Package 'circular'

Description

Operators act on vectors and matrices to extract or replace subsets, methods for Circular Data.

Usage

## S3 method for class 'circular'
x[i, ...]

Arguments

Author(s)

Claudio Agostinelli

Examples

x <- circular(matrix(rwrappednormal(n=100, mu=circular(0)), nrow=5))
dim(x)
x[1,]
x[,1]
x[,1, drop=FALSE]

Description

Evaluates the first and zeroth order Bessel functions of the first kind at a specified non-negative real number, and returns the ratio.

Usage

A1(kappa)

Arguments

Details

The function uses besselI.

Value

If I1(kappa) is the first order Bessel function and I0(kappa) is the zeroth order Bessel function, then A1(kappa) returns I1(kappa)/I0(kappa).

Author(s)

Claudio Agostinelli

See Also

besselI, A1inv.

Description

Evaluates the first derivative of the Ratio of First and Zeroth Order Bessel Functions

Usage

A1FirstDerivative(kappa)

Arguments

Details

The formula (3.48) of Fisher (1993), pag. 52 is implemented. The function uses A1 and besselI.

Value

The value of the first derivative of A1 function in the point kappa.

Author(s)

Claudio Agostinelli and Alessandro Gagliardi.

References

N.I. Fisher (1993) Statistical Analysis of Circular Data, Cambridge University Press.

See Also

A1, besselI, A1inv.

Description

Inverse function of the ratio of the first and zeroth order Bessel functions of the first kind. This function is used to compute the maximum likelihood estimate of the concentration parameter of a von Mises distribution.

Usage

A1inv(x)

Arguments

Details

A1inv(0) = 0 and A1inv(1) = Inf. This function is useful in estimating the concentration parameter of data from a von Mises distribution. Our function use the results in Best and Fisher (1981). Tables use tabulated values by Gumbel, Greenwood and Durand (1953).

Value

Returns the value k, such that A1inv(x) = k, i.e. A1(k) = x.

Author(s)

Claudio Agostinelli

References

BEST, D.J. and FISHER, N.I. 1981. The bias of the maximum likelihood estimators for the von Mises-Fisher concentration parameters. Communications in Statistics, 10, 493-502.

GUMBEL, E.J., GREENWOOD, J.A. AND DURAND, D. 1953. The circular normal distribution: theory and tables. J. Amer. Statis. Assoc., 48, 131-152.

See Also

mle.vonmises, A1, besselI.

Examples

#Generate data from a von Mises distribution
data <- rvonmises(n=50, mu=circular(pi), kappa=4)
#Estimate the concentration parameter
s <- sum(sin(data))
c <- sum(cos(data))
mean.dir <- atan2(s, c)
kappa <- A1inv(mean(cos(data - mean.dir)))

Description

Evaluates the second derivative of the second derivative of the Ratio of First and Zeroth Order Bessel Functions.

Usage

A1SecondDerivative(kappa)

Arguments

Details

Formula (3.49) of Fisher (1993), pag. 52 is implemented. The function uses A1, A1FirstDerivative and besselI.

Value

The value of the second derivative of A1 function in the point kappa.

Author(s)

Claudio Agostinelli and Alessandro Gagliardi.

References

N.I. Fisher (1993) Statistical Analysis of Circular Data, Cambridge University Press.

See Also

A1, A1FirstDerivative, besselI, A1inv.

Description

Returns the square root of twice one minus the mean resultant length divided by the sample size of a vector of circular data.

Usage

angular.deviation(x, na.rm = FALSE)

Arguments

Value

Returns the square root of twice one minus the mean resultant length divided by the sample size.

Author(s)

Claudio Agostinelli

References

Batschelet, E. (1981) Circular Statistics in Biology. Academic Press, London.

Jammalamadaka, S. Rao and SenGupta, A. (2001). Topics in Circular Statistics, Section 1.3, World Scientific Press, Singapore.

Zar, J.H. (2010) Biostatistical Analysis. Fifth edition. Pearson Educational International.

See Also

sd.circular, angular.variance, mean.circular, rho.circular and summary.circular.

Examples

x <- rvonmises(n=100, mu=circular(0), kappa=1)
angular.deviation(x)

Description

Returns twice one minus the mean resultant length divided by the sample size of a vector of circular data.

Usage

angular.variance(x, na.rm = FALSE)

Arguments

Value

Returns twice one minus the mean resultant length divided by the sample size.

Author(s)

Claudio Agostinelli

References

Batschelet, E. (1981) Circular Statistics in Biology. Academic Press, London.

Jammalamadaka, S. Rao and SenGupta, A. (2001). Topics in Circular Statistics, Section 1.3, World Scientific Press, Singapore.

Zar, J.H. (2010) Biostatistical Analysis. Fifth edition. Pearson Educational International.

See Also

var.circular, angular.deviation, mean.circular, rho.circular and summary.circular.

Examples

x <- rvonmises(n=100, mu=circular(0), kappa=1)
angular.variance(x)

Description

One Critrion Analysis of Variance for circular data

Usage

aov.circular(x, group, kappa = NULL,
    method = c("F.test", "LRT"), F.mod = TRUE, control.circular=list())
## S3 method for class 'aov.circular'
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

Details

The samples are assumed to have been drawn from von Mises populations with equal concentration parameter, kappa. The null hypothesis being tested is that all populations also have the same mean direction.

If method is "F.test" a high concentration F-test makes use of a decomposition of total sample variation into between groups and within groups variation, analogous to the one-way classification analysis of variance for linear data. Stephens (1972) presented an improved modification to the F-test derived from this decomposition. This is implemented when F.mod is TRUE.

A likelihood ratio test is performed when method is "LRT". This function uses the test statistic presented by Cordeiro, Paula, and Botter (1994) which has an approximate chi-squared distribution. If the common concentration parameter is known, it can be specified and used in the computation of the test statistic. Otherwise, the maximum likelihood estimate of the common concentration parameter is used.

Value

An object of class aov.circular with the following components:

If the method is "F.test" then the object contains also:

Author(s)

Claudio Agostinelli and Ulric Lund

References

Cordeiro, G., Paula, G. and Botter, D. (1994). Improved likelihood ratio tests for dispersion models. International Statistical Review, 62, 257-274.

Jammalamadaka, S. Rao and SenGupta, A. (2001). Topics in Circular Statistics, Section 5.3, World Scientific Press, Singapore.

Mardia, K. and Jupp, P. (1999). Directional Statistics, Section 7.4, John Wiley and Sons, England.

Stephens, M. (1972). Multi-sample tests for the von Mises distribution. Technical Report 190, Department of Statistics, Stanford University.

Examples

x <- c(rvonmises(50, circular(0), 1), rvonmises(100, circular(pi/3), 10))
group <- c(rep(0, 50), rep(1, 100))

aov.circular(x, group)
aov.circular(x, group, method="LRT")

Description

Draw arrows in a circular plot.

Usage

arrows.circular(x, y = NULL, x0 = 0, y0 = 0, na.rm = FALSE, 
  shrink = 1, plot.info = NULL, zero = NULL, rotation = NULL, ...)

Arguments

Note

The function call arrows and it is not a method of arrows.

Author(s)

Claudio Agostinelli

See Also

arrows

Examples

plot(rvonmises(10, circular(0), kappa=1))
  arrows.circular(rvonmises(10, circular(0), kappa=1))
  arrows.circular(rvonmises(10, circular(0), kappa=1), y=runif(10), col=2)
  arrows.circular(rvonmises(10, circular(0), kappa=1), y=runif(10), 
    x0=runif(10, -1, 1), y0=runif(10, -1, 1), col=3)

Description

This function is a method of as.data.frame for a circular object.

