S3: (0, 0, 1), (0, 0, -1) (m); 10, 1 (k); 1/2, 1/2 (w).

S4: (0, 0, 1); (0, 1/\sqrt2, 1/\sqrt2) (m); 10, 10 (k); 1/2, 1/2 (w).

S5: (0, 0, 1); (0, 1/\sqrt2, 1/\sqrt2) (m); 10, 10 (k); 2/5, 3/5 (w).

S6: (0, 0, 1); (0, 1/\sqrt2, 1/\sqrt2 ) (m); 10, 5 (k); 1/5, 4/5 (w).

S7: (0, 0, 1), (0, 1, 0), (1, 0, 0) (m); 5, 5, 5 (k); 1/3, 1/3, 1/3 (w).

S8: (0, 0, 1), (0, 1, 0), (1, 0, 0) (m); 5, 5, 5 (k); 2/3, 1/6, 1/6 (w).

S9: (0, 0, 1); (0, 1/\sqrt 2, 1/\sqrt 2), (0, 1, 0) (m); 10, 10, 10 (k); 1/3, 1/3, 1/3 (w).

Value

A matrix with n unit length rows representing the generated values from a finite mixture of spherical von Mises-Fisher.

Author(s)

Paula Saavedra-Nieves and Rosa M. Crujeiras.

References

Hornik, K. and Grun, B. (2014). movMF: an R package for fitting mixtures of von Mises-Fisher distributions. Journal of Statistical Software, 58(10), 1-31.
Saavedra-Nieves, P. and Crujeiras, R. M. (2021). Nonparametric estimation of directional highest density regions. Advances in Data Analysis and Classification, 1-36.
Wood, A. T. (1994). Simulation of the von Mises Fisher distribution. Communications in Statistics-Simulation and Computation, 23(1), 157-164.

Examples

# Random generation from model 1 in library HDiR
data <- rspheremix(500, model=1)
library(Directional)
sphereplot(data)

Random generation of a sample on a d-dimensional sphere of radius r

Description

This function generates a random sample on the unit sphere.

Usage

runif_on_sphere(n, d, r = 1)

Arguments

Value

A matrix of n rows and d columns containing the sample points.

Author(s)

Paula Saavedra-Nieves and Rosa M. Crujeiras.

Behavioral plasticity of Talitrus saltator and Talorchestia brito

Description

Orientation measured under natural conditions and other variables of interest for analyzing the behavioral plasticity of two sympatric sandhoppers species, Talitrus saltator and Talorchestia brito. The experiment was carried out on the exposed nontidal sand of Zouara beach located in the Tunisian northwestern coast. More details can be found in Marchetti and Scapini (2003) or Scapini et al. (2002).

Usage

data("sandhoppers")

Format

A data frame with 1828 observations on the following 12 variables.

angle

Numeric vector containing the orientation angles in radians between 0 and 2\pi.

date

A factor where each level indicates the date when angles were measured.

month

A factor with two levels indicating the month when angles were measured. Experiments were performed in two different periods, April and October, which were chosen for the abundance of the populations, as well as for their non-extreme and changing climatic conditions.

time

A factor with levels afternoon, morning and noon.

azim

A numeric vector indicating the sun azimuth. The sun position was confounded with the time of the day (morning: 100-150, noon: az=151-210 and afternoon: az=211-260 experiments).

hour

A factor with hours when angles were measured.

species

A factor with three levels (brito, salt, ND) indicating the specie (brito, saltator, not determined).

sex

A factor with three levels (F, M, J) indicating the sex (female, male, J).

temp

A numeric vector indicating the temperature (degrees centigrade).

humid

A numeric vector indicating the air relative humidity (%).

land

A factor with two levels (no, yes) indicating landscape view was either permitted or screened.

trap

A numeric vector containing the traps identifier used for capturing the sandhoppers.

Details

Authors thank Prof. Felicita Scapini for providing the sandhoppers data (collected under the support of the European Project ERB ICI8-CT98-0270).

References

Marchetti, G. M. and Scapini, F., Use of multiple regression models in the study of sandhopper orientation under natural conditions, Estuarine, Coastal and Shelf Science, 58, 207-215 (2003).
Scapini, F., Aloia, A., Bouslama, M. F., Chelazzi, L., Colombini, I., ElGtari, M., Fallaci, M. and Marchetti, G. M. Multiple regression analysis of the sources of variation in orientation of two sympatric sandhoppers, Talitrus saltator and Talorchestia brito, from an exposed Mediterranean beach, Behavioral Ecology and Sociobiology, 51(5), 403-414 (2002).

Examples

data(sandhoppers)
attach(sandhoppers)
library(NPCirc)
saltator=circular(angle[(species=="salt")],type="angles",units="radians")
brito=circular(angle[(species=="brito")],type="angles",units="radians")
library(NPCirc)
oldpar<-par(mfrow=c(1,2))
plot(saltator)
plot(brito)
par(oldpar)

Spherical smoothing parameter for HDRs estimation

Description

This function provides the specific smoothing parameter for spherical HDRs estimation proposed in Saavedra-Nieves and Crujeiras (2021).

