| Title: | Circular-Linear Copulas with Angular Symmetry for Movement Data |
|---|---|
| Description: | Classes (S4) of circular-linear, symmetric copulas with corresponding methods, extending the 'copula' package. These copulas are especially useful for modeling correlation in discrete-time movement data. Methods for density, (conditional) distribution, random number generation, bivariate dependence measures and fitting parameters using maximum likelihood and other approaches. The package also contains methods for visualizing movement data and copulas. |
| Authors: | Florian Hodel [aut, cre]
|
| Maintainer: | Florian Hodel <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 0.2.0 |
| Built: | 2024-11-15 03:06:25 UTC |
| Source: | https://github.com/cran/cylcop |
The x-y-coordinates of a position in 2-D space is calculated from the angle between that position and the 2 previous ones in the trajectory and the distance between that position and the previous one.
angstep2xy(angle, steplength, prevp1, prevp2)angstep2xy(angle, steplength, prevp1, prevp2)
angle |
numeric value of the turn angle or a
|
steplength |
numeric value giving the distance between the position and the previous one. |
prevp1 |
numeric vector holding the x and y coordinates of the previous position. |
prevp2 |
numeric vector holding the x and y coordinates of the position before the previous one. |
The function returns a numeric vector holding the x and y coordinates of the position
angstep2xy(1.5*pi, 2, prevp1 = c(1, 4), prevp2 = c(2, 7.5)) angstep2xy(-0.5*pi, 2, c(1, 4), c(2, 7.5))angstep2xy(1.5*pi, 2, prevp1 = c(1, 4), prevp2 = c(2, 7.5)) angstep2xy(-0.5*pi, 2, c(1, 4), c(2, 7.5))
The angle between a line between 2 points in Euclidean 2-D space and the line from (0,0) to (0,1) is calculated. In other words, the compass bearing of a line between 2 points where north is 0. Angles increase in clockwise direction.
bearing(point1, point2, fullcirc = FALSE)bearing(point1, point2, fullcirc = FALSE)
point1 |
numeric vector holding the x and y coordinates of the first point. |
point2 |
numeric vector holding the x and y coordinates of the second point. |
fullcirc |
logical value indicating whether the output
should be an angle on |
If fullcirc = FALSE, the function returns a numeric
value (angle) from the interval .
If fullcirc = TRUE, the function returns a numeric value numeric from the interval
.
bearing(c(3,5), c(1,4)) bearing(c(3,5), c(1,4), fullcirc = TRUE)bearing(c(3,5), c(1,4)) bearing(c(3,5), c(1,4), fullcirc = TRUE)
Calculates the conditional distributions and their inverses of circular-linear copulas and 2-dimensional linear-linear copulas.
ccylcop(u, copula, cond_on = 2, inverse = FALSE, ...) ## S4 method for signature 'Copula' ccylcop(u, copula, cond_on, inverse) ## S4 method for signature 'cyl_cubsec' ccylcop(u, copula, cond_on = 2, inverse = FALSE) ## S4 method for signature 'cyl_quadsec' ccylcop(u, copula, cond_on = 2, inverse = FALSE) ## S4 method for signature 'cyl_rect_combine' ccylcop(u, copula, cond_on = 2, inverse = FALSE) ## S4 method for signature 'cyl_rot_combine' ccylcop(u, copula, cond_on = 2, inverse = FALSE) ## S4 method for signature 'cyl_vonmises' ccylcop(u, copula, cond_on = 2, inverse = FALSE)ccylcop(u, copula, cond_on = 2, inverse = FALSE, ...) ## S4 method for signature 'Copula' ccylcop(u, copula, cond_on, inverse) ## S4 method for signature 'cyl_cubsec' ccylcop(u, copula, cond_on = 2, inverse = FALSE) ## S4 method for signature 'cyl_quadsec' ccylcop(u, copula, cond_on = 2, inverse = FALSE) ## S4 method for signature 'cyl_rect_combine' ccylcop(u, copula, cond_on = 2, inverse = FALSE) ## S4 method for signature 'cyl_rot_combine' ccylcop(u, copula, cond_on = 2, inverse = FALSE) ## S4 method for signature 'cyl_vonmises' ccylcop(u, copula, cond_on = 2, inverse = FALSE)
u |
matrix (or vector) of numeric
values in |
copula |
R object of class ' |
cond_on |
column number of |
inverse |
logical indicating whether the inverse of the conditional copula is calculated. |
... |
additional arguments. |
This is a generic that calls the function copula::cCopula()
for 2-dimensional 'Copula' objects from the 'copula'
package for which copula::cCopula() is available. If
copula::cCopula() is not available, the conditional copula is
calculated numerically. For 'cyl_copula' objects,
the conditional copula is calculated analytically or numerically
(depending on the copula and the values of u).
Note that the input arguments and the
output of cylcop::ccylcop() differ from those of
copula::cCopula().
A vector containing the values of the distribution of the copula at
u[,-cond_on] conditional on the values of u[,cond_on].
Nelsen RB (2006). An Introduction to Copulas, volume 139 of Lecture Notes in Statistics. Springer New York, New York, NY. ISBN 978-0-387-98623-4, doi:10.1007/978-1-4757-3076-0, https://link.springer.com/book/10.1007/978-1-4757-3076-0.
Hodel FH, Fieberg JR (2021). “Cylcop: An R Package for Circular-Linear Copulae with Angular Symmetry.” bioRxiv. doi:10.1101/2021.07.14.452253, https://www.biorxiv.org/content/10.1101/2021.07.14.452253v3/.
copula::cCopula()
cop <- cyl_quadsec(0.1) #calculate C_u(v) with u = 0.1 and v = 0.5 cylcop::ccylcop(u = c(0.1, 0.5), copula = cop, cond_on = 1, inverse = FALSE) #calculate C^-1_v(u) with u = 0.1 and v = 0.5 and with u = 0.4 and v = 0.2 cylcop::ccylcop(u = rbind(c(0.1, 0.5), c(0.4, 0.2)), copula = cop, cond_on = 2, inverse = TRUE)cop <- cyl_quadsec(0.1) #calculate C_u(v) with u = 0.1 and v = 0.5 cylcop::ccylcop(u = c(0.1, 0.5), copula = cop, cond_on = 1, inverse = FALSE) #calculate C^-1_v(u) with u = 0.1 and v = 0.5 and with u = 0.4 and v = 0.2 cylcop::ccylcop(u = rbind(c(0.1, 0.5), c(0.4, 0.2)), copula = cop, cond_on = 2, inverse = TRUE)
The code is based on Mardia (1976), Solow et al. (1988) and Tu (2015). The function returns a numeric value between 0 and 1, not -1 and 1, positive and negative correlation cannot be discerned. Note also that the correlation coefficient is independent of the marginal distributions.
cor_cyl(theta, x)cor_cyl(theta, x)
theta |
numeric vector of angles (measurements of a circular variable). |
x |
numeric vector of step lengths (measurements of a linear variable). |
A numeric value between 0 and 1, the circular-linear correlation coefficient.
Mardia KV (1976). “Linear-Circular Correlation Coefficients and Rhythmometry.” Biometrika, 63(2), 403–405. ISSN 00063444, doi:10.2307/2335637.
Solow AR, Bullister JL, Nevison C (1988). “An application of circular-linear correlation analysis to the relationship between Freon concentration and wind direction in Woods Hole, Massachusetts.” Environmental Monitoring and Assessment, 10(3), 219–228. ISSN 1573-2959, doi:10.1007/BF00395081, https://doi.org/10.1007/BF00395081.
Tu R (2015). “A Study of the Parametric and Nonparametric Linear-Circular Correlation Coefficient.” California Polytechnic State University, San Luis Obispo, 1–24. https://digitalcommons.calpoly.edu/statsp/51/.
Hodel FH, Fieberg JR (2021). “Cylcop: An R Package for Circular-Linear Copulae with Angular Symmetry.” bioRxiv. doi:10.1101/2021.07.14.452253, https://www.biorxiv.org/content/10.1101/2021.07.14.452253v3/.
mi_cyl(), fit_cylcop_cor().
set.seed(123) cop <- cyl_quadsec(0.1) #draw samples and calculate the correlation coefficient sample <- rcylcop(100, cop) cor_cyl(theta = sample[,1], x = sample[,2]) #the correlation coefficient is independent of the marginal distribution. sample <- traj_sim(100, cop, marginal_circ = list(name = "vonmises", coef = list(0, 1)), marginal_lin = list(name = "weibull", coef = list(shape = 2)) ) cor_cyl(theta = sample$angle, x = sample$steplength) cor_cyl(theta = sample$cop_u, x = sample$cop_v) # Estimate correlation of samples drawn from circular-linear copulas with # perfect correlation cop <- cyl_rect_combine(copula::normalCopula(1)) sample <- rcylcop(100, cop) cor_cyl(theta = sample[,1], x = sample[,2])set.seed(123) cop <- cyl_quadsec(0.1) #draw samples and calculate the correlation coefficient sample <- rcylcop(100, cop) cor_cyl(theta = sample[,1], x = sample[,2]) #the correlation coefficient is independent of the marginal distribution. sample <- traj_sim(100, cop, marginal_circ = list(name = "vonmises", coef = list(0, 1)), marginal_lin = list(name = "weibull", coef = list(shape = 2)) ) cor_cyl(theta = sample$angle, x = sample$steplength) cor_cyl(theta = sample$cop_u, x = sample$cop_v) # Estimate correlation of samples drawn from circular-linear copulas with # perfect correlation cop <- cyl_rect_combine(copula::normalCopula(1)) sample <- rcylcop(100, cop) cor_cyl(theta = sample[,1], x = sample[,2])
Calculate the Cramér-von-Mises criterion with a p-value (via parametric bootstrapping) to assess the goodness of fit of a parametric copula compared to the empirical copula of the data.
cramer_vonmises( copula, theta, x, n_bootstrap = 1000, parameters = NULL, optim.method = "L-BFGS-B", optim.control = list(maxit = 100) )cramer_vonmises( copula, theta, x, n_bootstrap = 1000, parameters = NULL, optim.method = "L-BFGS-B", optim.control = list(maxit = 100) )
copula |
R object of class ' |
theta |
numeric vector of angles (measurements of a circular variable) or "circular" component of pseudo-observations. |
x |
numeric vector of step lengths (measurements of a linear variable) or "linear" component of pseudo-observations. |
n_bootstrap |
integer number of bootstrap replicates. If
|
parameters |
vector of character strings
holding the names of the parameters to be optimized when using the bootstrap
procedure.
These can be any parameters in |
optim.method |
character string, optimizer used in
|
optim.control |
The Cramér-von Misses criterion is calculated as the sum of the squared
differences between the empirical copula and the parametric copula, copula,
evaluated at the pseudo-observations obtained from theta and x.
If the bootstrap procedure is used, a random sample is drawn from copula
and converted to pseudo-observations. A new (set of) copula parameter(s) is then
fit to those pseudo-observations using maximum likelihood (function
cylcop::fit_cylcop_ml()).
A list of length 2 containing the Cramér-von Mises criterion and the p-value.
Genest C, Rémillard B (2008). “Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models.” Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 44(6), 1096 – 1127. doi:10.1214/07-AIHP148.
set.seed(1234) sample <- rcylcop(100,cyl_cubsec(0.1, 0.1)) opt_cop <- fit_cylcop_ml(copula = cyl_quadsec(), theta = sample[,1], x = sample[,2], parameters = "a", start = 0 )$copula cramer_vonmises(opt_cop, theta = sample[,1], x = sample[,2], n_bootstrap=5)set.seed(1234) sample <- rcylcop(100,cyl_cubsec(0.1, 0.1)) opt_cop <- fit_cylcop_ml(copula = cyl_quadsec(), theta = sample[,1], x = sample[,2], parameters = "a", start = 0 )$copula cramer_vonmises(opt_cop, theta = sample[,1], x = sample[,2], n_bootstrap=5)
The class 'cyl_copula' follows somewhat the structure of the class
'Copula' of the package 'copula'. It contains
circular-linear copulas.
namecharacter string holding the name of the copula.
parametersparam.namesparam.lowbndparam.upbnd'cyl_copula' is extended by the following classes:
'cyl_vonmises': von Mises copulas.
'cyl_quadsec': Copulas with quadratic sections.
'cyl_cubsec': Copulas with cubic sections.
'cyl_rot_combine': Linear combinations of copulas and their
180 degree rotations.
'cyl_rect_combine': Rectangular patchwork copulas.
Objects are created by the functions cyl_vonmises(),
cyl_quadsec(), cyl_cubsec(), cyl_rot_combine(),
and cyl_rect_combine().
Hodel FH, Fieberg JR (2022). “Circular-Linear Copulae for Animal Movement Data.” Methods in Ecology and Evolution. doi:10.1111/2041-210X.13821.