Usage

## S3 method for class 'circular'
as.data.frame(x, row.names = NULL, optional = FALSE, ...)

Arguments

Author(s)

Claudio Agostinelli

Description

Density the Asymmetric Triangular circular distribution.

Usage

dasytriangular(x, rho)

Arguments

Value

dasytriangular gives the density.

Author(s)

Claudio Agostinelli

References

Mardia (1972) Statistics for Directional Data, Wiley. Pag. 52

Examples

ff <- function(x) dasytriangular(x, rho=0.3)
curve.circular(ff, shrink=1.2, join=TRUE)

Description

Density for the axial von Mises circular distribution.

Usage

daxialvonmises(x, mu, kappa, l = 2)

Arguments

Value

daxialvonmises gives the density.

Author(s)

Claudio Agostinelli

References

Jammalamadaka, S. Rao and SenGupta, A. (2001). Topics in Circular Statistics, Section 2.2.4, World Scientific Press, Singapore.

Description

Add axis to a plot of circular data points on the current graphics device.

Usage

axis.circular(at=NULL, labels=NULL,  units = NULL, template=NULL,  
          modulo=NULL, zero=NULL, rotation=NULL, tick=TRUE, lty, lwd, 
          cex, col, font, tcl=0.025, tcl.text=0.125, digits=2)

Arguments

Author(s)

Claudio Agostinelli

See Also

plot.circular and ticks.circular.

Examples

data.vm <- rvonmises(n=100, mu=circular(0), kappa=3) 
plot(data.vm, axes=FALSE, ticks=FALSE)
axis.circular(at=circular(seq(0, 11/6*pi, pi/6)), labels=c("0",
expression(frac(pi,6)), expression(paste(frac(1,3), pi)),
expression(frac(pi,2)), expression(paste(frac(2,3), pi)),
expression(paste(frac(5,6), pi)), expression(pi),
expression(paste(frac(7,6), pi)), expression(paste(frac(4,3), pi)),
expression(paste(frac(3,2), pi)), expression(paste(frac(5,3), pi)),
expression(paste(frac(11,6), pi))))

Description

Bandwidth selectors for circular kernels in density.circular.

Usage

bw.cv.mse.circular(x, lower=NULL, upper=NULL, tol = 1e-4,
  kernel = c("vonmises", "wrappednormal"), K = NULL, min.k = 10)

bw.cv.ml.circular(x, lower=NULL, upper=NULL, tol = 1e-4,
  kernel = c("vonmises", "wrappednormal"), K = NULL, min.k = 10)

bw.nrd.circular(x, lower=NULL, upper=NULL,
  kappa.est=c("ML","trigmoments"), kappa.bias=FALSE, P=3)

Arguments

Details

bw.cv.mse.circular and bw.cv.ml.circular implement cross validatory bandwidths minimizing squared–error loss and Kullback–Leibler loss, respectively. This is done by minimizing the second and third equations in section 5 of Hall, Watson and Cabrera (1987). Kullback–Leibler loss is equivalent to maximize the cross validation log–likelihood with respect to the bandwidth parameter.

bw.nrd.circular implements a rule-of-thumb for choosing the bandwidth of a von Mises kernel density estimator with underlying population von Mises. It was proposed by Taylor (2008, equation (7)) and is the circular analogue of the usual rule of thumb used for the normal distribution. The only remarkable difference between them is that Taylor's bandwidth supposes a von Mises population for the derivation of AMISE, while normal rule of thumb only introduces distribution assumption to compute the density curvature. Estimation of the spread is done by maximum likelihood. The "trigmoments" method for the estimation of kappa is implemented as follows. Let $\mu_p$ be the p-th sample trigonometric moment. Let $k_p$ be the estimates of kappa using the p-th sample trigonometric moment, as solution (using uniroot function) of the equation $A_p(k) = \frac{1}{n} \sum_{i=1}^n \cos(p x_i - \mu_p)$. We let kappa equal to $max(k_1, k_2, \cdots, k_P)$, see Taylor (2008) for further details.

Note that circular bandwidth has a different scale from linear bandwidth (see Hall, Watson and Cabrera (1987)). The behaviour of the circular bandwidth is the inverse of the linear: large values overestimate the density, whereas small values underestimate.

Value

A bandwidth on a scale suitable for the bw argument of density.circular.

Warning

Plug-in bandwidth selector bw.nrd.circular assumes that the underlying population is von Mises. If this is not true, it might lead to serious misestimations of the circular bandwidth. Example 2 below shows how this behaviour can appear with multimodality populations. In those cases, the use of kappa.est="trigmoments" could be of help.

Author(s)

Claudio Agostinelli and Eduardo Garcia–Portugues

References

P. Hall and G.S. Watson and J. Cabrera (1987). Kernel Density Estimation with Spherical Data, Biometrika, 74, 4, 751–762.

C.C Taylor (2008). Automatic bandwidth selection for circular density estimation. Computational Statistics and Data Analysis, 52, 7, 3493–3500.

See Also

density.circular

Examples

set.seed(12345)

## Example 1: von Mises ##
theta1 <- rvonmises(n=150,mu=circular(pi),kappa=2)

bw.nrd1 <- bw.nrd.circular(theta1)
bw.cv.mse1 <- bw.cv.mse.circular(theta1)
bw.cv.ml1 <- bw.cv.ml.circular(theta1)

## Linear plot
plot(function(x) dvonmises(circular(x), mu=circular(pi), kappa=2),
type="l", lwd=2, col=1, main="von Mises", xlab=expression(theta),
ylab="Density", from=0, to=2*pi)
plot(approxfun(density.circular(x=theta1, bw=bw.nrd1)), col=2, from=0, to=2*pi, add=TRUE)
plot(approxfun(density.circular(x=theta1, bw=bw.cv.mse1)), col=3,
from=0, to=2*pi, add=TRUE)
plot(approxfun(density.circular(x=theta1, bw=bw.cv.ml1)), col=4, from=0,
to=2*pi, add=TRUE)
legend("topright", legend=c("True", "Taylor", "LSCV", "MLCV"), col=1:4, lwd=2)
rug(theta1)

## Circular plot
dvonmises1 <- function(x) dvonmises(circular(x), mu=circular(pi), kappa=2)
curve.circular(dvonmises1, lwd=2, col=1, main="von Mises", xlim=c(-1.5,
1.5), ylim=c(-1.5,1.5))
lines(density.circular(x=theta1, bw=bw.nrd1), col=2)
lines(density.circular(x=theta1, bw=bw.cv.mse1), col=3)
lines(density.circular(x=theta1, bw=bw.cv.ml1), col=4)
legend("topright", legend=c("True", "Taylor", "LSCV", "MLCV"), col=1:4, lwd=2)
points(theta1)

## Example 2: mixture of von Mises ##

theta2 <- rmixedvonmises(n=150, mu1=circular(pi/2),
mu2=circular(3*pi/2), kappa1=5, kappa2=5,p=0.5)

bw.nrd2 <- bw.nrd.circular(theta2)
bw.cv.mse2 <- bw.cv.mse.circular(theta2)
bw.cv.ml2 <- bw.cv.ml.circular(theta2)