Usage

sphere.boot.bw(sample,bw="none",tau=0.5,ngrid=500,
               B=50,nborder=500,upper=NULL)

Arguments

Details

Saavedra-Nieves and Crujeiras (2021) propose a specific smoothing parameter for HDRs estimation based on the minimization of the Hausdorff distance between the boundaries of the theoretical HDR and the plug-in estimator.

Value

A numeric value corresponding to the selected smoothing parameter.

Author(s)

Paula Saavedra-Nieves and Rosa M. Crujeiras.

References

García-Portugués, E. (2013). Exact risk improvement of bandwidth selectors for kernel density estimation with directional data. Electronic Journal of Statistics, 7, 1655-1685.
Saavedra-Nieves, P. and Crujeiras, R. M. (2021). Nonparametric estimation of directional highest density regions. Advances in Data Analysis and Classification, 1-36.

Examples

# HDR selector from a sample of size 1000 of model 4 in library HDiR
set.seed(1)
sample=rspheremix(500,model=4)
sphere.boot.bw(sample,tau=0.8,B=2)

Bootstrap samples from a spherical kernel estimator

Description

This function generates bootstrap samples from a spherical kernel estimator.

Usage

sphere.boot.sample(sample, n, bw)

Arguments

Value

A matrix containing a bootstrap sample on the unit sphere.

Author(s)

Paula Saavedra-Nieves and Rosa M. Crujeiras.

Euclidean and Hausdorff distances between two sets of points on the unit sphere

Description

This function determines the Euclidean and Hausdorff distances between two sets of points on the unit sphere.

Usage

sphere.distances(x, y)

Arguments

Details

If x and y correspond to two HDRs boundaries, this function returns the Euclidean and Hausdorff distances between the HDR frontiers, but the function computes the Euclidean and Hausdorff distance for two sets of points on the sphere, no matter their nature. See Saavedra-Nieves and Crujeiras (2021) for more details on these two distances.

Value

A list with two components:

Author(s)

Paula Saavedra-Nieves and Rosa M. Crujeiras.

References

Saavedra-Nieves, P. and Crujeiras, R. M. (2021). Nonparametric estimation of directional highest density regions. Advances in Data Analysis and Classification, 1-36.

Examples

# Distances between boundaries of two plug-in HDR estimators for spherical model 9 in HDiR package
set.seed(1)
sample=rspheremix(1000, model =9)
x<-sphere.plugin.hdr(sample,tau=0.8,plot.hdr=FALSE)$hdr
y<-sphere.plugin.hdr(sample,tau=0.5,plot.hdr=FALSE)$hdr
sphere.distances(x, y)

Computation of HDRs for a spherical density and of general level sets for spherical real-valued functions

Description

This function computes HDRs of a spherical density and general level sets for real-valued functions defined on the unit sphere.

Usage

sphere.hdr(f,tau=NULL,level=NULL,nborder=1000,tol=0.1,
           mesh=40,deg=6,plot.hdr=TRUE,col=NULL)

Arguments

Details

A detailed definition of directional HDRs for a density is given in Saavedra-Nieves and Crujeiras (2021). Note that numerical integration on the sphere is used to compute the threshold of HDR when tau is provided.

Value

If tau is provided, a list with the next components:

If level is provided, a list with the next components:

Author(s)

Paula Saavedra-Nieves, Rosa M. Crujeiras and Andrés Prieto.

References

Saavedra-Nieves, P. and Crujeiras, R. M. (2021). Nonparametric estimation of directional highest density regions. Advances in Data Analysis and Classification, 1-36.

Examples

#HDR of model 8 in library HDiR
f1<-function(x){return(dspheremix(x,model=8))}
sphere.hdr(f1,tau=0.5,mesh=20,deg=3)


# Density level set plug-in reconstruction from a sample
# of size 500 (model 8) by using a kernel density
# estimator with uniform kernel
library(DirStats)
f2<-function(x){
  set.seed(1)
  sample<-rspheremix(500, model = 3)
  return(kde_dir(x, data = sample, h = 0.4,
  L = function(x) dunif(x)))
}
sphere.hdr(f2,level=0.3)

Numerical integration on the unit sphere

Description

This function calculates the integral of a function f on the unit sphere.

Usage

sphere.integration(f, mesh = 40, deg = 3)

Arguments

Value

A numeric value of the integral.

Author(s)

Andrés Prieto, Rosa M. Crujeiras and Paula Saavedra-Nieves.

Spherical plug-in estimation of HDRs

Description

This function computes the spherical plug-in estimator of HDRs.

Usage

sphere.plugin.hdr(sample,bw="none",ngrid=500,
                  tau=NULL,level=NULL,nborder=1000,tol=0.01,
                  mesh=40,deg=3,plot.hdr=TRUE, col=NULL)

Arguments

Details

A detailed definition of plug-in estimators for directional HDRs is given in Saavedra-Nieves and Crujeiras (2021). Moreover, the density quantile algorithm proposed in Hyndman (1996) is used to compute the threshold of HDR.