Hodel FH, Fieberg JR (2021). “Cylcop: An R Package for Circular-Linear Copulae with Angular Symmetry.” bioRxiv. doi:10.1101/2021.07.14.452253, https://www.biorxiv.org/content/10.1101/2021.07.14.452253v3/.
cop <- cyl_quadsec(0.1) is(cop)cop <- cyl_quadsec(0.1) is(cop)
cyl_cubsec' ObjectsConstructs a circular-linear copula with cubic sections of class
'cyl_cubsec'.
cyl_cubsec(a = 1/(2 * pi), b = 1/(2 * pi))cyl_cubsec(a = 1/(2 * pi), b = 1/(2 * pi))
a |
numeric value of the first parameter of the copula.
It must be in |
b |
numeric value of the second parameter of the copula.
It must be in |
An R object of class 'cyl_cubsec'.
Nelsen RB, Quesada-Molina JJ, Rodríguez-Lallena JA (1997). “Bivariate copulas with cubic sections.” Journal of Nonparametric Statistics, 7(3), 205–220. ISSN 10485252, doi:10.1080/10485259708832700.
Hodel FH, Fieberg JR (2022). “Circular-Linear Copulae for Animal Movement Data.” Methods in Ecology and Evolution. doi:10.1111/2041-210X.13821.
Hodel FH, Fieberg JR (2021). “Cylcop: An R Package for Circular-Linear Copulae with Angular Symmetry.” bioRxiv. doi:10.1101/2021.07.14.452253, https://www.biorxiv.org/content/10.1101/2021.07.14.452253v3/.
cop <- cyl_cubsec(a = 0.1, b = -0.1) if(interactive()){ plot_cop_surf(copula = cop, type = "pdf", plot_type = "ggplot") }cop <- cyl_cubsec(a = 0.1, b = -0.1) if(interactive()){ plot_cop_surf(copula = cop, type = "pdf", plot_type = "ggplot") }
This class contains bivariate circular-linear copulas with cubic sections in the linear dimension.
They are periodic in the circular dimension, , and symmetric with respect
to . Therefore,
they can capture correlation in data where there is symmetry between positive and negative angles.
These copulas are described by two parameters, a and b.
namecharacter string holding the name of the copula.
parametersparam.namesparam.lowbndparam.upbndObjects are created by cyl_cubsec().
Class 'cyl_cubsec' extends class 'cyl_copula'.
Nelsen RB, Quesada-Molina JJ, Rodríguez-Lallena JA (1997). “Bivariate copulas with cubic sections.” Journal of Nonparametric Statistics, 7(3), 205–220. ISSN 10485252, doi:10.1080/10485259708832700.
Hodel FH, Fieberg JR (2022). “Circular-Linear Copulae for Animal Movement Data.” Methods in Ecology and Evolution. doi:10.1111/2041-210X.13821.
Hodel FH, Fieberg JR (2021). “Cylcop: An R Package for Circular-Linear Copulae with Angular Symmetry.” bioRxiv. doi:10.1101/2021.07.14.452253, https://www.biorxiv.org/content/10.1101/2021.07.14.452253v3/.
cyl_quadsec' ObjectsConstructs a circular-linear copula with cubic sections of class
'cyl_quadsec'.
cyl_quadsec(a = 1/(2 * pi))cyl_quadsec(a = 1/(2 * pi))
a |
numeric value of the parameter of the copula. It must be in
|
An R object of class 'cyl_quadsec'.
Quesada-Molina JJ, Rodríguez-Lallena JA (1995). “Bivariate copulas with quadratic sections.” Journal of Nonparametric Statistics, 5(4), 323–337. ISSN 10290311, doi:10.1080/10485259508832652.
Hodel FH, Fieberg JR (2022). “Circular-Linear Copulae for Animal Movement Data.” Methods in Ecology and Evolution. doi:10.1111/2041-210X.13821.
Hodel FH, Fieberg JR (2021). “Cylcop: An R Package for Circular-Linear Copulae with Angular Symmetry.” bioRxiv. doi:10.1101/2021.07.14.452253, https://www.biorxiv.org/content/10.1101/2021.07.14.452253v3/.
cop <- cyl_quadsec(a = 0.1) if(interactive()){ plot_cop_surf(copula = cop, type = "pdf", plot_type = "ggplot") }cop <- cyl_quadsec(a = 0.1) if(interactive()){ plot_cop_surf(copula = cop, type = "pdf", plot_type = "ggplot") }
This class contains bivariate circular-linear copulas with quadratic sections
in the linear dimension. They are periodic in the circular dimension, ,
and symmetric with respect to . Therefore,
they can capture correlation in data where there is symmetry between positive
and negative angles. These copulas are described by one parameter, a.
namecharacter string holding the name of the copula.
parametersparam.namesparam.lowbndparam.upbndObjects are created by cyl_quadsec().
Class 'cyl_quadsec' extends class 'cyl_copula'.
Quesada-Molina JJ, Rodríguez-Lallena JA (1995). “Bivariate copulas with quadratic sections.” Journal of Nonparametric Statistics, 5(4), 323–337. ISSN 10290311, doi:10.1080/10485259508832652.
Hodel FH, Fieberg JR (2022). “Circular-Linear Copulae for Animal Movement Data.” Methods in Ecology and Evolution. doi:10.1111/2041-210X.13821.
Hodel FH, Fieberg JR (2021). “Cylcop: An R Package for Circular-Linear Copulae with Angular Symmetry.” bioRxiv. doi:10.1101/2021.07.14.452253, https://www.biorxiv.org/content/10.1101/2021.07.14.452253v3/.
cyl_rect_combine' ObjectsConstructs a circular-linear copula of class
'cyl_rect_combine' from a rectangular patchwork of copulas.
cyl_rect_combine( copula, background = indepCopula(), low_rect = c(0, 0.5), up_rect = "symmetric", flip_up = TRUE )cyl_rect_combine( copula, background = indepCopula(), low_rect = c(0, 0.5), up_rect = "symmetric", flip_up = TRUE )
copula |
' |
background |
' |
low_rect |
numeric vector of length 2 containing the lower and upper edge (u-value) of the lower rectangle. |
up_rect |
numeric vector of length 2 containing the lower and upper edge (u-value) of the upper rectangle, or the character string "symmetric" if it should be the mirror image (with respect to u=0.5) of the lower rectangle. |
flip_up |
logical value indicating whether the copula ( |
An R object of class 'cyl_rect_combine'.
Durante F, Saminger-Platz S, Sarkoci P (2009). “Rectangular patchwork for bivariate copulas and tail dependence.” Communications in Statistics - Theory and Methods, 38(15), 2515–2527. ISSN 03610926, doi:10.1080/03610920802571203.
Hodel FH, Fieberg JR (2022). “Circular-Linear Copulae for Animal Movement Data.” Methods in Ecology and Evolution. doi:10.1111/2041-210X.13821.
Hodel FH, Fieberg JR (2021). “Cylcop: An R Package for Circular-Linear Copulae with Angular Symmetry.” bioRxiv. doi:10.1101/2021.07.14.452253, https://www.biorxiv.org/content/10.1101/2021.07.14.452253v3/.
#symmetric rectangles spanning entire unit square cop <- cyl_rect_combine(copula::frankCopula(2)) if(interactive()){ plot_cop_surf(copula = cop, type = "pdf", plot_type = "ggplot", resolution = 20) } #symmetric rectangles, independence copula as background cop <- cyl_rect_combine(copula::frankCopula(2), low_rect = c(0, 0.3), up_rect = "symmetric", flip_up = FALSE ) if(interactive()){ plot_cop_surf(copula = cop, type = "pdf", plot_type = "ggplot", resolution = 20) } #symmetric rectangles, cy_quadsec-copula as background cop <- cyl_rect_combine(copula::normalCopula(0.3), low_rect = c(0.1, 0.4), up_rect = "symmetric", background = cyl_quadsec(-0.1) ) if(interactive()){ plot_cop_surf(copula = cop, type = "pdf", plot_type = "ggplot", resolution = 20) } #asymmetric rectangles, von Mises copula as background. #!!Not a symmetric circular linear copula!! cop <- cyl_rect_combine(copula::normalCopula(0.3), low_rect = c(0.1, 0.4), up_rect = c(0.5, 0.7), background = cyl_vonmises(mu = pi, kappa = 0.3) ) if(interactive()){ plot_cop_surf(copula = cop, type = "pdf", plot_type = "ggplot", resolution = 20) }#symmetric rectangles spanning entire unit square cop <- cyl_rect_combine(copula::frankCopula(2)) if(interactive()){ plot_cop_surf(copula = cop, type = "pdf", plot_type = "ggplot", resolution = 20) } #symmetric rectangles, independence copula as background cop <- cyl_rect_combine(copula::frankCopula(2), low_rect = c(0, 0.3), up_rect = "symmetric", flip_up = FALSE ) if(interactive()){ plot_cop_surf(copula = cop, type = "pdf", plot_type = "ggplot", resolution = 20) } #symmetric rectangles, cy_quadsec-copula as background cop <- cyl_rect_combine(copula::normalCopula(0.3), low_rect = c(0.1, 0.4), up_rect = "symmetric", background = cyl_quadsec(-0.1) ) if(interactive()){ plot_cop_surf(copula = cop, type = "pdf", plot_type = "ggplot", resolution = 20) } #asymmetric rectangles, von Mises copula as background. #!!Not a symmetric circular linear copula!! cop <- cyl_rect_combine(copula::normalCopula(0.3), low_rect = c(0.1, 0.4), up_rect = c(0.5, 0.7), background = cyl_vonmises(mu = pi, kappa = 0.3) ) if(interactive()){ plot_cop_surf(copula = cop, type = "pdf", plot_type = "ggplot", resolution = 20) }
This class contains bivariate circular-linear copulas generated from
linear-linear bivariate 'Copula' objects of the package
'copula' or circular-linear copulas of class 'cyl_copula'.
2 non-overlapping rectangles are laid over the unit square, both have width
1 in v-direction. In the area covered by the first rectangle, the copula is
derived from a linear-linear bivariate 'Copula' object.
Rectangle 2 contains the same copula as rectangle 1, but 90 degrees rotated.
In the area not covered by the rectangles, the "background", the copula is
derived from a circular-linear 'cyl_copula' object.
The copula regions are combined in a way that the overall result on the entire
unit square is also a copula.
With appropriate choices of the rectangles this results in copulas
that are periodic in u-direction (and not in v-direction) and therefore are
circular-linear. When the 2 rectangles are mirror images with
respect to , the resulting overall copula is symmetric with respect
to , i.e. there is symmetry between positive and negative angles.
Note that as "background copula", we can also chose a linear-linear copula, the overall result will then, however, not be a symmetric circular linear copula.
namecharacter string holding the name of the copula.
parametersparam.namesparam.lowbndparam.upbndsym.cop'Copula' object of the package
'copula' or 'cyl_vonmises' object. The copula in
the rectangles.
background.cop'cyl_vonmises' or
'Copula' object of the package 'copula',
the copula where no rectangles overlay the unit square. If this copula is not
symmetric, the overall cyl_rect_combine-copula will also not be symmetric.
flip_uplogical value indicating whether the copula (sym.cop) is
rotated 90 degrees in the upper or lower rectangle.
sym_rectlogical value indicating whether the upper rectangle was forced to be a mirror image of the lower one with respect to u=0.5 at the construction of the object.
Objects are created by
cyl_rect_combine().
Class 'cyl_rect_combine' extends class 'Copula'.
Durante F, Saminger-Platz S, Sarkoci P (2009). “Rectangular patchwork for bivariate copulas and tail dependence.” Communications in Statistics - Theory and Methods, 38(15), 2515–2527. ISSN 03610926, doi:10.1080/03610920802571203.
Hodel FH, Fieberg JR (2022). “Circular-Linear Copulae for Animal Movement Data.” Methods in Ecology and Evolution. doi:10.1111/2041-210X.13821.
Hodel FH, Fieberg JR (2021). “Cylcop: An R Package for Circular-Linear Copulae with Angular Symmetry.” bioRxiv. doi:10.1101/2021.07.14.452253, https://www.biorxiv.org/content/10.1101/2021.07.14.452253v3/.
cyl_rot_combine' ObjectsConstructs a circular-linear copula of class
'cyl_rot_combine' from linear combinations
of copulas.
cyl_rot_combine(copula, shift = FALSE)cyl_rot_combine(copula, shift = FALSE)
copula |
linear-linear 2-dimensional ' |
shift |
logical value indicating whether the (u-periodic) copula should be shifted by 0.5 in u direction. |
An R object of class 'cyl_rot_combine'.
Nelsen RB (2006). An Introduction to Copulas, volume 139 of Lecture Notes in Statistics. Springer New York, New York, NY. ISBN 978-0-387-98623-4, doi:10.1007/978-1-4757-3076-0, https://link.springer.com/book/10.1007/978-1-4757-3076-0.
Hodel FH, Fieberg JR (2022). “Circular-Linear Copulae for Animal Movement Data.” Methods in Ecology and Evolution. doi:10.1111/2041-210X.13821.