## Linear plot
plot(function(x) dmixedvonmises(circular(x), mu1=circular(pi/2),
mu2=circular(3*pi/2), kappa1=5, kappa2=5, p=0.5), type="l", lwd=2,
col=1, main="mixture of von Mises", xlab=expression(theta),
ylab="Density", from=0, to=2*pi)
lines(density.circular(x=theta2, bw=bw.nrd2), plot.type='line', col=2)
lines(density.circular(x=theta2, bw=bw.cv.mse2), plot.type='line',
col=3)
lines(density.circular(x=theta2, bw=bw.cv.ml2), plot.type='line', col=4)
rug(theta2)
legend("topright", legend=c("True", "Taylor", "LSCV", "MLCV"), col=1:4, lwd=2)

## Circular plot
dmixedvonmises1 <- function(x) dmixedvonmises(circular(x), mu1=circular(pi/2),
mu2=circular(3*pi/2), kappa1=5, kappa2=5, p=0.5)
curve.circular(dmixedvonmises1, join=TRUE,
xlim=c(-1.5, 1.5), ylim=c(-1.5, 1.5), lwd=2, col=1, main="mixture of von
Mises")
lines(density.circular(x=theta2, bw=bw.nrd2), col=2)
lines(density.circular(x=theta2, bw=bw.cv.mse2), col=3)
lines(density.circular(x=theta2, bw=bw.cv.ml2), col=4)
points(theta2)
legend("topright", legend=c("True", "Taylor", "LSCV", "MLCV"), col=1:4, lwd=2)

## Example 3: mixture of von Mises and Wrapped Cauchy ##

rmixture <- function(n){
  x <- circular(sapply(runif(n), function(u) ifelse(u>0.5,
  rvonmises(n=1, mu=circular(pi),kappa=10),
  rwrappedcauchy(n=1,mu=circular(pi/2),rho=0.75))))
  return(x)
}

theta3 <- rmixture(n=150)

bw.nrd3 <- bw.nrd.circular(theta3)
bw.cv.mse3 <- bw.cv.mse.circular(theta3, lower=0.1, upper=100)
bw.cv.ml3 <- bw.cv.ml.circular(theta3, lower=0.1, upper=100)

dmixture <- function(x) (dvonmises(x, mu=circular(pi),
kappa=10)+dwrappedcauchy(x, mu=circular(pi/2), rho=0.75))/2
curve.circular(dmixture, join=TRUE, xlim=c(-1.5, 1.5), ylim=c(-1.5,
1.5), lwd=2, col=1, main="mixture of von Mises and Wrapped Normal")
lines(density.circular(x=theta3, bw=bw.nrd3), col=2)
lines(density.circular(x=theta3, bw=bw.cv.mse3), col=3)
lines(density.circular(x=theta3, bw=bw.cv.ml3), col=4)
legend("topright", legend=c("True", "Taylor", "LSCV", "MLCV"), col=1:4, lwd=2)
points(theta3)

Description

A method for circular object, which combines its arguments

Usage

## S3 method for class 'circular'
c(..., recursive = FALSE)

Arguments

Author(s)

Claudio Agostinelli

See Also

c

Examples

x <- rvonmises(10, circular(0), 10)
y <- rvonmises(10, circular(0), 10, control.circular=list(units="degrees"))
z <- runif(10, 0, 20) # here you do not use circular properties, 
#####but you mean it is measured in degrees
c(x, y, z) # While y is converted in radians, z is treated as it was!

Description

Density and random generation for the Cardioid circular distribution.

Usage

dcardioid(x, mu = circular(0), rho = 0)
rcardioid(n, mu = circular(0), rho = 0, control.circular=list())

Arguments

Value

dcardioid gives the density and rcardioid generates random deviates.

Author(s)

Claudio Agostinelli and Ulric Lund

References

Jammalamadaka, S. Rao and SenGupta, A. (2001). Topics in Circular Statistics, Section 2.2.2, World Scientific Press, Singapore.

Examples

set.seed(1234) 
  resrad <- rcardioid(n=10)
  set.seed(1234)
  resdeg <- rcardioid(n=10, control.circular=list(units="radians", zero=pi))  
  max(abs(resrad - conversion.circular(resdeg, zero=0)))

Description

Density for the Carthwrite's power-of-cosine distribution.

Usage

dcarthwrite(x, mu, psi)

Arguments

Details

The Carthwrite's power-of-cosine distribution has density

$f(x)=\frac{2^{(1/\psi)-1} \Gamma^2((1/\psi)+1) (1+\cos(x-\mu))^{1/\psi}} {\pi\Gamma((2/\psi)+1)},$

for $0 \le x < 2\pi$.

Value

The density

Author(s)

Federico Rotolo

References

Carthwrite, D.E. (1963). The use of directional spectra in studying the output of a wave recorder on a moving ship. Ocean Wave Spectra , 203-218.

Description

Tests for a change in mean direction, concentration, or both, given a set of directional data points.

Usage

change.point(x)

Arguments

Details

In either context, the user can choose which statistic (max or ave) to use, and then consult the appropriate table provided in the book referenced below. The critical values for these 4 statistics are to be found in Table 11.3 (or Figure 11.3) for rmax, Table 11.4 (or Figure 11.4) for rave, Figure 11.5 for tmax and Figure 11.6 for tave.

Value

Returns a list with variables n, rho, rmax, k.r, rave, tmax, k.t, and tave. The first of these is the sample size, followed by the overall mean resultant length. Both of these are needed to enter any of the tables or nomograms (see under Details). The other values represent the change point test statistics. While rmax and rave test for a change in mean direction (with unknown concentration), tmax and tave are useful in the context of testing more generally, for a change in mean direction and/or concentration. k.r and k.t are the observation numbers for which rmax and tmax attain their maximum value and indicate the observation at which the change is most likely to have occurred, when the tables or nomograms indicate significance.

Author(s)

Claudio Agostinelli and Ulric Lund

See Also

Jammalamadaka, S. Rao and SenGupta, A. (2001). Topics in Circular Statistics, Chapter 11, World Scientific Press, Singapore.

Description

Auxiliary function as user interface for circular plots. Typically only used when calling plot.circular.

Usage

circle.control(n = 1000, type = "l", col = 1, bg = par("bg"), 
  pch = 1, cex = 1, lty = 1, lwd = 1)

Arguments

Author(s)

Claudio Agostinelli

See Also

plot.circular

Examples

plot(rvonmises(10, circular(0), 1), control.circle=circle.control(col=2, lty=2))

Description

The function circular is used to create circular objects. as.circular and is.circular coerce an object to a circular and test whether an object is a circular data.

Usage

circular(x, type = c("angles", "directions"), 
  units = c("radians", "degrees", "hours"),
  template = c("none", "geographics", "clock12", "clock24"),
  modulo = c("asis", "2pi", "pi"), 
  zero = 0, rotation = c("counter", "clock"), names)
## S3 method for class 'circular'
as(x, control.circular=list(), ...)
## S3 method for class 'circular'
is(x)
## S3 method for class 'circular'
print(x, info=TRUE, ...)

Arguments

Value

an object of class circular. Since version 0.3-5 the previous class of the object is retain.