Value

If tau is provided, a list with the next components:

If level is provided, a list with the next components:

Author(s)

Paula Saavedra-Nieves and Rosa M. Crujeiras.

References

García-Portugués, E. (2013). Exact risk improvement of bandwidth selectors for kernel density estimation with directional data. Electronic Journal of Statistics, 7, 1655-1685.
Hyndman, R.J. (1996). Computing and graphing highest density regions, The American Statistician, 50, 120-126.
Saavedra-Nieves, P. and Crujeiras, R. M. (2021). Nonparametric estimation of directional highest density regions. Advances in Data Analysis and Classification, 1-36.

Examples

# Plug-in HDR estimator for spherical model 9 in HDiR package
set.seed(1)
sample=rspheremix(1000, model =9)
sphere.plugin.hdr(sample,tau=0.8,col="red")

#Plug-in HDR estimator for data on earthquakes on Earth
if (requireNamespace("ggplot2", quietly = TRUE)) {
library(ggplot2)
}
if (requireNamespace("maps", quietly = TRUE)) {
library(maps)
}
if (requireNamespace("mapproj", quietly = TRUE)) {
library(mapproj)
}
data(earthquakes)
library(Directional)
hdr08<-as.data.frame(euclid.inv(sphere.plugin.hdr(euclid(earthquakes),tau=0.8,
plot.hdr=FALSE)$hdr))
world <- map_data("world")
g.earthquakes <- ggplot() +
  geom_map(data = world, map = world,
           mapping = aes(map_id = region),
           color = "grey90", fill = "grey80") +
  geom_point(data = earthquakes,
            mapping = aes(x = Longitude, y = Latitude),
           color = "red",alpha=.2,size=.75,stroke=0) +
  geom_point(data = hdr08,
            mapping = aes(x = Long, y = Lat),
            color = "darkblue", size = 1) +
  scale_y_continuous(breaks = NULL, limits = c(-90, 90)) +
 scale_x_continuous(breaks = NULL, limits = c(-180, 180)) +
  coord_map("mercator")
g.earthquakes

Spherical scatterplot for plug-in HDRs

Description

This function produces a spherical scatterplot with points coloured according to the HDRs in which they fall.

Usage

sphere.scatterplot(sample,tau=c(0.25,0.5,.75),bw="none",
                   ngrid=500,nborder=1000,tol=0.1, col=NULL)

Arguments

Details

A detailed definition of directional HDRs and of their plug-in estimators is given in Saavedra-Nieves and Crujeiras (2021).
Package Directional is used to compute tha von Mises-Fisher kernel density estimate.
The density quantile algorithm proposed in Hyndman (1996) is used to calculate the threshold of HDR.
The scatterplot is created colouring the sample points according to which HDR they fall.

Value

A scatterplot showing the points coloured according to which HDR they fall. Futhermore, a list where the number of components is equal to the number HDR estimated or, equivalently, to the length of tau vector. Each component contains the sample points in each HDR from the smallest value of tau to the biggest one.

Author(s)

Paula Saavedra-Nieves and Rosa M. Crujeiras.

References

García-Portugués, E. (2013). Exact risk improvement of bandwidth selectors for kernel density estimation with directional data. Electronic Journal of Statistics, 7, 1655-1685.
Tsagris, M., Athineou, G., Sajib, A., Tsagris, M. M. and Imports, M. A. S. S. (2016). Package Directional. https://cran.r-project.org/package=Directional.
Saavedra-Nieves, P. and Crujeiras, R. M. (2021). Nonparametric estimation of directional highest density regions. Advances in Data Analysis and Classification, 1-36.

Examples

# Scatterplot of model 4 in library HDiR
set.seed(1)
sample=rspheremix(1000,model=4)
sphere.scatterplot(sample,tau=c(.2,.5,.8))
#Scatterplot of model 9 in library HDiR
set.seed(1)
sample=rspheremix(1000,model=9)
sphere.scatterplot(sample)

Trapezoidal rule for numerical integration over a threshold

Description

This functions implements the trapezoidal rule for numerical integration of a function f over a threshold.

Usage

trapezoidal.rule(f, step, y)

Arguments

Value

A numeric value of the integral.

Author(s)

Paula Saavedra-Nieves and Rosa M. Crujeiras.

Von Mises-Fisher kernel density estimator with numerical smoothing parameter

Description

This function calculates the Von Mises-Fisher kernel density estimator is a slight modification of function vmf.kerncontour in Directional package. In this new version, it is possible to provide a numerical value for the bandwidth.

Usage

vmf.kerncontour2(u, h, full = FALSE, ngrid = 100)

Arguments

Value

A list with kernel density estimation. For more details, see function vmf.kerncontour in Directional package.

Author(s)

Paula Saavedra-Nieves and Rosa M. Crujeiras.