Hodel FH, Fieberg JR (2021). “Cylcop: An R Package for Circular-Linear Copulae with Angular Symmetry.” bioRxiv. doi:10.1101/2021.07.14.452253, https://www.biorxiv.org/content/10.1101/2021.07.14.452253v3/.
cop <- cyl_rot_combine(copula = copula::frankCopula(param = 3), shift = TRUE) if(interactive()){ plot_cop_surf(copula = cop, type = "pdf", plot_type = "ggplot", resolution = 20) } cop <- cyl_rot_combine(copula = copula::claytonCopula(param = 10), shift = FALSE) if(interactive()){ plot_cop_surf(copula = cop, type = "pdf", plot_type = "ggplot", resolution = 20) }cop <- cyl_rot_combine(copula = copula::frankCopula(param = 3), shift = TRUE) if(interactive()){ plot_cop_surf(copula = cop, type = "pdf", plot_type = "ggplot", resolution = 20) } cop <- cyl_rot_combine(copula = copula::claytonCopula(param = 10), shift = FALSE) if(interactive()){ plot_cop_surf(copula = cop, type = "pdf", plot_type = "ggplot", resolution = 20) }
This class contains bivariate circular-linear copulas, generated from
linear-linear bivariate 'Copula' objects of the package
'copula', by taking the arithmetic mean of the original copula and
the 90 deg rotated copula. This results in copulas that are periodic in the
circular dimension, u, and symmetric with respect to , i.e. positive
and negative angles.
namecharacter string holding the name of the copula.
parametersparam.namesparam.lowbndparam.upbndorig.coplinear-linear 2-dimensional 'Copula'
object of the package 'copula'.
shiftlogical value indicating whether the (u-periodic) copula should be shifted by 0.5 in u direction.
Objects are created by cyl_rot_combine().
Class 'cyl_rot_combine' extends class 'Copula'.
Nelsen RB (2006). An Introduction to Copulas, volume 139 of Lecture Notes in Statistics. Springer New York, New York, NY. ISBN 978-0-387-98623-4, doi:10.1007/978-1-4757-3076-0, https://link.springer.com/book/10.1007/978-1-4757-3076-0.
Hodel FH, Fieberg JR (2022). “Circular-Linear Copulae for Animal Movement Data.” Methods in Ecology and Evolution. doi:10.1111/2041-210X.13821.
Hodel FH, Fieberg JR (2021). “Cylcop: An R Package for Circular-Linear Copulae with Angular Symmetry.” bioRxiv. doi:10.1101/2021.07.14.452253, https://www.biorxiv.org/content/10.1101/2021.07.14.452253v3/.
cyl_vonmises' ObjectsConstructs a circular-linear von Mises copula according to
Johnson and Wehrly (1978) of class
'cyl_vonmises'.
cyl_vonmises(mu = 0, kappa = 1, flip = FALSE)cyl_vonmises(mu = 0, kappa = 1, flip = FALSE)
mu |
numeric value giving the mean of the von Mises function used to construct the copula. |
kappa |
numeric value giving the concentration of the von Mises function used to construct the copula. |
flip |
logical value indicating whether the copula should be rotated 90 degrees to capture negative correlation. |
An R object of class 'cyl_vonmises'.
Johnson RA, Wehrly TE (1978). “Some Angular-Linear Distributions and Related Regression Models.” Journal of the American Statistical Association ISSN:, 73(363), 602–606. ISSN 00401706, doi:10.2307/1270921.
Hodel FH, Fieberg JR (2022). “Circular-Linear Copulae for Animal Movement Data.” Methods in Ecology and Evolution. doi:10.1111/2041-210X.13821.
Hodel FH, Fieberg JR (2021). “Cylcop: An R Package for Circular-Linear Copulae with Angular Symmetry.” bioRxiv. doi:10.1101/2021.07.14.452253, https://www.biorxiv.org/content/10.1101/2021.07.14.452253v3/.
cop <- cyl_vonmises(mu=pi, kappa=10, flip = TRUE) if(interactive()){ plot_cop_surf(copula = cop, type = "pdf", plot_type = "ggplot", resolution = 20) } cop <- cyl_vonmises(mu=0, kappa=8, flip = FALSE) if(interactive()){ plot_cop_surf(copula = cop, type = "pdf", plot_type = "ggplot", resolution = 20) }cop <- cyl_vonmises(mu=pi, kappa=10, flip = TRUE) if(interactive()){ plot_cop_surf(copula = cop, type = "pdf", plot_type = "ggplot", resolution = 20) } cop <- cyl_vonmises(mu=0, kappa=8, flip = FALSE) if(interactive()){ plot_cop_surf(copula = cop, type = "pdf", plot_type = "ggplot", resolution = 20) }
This class contains circular-linear copulas that are based on the approach by
Johnson and Wehrly (1978) with a von Mises periodic function.
They are periodic in the circular dimension, u, but not symmetric with
respect to i.e. there is no symmetry between positive and negative angles.
namecharacter string holding the name of the copula.
parametersparam.namesparam.lowbndparam.upbndfliplogical value indicating whether the copula should be rotated 90 degrees to capture negative correlation.
Objects are created by cyl_vonmises().
Class 'cyl_vonmises' extends class 'cyl_copula'.
Johnson RA, Wehrly TE (1978). “Some Angular-Linear Distributions and Related Regression Models.” Journal of the American Statistical Association ISSN:, 73(363), 602–606. ISSN 00401706, doi:10.2307/1270921.
Hodel FH, Fieberg JR (2022). “Circular-Linear Copulae for Animal Movement Data.” Methods in Ecology and Evolution. doi:10.1111/2041-210X.13821.
Hodel FH, Fieberg JR (2021). “Cylcop: An R Package for Circular-Linear Copulae with Angular Symmetry.” bioRxiv. doi:10.1101/2021.07.14.452253, https://www.biorxiv.org/content/10.1101/2021.07.14.452253v3/.
Calculate the distribution (pcylcop()), the density (dcylcop()),
and generate random
samples (rcylcop()) of a 'cyl_copula' object or a
'Copula' object (package 'copula', only 2-dimensional).
pcylcop(u, copula) rcylcop(n, copula) dcylcop(u, copula, log = FALSE) ## S4 method for signature 'matrix,Copula' dcylcop(u, copula) ## S4 method for signature 'numeric,Copula' rcylcop(n, copula) ## S4 method for signature 'matrix,Copula' pcylcop(u, copula) ## S4 method for signature 'numeric,cyl_cubsec' rcylcop(n, copula) ## S4 method for signature 'matrix,cyl_cubsec' dcylcop(u, copula) ## S4 method for signature 'matrix,cyl_cubsec' pcylcop(u, copula) ## S4 method for signature 'numeric,cyl_quadsec' rcylcop(n, copula) ## S4 method for signature 'matrix,cyl_quadsec' dcylcop(u, copula) ## S4 method for signature 'matrix,cyl_quadsec' pcylcop(u, copula) ## S4 method for signature 'numeric,cyl_rect_combine' rcylcop(n, copula) ## S4 method for signature 'matrix,cyl_rect_combine' dcylcop(u, copula) ## S4 method for signature 'matrix,cyl_rect_combine' pcylcop(u, copula) ## S4 method for signature 'numeric,cyl_rot_combine' rcylcop(n, copula) ## S4 method for signature 'matrix,cyl_rot_combine' dcylcop(u, copula) ## S4 method for signature 'matrix,cyl_rot_combine' pcylcop(u, copula) ## S4 method for signature 'numeric,cyl_vonmises' rcylcop(n, copula) ## S4 method for signature 'matrix,cyl_vonmises' dcylcop(u, copula) ## S4 method for signature 'matrix,cyl_vonmises' pcylcop(u, copula)pcylcop(u, copula) rcylcop(n, copula) dcylcop(u, copula, log = FALSE) ## S4 method for signature 'matrix,Copula' dcylcop(u, copula) ## S4 method for signature 'numeric,Copula' rcylcop(n, copula) ## S4 method for signature 'matrix,Copula' pcylcop(u, copula) ## S4 method for signature 'numeric,cyl_cubsec' rcylcop(n, copula) ## S4 method for signature 'matrix,cyl_cubsec' dcylcop(u, copula) ## S4 method for signature 'matrix,cyl_cubsec' pcylcop(u, copula) ## S4 method for signature 'numeric,cyl_quadsec' rcylcop(n, copula) ## S4 method for signature 'matrix,cyl_quadsec' dcylcop(u, copula) ## S4 method for signature 'matrix,cyl_quadsec' pcylcop(u, copula) ## S4 method for signature 'numeric,cyl_rect_combine' rcylcop(n, copula) ## S4 method for signature 'matrix,cyl_rect_combine' dcylcop(u, copula) ## S4 method for signature 'matrix,cyl_rect_combine' pcylcop(u, copula) ## S4 method for signature 'numeric,cyl_rot_combine' rcylcop(n, copula) ## S4 method for signature 'matrix,cyl_rot_combine' dcylcop(u, copula) ## S4 method for signature 'matrix,cyl_rot_combine' pcylcop(u, copula) ## S4 method for signature 'numeric,cyl_vonmises' rcylcop(n, copula) ## S4 method for signature 'matrix,cyl_vonmises' dcylcop(u, copula) ## S4 method for signature 'matrix,cyl_vonmises' pcylcop(u, copula)
u |
matrix (or vector) of numeric
values in |
copula |
R object of class ' |
n |
number of random samples to be generated with |
log |
logical indicating if the logarithm of the density
should be returned ( |
For 'Copula' objects, pcylcop() and rcylcop()
just call the functions of the 'copula' package
pCopula() and rCopula(), respectively.
The density is, however, calculated differently in dcylcop() and
dCopula(). The difference is
that copula::dCopula()
will return a density of 0 for points on the boundary of the unit square,
whereas dcylcop() will return the correct density on the boundaries
for both 'cyl_copula' and 'Copula' objects.
The functions pcylcop() and dcylcop() give a vector of
length nrow(u) containing the distribution and the density, respectively,
at the corresponding values of u. The function rcylcop() generates a
matrix with 2 columns and n rows containing
the random samples.
Nelsen RB (2006). An Introduction to Copulas, volume 139 of Lecture Notes in Statistics. Springer New York, New York, NY. ISBN 978-0-387-98623-4, doi:10.1007/978-1-4757-3076-0, https://link.springer.com/book/10.1007/978-1-4757-3076-0.
Hodel FH, Fieberg JR (2021). “Cylcop: An R Package for Circular-Linear Copulae with Angular Symmetry.” bioRxiv. doi:10.1101/2021.07.14.452253, https://www.biorxiv.org/content/10.1101/2021.07.14.452253v3/.
copula::dCopula(),
copula::pCopula(),
copula::rCopula().
set.seed(123) cop <- cyl_quadsec(0.1) rcylcop(5, cop) pcylcop(c(0.3, 0.1), cop) pcylcop(rbind(c(0.3, 0.1), c(0.2, 1)), cop) cop <- cyl_rot_combine(copula::frankCopula(2), shift = TRUE) dcylcop(u = rbind(c(0.1, 0.4), c(1.0, 0.2)), copula = cop) dcylcop(c(0.1, 0.3), cyl_quadsec(0.1), log = TRUE) cop <- copula::normalCopula(0.3) copula::dCopula(c(.Machine$double.eps,0.2),cop) copula::dCopula(c(0,0.2),cop) dcylcop(c(.Machine$double.eps,0.2),cop) dcylcop(c(0,0.2),cop)set.seed(123) cop <- cyl_quadsec(0.1) rcylcop(5, cop) pcylcop(c(0.3, 0.1), cop) pcylcop(rbind(c(0.3, 0.1), c(0.2, 1)), cop) cop <- cyl_rot_combine(copula::frankCopula(2), shift = TRUE) dcylcop(u = rbind(c(0.1, 0.4), c(1.0, 0.2)), copula = cop) dcylcop(c(0.1, 0.3), cyl_quadsec(0.1), log = TRUE) cop <- copula::normalCopula(0.3) copula::dCopula(c(.Machine$double.eps,0.2),cop) copula::dCopula(c(0,0.2),cop) dcylcop(c(.Machine$double.eps,0.2),cop) dcylcop(c(0,0.2),cop)
Currently the only option ("silent") is to toggle verbosity on or off.
cylcop_get_option(option = NULL)cylcop_get_option(option = NULL)
option |
character string, the name of the option. |
The numeric value of option. If no argument is provided, a list of all options is printed.
cylcop_get_option("silent") cylcop_get_option()cylcop_get_option("silent") cylcop_get_option()
Currently the only option is to toggle verbosity on or off.
cylcop_set_option(silent = FALSE)cylcop_set_option(silent = FALSE)
silent |
logical, suppress all sounds and messages. |
No output, only side effects.
cylcop_set_option(silent = FALSE)cylcop_set_option(silent = FALSE)
These functions are provided for compatibility with older version of the cylcop package. They may eventually be completely
scat_plot(...)scat_plot(...)