Author(s)

Claudio Agostinelli

See Also

conversion.circular

Examples

x <- circular(c(pi, pi/3, pi/4))
print(x)
is.circular(x)

x <- circular(runif(10, -pi/2, pi/2), template="geographics")
plot(x)
class(x)

x <- circular(data.frame(runif(10, -pi/2, pi/2)))
plot(x)
class(x)

cbind(x, x) # the matrix, cbind, rbind functions unclass and lost attributes! 
########Use it with care.

x <- c(pi/12,2*pi+pi/12)
print(x)
x <- unique(x)
print(x)

x[1]==x[2]

all.equal(x[1], x[2])

x <- as.circular(pi, control.circular=list(units="radians", zero=pi))
y <- conversion.circular(circular(pi), zero=pi)
res <- plot(x)
points(y, col=2, plot.info=res)

Description

The package ‘circular’ provides functions for the statistical analysis and graphics representation of circular data (observations which are angles). It originally started as a porting from S-plus to R of functions developed for the book: Circular Statistics, from "Topics in circular Statistics" (2001) S. Rao Jammalamadaka and A. SenGupta, World Scientific. Now, it has an S3 implementation and several new functions and datasets.

Version

The version level of the package is given by the command packageDescription("circular"). The most recent version of the package can be obtained from the R-Forge repository at https://r-forge.r-project.org/projects/circular/

Author

Claudio Agostinelli, Department of Mathematics University of Trento, Italy (http://datascience.maths.unitn.it/~claudio/)

Ulric Lund, Department of Statistics, California Polytechnic State University, San Luis Obispo, California, USA (https://statistics.calpoly.edu/ulric-lund)

Licence

This package and its documentation are usable under the terms of the "GNU General Public License", a copy of which is distributed with the package. While the software is freely usable, it would be appreciated if a reference is inserted in publications or other work which makes use of it; for this purpose, see the command citation("circular").

Acknowledgements

The package has evolved through several versions, developed over some years.

Many thanks to all that points out bugs, provide suggestions and comments.

The functions median and medianHS are developed together with Alessandro Gagliardi mailto:[email protected]

The functions watson.wiliams.test and wallraff.test are developed by Jean-Olivier Irisson (https://www.obs-vlfr.fr/~irisson/)

The functions dcarthwrite, dgenvonmises, (d,r)katojones, djonespewsey are developed by Federico Rotolo

The function rose.diag has contribution by Hiroyoshi Arai (mailto:[email protected])

The function windrose is developed by Matthew Pocernich

Dataset swallows is kindly provided by Dimitri Giunchi http://unimap.unipi.it/cercapersone/dettaglio.php?ri=2504&template=dettaglio.tpl

The function bw.circular is developed together with Eduardo Garcia Portugues https://egarpor.github.io/

If I miss to report your contribution please let me know by email at mailto:[email protected]

Description

Density and random generation for the Circular Uniform distribution on the whole circle.

Usage

dcircularuniform(x)
rcircularuniform(n, control.circular=list())

Arguments

Value

dcircularuniform gives the density and rcircularuniform generates random deviates.

Author(s)

Claudio Agostinelli

References

Jammalamadaka, S. Rao and SenGupta, A. (2001). Topics in Circular Statistics, Section 2.2.1, World Scientific Press, Singapore.

Examples

data1 <- rcircularuniform(100, control.circular=list(units="degrees"))
plot(data1)

curve.circular(dcircularuniform, join=TRUE, xlim=c(-1.2, 1.2), 
  ylim=c(-1.2, 1.2), main="Density of a Circular Uniform Distribution")

Description

Create a vector of n contiguous colors.

Usage

circular.colors(n, m = 0, M = 2 * pi, offset = 0, ...)

Arguments

Value

a vector of length n.

Author(s)

Claudio Agostinelli

See Also

hsv, colors

Examples

circular.colors(n=10, m=0, M=2*pi)

Description

‘circularp’ returns the ‘circularp’ attribute (or ‘NULL’). ‘circularp<-’ sets the ‘circularp’ attribute.

Usage

circularp(x)
circularp(x) <- value

Arguments

Details

The circularp attribute is a list of six elements: type, units, template, modulo, zero and rotation; see circular for their meaning.

Assignments are checked for consistency.

Assigning NULL removes the circularp attribute and any "circular" class of x.

Author(s)

Claudio Agostinelli

See Also

circular

Examples

x <- pi
circularp(x) # now NULL
circularp(x) <- list(type="angles", units="radians", template="none", 
  modulo="asis", zero=0, rotation="counter")
circularp(x)
x
class(x) <- "circular" # now we set also the class so that print.circular is used
x

Description

Conversion for Circular Data from one coordinate/units system to another one. For back compatibility, without arguments the function converts data from degrees to radians.

Usage

conversion.circular(x, units = c("radians", "degrees", "hours"), type = NULL, 
  template = NULL, modulo = NULL, zero = NULL, rotation = NULL)

Arguments

Value

an object of class circular with the specified unit of measure, modulo, zero and rotation.

Author(s)

Claudio Agostinelli

See Also

deg and rad. If you want to set the properties of an object instead to transform it, you can use circular or circularp<-.

Examples

x <- rvonmises(n=10, mu=circular(0), kappa=9, control.circular=list(units="degrees"))
par(mfcol=c(2, 2))
plot(x)
y <- conversion.circular(x) # only the unit is changed (to radians) and 
####### the data converted.
plot(y)
z <- conversion.circular(x, units="degrees", zero=pi) # only the zero is changed and 
####### the data converted.
plot(z)
w <- conversion.circular(x, zero=pi, rotation="clock") # zero and rotation is 
####### changed and the data converted.
plot(w)

Description

A dataset taken from the paper of Coope (1993).

Usage

data(coope)

Format

x.coope and y.coope are vectors of length 8.

Source

Coope, I. (1993). Circle fitting by linear and non-linear least squares. Journal of Optimization Theory and Applications, 76, 381-388.

Description

From coordinates of the end point of a vector in 2 dimensions to the angle between this vector and the x-axis

Usage

coord2rad(x, y = NULL, control.circular = list())

Arguments

Value

an object of class circular

Author(s)

Claudio Agostinelli and Frederick T. Wehrle

See Also

circular

Examples

set.seed(1234)
x <- cbind(rnorm(20), rnorm(20))
y <- coord2rad(x)

Description

Computes a circular version of the Pearson's product moment correlation, and performs a significance test if requested.

Usage

cor.circular(x, y=NULL, test=FALSE)

Arguments

Details

The correlation coefficient is computed like Pearson's product moment correlation for two linear variables X and Y. In the computational formula, however, (xi - xbar) and (yi - ybar) are replaced by sin(xi - xbar) and sin(yi - ybar), where xbar and ybar in the second two expressions are the mean directions of the samples.

Value

Returns a vector or a matrix of a circular version of the Pearson's product moment correlation, if test == TRUE then a list is reported with statistic and p.value, the test statistic and p-value respectively, for testing significance of the correlation coefficient.

Author(s)

Claudio Agostinelli and Ulric Lund

References

Jammalamadaka, S. Rao and SenGupta, A. (2001). Topics in Circular Statistics, Section 8.2, World Scientific Press, Singapore.

Jammalamadaka, S. and Sarma, Y. (1988). A correlation coefficient for angular variables. Statistical Theory and Data Analysis 2. North Holland: New York.

Examples

# Generate two circular data sets, and compute their correlation.
x <- rvonmises(n=50, mu=circular(0), kappa=3)
y <- x + rvonmises(n=50, mu=circular(pi), kappa=10)
cor.circular(x, y, test=TRUE)

Description

Draws a curve corresponding to the given function or expression (in x) over the interval [from,to] in a circle. Mainly used to plot circular density functions.