... |
Parameters to be passed to the new versions of the functions |
scat_plot() |
is replaced by plot_joint_scat()
|
traj_plot() |
is replaced by plot_track()
|
circ_plot() |
is replaced by plot_joint_circ()
|
cop_scat_plot() |
is replaced by plot_cop_scat()
|
cop_plot() |
is replaced by plot_cop_surf()
|
make_traj() |
is replaced by traj_sim()
|
qmixedvonmises() |
is replaced by qvonmisesmix()
|
mle.mixedvonmises() |
is replaced by mle.vonmisesmix()
|
Calculate the density (ddens()), the distribution (pdens()),
the quantiles (qdens()) and generate random
samples (rdens()) of a kernel density estimate as returned by
fit_angle() or fit_steplength().
rdens(n, density) ddens(x, density) pdens(x, density) qdens(p, density)rdens(n, density) ddens(x, density) pdens(x, density) qdens(p, density)
n |
integer value, the number of random samples to be
generated with |
density |
a ' |
x |
numeric vector giving the points where the density or distribution function is evaluated. |
p |
numeric vector giving the probabilities where the quantile function is evaluated. |
ddens() and pdens() give a vector of length length(x) containing
the density or distribution function at the corresponding values of x.
qdens() gives a vector of length length(p) containing
the quantiles at the corresponding values of p. The function rdens()
generates a vector of length n containing the random samples.
fit_angle(), fit_steplength(),
fit_steplength().
set.seed(123) steps <- rweibull(10, shape=3) dens <- fit_steplength(x = steps, parametric = FALSE) ddens(c(0.1,0.3), dens) pdens(c(0.1,0.3), dens) qdens(c(0.1,0.3), dens) rdens(4, dens) angles <- full2half_circ( circular::rvonmises(10, mu = circular::circular(0), kappa = 2) ) dens <- fit_angle(theta = angles, parametric = FALSE) ddens(c(0.1,0.3), dens) pdens(c(0.1,0.3), dens) qdens(c(0.1,0.3), dens) rdens(4, dens)set.seed(123) steps <- rweibull(10, shape=3) dens <- fit_steplength(x = steps, parametric = FALSE) ddens(c(0.1,0.3), dens) pdens(c(0.1,0.3), dens) qdens(c(0.1,0.3), dens) rdens(4, dens) angles <- full2half_circ( circular::rvonmises(10, mu = circular::circular(0), kappa = 2) ) dens <- fit_angle(theta = angles, parametric = FALSE) ddens(c(0.1,0.3), dens) pdens(c(0.1,0.3), dens) qdens(c(0.1,0.3), dens) rdens(4, dens)
This function finds parameter estimates of the marginal circular distribution (with potentially fixed mean), or gives a kernel density estimate using a von Mises smoothing kernel.
fit_angle( theta, parametric = c("vonmises", "wrappedcauchy", "vonmisesmix", FALSE), bandwidth = NULL, mu = NULL, ncomp = 2 )fit_angle( theta, parametric = c("vonmises", "wrappedcauchy", "vonmisesmix", FALSE), bandwidth = NULL, mu = NULL, ncomp = 2 )
theta |
|
parametric |
either a character string describing what distribution
should be fitted ( |
bandwidth |
If |
mu |
(optional) numeric vector, fixed mean direction(s) of the parametric distribution. |
ncomp |
integer, number of components of the mixed von Mises distribution.
Only has an effect if |
If a parametric estimate is made, a list is returned
containing the estimated parameters, their
standard errors (if available), the log-likelihood,
the AIC and the name of the distribution.
If a non-parametric estimate is made, the output is a 'density.circular' object
obtained with the function circular::density.circular() of the 'circular'
package.
circular::density.circular(),
fit_angle(), opt_circ_bw().
set.seed(123) silent_curr <- cylcop_get_option("silent") cylcop_set_option(silent = TRUE) n <- 10 #n (number of samples) is set small for performance. angles <- rvonmisesmix(n, mu = c(0, pi), kappa = c(2,1), prop = c(0.5, 0.5) ) bw <- opt_circ_bw(theta = angles, method="nrd", kappa.est = "trigmoments" ) dens_non_param <- fit_angle(theta = angles, parametric = FALSE, bandwidth = bw ) param_estimate <- fit_angle(theta = angles, parametric = "vonmisesmix" ) param_estimate_fixed_mean <- fit_angle(theta = angles, parametric = "vonmisesmix", mu = c(0, pi), ncomp =2 ) cylcop_set_option(silent = silent_curr)set.seed(123) silent_curr <- cylcop_get_option("silent") cylcop_set_option(silent = TRUE) n <- 10 #n (number of samples) is set small for performance. angles <- rvonmisesmix(n, mu = c(0, pi), kappa = c(2,1), prop = c(0.5, 0.5) ) bw <- opt_circ_bw(theta = angles, method="nrd", kappa.est = "trigmoments" ) dens_non_param <- fit_angle(theta = angles, parametric = FALSE, bandwidth = bw ) param_estimate <- fit_angle(theta = angles, parametric = "vonmisesmix" ) param_estimate_fixed_mean <- fit_angle(theta = angles, parametric = "vonmisesmix", mu = c(0, pi), ncomp =2 ) cylcop_set_option(silent = silent_curr)
This function implements a simple search of the parameter space of a
'cyl_copula' object to find the
parameter values that lead to a correlation that is closest to the correlation
in the data (theta and x). In some special cases of
'cyl_rect_combine' copulas, the parameter can be
obtained analytically from Kendall's tau of the data.
fit_cylcop_cor(copula, theta, x, acc = NULL, n = 10000, method, ...) ## S4 method for signature 'cyl_vonmises' fit_cylcop_cor(copula, theta, x, acc, n, method = "cor_cyl") ## S4 method for signature 'cyl_quadsec' fit_cylcop_cor(copula, theta, x, acc, n, method = "cor_cyl") ## S4 method for signature 'cyl_cubsec' fit_cylcop_cor( copula, theta, x, acc, n, method = "cor_cyl", parameter = "both" ) ## S4 method for signature 'cyl_rot_combine' fit_cylcop_cor(copula, theta, x, acc, n, method = "mi_cyl") ## S4 method for signature 'cyl_rect_combine' fit_cylcop_cor(copula, theta, x, acc, n, method = "tau", background = FALSE) optCor(copula, theta, x, acc = NULL, n = 10000, method, ...)fit_cylcop_cor(copula, theta, x, acc = NULL, n = 10000, method, ...) ## S4 method for signature 'cyl_vonmises' fit_cylcop_cor(copula, theta, x, acc, n, method = "cor_cyl") ## S4 method for signature 'cyl_quadsec' fit_cylcop_cor(copula, theta, x, acc, n, method = "cor_cyl") ## S4 method for signature 'cyl_cubsec' fit_cylcop_cor( copula, theta, x, acc, n, method = "cor_cyl", parameter = "both" ) ## S4 method for signature 'cyl_rot_combine' fit_cylcop_cor(copula, theta, x, acc, n, method = "mi_cyl") ## S4 method for signature 'cyl_rect_combine' fit_cylcop_cor(copula, theta, x, acc, n, method = "tau", background = FALSE) optCor(copula, theta, x, acc = NULL, n = 10000, method, ...)
copula |
R object of class ' |
theta |
numeric vector of angles (measurements of a circular variable). |
x |
numeric vector of step lengths (measurements of a linear variable). |
acc |
numeric value, the interval of the copula parameter at which to evaluate the correlation. |
n |
numeric value, the number of sample points at each optimization step. |
method |
character string describing what correlation metric
to use. Either a rank-based circular-linear correlation coefficient ( |
... |
Additional parameters (see individual methods). |
parameter |
For ' |
background |
For ' |
The code assumes that the correlation captured by the copula increases
monotonously with the copula parameter values. It starts with a parameter value close
to the minimum for that copula and calculates the correlation for a sample of size n
from that copula. Next, the parameter is doubled and again the correlation for a sample
of size n calculated. After this exponential search pattern, a binary search
is implemented similarly between the bounds found with the exponential search. For this
binary search, the interval between those bounds is split into small intervals of length
acc. Thus, smaller values of acc lead to higher accuracy.
If a 'cyl_rect_combine' copula has rectangles spanning
the entire unit square and as background the independence copula, Kendall's tau can be used
to analytically calculate the parameter value leading to the correlation of the data.
No search is necessary in this case. This makes it the recommended method to use
for those 'cyl_rect_combine' copulas.
optCor() is an alias for fit_cylcop_cor.
See also individual methods (below) for more detailed explanations.
numeric vector containing the estimated parameter value(s).
fit_cylcop_cor(cyl_vonmises): only parameter "kappa" can be optimized, since parameter
"mu" does not influence the correlation.
fit_cylcop_cor(cyl_quadsec): the absolute value of the parameter is optimized, positive
and negative values give the same correlation.
fit_cylcop_cor(cyl_cubsec): optimization of parameters, "a" and "b",
can be done separately or simultaneously.
fit_cylcop_cor(cyl_rot_combine): the circular-linear correlation coefficient will give a
value close to 0 for any parameter value. It therefore only makes sense to
use method = "mi_cyl" for the optimization.
fit_cylcop_cor(cyl_rect_combine): if the rectangles span the entire unit square and the background
is the independence copula, it is recommended to use method = "tau", since this
calculates the copula parameter analytically. If there is a background copula,
other than the independence copula, its parameter can be optimized by setting
background=TRUE.
Hodel FH, Fieberg JR (2022). “Circular-Linear Copulae for Animal Movement Data.” Methods in Ecology and Evolution. doi:10.1111/2041-210X.13821.
Hodel FH, Fieberg JR (2021). “Cylcop: An R Package for Circular-Linear Copulae with Angular Symmetry.” bioRxiv. doi:10.1101/2021.07.14.452253, https://www.biorxiv.org/content/10.1101/2021.07.14.452253v3/.
mi_cyl(), cor_cyl(), fit_cylcop_ml(),
opt_auto(), copula::fitCopula().
set.seed(123) sample <- rcylcop(100, cyl_rect_combine(copula::frankCopula(2))) fit_cylcop_cor(cyl_rect_combine(copula::frankCopula()), theta = sample[,1], x = sample[,2], method = "tau" ) fit_cylcop_cor(cyl_rect_combine(copula::frankCopula()), theta = sample[,1], x = sample[,2], method = "mi_cyl", n = 100 ) fit_cylcop_cor(cyl_rect_combine(copula::claytonCopula()), theta = sample[,1], x = sample[,2], method = "tau" ) fit_cylcop_cor(cyl_quadsec(), theta = sample[,1], x = sample[,2], method = "mi_cyl") fit_cylcop_cor(cyl_quadsec(), theta = sample[,1], x = sample[,2], method = "cor_cyl") fit_cylcop_cor(cyl_quadsec(), theta = sample[,1], x = sample[,2], method = "cor_cyl", n = 100, acc = 0.001 ) optCor(cyl_quadsec(), theta = sample[,1], x = sample[,2], method = "mi_cyl")set.seed(123) sample <- rcylcop(100, cyl_rect_combine(copula::frankCopula(2))) fit_cylcop_cor(cyl_rect_combine(copula::frankCopula()), theta = sample[,1], x = sample[,2], method = "tau" ) fit_cylcop_cor(cyl_rect_combine(copula::frankCopula()), theta = sample[,1], x = sample[,2], method = "mi_cyl", n = 100 ) fit_cylcop_cor(cyl_rect_combine(copula::claytonCopula()), theta = sample[,1], x = sample[,2], method = "tau" ) fit_cylcop_cor(cyl_quadsec(), theta = sample[,1], x = sample[,2], method = "mi_cyl") fit_cylcop_cor(cyl_quadsec(), theta = sample[,1], x = sample[,2], method = "cor_cyl") fit_cylcop_cor(cyl_quadsec(), theta = sample[,1], x = sample[,2], method = "cor_cyl", n = 100, acc = 0.001 ) optCor(cyl_quadsec(), theta = sample[,1], x = sample[,2], method = "mi_cyl")
The code of this function is based on copula::fitCopula().
A circular-linear copula is fit to a set of bivariate observations.
fit_cylcop_ml( copula, theta, x, parameters = NULL, start = NULL, lower = NULL, upper = NULL, optim.method = "L-BFGS-B", optim.control = list(maxit = 100), estimate.variance = FALSE, traceOpt = FALSE ) optML( copula, theta, x, parameters = NULL, start = NULL, lower = NULL, upper = NULL, optim.method = "L-BFGS-B", optim.control = list(maxit = 100), estimate.variance = FALSE, traceOpt = FALSE )fit_cylcop_ml( copula, theta, x, parameters = NULL, start = NULL, lower = NULL, upper = NULL, optim.method = "L-BFGS-B", optim.control = list(maxit = 100), estimate.variance = FALSE, traceOpt = FALSE ) optML( copula, theta, x, parameters = NULL, start = NULL, lower = NULL, upper = NULL, optim.method = "L-BFGS-B", optim.control = list(maxit = 100), estimate.variance = FALSE, traceOpt = FALSE )
copula |
R object of class ' |
theta |
numeric vector of angles (measurements of a circular variable) or "circular" component of pseudo-observations. |
x |
numeric vector of step lengths (measurements of a linear variable) or "linear" component of pseudo-observations. |
parameters |
vector of character strings
holding the names of the parameters to be optimized.