Usage

## S3 method for class 'circular'
curve(expr, from=NULL, to=NULL, n=101, add=FALSE, 
  cex=1, axes=TRUE, ticks=FALSE, shrink=1, tcl=0.025, 
  tcl.text=0.125, tol=0.04, uin=NULL, xlim=c(-1, 1), 
  ylim=c(-1, 1), digits=2, modulo=c("2pi", "asis", "pi"), 
  main=NULL, sub=NULL, xlab="", ylab="", 
  control.circle=circle.control(), ...)
## S3 method for class 'function.circular'
plot(x, from=0, to=2*pi, ...)

Arguments

Details

For now, curve circular draws functions defined in radians, counterclockwise coordinate and zero at 0.

Value

A list with information on the plot: zero, rotation and next.points.

Author(s)

Claudio Agostinelli

See Also

lines.circular and circle.control

Examples

ff <- function(x) sqrt(x)/20
curve.circular(ff)
curve.circular(ff, to=6*pi, join=FALSE, nosort=TRUE, n=1001, modulo="asis",
  shrink=1.2)

plot.function.circular(function(x) dvonmises(x, circular(0), 10), xlim=c(-1, 2.2))

Description

Converts radians to degrees.

Usage

deg(x)

Arguments

Details

This function is available for compatibility with the CircStats package; please use conversion.circular.

Value

Returns a vector or matrix of degree measurements corresponding to the data in radians.

Author(s)

Claudio Agostinelli and Ulric Lund

See Also

conversion.circular and rad

Description

The function density.circular computes kernel density estimates with the given kernel and bandwidth for circular data.

Usage

## S3 method for class 'circular'
density(x, z=NULL, bw, adjust = 1, type = c("K", "L"),
  kernel = c("vonmises", "wrappednormal"), na.rm = FALSE, 
  from = circular(0), to = circular(2 * pi), n = 512, K = NULL, min.k=10, 
  control.circular=list(), ...)
## S3 method for class 'density.circular'
print(x, digits = NULL, ...)

Arguments

Value

an object with class "density.circular" whose underlying structure is a list containing the following components.

Author(s)

Claudio Agostinelli

References

Z.D. Bai and C.R. Rao and L.C. Zhao (1988). Kernel Estimators of Density Function of Directional Data, Journal of Multivariate Analysis, 27, 24-39.

J. Klemel\"a (2000). Estimation of densities and derivatives of densities with directional data, Journal of Multivariate Analysis, 73, 18-40.

V.R. Prayag and A.P. Gore (1990). Density Estimation for Randomly Distributed Circular Objects, Metrika, 1990, 37, 63-69.

P. Hall and G.S. Watson and J. Cabrera (1987). Kernel Density Estimation with Spherical Data, Biometrika, 74, 4, 751–762.

See Also

plot.density.circular and lines.density.circular

Examples

x <- rvonmises(n=100, mu=circular(pi), kappa=2)
res25 <- density(x, bw=25, control.circular=list(units="degrees"))
circularp(res25$x)
plot(res25, points.plot=TRUE, xlim=c(-1.6,1))
res50 <- density(x, bw=25, adjust=2)
lines(res50, col=2)
lines(res50, col=3, shrink=0.9) #shrink the plot wrt the function :-)
lines(res50, col=4, offset=0.5) #draw it with a reference circle of 0.5

Description

This function computes and returns the distance matrix computed by using the specified distance measure to compute the distances between the rows of a data matrix containing circular data.

Usage

dist.circular(x, method = "correlation", diag = FALSE, upper = FALSE)

Arguments

Details

Available distance measures are (written for two vectors $x$ and $y$):

correlation:

$\sqrt{1 - \rho}$ where $\rho$ is the Circular Correlation coefficient defined as

$\frac{\sum_{i=1}^n \sin(x_i - \mu_x) \sin(y_i - \mu_y)}{\sqrt{\sum_{i=1}^n \sin^2(x_i - \mu_x) \sum_{i=1}^n \sin^2(y_i - \mu_y)}}$

and $\mu_x$, $\mu_y$ are the mean direction of the two vectors

angularseparation:

$\sum_{i=1}^n 1 - cos(x_i - y_i)$

chord:

$\sum_{i=1}^n \sqrt{2 (1 - \cos(x_i - y_i))}$

geodesic:

$\sum_{i=1}^n \pi - |\pi - |x_i - y_i||$ where the abs(x - y) is expressed with an angle in [-pi,pi]

Missing values are allowed, and are excluded from all computations involving the rows within which they occur. Further, when Inf values are involved, all pairs of values are excluded when their contribution to the distance gave NaN or NA.
If some columns are excluded in calculating the sum is scaled up proportionally to the number of columns used. If all pairs are excluded when calculating a particular distance, the value is NA.

Value

dist.circular returns an object of class "dist".

The lower triangle of the distance matrix stored by columns in a vector, say do. If n is the number of observations, i.e., n <- attr(do, "Size"), then for $i < j <= n$, the dissimilarity between (row) i and j is do[n*(i-1) - i*(i-1)/2 + j-i]. The length of the vector is $n*(n-1)/2$, i.e., of order $n^2$.

The object has the following attributes (besides "class" equal to "dist"):

See Also

dist

Description

This function tests for the homogeneity of concentration parameters for multiple samples of directional data.

Usage

equal.kappa.test(x, group)
## S3 method for class 'equal.kappa.test'
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

Details

The samples are assumed to have been drawn from von Mises populations. The null hypothesis tested is that all populations sampled have the same concentration parameter, kappa.

When the pooled data has high concentration, sample mean resultant length above 0.70, Bartlett's test is used. For less concentrated pooled data, variance-stabilizing transformations are used to improve normal approximations needed to arrive at an approximate chi-squared test statistic (see references below). For pooled sample mean resultant length below 0.45, it is possible that individually a sample may in fact have quite a large sample mean resultant length. In this case, it is possible that the variance-stabilizing transformation involving the inverse sine function is passed a value outside of -1,1. If this occurs, the function will automatically use Bartlett's test and issue a warning to that effect.

Value

An object of class equal.kappa.test with the following components:

Author(s)

Claudio Agostinelli and Ulric Lund

References

Jammalamadaka, S. Rao and SenGupta, A. (2001). Topics in Circular Statistics, Section 5.3, World Scientific Press, Singapore.

Mardia, K. and Jupp, P. (1999). Directional Statistics, Section 7.4, John Wiley and Sons, England.

Examples

x <- c(rvonmises(50, circular(0), 1), rvonmises(100, circular(pi/3), 10))
group <- c(rep(0, 50), rep(1, 100)) 

equal.kappa.test(x, group)

Description

Arrival time on a 24-hour clock of 254 patients at an intensive care unit, over a period of about 12 months.

Usage

data(fisherB1)
data(fisherB1c)

Format

fisherB1 is a vector of 254 observations (in the format hours.minutes). fisherB1c contains the same observations in a circular objects (minutes are expressed as decimals).

Source

Cox, D.R. and Lewis, P.A.W. (1966) The Statistical Analysis of Series of Events. London : Methuen & CO. Ltd. pp. 254-255

See Also

N.I. Fisher (1993) Statistical analysis of circular data. Cambridge University Press. Pag. 239.

Examples

data(fisherB1c)
par(mfcol=c(1,2))
plot(fisherB1c, main="Clock 24", shrink=1.5)
plot(fisherB1c, template="clock12", main="Clock 12", shrink=1.5)

Description

Directions of 11 long-legged desert ants (Cataglyphis fortis) after one eye on each ant was 'trained' to learn the ant's home direction, then covered and the other eye uncovered.

Usage

data(fisherB10)
data(fisherB10c)

Format

fisherB10 is a list (in degrees). fisherB10c contains the same observations in a circular objects.