These can be any parameters in |
start |
vector of starting values of the parameters. Default is
to take the starting values from |
lower |
(optional) vector of lower bounds of the parameters. |
upper |
(optional) vector of upper bounds of the parameters. |
optim.method |
character string, optimizer used in
|
optim.control |
|
estimate.variance |
logical value, denoting whether to include an estimate of the variance (NOT YET IMPLEMENTED). |
traceOpt |
logical value, whether to print information regarding convergence, current values, etc. during the optimization process. |
The data is first converted to pseudo observations to which
the copula is then fit. Therefore, the result of the optimization will be
exactly the same whether measurements (theta=theta and x=x)
or pseudo observations (theta=copula::pobs(theta,x)[,1]
and x=copula::pobs(theta,x)[,2]) are provided.
If you wish to fit parameters of a 'Copula' object
(package 'copula'), use the function copula::fitCopula().
optML() is an alias for fit_cylcop_ml.
A list of length 3 containing the same type of 'cyl_copula'
object as copula, but with optimized parameters, the log-likelihood
and the AIC.
Hodel FH, Fieberg JR (2022). “Circular-Linear Copulae for Animal Movement Data.” Methods in Ecology and Evolution. doi:10.1111/2041-210X.13821.
Hodel FH, Fieberg JR (2021). “Cylcop: An R Package for Circular-Linear Copulae with Angular Symmetry.” bioRxiv. doi:10.1101/2021.07.14.452253, https://www.biorxiv.org/content/10.1101/2021.07.14.452253v3/.
copula::fitCopula(), fit_cylcop_cor(),
opt_auto().
set.seed(123) sample <- rcylcop(100,cyl_quadsec(0.1)) fit_cylcop_ml(copula = cyl_quadsec(), theta = sample[,1], x = sample[,2], parameters = "a", start = 0 ) fit_cylcop_ml(copula = cyl_rect_combine(copula::frankCopula()), theta = sample[,1], x = sample[,2], parameters = "alpha", start = 1 ) sample <- rjoint( n = 100, copula = cyl_cubsec(0.1, -0.08), marginal_1 = list(name = "vonmisesmix", coef = list( mu = c(pi, 0), kappa = c(2, 5), prop = c(0.3, 0.7) )), marginal_2 = list(name = "exp", coef = list(0.3)) ) fit_cylcop_ml(copula = cyl_cubsec(), theta = sample[,1], x = sample[,2], parameters = c("a","b"), start = c(0,0), upper= c(0.1, 1/(2*pi)) ) optML(copula = cyl_quadsec(), theta = sample[,1], x = sample[,2], parameters = "a", start = 0 )set.seed(123) sample <- rcylcop(100,cyl_quadsec(0.1)) fit_cylcop_ml(copula = cyl_quadsec(), theta = sample[,1], x = sample[,2], parameters = "a", start = 0 ) fit_cylcop_ml(copula = cyl_rect_combine(copula::frankCopula()), theta = sample[,1], x = sample[,2], parameters = "alpha", start = 1 ) sample <- rjoint( n = 100, copula = cyl_cubsec(0.1, -0.08), marginal_1 = list(name = "vonmisesmix", coef = list( mu = c(pi, 0), kappa = c(2, 5), prop = c(0.3, 0.7) )), marginal_2 = list(name = "exp", coef = list(0.3)) ) fit_cylcop_ml(copula = cyl_cubsec(), theta = sample[,1], x = sample[,2], parameters = c("a","b"), start = c(0,0), upper= c(0.1, 1/(2*pi)) ) optML(copula = cyl_quadsec(), theta = sample[,1], x = sample[,2], parameters = "a", start = 0 )
This function finds parameter estimates of the marginal linear distribution, or gives a kernel density estimate using a Gaussian smoothing kernel.
fit_steplength( x, parametric = c("beta", "cauchy", "chi-squared", "chisq", "exponential", "exp", "gamma", "lognormal", "lnorm", "lognorm", "logistic", "normal", "t", "weibull", "normalmix", "weibullmix", "gammamix", "lnormmix", FALSE), start = NULL, bandwidth = NULL, ncomp = 2 )fit_steplength( x, parametric = c("beta", "cauchy", "chi-squared", "chisq", "exponential", "exp", "gamma", "lognormal", "lnorm", "lognorm", "logistic", "normal", "t", "weibull", "normalmix", "weibullmix", "gammamix", "lnormmix", FALSE), start = NULL, bandwidth = NULL, ncomp = 2 )
x |
numeric vector of measurements of a linear
random variable in |
parametric |
either a character string describing what distribution
should be fitted ( |
start |
(optional, except when |
bandwidth |
numeric value for the kernel density bandwidth.
Default is |
ncomp |
integer, number of components of the mixed distribution.
Only has an effect if |
If a parametric estimate is made, a list is returned
containing the estimated parameters, their standard errors,
the log-likelihood, the AIC and the name of the distribution.
If a non-parametric estimate is made, the output is a a 'density' object,
which is obtained with the function
GoFKernel::density.reflected() of the 'GoFKernel'
package.
GoFKernel::density.reflected(),
fit_angle(), opt_lin_bw().
require(graphics) set.seed(123) silent_curr <- cylcop_get_option("silent") cylcop_set_option(silent = TRUE) n <- 100 #n (number of samples) is set small for performance. x <- rweibull(n, shape = 10) dens_non_param <- fit_steplength(x = x, parametric = FALSE) weibull <- fit_steplength(x = x, parametric = "weibull") gamma <- fit_steplength(x = x, parametric = "gamma") chisq <- fit_steplength(x = x, parametric = "chi-squared", start = list(df = 1)) true_dens <- dweibull(seq(0, max(x), length.out = 200), shape = 10 ) dens_weibull <- dweibull(seq(0, max(x),length.out = 200), shape = weibull$coef$shape, scale = weibull$coef$scale ) dens_gamma <- dgamma(seq(0, max(x),length.out = 200), shape = gamma$coef$shape, rate = gamma$coef$rate ) dens_chisq <- dchisq(seq(0, max(x),length.out = 200), df = chisq$coef$df ) plot(seq(0,max(x),length.out = 200), true_dens, type = "l") lines(dens_non_param$x, dens_non_param$y, col = "red") lines(seq(0,max(x),length.out = 200), dens_weibull, col = "green") lines(seq(0,max(x),length.out = 200), dens_gamma, col = "blue") lines(seq(0,max(x),length.out = 200), dens_chisq, col = "cyan") cylcop_set_option(silent = silent_curr)require(graphics) set.seed(123) silent_curr <- cylcop_get_option("silent") cylcop_set_option(silent = TRUE) n <- 100 #n (number of samples) is set small for performance. x <- rweibull(n, shape = 10) dens_non_param <- fit_steplength(x = x, parametric = FALSE) weibull <- fit_steplength(x = x, parametric = "weibull") gamma <- fit_steplength(x = x, parametric = "gamma") chisq <- fit_steplength(x = x, parametric = "chi-squared", start = list(df = 1)) true_dens <- dweibull(seq(0, max(x), length.out = 200), shape = 10 ) dens_weibull <- dweibull(seq(0, max(x),length.out = 200), shape = weibull$coef$shape, scale = weibull$coef$scale ) dens_gamma <- dgamma(seq(0, max(x),length.out = 200), shape = gamma$coef$shape, rate = gamma$coef$rate ) dens_chisq <- dchisq(seq(0, max(x),length.out = 200), df = chisq$coef$df ) plot(seq(0,max(x),length.out = 200), true_dens, type = "l") lines(dens_non_param$x, dens_non_param$y, col = "red") lines(seq(0,max(x),length.out = 200), dens_weibull, col = "green") lines(seq(0,max(x),length.out = 200), dens_gamma, col = "blue") lines(seq(0,max(x),length.out = 200), dens_chisq, col = "cyan") cylcop_set_option(silent = silent_curr)
Converts an angle from the full circle (i.e. in the interval )
to an angle on the half circle (i.e. in the interval ).
full2half_circ(angle)full2half_circ(angle)
angle |
The numeric value of the angle in .
full2half_circ(0 * pi) / pi full2half_circ(0.5 * pi) / pi full2half_circ(1 * pi) / pi full2half_circ(1.5 * pi) / pi full2half_circ(2 * pi) / pifull2half_circ(0 * pi) / pi full2half_circ(0.5 * pi) / pi full2half_circ(1 * pi) / pi full2half_circ(1.5 * pi) / pi full2half_circ(2 * pi) / pi
The number of components in the mixed gamma distribution is specified by the length of the parameter vectors. The quantiles are numerically obtained from the distribution function using monotone cubic splines.
rgammamix(n, shape, rate = 1, scale = 1/rate, prop) dgammamix(x, shape, rate = 1, scale = 1/rate, prop) pgammamix(q, shape, rate = 1, scale = 1/rate, prop) qgammamix(p, shape, rate = 1, scale = 1/rate, prop)rgammamix(n, shape, rate = 1, scale = 1/rate, prop) dgammamix(x, shape, rate = 1, scale = 1/rate, prop) pgammamix(q, shape, rate = 1, scale = 1/rate, prop) qgammamix(p, shape, rate = 1, scale = 1/rate, prop)
n |
integer value, the number of random samples to be
generated with |
shape |
numeric vector holding the shape parameter of the components. |
rate |
numeric vector an alternative way to specify the scale
( |
scale |
numeric vector holding the scale parameter of the components. |
prop |
numeric vector, holding the mixing proportions of the components. |
x |
numeric vector giving the points where the density function is evaluated. |
q |
numeric vector giving the quantiles where the distribution function is evaluated. |
p |
numeric vector giving the probabilities where the quantile function is evaluated. |
dgammamix() gives a vector of length length(x)
containing the density at x.
pgammamix() gives a
vector of length length(q) containing
the distribution function at the corresponding values of q.
qgammamix() gives a vector of length length(p)
containing the quantiles at the corresponding values of p.
rgammamix() generates a vector of length n
containing the random samples.
rgammamix(10, shape = c(1, 3, 7), scale = c(2, 2, 4), prop = c(0.6, 0.3, 0.1)) dgammamix(c(0, 2, 1), shape = c(1, 3), rate = c(2, 2), prop = c(0.6, 0.4)) prob <- pgammamix(c(0.1, 7), shape = c(1, 3, 7), scale = c(2, 2, 4), prop = c(0.6, 0.3, 0.1)) prob qgammamix(prob, shape = c(1, 3, 7), scale = c(2, 2, 4), prop = c(0.6, 0.3, 0.1))rgammamix(10, shape = c(1, 3, 7), scale = c(2, 2, 4), prop = c(0.6, 0.3, 0.1)) dgammamix(c(0, 2, 1), shape = c(1, 3), rate = c(2, 2), prop = c(0.6, 0.4)) prob <- pgammamix(c(0.1, 7), shape = c(1, 3, 7), scale = c(2, 2, 4), prop = c(0.6, 0.3, 0.1)) prob qgammamix(prob, shape = c(1, 3, 7), scale = c(2, 2, 4), prop = c(0.6, 0.3, 0.1))
Converts an angle from the half circle (i.e. in the interval )
to an angle on the full circle (i.e. in the interval ).
half2full_circ(angle)half2full_circ(angle)
angle |
The numeric value of the angle in .
half2full_circ(-1 * pi) / pi half2full_circ(-0.5 * pi) / pi half2full_circ(-0 * pi) / pi half2full_circ(0.5 * pi) / pihalf2full_circ(-1 * pi) / pi half2full_circ(-0.5 * pi) / pi half2full_circ(-0 * pi) / pi half2full_circ(0.5 * pi) / pi
The bivariate joint distributions are described in terms of two marginal distributions and a copula
rjoint(n, copula, marginal_1, marginal_2) djoint(x, copula, marginal_1, marginal_2) pjoint(q, copula, marginal_1, marginal_2)rjoint(n, copula, marginal_1, marginal_2) djoint(x, copula, marginal_1, marginal_2) pjoint(q, copula, marginal_1, marginal_2)
n |
integer value, the number of random samples to be
generated with |
copula |
R object of class ' |
marginal_1 |
named list (for parametric estimates) or
a ' |
marginal_2 |
This input is similar to |
x |
matrix (or vector) of numeric values giving the points (in 2 dimensions) where the density function is evaluated. |
q |
matrix (or vector) of numeric values giving the points (in 2 dimensions) where the distribution function is evaluated. |
If entered "by hand", the named lists describing the parametric distributions
(marginal_1 and marginal_2) must contain 2 entries:
name:
a character string denoting the name of the distribution.