Source

Personal communication of Prof. Dr. R. Wehner to Prof. N.I. Fisher, experiment described in

R. Wehner and M. Muller (1985) Does interocular transfer occur in visual navigation by ants? Nature, 315, 228-9.

See Also

N.I. Fisher (1993) Statistical analysis of circular data. Cambridge University Press. Pag. 244-245.

Examples

data(fisherB10c)
res <- plot(fisherB10c$set1)
points(fisherB10c$set2, col=2, plot.info=res)
points(fisherB10c$set3, col=3, plot.info=res)

Description

Resultant directions of 22 sea stars 11 days after being displaced from their natural habitat.

Usage

data(fisherB11)
data(fisherB11c)

Format

fisherB11 a vector of 22 observations (in degrees). fisherB11c contains the same observations in a circular objects.

Source

G.J.G. Upton and B. Fingleton (1989) Spatial Data Analysis by Example. Volume 2. Categorical and Directional Data. New York: John Wiley as adapted from B. Pabst and H. Vicentini (1978) Dislocation experiments in the migrating seastar. Astropecten jonstoni. Marine Biology 48, 271-8.

See Also

N.I. Fisher (1993) Statistical analysis of circular data. Cambridge University Press. Pag. 245.

Examples

data(fisherB11c)
plot(fisherB11c, stack=TRUE, shrink=1.5)

Description

Vanishing directions of 15 homing pigeons, released just over 16 kilometres Northwest of their loft.

Usage

data(fisherB12)
data(fisherB12c)

Format

fisherB12 a vector of 15 observations (in degrees). fisherB12c contains the same observations in a circular objects.

Source

Schmidt-Koenig (1963) On the role of the loft, the distance and site of release in pigeon homing (the "cross-loft experiment"). Biol. Bull. (125)154-164.

References

N.I. Fisher (1993) Statistical analysis of circular data. Cambridge University Press. Pag. 245.

Examples

data(fisherB12c)
plot(fisherB12c, stack=TRUE, shrink=1.5)

Description

Orientations of termite mounds of Amitermes laurensis at 14 sites in Cape York Penisula, North Queensland.

Usage

data(fisherB13)
data(fisherB13c)

Format

fisherB13 a list of 14 datasets (axes in degrees) at several locations. fisherB13c contains the same observations in a circular objects.

Details

Set 1: n=100, Latitude -15'43”, Longitude 144'42” Set 2: n=50, Latitude -15'32”, Longitude 144'17”

Source

A.V. Spain, T. Okello-Oloya and R.D. John (1983) Orientation of the termitaria of two species of Amitermes (Isoptera:Termitinae) from Northern Queensland. Aust. J. Zoo. (31):167-177.

References

N.I. Fisher (1993) Statistical analysis of circular data. Cambridge University Press. Pag. 246.

Examples

data(fisherB13c)
plot(fisherB13c$set1, stack=TRUE, shrink=1.5)

Description

19 measurements of wind direction 'theta' and ozone level 'x' taken at 6.00am at four-day intervals between April 18th and June 29th, 1975 at a weather station in Milwaukee.

Usage

data(fisherB18)
data(fisherB18c)

Format

fisherB18 is a data.frame of integer value. fisherB18c is a data.frame that contains the same observations, but in the first column, the data is a circular object.

Source

N.I. Fisher (1993) pag. 251. Johnson & Wehrly (1977, Table 1).

References

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

Examples

data(fisherB18)
data(fisherB18c)
par(mfcol=c(1,3))
plot(fisherB18c$theta, xlab=expression(theta))
boxplot(fisherB18c$x, xlab="x")
plot(c(fisherB18$x, fisherB18$x), c(fisherB18$theta,
  fisherB18$theta+360), xlab="x", ylab=expression(theta))

Description

Measurements of long-axis orientation of 133 feldspar laths in basalt

Usage

data(fisherB2)
data(fisherB2c)

Format

fisherB2 is a vector of 133 observations (in degrees). fisherB2c contains the same observations in a circular objects.

Source

Smith, N.M. (1988) Reconstruction of the Tertiary drainage systems of the Inverell region. Unpublished B.Sc. (Hons.) thesis, Department of Geography, University of Sydney, Australia.

This dataset (set 28-6-1co.prn) was kindly supplied by Ms Nicola Smith to Prof. N.I. Fisher.

See Also

N.I. Fisher (1993) Statistical analysis of circular data. Cambridge University Press. Pag. 240.

Examples

data(fisherB2c)
plot(fisherB2c)

Description

Distances 'x' and directions 'theta' by small blue periwinkles, Nodilittorina unifasciata, after they had been transplanted downshore from the height at which they normally live.

Usage

data(fisherB20)
data(fisherB20c)

Format

fisherB20 is a data.frame of integer value. fisherB20c is a data.frame that contains the same observations, but in the first column, the data is a circular object.

Source

N.I. Fisher (1993) pag. 252-253. Data kindly supplied by Dr A. Underwood and Ms G. Chapman.

References

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

Examples

data(fisherB20)
data(fisherB20c)
par(mfcol=c(1,3))
plot(fisherB20c$theta, xlab=expression(theta))
boxplot(fisherB20c$x, xlab="x")
plot(c(fisherB20$x, fisherB20$x), c(fisherB20$theta,
  fisherB20$theta+360), xlab="x", ylab=expression(theta))

Description

Measurements of the directions taken by 76 turtles after treatment.

Usage

data(fisherB3)
data(fisherB3c)

Format

fisherB3 is a vector of 76 observations (in degrees). fisherB3c contains the same observations in a circular objects.

Source

Stephens, M.A. (1969) Techniques for directional data. Technical Report #150, Department of Statistics, Stanford University, Stanford, CA.

See Also

N.I. Fisher (1993) Statistical analysis of circular data. Cambridge University Press. Pag. 241.

Examples

data(fisherB3c)
plot(fisherB3c)

Description

Sun compass orientations of 50 starhead topminnows, measured under heavily overcast conditions.

Usage

data(fisherB4)
data(fisherB4c)

Format

fisherB4 is a vector of 50 observations (in degrees). fisherB4c contains the same observations in a circular objects.

Source

Goodyear (1970) Terrestrial and aquatic orientation in the Starhead Topminnow, Fundulus notti. Science 168, 603-5. Figure 1D.

See Also

N.I. Fisher (1993) Statistical analysis of circular data. Cambridge University Press. Pag. 241.

Examples

data(fisherB3c)
plot(fisherB3c)

Description

Measurements of long-axis orientation of 164 feldspar laths in basalt

Usage

data(fisherB5)
data(fisherB5c)

Format

fisherB5 is a vector of 164 observations (in degrees). fisherB5c contains the same observations in a circular objects.

Source

Smith, N.M. (1988) Reconstruction of the Tertiary drainage systems of the Inverell region. Unpublished B.Sc. (Hons.) thesis, Department of Geography, University of Sydney, Australia.

This dataset (set 24-6-5co.prn) was kindly supplied by Ms Nicola Smith to Prof. N.I. Fisher.

See Also

N.I. Fisher (1993) Statistical analysis of circular data. Cambridge University Press. Pag. 242.

Examples

data(fisherB5c)
plot(fisherB5c)

Description

Set of cross-bed azimuths of palaeocurrents measured in the Belford Anticline (New South Wales).

Usage

data(fisherB6)
data(fisherB6c)

Format

fisherB6 is a list (in degrees). fisherB6c contains the same observations in a circular objects.