For a circular distribution, it can be "vonmises", "vonmisesmix", or
"wrappedcauchy". For a linear distribution, it must be a
string denoting the name of a linear distribution in the environment, i.e. the name of its
distribution function without the "p",
e.g. "norm" for normal distribution
coef: For a circular distribution coef is a (named) list of
parameters of the circular
marginal distribution as taken by the functions
qvonmises(), qvonmisesmix(),
or qwrappedcauchy(). For a linear distribution, coef is
a named list containing the parameters of the distribution given in "name".
djoint() gives a vector of length length(x)
containing the density at x.
pjoint() gives a
vector of length length(q) containing
the distribution function at the corresponding values of q.
rjoint() generates a vector of length n
containing the random samples.
cop <- copula::normalCopula(0.6) marginal_1 <- list(name="exp",coef=list(rate=2)) marginal_2 <- list(name="lnorm", coef=list(0,0.1)) sample <- rjoint(10,cop,marginal_1,marginal_2) pjoint(sample,cop,marginal_1,marginal_2) djoint(sample,cop,marginal_1,marginal_2) cop <- cyl_quadsec() marginal_1 <- list(name="wrappedcauchy", coef=list(location=0,scale=0.3)) marginal_2 <- list(name="weibull",coef=list(shape=3)) sample <- rjoint(10,cop,marginal_1,marginal_2) marginal_1 <- fit_angle(theta=sample[,1], parametric=FALSE) marginal_2 <- fit_steplength(x=sample[,2],parametric="lnorm") pjoint(c(0.3*pi,4),cop,marginal_1,marginal_2) djoint(c(0,2),cop,marginal_1,marginal_2)cop <- copula::normalCopula(0.6) marginal_1 <- list(name="exp",coef=list(rate=2)) marginal_2 <- list(name="lnorm", coef=list(0,0.1)) sample <- rjoint(10,cop,marginal_1,marginal_2) pjoint(sample,cop,marginal_1,marginal_2) djoint(sample,cop,marginal_1,marginal_2) cop <- cyl_quadsec() marginal_1 <- list(name="wrappedcauchy", coef=list(location=0,scale=0.3)) marginal_2 <- list(name="weibull",coef=list(shape=3)) sample <- rjoint(10,cop,marginal_1,marginal_2) marginal_1 <- fit_angle(theta=sample[,1], parametric=FALSE) marginal_2 <- fit_steplength(x=sample[,2],parametric="lnorm") pjoint(c(0.3*pi,4),cop,marginal_1,marginal_2) djoint(c(0,2),cop,marginal_1,marginal_2)
The number of components in the mixed log-normal distribution is specified by the length of the parameter vectors. The quantiles are numerically obtained from the distribution function using monotone cubic splines.
rlnormmix(n, meanlog, sdlog, prop) dlnormmix(x, meanlog, sdlog, prop) plnormmix(q, meanlog, sdlog, prop) qlnormmix(p, meanlog, sdlog, prop)rlnormmix(n, meanlog, sdlog, prop) dlnormmix(x, meanlog, sdlog, prop) plnormmix(q, meanlog, sdlog, prop) qlnormmix(p, meanlog, sdlog, prop)
n |
integer value, the number of random samples to be
generated with |
meanlog |
numeric vector holding the means of the components on the log scale. |
sdlog |
numeric vector holding the standard deviations of the components on the log scale. |
prop |
numeric vector, holding the mixing proportions of the components. |
x |
numeric vector giving the points where the density function is evaluated. |
q |
numeric vector giving the quantiles where the distribution function is evaluated. |
p |
numeric vector giving the probabilities where the quantile function is evaluated. |
dlnormmix() gives a vector of length length(x)
containing the density at x.
plnormmix() gives a
vector of length length(q) containing
the distribution function at the corresponding values of q.
qlnormmix() gives a vector of length length(p)
containing the quantiles at the corresponding values of p.
rlnormmix() generates a vector of length n
containing the random samples.
rlnormmix(10, meanlog = c(1, 3, 7), sdlog = c(2, 2, 4), prop = c(0.6, 0.3, 0.1)) dlnormmix(c(0, 2, 1), meanlog = c(1, 3), sdlog = c(2, 2), prop = c(0.6, 0.4)) prob <- plnormmix(c(0.1, 7), meanlog = c(1, 3, 7), sdlog = c(2, 2, 4), prop = c(0.6, 0.3, 0.1)) prob qlnormmix(prob, meanlog = c(1, 3, 7), sdlog = c(2, 2, 4), prop = c(0.6, 0.3, 0.1))rlnormmix(10, meanlog = c(1, 3, 7), sdlog = c(2, 2, 4), prop = c(0.6, 0.3, 0.1)) dlnormmix(c(0, 2, 1), meanlog = c(1, 3), sdlog = c(2, 2), prop = c(0.6, 0.4)) prob <- plnormmix(c(0.1, 7), meanlog = c(1, 3, 7), sdlog = c(2, 2, 4), prop = c(0.6, 0.3, 0.1)) prob qlnormmix(prob, meanlog = c(1, 3, 7), sdlog = c(2, 2, 4), prop = c(0.6, 0.3, 0.1))
The empirical copula is obtained from the data (theta and x),
and the mutual information of the 2 components is calculated. This gives a
non-negative number that can be normalized to lie between 0 and 1.
mi_cyl(theta, x, normalize = TRUE, symmetrize = FALSE)mi_cyl(theta, x, normalize = TRUE, symmetrize = FALSE)
theta |
numeric vector of angles (measurements of a circular variable). |
x |
numeric vector of step lengths (measurements of a linear variable). |
normalize |
logical value whether the mutual information should be
normalized to lie within |
symmetrize |
logical value whether it should be assumed that right and left
turns are equivalent. If |
First, the two components of the empirical copula, and
are obtained. Then the mutual information is calculated via discretizing and
into length(theta)^(1/3) bins. The mutual information can be
normalized to lie between 0 and 1 by dividing by the product of the entropies
of u and v. This is done using functions from the 'infotheo'
package.
Even if u and v are perfectly correlated
(i.e. cor_cyl goes to 1 with large sample sizes),
the normalized mutual information will not be 1 if the underlying copula is periodic and
symmetric. E.g. while normalCopula(1) has a correlation of 1 and a density
that looks like a line going from to ,
cyl_rect_combine(normalCopula(1))
has a density that looks like "<". The mutual information will be 1 in the first case,
but not in the second. Therefore, we can set symmetrize = TRUE to first
convert (if necessary) theta to lie in and then multiply all angles
larger than 0 with -1. The empirical copula is then calculated and the mutual information
is obtained from those values. It is exactly 1 in the case of
perfect correlation as captured by e.g.
cyl_rect_combine(normalCopula(1)).
Note also that the mutual information is independent of the marginal distributions.
However, symmetrize=TRUE only works with angles, not with pseudo-observations.
When x and theta are pseudo-observations, information is lost
due to the ranking, and symmetrization will fail.
A numeric value, the mutual information between theta and x
in nats.
Ma J, Sun Z (2011). “Mutual Information Is Copula Entropy.” Tsinghua Science and Technology, 16(1), 51-54. ISSN 1007-0214, doi:10.1016/S1007-0214(11)70008-6, https://www.sciencedirect.com/science/article/pii/S1007021411700086/.
Calsaverini RS, Vicente R, Systems C, Artes ED (2009). “An information-theoretic approach to statistical dependence: Copula information.” Europhysics Letters, 88(6), 1–6. doi:10.1209/0295-5075/88/68003, https://iopscience.iop.org/article/10.1209/0295-5075/88/68003/.
Hodel FH, Fieberg JR (2021). “Cylcop: An R Package for Circular-Linear Copulae with Angular Symmetry.” bioRxiv. doi:10.1101/2021.07.14.452253, https://www.biorxiv.org/content/10.1101/2021.07.14.452253v3/.
cor_cyl(), fit_cylcop_cor().
set.seed(123) cop <- cyl_quadsec(0.1) marg1 <- list(name="vonmises",coef=list(0,4)) marg2 <- list(name="lnorm",coef=list(2,3)) #draw samples and calculate the mutual information. sample <- rjoint(100,cop,marg1,marg2) mi_cyl(theta = sample[,1], x = sample[,2], normalize = TRUE, symmetrize = FALSE ) #the correlation coefficient is independent of the marginal distribution. sample <- traj_sim(100, cop, marginal_circ = list(name = "vonmises", coef = list(0, 1)), marginal_lin = list(name = "weibull", coef = list(shape = 2)) ) mi_cyl(theta = sample$angle, x = sample$steplength, normalize = TRUE, symmetrize = FALSE) mi_cyl(theta = sample$cop_u, x = sample$cop_v, normalize = TRUE, symmetrize = FALSE) # Estimate correlation of samples drawn from circular-linear copulas # with perfect correlation. cop <- cyl_rect_combine(copula::normalCopula(1)) sample <- rjoint(100,cop,marg1,marg2) # without normalization mi_cyl(theta = sample[,1], x = sample[,2], normalize = FALSE, symmetrize = FALSE ) #with normalization mi_cyl(theta = sample[,1], x = sample[,2], normalize = TRUE, symmetrize = FALSE ) #only with normalization and symmetrization do we get a value of 1 mi_cyl(theta = sample[,1], x = sample[,2], normalize = TRUE, symmetrize = TRUE )set.seed(123) cop <- cyl_quadsec(0.1) marg1 <- list(name="vonmises",coef=list(0,4)) marg2 <- list(name="lnorm",coef=list(2,3)) #draw samples and calculate the mutual information. sample <- rjoint(100,cop,marg1,marg2) mi_cyl(theta = sample[,1], x = sample[,2], normalize = TRUE, symmetrize = FALSE ) #the correlation coefficient is independent of the marginal distribution. sample <- traj_sim(100, cop, marginal_circ = list(name = "vonmises", coef = list(0, 1)), marginal_lin = list(name = "weibull", coef = list(shape = 2)) ) mi_cyl(theta = sample$angle, x = sample$steplength, normalize = TRUE, symmetrize = FALSE) mi_cyl(theta = sample$cop_u, x = sample$cop_v, normalize = TRUE, symmetrize = FALSE) # Estimate correlation of samples drawn from circular-linear copulas # with perfect correlation. cop <- cyl_rect_combine(copula::normalCopula(1)) sample <- rjoint(100,cop,marg1,marg2) # without normalization mi_cyl(theta = sample[,1], x = sample[,2], normalize = FALSE, symmetrize = FALSE ) #with normalization mi_cyl(theta = sample[,1], x = sample[,2], normalize = TRUE, symmetrize = FALSE ) #only with normalization and symmetrization do we get a value of 1 mi_cyl(theta = sample[,1], x = sample[,2], normalize = TRUE, symmetrize = TRUE )
Computes the maximum likelihood estimates for the parameters of a mixed
von Mises distribution: the mean directions, the concentration parameters,
and the proportions of the distributions. The code is a simplified version of
movMF::movMF() with the added
feature of optionally fixed mean directions (Hornik and Grün 2014).
mle.vonmisesmix(theta, mu = NULL, ncomp = 2)mle.vonmisesmix(theta, mu = NULL, ncomp = 2)
theta |
|
mu |
(optional) numeric vector of length |
ncomp |
positive integer specifying the number of components of the mixture model. |
The function complements the 'circular' package, which
provides functions to make maximum likelihood estimates of e.g. von Mises
(circular::mle.vonmises()), or wrapped Cauchy distributions
(circular::mle.wrappedcauchy())
A list containing the optimized parameters mu, kappa,
and prop.