Source

Fisher, N.I. & Powell C. McA. (1989) Statistical analysis of two-dimensional palaeocurrent data: Methods and examples. Aust. J. Earth Sci. 36, 91-107.

See Also

N.I. Fisher (1993) Statistical analysis of circular data. Cambridge University Press. Pag. 242.

Examples

data(fisherB6c)
res <- plot(fisherB6c$set1)
points(fisherB6c$set2, col=2, plot.info=res)
points(fisherB6c$set3, col=3, plot.info=res)

Description

Directions chosen by 100 ants in response to an evenly illuminated black targets placed as shown.

Usage

data(fisherB7)
data(fisherB7c)

Format

fisherB7 a vector of 100 observations (in degrees). fisherB7c contains the same observations in a circular objects.

Source

Randomly selected values from Jander, R. (1957) Die optische Richtangsorientierung der roten Waldameise (Formica rufa. L.) Z. vergl. Physiologie 40, 162-238. Figure 18A.

See Also

N.I. Fisher (1993) Statistical analysis of circular data. Cambridge University Press. Pag. 243.

Examples

data(fisherB7c)
plot(fisherB7c, zero=pi/2, rotation='clock', stack=TRUE)

Description

Horizontal axes of 100 outwash pebbles fromo a late Wisconsin outwash terrace along Fox river, near Cary, Illinois

Usage

data(fisherB8)
data(fisherB8c)

Format

fisherB8 a vector of 100 observations (in degrees). fisherB8c contains the same observations in a circular objects.

Source

Mardia, K.V. (1972) Statistics of Directional Data. London: Academic Press. Table 1.6 adapted from Krumbein W.C. (1939) Preferred orientations of pebbles in sedimentary deposits. J. Geol. 47, 673-706.

See Also

N.I. Fisher (1993) Statistical analysis of circular data. Cambridge University Press. Pag. 243.

Examples

data(fisherB8c)
plot(fisherB8c, stack=TRUE, shrink=1.5)

Description

Dance directions of 279 honey bees viewing a zenith patch of artificially polarised light.

Usage

data(fisherB9)
data(fisherB9c)

Format

fisherB9 a vector of 279 observations (in degrees). fisherB9c contains the same observations in a circular objects.

Source

Adapted by Prof. N.I. Fisher from R. Wehner & S. Strasser (1985) The POL area of the honey bee's eye: behavioural evidence. Physiol. Entomol. 10, 337-49. Pag. 346.

See Also

N.I. Fisher (1993) Statistical analysis of circular data. Cambridge University Press. Pag. 244.

Examples

data(fisherB9c)
plot(fisherB9c, stack=TRUE, shrink=1.5)

Description

Density for the Generalized von Mises circular distribution.

Usage

dgenvonmises(x, mu1, mu2, kappa1, kappa2)

Arguments

Details

The Generalized von Mises distribution has density

$f(x)=\frac1{2\pi G_0(\delta,\kappa_1,\kappa_2)} \exp\{\kappa_1 \cos(x-\mu_1) + \kappa_2 \cos2(x-\mu_2)\},$

for $0 \le x < 2\pi$, where $\delta=(\mu_1-\mu_2)$ and $G_0$ is the normalizing constant.

Value

The density

Author(s)

Federico Rotolo

References

Gatto , R. & Jammalamadaka , S.R. (2007). The generalized von Mises distribution. Statistical Methodology 4, 341-353.

Examples

ff <- function(x) dgenvonmises(x, mu1=circular(5*pi/4), mu2=circular(pi/4), kappa1=.3, kappa2=1)
curve.circular(ff, join=TRUE, xlim=c(-1, 1), ylim=c(-1.2, 1.2),
  main="Density of a Generalized von Mises Distribution",
  xlab=expression(paste(mu,"1=5/4",pi,", ",mu2,"=",pi/4,", ",kappa,"1=0.3, ",kappa,"2=1"))
  )

Description

A heat map is a false color image (basically image(t(x))) with a dendrogram added to the left side and to the top. Typically, reordering of the rows and columns according to some set of values (row or column means) within the restrictions imposed by the dendrogram is carried out. See also heatmap.

Usage

heatmap.circular(x, Rowv = NULL, Colv = if (symm) "Rowv" else NULL, 
distfun = dist.circular, hclustfun = hclust, 
reorderfun = function(d, w) reorder(d, w), add.expr, symm = FALSE, 
revC = identical(Colv, "Rowv"), na.rm = TRUE, margins = c(5, 5), 
lwid = c(1, 4), lhei = c(1, 4), ColSideColors, RowSideColors, 
NAColors = "black", cexRow = 0.2 + 1/log10(nr), cexCol = 0.2 + 1/log10(nc), 
labRow = NULL, labCol = NULL, main = NULL, xlab = NULL, ylab = NULL, 
keep.dendro = FALSE, annotate.expr, annotate = rep(NA, 4), 
verbose = getOption("verbose"), ...)

Arguments

Details

If either Rowv or Colv are dendrograms they are honored (and not reordered). Otherwise, dendrograms are computed as dd <- as.dendrogram(hclustfun(distfun(X))) where X is either x or t(x).

If either is a vector (of ‘weights’) then the appropriate dendrogram is reordered according to the supplied values subject to the constraints imposed by the dendrogram, by reorder(dd, Rowv), in the row case. If either is missing, as by default, then the ordering of the corresponding dendrogram is by the mean direction value of the rows/columns, i.e., in the case of rows, Rowv <- rowMeans(x, na.rm=na.rm). If either is NULL, no reordering will be done for the corresponding side.

Unless Rowv = NA (or Colw = NA), the original rows and columns are reordered in any case to match the dendrogram, e.g., the rows by order.dendrogram(Rowv) where Rowv is the (possibly reorder()ed) row dendrogram.

heatmap() uses layout and draws the image in the lower right corner of a 2x2 layout. Consequentially, it can not be used in a multi column/row layout, i.e., when par(mfrow= *) or (mfcol= *) has been called.

Value

par(mfrow= *) or (mfcol= *) has been called.

Author(s)

Claudio Agostinelli using the code from heatmap.

See Also

dist.circular, heatmap, image, hclust

Description

An alias of besselI(x, nu=0).

Usage

I.0(x)

Arguments

Value

Returns the zeroth order Bessel function of the first kind evaluated at a specified real number.

See Also

besselI.

Description

An alias of besselI(x, nu=1).

Usage

I.1(x)

Arguments

Value

Returns the first order Bessel function of the first kind, evaluated at a specified real number.

See Also

besselI.

Description

An alias of besselI(x, nu=p).

Usage

I.p(p, x)

Arguments

Value

Returns the p-th order Bessel function of the first kind, evaluated at a specified real number.

See Also

besselI.

Description

Find an estimates of the probability of the intersection between a modal region and a given interval.

Usage

intersect.modal.region(x, ...)
## Default S3 method:
intersect.modal.region(x, ...)
## S3 method for class 'circular'
intersect.modal.region(x, breaks, z = NULL, q = 0.95, bw,
  adjust = 1, type = c("K", "L"), kernel = c("vonmises", "wrappednormal"),
  na.rm = FALSE, step = 0.01, eps.lower = 10^(-4), eps.upper = 10^(-4), ...)

Arguments

Details

Only the version for circular data is actually implemented.