Hornik K, Grün B (2014). “movMF : An R Package for Fitting Mixtures of von Mises-Fisher Distributions.” Journal of Statistical Software, 58. doi:10.18637/jss.v058.i10..
movMF::movMF(),
circular::mle.vonmises(),
dvonmisesmix(),
qvonmisesmix().
set.seed(123) n <- 10 angles <- rvonmisesmix(n, mu = c(0, pi), kappa = c(2, 1), prop = c(0.4,0.6) ) mle.vonmisesmix(theta = angles) mle.vonmisesmix(theta = angles, mu = c(0, pi))set.seed(123) n <- 10 angles <- rvonmisesmix(n, mu = c(0, pi), kappa = c(2, 1), prop = c(0.4,0.6) ) mle.vonmisesmix(theta = angles) mle.vonmisesmix(theta = angles, mu = c(0, pi))
The number of components in the mixed normal distribution is specified by the length of the parameter vectors. The quantiles are numerically obtained from the distribution function using monotone cubic splines.
rnormmix(n, mu, sigma, prop) dnormmix(x, mu, sigma, prop) pnormmix(q, mu, sigma, prop) qnormmix(p, mu, sigma, prop)rnormmix(n, mu, sigma, prop) dnormmix(x, mu, sigma, prop) pnormmix(q, mu, sigma, prop) qnormmix(p, mu, sigma, prop)
n |
integer value, the number of random samples to be
generated with |
mu |
|
sigma |
numeric vector holding the standard deviations of the components. |
prop |
numeric vector, holding the mixing proportions of the components. |
x |
numeric vector giving the points where the density function is evaluated. |
q |
numeric vector giving the quantiles where the distribution function is evaluated. |
p |
numeric vector giving the probabilities where the quantile function is evaluated. |
dnormmix() gives a vector of length length(x)
containing the density at x.
pnormmix() gives a
vector of length length(q) containing
the distribution function at the corresponding values of q.
qnormmix() gives a vector of length length(p)
containing the quantiles at the corresponding values of p.
rnormmix() generates a vector of length n
containing the random samples.
rnormmix(10, mu = c(0, 3, 7), sigma = c(2, 2, 4), prop = c(0.6, 0.3, 0.1)) dnormmix(c(0, 2, 1), mu = c(0, 3), sigma = c(2, 2), prop = c(0.6, 0.4)) prob <- pnormmix(c(0.1, 7), mu = c(0, 3, 7), sigma = c(2, 2, 4), prop = c(0.6, 0.3, 0.1)) prob qnormmix(prob, mu = c(0, 3, 7), sigma = c(2, 2, 4), prop = c(0.6, 0.3, 0.1))rnormmix(10, mu = c(0, 3, 7), sigma = c(2, 2, 4), prop = c(0.6, 0.3, 0.1)) dnormmix(c(0, 2, 1), mu = c(0, 3), sigma = c(2, 2), prop = c(0.6, 0.4)) prob <- pnormmix(c(0.1, 7), mu = c(0, 3, 7), sigma = c(2, 2, 4), prop = c(0.6, 0.3, 0.1)) prob qnormmix(prob, mu = c(0, 3, 7), sigma = c(2, 2, 4), prop = c(0.6, 0.3, 0.1))
Numerically Calculate the Conditional Copula
numerical_conditional_cop(u, copula, cond_on)numerical_conditional_cop(u, copula, cond_on)
u |
matrix or vector of numeric
values in |
copula |
R object of class ' |
cond_on |
column number of |
A vector containing the values of the distribution of the copula at
u[,-cond_on] conditional on the values of u[,cond_on].
Hodel FH, Fieberg JR (2021). “Cylcop: An R Package for Circular-Linear Copulae with Angular Symmetry.” bioRxiv. doi:10.1101/2021.07.14.452253, https://www.biorxiv.org/content/10.1101/2021.07.14.452253v3/.
Hodel FH, Fieberg JR (2022). “Circular-Linear Copulae for Animal Movement Data.” Methods in Ecology and Evolution. doi:10.1111/2041-210X.13821.
ccylcop(), numerical_inv_conditional_cop().
cop <- cyl_quadsec(0.1) u <- cbind(c(0.3, 0.1), c(0.7, 0.3)) numerical_conditional_cop(u = u, cop = cop, cond_on = 1)cop <- cyl_quadsec(0.1) u <- cbind(c(0.3, 0.1), c(0.7, 0.3)) numerical_conditional_cop(u = u, cop = cop, cond_on = 1)
Numerically calculate the inverse of the conditional copula
numerical_inv_conditional_cop(u, copula, cond_on)numerical_inv_conditional_cop(u, copula, cond_on)
u |
matrix or vector of numeric
values in |
copula |
R object of class ' |
cond_on |
column number of |
A vector containing the values of the inverse distribution of the copula at
[u,-cond_on] conditional on the values of [u,cond_on].
Hodel FH, Fieberg JR (2021). “Cylcop: An R Package for Circular-Linear Copulae with Angular Symmetry.” bioRxiv. doi:10.1101/2021.07.14.452253, https://www.biorxiv.org/content/10.1101/2021.07.14.452253v3/.
Hodel FH, Fieberg JR (2022). “Circular-Linear Copulae for Animal Movement Data.” Methods in Ecology and Evolution. doi:10.1111/2041-210X.13821.
ccylcop(), numerical_conditional_cop().
cop <- cyl_quadsec(0.1) u <- cbind(c(0.3, 0.1), c(0.7, 0.3)) numerical_inv_conditional_cop(u = u, cop = cop, cond_on = 1)cop <- cyl_quadsec(0.1) u <- cbind(c(0.3, 0.1), c(0.7, 0.3)) numerical_inv_conditional_cop(u = u, cop = cop, cond_on = 1)
The parameters of 15 different circular-linear copulas are fitted to data
and sorted
according to AIC. For each copula, first, a starting value for the maximum
likelihood estimation (MLE) is found using fit_cylcop_cor().
Then, MLE is carried out with a "reasonable" setup using fit_cylcop_ml().
If MLE fails, parameters obtained with fit_cylcop_cor() are reported.
opt_auto(theta, x)opt_auto(theta, x)
theta |
numeric vector of angles (measurements of a circular variable). |
x |
numeric vector of step lengths (measurements of a linear variable). |
A list containing 3 lists: Descriptions of the copulas, the
'cyl_copula' objects with fitted parameters, and the AIC.
The lists are sorted by ascending AIC.
If fit_cylcop_ml() has failed, the reported parameters are the ones obtained
with fit_cylcop_cor() and the AIC is set to NA.
Hodel FH, Fieberg JR (2022). “Circular-Linear Copulae for Animal Movement Data.” Methods in Ecology and Evolution. doi:10.1111/2041-210X.13821.
Hodel FH, Fieberg JR (2021). “Cylcop: An R Package for Circular-Linear Copulae with Angular Symmetry.” bioRxiv. doi:10.1101/2021.07.14.452253, https://www.biorxiv.org/content/10.1101/2021.07.14.452253v3/.
fit_cylcop_cor(), fit_cylcop_ml()
set.seed(123) #Optimal copula is independent of marginals. data <- rcylcop(100,cyl_quadsec(0.1)) #This takes a few seconds to run. copula_lst <- opt_auto(theta = data[,1], x = data[,2])set.seed(123) #Optimal copula is independent of marginals. data <- rcylcop(100,cyl_quadsec(0.1)) #This takes a few seconds to run. copula_lst <- opt_auto(theta = data[,1], x = data[,2])
This function basically wraps circular::bw.cv.ml.circular()
and circular::bw.nrd.circular() of the 'circular'
package, simplifying their inputs. For more control,
these 'circular' functions could be used directly.
The normal reference distribution ("nrd") method of finding the bandwidth
parameter might give very bad results,
especially for multi-modal population distributions.
In these cases it can help to set kappa.est = "trigmoments".
opt_circ_bw(theta, method = c("cv", "nrd"), kappa.est = "trigmoments")opt_circ_bw(theta, method = c("cv", "nrd"), kappa.est = "trigmoments")
theta |
|
method |
character string describing the method,
either |
kappa.est |
character string describing how the spread is estimated.
Either maximum likelihood |
method="nrd" is somewhat similar to the linear case (see
fit_steplength()). Instead of matching a normal distribution to
the data and then calculating its optimal bandwidth, a von Mises distribution is used.
To match that von Mises distribution to the data we can either find its concentration
parameter kappa using maximum likelihood (kappa.est="ML") or by trigonometric moment
matching (kappa.est="trigmoments"). When the data is multimodal, fitting a
(unimodal) von Mises distribution using maximum likelihood will probably give bad
results. Using kappa.est="trigmoments" potentially works better in those cases.
As an alternative, the bandwidth can be found by maximizing the cross-validation likelihood
(method="cv"). However, with this leave-one-out cross-validation scheme, at every
likelihood optimization step, von Mises densities need to be calculated, where
length(theta). Therefore, this method can become quite slow with
large sample sizes.
A numeric value, the optimized bandwidth.
circular::bw.cv.ml.circular(),
circular::bw.nrd.circular(),
opt_circ_bw().
require(circular) require(graphics) set.seed(123) n <- 10 #n (number of samples) is set small for performance. Increase n to # a value larger than 1000 to see the effects of multimodality angles <- rvonmisesmix(n, mu = c(0,pi), kappa = c(2,1), prop = c(0.5,0.5) ) bw1 <- opt_circ_bw(theta = angles, method="nrd", kappa.est = "ML") bw2 <- opt_circ_bw(theta = angles, method="nrd", kappa.est = "trigmoments") bw3 <- opt_circ_bw(theta = angles, method="cv") dens1 <- fit_angle(theta = angles, parametric = FALSE, bandwidth = bw1) dens2 <- fit_angle(theta = angles, parametric = FALSE, bandwidth = bw2) dens3 <- fit_angle(theta = angles, parametric = FALSE, bandwidth = bw3) true_dens <- dvonmisesmix( seq(-pi,pi,0.001), mu = c(0,pi), kappa = c(2,1), prop = c(0.5,0.5) ) if(interactive()){ plot(seq(-pi, pi, 0.001), true_dens, type = "l") lines(as.double(dens1$x), as.double(dens1$y), col = "red") lines(as.double(dens2$x), as.double(dens2$y), col = "green") lines(as.double(dens3$x), as.double(dens3$y), col = "blue") }require(circular) require(graphics) set.seed(123) n <- 10 #n (number of samples) is set small for performance. Increase n to # a value larger than 1000 to see the effects of multimodality angles <- rvonmisesmix(n, mu = c(0,pi), kappa = c(2,1), prop = c(0.5,0.5) ) bw1 <- opt_circ_bw(theta = angles, method="nrd", kappa.est = "ML") bw2 <- opt_circ_bw(theta = angles, method="nrd", kappa.est = "trigmoments") bw3 <- opt_circ_bw(theta = angles, method="cv") dens1 <- fit_angle(theta = angles, parametric = FALSE, bandwidth = bw1) dens2 <- fit_angle(theta = angles, parametric = FALSE, bandwidth = bw2) dens3 <- fit_angle(theta = angles, parametric = FALSE, bandwidth = bw3) true_dens <- dvonmisesmix( seq(-pi,pi,0.001), mu = c(0,pi), kappa = c(2,1), prop = c(0.5,0.5) ) if(interactive()){ plot(seq(-pi, pi, 0.001), true_dens, type = "l") lines(as.double(dens1$x), as.double(dens1$y), col = "red") lines(as.double(dens2$x), as.double(dens2$y), col = "green") lines(as.double(dens3$x), as.double(dens3$y), col = "blue") }
This function wraps stats::bw.ucv()
and stats::bw.nrd() of the 'stats'
package, simplifying their inputs. For more control,
these 'stats' functions could be used directly.
opt_lin_bw(x, method = c("cv", "nrd"))opt_lin_bw(x, method = c("cv", "nrd"))
x |
|
method |
character string describing the method used to find
the optimal bandwidth. Either |
The normal reference distribution (nrd) method involves
matching a normal distribution to the data using an empirical measure of spread.
The optimal bandwidth for that normal distribution can then be exactly calculated
by minimizing the mean integrated square error.
method="cv" finds the optimal bandwidth using unbiased cross-validation.
A numeric value, the optimized bandwidth.
stats::bw.ucv(),
stats::bw.nrd()
opt_lin_bw().
require(graphics) set.seed(123) n <- 1000 x <- rweibull(n, shape = 10) bw1 <- opt_lin_bw(x = x, method="nrd") bw2 <- opt_lin_bw(x = x, method="cv") dens1 <- fit_steplength(x = x, parametric = FALSE, bandwidth = bw1) dens2 <- fit_steplength(x = x, parametric = FALSE, bandwidth = bw2) true_dens <- dweibull(seq(0,max(x),length.out = 200), shape = 10) plot(seq(0,max(x),length.out = 200), true_dens, type = "l") lines(dens1$x, dens1$y, col = "red") lines(dens2$x, dens2$y, col = "green")require(graphics) set.seed(123) n <- 1000 x <- rweibull(n, shape = 10) bw1 <- opt_lin_bw(x = x, method="nrd") bw2 <- opt_lin_bw(x = x, method="cv") dens1 <- fit_steplength(x = x, parametric = FALSE, bandwidth = bw1) dens2 <- fit_steplength(x = x, parametric = FALSE, bandwidth = bw2) true_dens <- dweibull(seq(0,max(x),length.out = 200), shape = 10) plot(seq(0,max(x),length.out = 200), true_dens, type = "l") lines(dens1$x, dens1$y, col = "red") lines(dens2$x, dens2$y, col = "green")
This function produces a circular histogram of turn angles, i.e. angles on the the half-circle between -pi and pi.
plot_circ_hist(theta, nbars = 20)plot_circ_hist(theta, nbars = 20)
theta |
numeric vector of angles
(measurements of a circular variable) or "circular" component of pseudo-observations.
They must be on the half-circle, i.e. theta must be in |
nbars |
numeric integer, the number of bins (bars) in the histogram. |
A 'ggplot' object.