Value

For the circular method a list with the following three components

Author(s)

Claudio Agostinelli

See Also

modal.region

Examples

x <- rvonmises(100, circular(pi), 10)  
  res <- intersect.modal.region(x, breaks=circular(matrix(c(pi,pi+pi/12,
    pi-pi/12, pi), ncol=2, byrow=TRUE)), bw=50)
  res$tot

  x <- rvonmises(100, circular(0), 10)
  res <- intersect.modal.region(x, breaks=circular(matrix(c(pi,pi+pi/12),
    ncol=2)), bw=50)
  res$tot
  
  res <- intersect.modal.region(x, breaks=circular(matrix(c(pi/12,
    2*pi-pi/12), ncol=2, byrow=TRUE)), bw=50)
  res$tot

Description

Density for the Jones and Pewsey circular distribution.

Usage

djonespewsey(x, mu, kappa, psi)

Arguments

Details

The JonesPewsey distribution has density

$f(x)=\frac{(\cosh(\kappa\psi) + \sinh(\kappa\psi)\cos(x-\mu))^{1/\psi}} {2\pi P_{1/\psi}(\cosh(\kappa\psi))},$

for $0 \le x < 2\pi$, where $P_{1/\psi}(\cdot)$ is the associated Legendre function of the first kind, degree $1/\psi$ and order 0.

Value

The density

Author(s)

Federico Rotolo

References

Jones , M.C. and Pewsey, A. (2005). A family of symmetric distributions on the circle. J. Am. Statist. Assoc. 100, 1422-1428

Examples

ff <- function(x) djonespewsey(x, mu=circular(4), kappa=1.8, psi=-.6)
curve.circular(ff, join=TRUE, xlim=c(-1, 1), ylim=c(-1.2, 1.2),
  main="Density of a JonesPewsey Distribution",
  xlab=expression(paste(mu,"=1.3",pi,", ",kappa,"=1.8, ",psi,"=-0.6"))
  )

Description

Density and random generation for the Kato and Jones distribution.

Usage

rkatojones(n, mu, nu, r, kappa, control.circular=list())
dkatojones(x, mu, nu, r, kappa)

Arguments

Details

The Kato and Jones distribution has density

$f(x)= \frac{1-r^2}{2\pi\mathcal I_0(\kappa)} \exp\left[ \frac{\kappa\{ \xi\cos(x-\eta)-2r\cos\nu \}} {1+r^2-2r\cos(x -\gamma)} \right]\\ \phantom{\exp[]} \times \frac1{1+r^2-2r\cos(x -\gamma)},$

for $0 \le x < 2\pi$, where $\gamma=\mu+\nu$, $\xi=\{r^4+2r^2\cos(2\nu)+1\}^{1/2}$ and $\eta=\mu+\arg[ r^2\{\cos(2\nu)+i\sin(2\nu)\}+1 ]$.

Original code for random generation is by Kato, S. and Jones, M.C. and can be found at the address http://pubs.amstat.org/doi/suppl/10.1198/jasa.2009.tm08313/suppl_file/t08-313code.txt.

Value

The density. dkatojones gives the density and rkatojones generates random deviates.

Author(s)

Federico Rotolo

References

Kato , S. and Jones, M.C. (2010). A family of distributions on the circle with links to, and applications arising from, Mobius transformation. J. Am. Statist. Assoc. 105, 249-262.

Examples

data1 <- rkatojones(n=100, mu=circular(0), nu=circular(pi/4), r=.2, kappa=1)
plot(data1)

data1 <- rkatojones(n=100, mu=circular(pi/3), nu=circular(pi), r=.7, kappa=2.3)
plot(data1)

ff <- function(x) dkatojones(x, mu=circular(pi/3), nu=circular(pi), r=.7, kappa=2.3)
curve.circular(ff, join=TRUE, xlim=c(-1, 1), ylim=c(-1.2, 1.2),
  main="Density of a KatoJones Distribution",
  xlab=expression(paste(mu,"=",pi,"/3, ",nu,"=",pi,", r=0.7, ",kappa,"=2.3"))
  )

Description

Performs Kuiper's one-sample test of uniformity on the circle.

Usage

kuiper.test(x, alpha=0)
## S3 method for class 'kuiper.test'
print(x, digits = 4, ...)

Arguments

Details

Kuiper's test statistic is a rotation-invariant Kolmogorov-type test statistic. The critical values of a modified Kuiper's test statistic are used according to the tabulation given in Stephens (1970).

Value

A list with the statistic and alpha value.

Note

Kuiper's one-sample test of uniformity is performed, and the results are printed to the screen. If alpha is specified and non-zero, the test statistic is printed along with the critical value and decision. If alpha is omitted, the test statistic is printed and a range for the p-value of the test is given.

Author(s)

Claudio Agostinelli and Ulric Lund

References

Jammalamadaka, S. Rao and SenGupta, A. (2001). Topics in Circular Statistics, Section 7.2, World Scientific Press, Singapore.

Stephens, M. (1970). Use of the Kolmogorov-Smirnov, Cramer-von Mises and related statistics without extensive tables. Journal of the Royal Statistical Society, B32, 115-122.

See Also

range.circular, rao.spacing.test, rayleigh.test and watson.test

Examples

# Generate data from the uniform distribution on the circle.
data <- circular(runif(100, 0, 2*pi))
kuiper.test(data)
# Generate data from the von Mises distribution.
data <- rvonmises(n=100, mu=circular(0), kappa=3)
kuiper.test(data, alpha=0.01)

Description

A method taking coordinates in a polar system and joining the corresponding points with line segments.

Usage

## S3 method for class 'circular'
lines(x, y, join = FALSE, nosort = FALSE, offset=1, shrink=1,
  plot.info = NULL, zero = NULL, rotation = NULL, modulo = NULL, ...)

Arguments

Value

A list with information on the plot: zero, rotation and next.points.

Author(s)

Claudio Agostinelli

See Also

plot.circular

Examples

x <- rvonmises(20, circular(0), 10)
y <- runif(20, 0.5, 1)

plot(x, shrink=2)
lines(x, y)

Description

The lines add a plot for density.circular objects.

Usage

## S3 method for class 'density.circular'
lines(x, type = "l", zero.line = TRUE,
  points.plot = FALSE, points.col = 1, points.pch = 1, points.cex = 1,
  plot.type = c("circle", "line"), bins = NULL, offset=1, shrink = 1,
  tcl = 0.025, sep = 0.025, join = TRUE, nosort = FALSE,
  plot.info = NULL, zero = NULL, rotation = NULL, ...)

Arguments

Value

A list with information on the plot: zero, rotation and next.points and, if available, the coordinates x and y.

Author(s)

Claudio Agostinelli

See Also

density.circular and plot.density.circular

Examples

set.seed(1234)
x <- rvonmises(n=100, mu=circular(pi), kappa=2)
y <- rvonmises(n=100, mu=circular(pi/2), kappa=2)
resx <- density(x, bw=25)
res <- plot(resx, points.plot=TRUE, xlim=c(-1.5,1), ylim=c(-1.1, 1.5))
resy <- density(y, bw=25)
lines(resy, points.plot=TRUE, col=2, points.col=2, plot.info=res)

Description

Fits a regression model for a circular dependent and circular independent variable or for a circular dependent and linear independent variables.

Usage

lm.circular(..., type=c("c-c", "c-l"))
lm.circular.cc(y, x, order = 1, level = 0.05, control.circular = list())
lm.circular.cl(y, x, init = NULL, verbose = FALSE, tol = 1e-10, 
  control.circular = list())
## S3 method for class 'lm.circular.cl'
print(x, digits = max(3, getOption("digits") - 3), 
  signif.stars= getOption("show.signif.stars"), ...)