Hodel FH, Fieberg JR (2021). “Cylcop: An R Package for Circular-Linear Copulae with Angular Symmetry.” bioRxiv. doi:10.1101/2021.07.14.452253, https://www.biorxiv.org/content/10.1101/2021.07.14.452253v3/.
set.seed(123) theta <- cylcop::rvonmisesmix(n = 100, mu = c(0, pi), kappa = c(5, 2), prop = c(4, 2) ) plot1 <- plot_circ_hist(theta)set.seed(123) theta <- cylcop::rvonmisesmix(n = 100, mu = c(0, pi), kappa = c(5, 2), prop = c(4, 2) ) plot1 <- plot_circ_hist(theta)
This function produces a scatterplot ('ggplot' object) of
a sample from a copula. Either a sample is provided as input, or a sample
is drawn from a copula to quickly visualize it.
plot_cop_scat(traj = NULL, u = NULL, v = NULL)plot_cop_scat(traj = NULL, u = NULL, v = NULL)
traj |
a data.frame containing the trajectory produced by e.g.
|
u |
(alternatively) numeric vector of first components of pseudo-observations or draws from a copula. |
v |
(alternatively) numeric vector of second components of pseudo-observations or draws from a copula. |
Alternatively, instead of plotting a sample from a copula cop
using scatterplot(copula=cop), you can also use plot(cop).
If a trajectory is provided and n is smaller than nrow(traj),
n steps are randomly selected from the trajectory and plotted.
A 'ggplot' object, the scatterplot.
Hodel FH, Fieberg JR (2022). “Circular-Linear Copulae for Animal Movement Data.” Methods in Ecology and Evolution. doi:10.1111/2041-210X.13821.
Hodel FH, Fieberg JR (2021). “Cylcop: An R Package for Circular-Linear Copulae with Angular Symmetry.” bioRxiv. doi:10.1101/2021.07.14.452253, https://www.biorxiv.org/content/10.1101/2021.07.14.452253v3/.
plot_track(),
plot_joint_circ(), plot_cop_surf(), plot_joint_scat().
set.seed(123) traj <- traj_sim(100, copula = cyl_quadsec(0.1), marginal_circ = list(name = "vonmises", coef = list(0, 1)), marginal_lin = list(name = "weibull", coef = list(shape = 3)) ) plot_cop_scat(traj = traj) sample <- rcylcop(100,cyl_quadsec(0.1)) plot_cop_scat(u = sample[,1], v = sample[,2])set.seed(123) traj <- traj_sim(100, copula = cyl_quadsec(0.1), marginal_circ = list(name = "vonmises", coef = list(0, 1)), marginal_lin = list(name = "weibull", coef = list(shape = 3)) ) plot_cop_scat(traj = traj) sample <- rcylcop(100,cyl_quadsec(0.1)) plot_cop_scat(u = sample[,1], v = sample[,2])
This function plots the distribution or the density of a copula. It can produce a surface plot using either functions from the 'rgl' or from the 'plotly' package, or it can produce a heat map using functions from 'ggplot2'.
plot_cop_surf( copula, type = "pdf", plot_type = "rgl", resolution = 50, n_gridlines = 11 )plot_cop_surf( copula, type = "pdf", plot_type = "rgl", resolution = 50, n_gridlines = 11 )
copula |
' |
type |
character string describing what is plotted,
either |
plot_type |
character string describing what type of plot
is produced. Available plot types are:
|
resolution |
numeric value. The density or distribution
will be calculated at |
n_gridlines |
numeric value giving the number of grid lines drawn in u and v direction. |
Depending on plot_type, a 'ggplot' object
is returned, or a 'plotly' visualization or 'rgl' plot is produced.
Hodel FH, Fieberg JR (2022). “Circular-Linear Copulae for Animal Movement Data.” Methods in Ecology and Evolution. doi:10.1111/2041-210X.13821.
Hodel FH, Fieberg JR (2021). “Cylcop: An R Package for Circular-Linear Copulae with Angular Symmetry.” bioRxiv. doi:10.1101/2021.07.14.452253, https://www.biorxiv.org/content/10.1101/2021.07.14.452253v3/.
plot_cop_scat(), plot_track(),
plot_joint_circ(), plot_joint_scat().
if(interactive()){ plot_cop_surf(copula::frankCopula(2), type="pdf", plot_type="ggplot", resolution = 5 ) plot_cop_surf(copula::frankCopula(2), type="cdf", plot_type="ggplot", resolution = 5 ) #opens a new window plot_cop_surf(cyl_quadsec(0.1), type="pdf", plot_type="rgl" ) plot_cop_surf(cyl_quadsec(0.1), type="pdf", plot_type="rgl", n_gridlines = 60 ) plot_cop_surf(cyl_quadsec(0.1), type="pdf", plot_type="plotly", n_gridlines = 10, resolution = 10 ) }if(interactive()){ plot_cop_surf(copula::frankCopula(2), type="pdf", plot_type="ggplot", resolution = 5 ) plot_cop_surf(copula::frankCopula(2), type="cdf", plot_type="ggplot", resolution = 5 ) #opens a new window plot_cop_surf(cyl_quadsec(0.1), type="pdf", plot_type="rgl" ) plot_cop_surf(cyl_quadsec(0.1), type="pdf", plot_type="rgl", n_gridlines = 60 ) plot_cop_surf(cyl_quadsec(0.1), type="pdf", plot_type="plotly", n_gridlines = 10, resolution = 10 ) }
This function produces circular boxplots (a 'ggplot' object)
of the turn angles corresponding to specific quantiles of the step lengths.
plot_joint_box( traj = NULL, theta = NULL, x = NULL, levels = 5, marginal_lin = NULL, spacing = 0.3, legend_pos = "right" )plot_joint_box( traj = NULL, theta = NULL, x = NULL, levels = 5, marginal_lin = NULL, spacing = 0.3, legend_pos = "right" )
traj |
data.frame containing the trajectory produced by e.g.
|
theta |
(alternatively) numeric vector of angles (measurements of a circular variable) or "circular" component of pseudo-observations. |
x |
(alternatively) numeric vector of step lengths (measurements of a linear variable) or "linear" component of pseudo-observations. |
levels |
integer value between 1 and 15, the number of quantiles into which the step lengths are split. |
marginal_lin |
named list (for parametric estimates) or
a ' |
spacing |
numeric value between 0 and 10 determining the spacing between the boxplots. |
legend_pos |
character string denoting the position of the legend (limits
of the step length quantiles). Either |
The step lengths are split into quantiles. For each quantile a boxplot of the corresponding turn angles is produced and wrapped around the circle. The turn angle values are plotted as scatter plot overlaying the boxplot. Outliers are plotted in red. The median of the turn angles is defined as the center of the shortest arc that connects all points. The length of the whiskers is 1.5 times the interquartile range.
You can either specify traj or the angels (theta)
and step lengths (codex).
If entered "by hand", the named list describing the marginal linear distribution
(for marginal_lin) must contain 2 entries:
name:
a character string denoting the name of the linear distribution,
i.e. the name of its
distribution function without the "p",
e.g. "norm" for normal distribution.
coef: a named list containing the parameters of the distribution
given in "name".
A 'ggplot' object, the circular boxplot.
Hodel FH, Fieberg JR (2022). “Circular-Linear Copulae for Animal Movement Data.” Methods in Ecology and Evolution. doi:10.1111/2041-210X.13821.
Hodel FH, Fieberg JR (2021). “Cylcop: An R Package for Circular-Linear Copulae with Angular Symmetry.” bioRxiv. doi:10.1101/2021.07.14.452253, https://www.biorxiv.org/content/10.1101/2021.07.14.452253v3/.
plot_cop_scat(), plot_track(),
plot_joint_circ(), plot_cop_surf().
set.seed(1234) traj <- traj_sim(100, copula = cyl_rect_combine(copula::frankCopula(6)), marginal_circ = list(name= "vonmises", coef=list(0, 2)), marginal_lin = list(name = "weibull", coef=list(shape=3)) ) plot1 <- plot_joint_box(traj) plot2 <- plot_joint_box(traj, marginal_lin=list(name = "weibull", coef=list(shape=3)) )set.seed(1234) traj <- traj_sim(100, copula = cyl_rect_combine(copula::frankCopula(6)), marginal_circ = list(name= "vonmises", coef=list(0, 2)), marginal_lin = list(name = "weibull", coef=list(shape=3)) ) plot1 <- plot_joint_box(traj) plot2 <- plot_joint_box(traj, marginal_lin=list(name = "weibull", coef=list(shape=3)) )
This function produces a circular scatterplot with the step lengths plotted as distance from the center of a circle and the turn angles as angles (polar coordinates).
plot_joint_circ(traj = NULL, theta = NULL, x = NULL)plot_joint_circ(traj = NULL, theta = NULL, x = NULL)
traj |
data.frame containing the trajectory produced by e.g.
|
theta |
(alternatively) numeric vector of angles (measurements of a circular variable) or "circular" component of pseudo-observations. |
x |
(alternatively) numeric vector of step lengths (measurements of a linear variable) or "linear" component of pseudo-observations. |
You can either specify traj or the angels and step lengths
theta and x.
A 'ggplot' object.
Hodel FH, Fieberg JR (2021). “Cylcop: An R Package for Circular-Linear Copulae with Angular Symmetry.” bioRxiv. doi:10.1101/2021.07.14.452253, https://www.biorxiv.org/content/10.1101/2021.07.14.452253v3/.
plot_cop_scat(), plot_track(),
plot_cop_surf(), plot_joint_scat().
set.seed(123) traj <- traj_sim(100, copula = cyl_quadsec(0.1), marginal_circ = list(name="vonmises",coef=list(0, 1)), marginal_lin = list(name="weibull", coef=list(shape=3)) ) plot1 <- plot_joint_circ(traj)set.seed(123) traj <- traj_sim(100, copula = cyl_quadsec(0.1), marginal_circ = list(name="vonmises",coef=list(0, 1)), marginal_lin = list(name="weibull", coef=list(shape=3)) ) plot1 <- plot_joint_circ(traj)
This function produces a scatterplot ('ggplot' object) of
the turn angles and step lengths.
plot_joint_scat( traj = NULL, theta = NULL, x = NULL, periodic = FALSE, plot_margins = FALSE )plot_joint_scat( traj = NULL, theta = NULL, x = NULL, periodic = FALSE, plot_margins = FALSE )
traj |
data.frame containing the trajectory produced by e.g.
|
theta |
(alternatively) numeric vector of angles (measurements of a circular variable). |
x |
(alternatively) numeric vector of step lengths (measurements of a linear variable). |
periodic |
logical value denoting whether the plot should be periodically extended past -pi and pi. |
plot_margins |
logical determining whether the marginal kernel
density estimates are computed and plotted. Alternatively, |
You can either specify traj or the angels and step lengths
(theta and x).
If plot_margins=T, the code will attempt to find appropriate bandwidths for
the kernel density estimate autonomously, also taking into account computational time.
For more control over the actual method and parameters used to obtain the kernel
density estimates, you can calculate them "by hand" using e.g.
fit_angle(theta, parametric=FALSE)
and fit_steplength(x, parametric=FALSE)).
A 'ggplot' object, the scatterplot.
Hodel FH, Fieberg JR (2022). “Circular-Linear Copulae for Animal Movement Data.” Methods in Ecology and Evolution. doi:10.1111/2041-210X.13821.
Hodel FH, Fieberg JR (2021). “Cylcop: An R Package for Circular-Linear Copulae with Angular Symmetry.” bioRxiv. doi:10.1101/2021.07.14.452253, https://www.biorxiv.org/content/10.1101/2021.07.14.452253v3/.
plot_cop_scat(), plot_track(),
plot_joint_circ(), plot_cop_surf().
set.seed(123) traj <- traj_sim(100, copula = cyl_quadsec(0.1), marginal_circ = list(name = "vonmises", coef = list(0, 1)), marginal_lin = list(name = "weibull", coef = list(shape = 3)) ) plot1 <- plot_joint_scat(traj) plot2 <- plot_joint_scat(traj, periodic = TRUE) plot3 <- plot_joint_scat(theta=traj$angle, x=traj$steplength, periodic = TRUE, plot_margins=TRUE) bw <- opt_circ_bw(theta = traj$angle, method = "nrd",kappa.est = "trigmoments") ang_dens <- fit_angle(theta=traj$angle, parametric=FALSE, bandwidth=bw) step_dens <- fit_steplength(x=traj$steplength, parametric=FALSE) plot4 <- plot_joint_scat(traj, periodic = TRUE, plot_margins=list(ang_dens, step_dens))set.seed(123) traj <- traj_sim(100, copula = cyl_quadsec(0.1), marginal_circ = list(name = "vonmises", coef = list(0, 1)), marginal_lin = list(name = "weibull", coef = list(shape = 3)) ) plot1 <- plot_joint_scat(traj) plot2 <- plot_joint_scat(traj, periodic = TRUE) plot3 <- plot_joint_scat(theta=traj$angle, x=traj$steplength, periodic = TRUE, plot_margins=TRUE) bw <- opt_circ_bw(theta = traj$angle, method = "nrd",kappa.est = "trigmoments") ang_dens <- fit_angle(theta=traj$angle, parametric=FALSE, bandwidth=bw) step_dens <- fit_steplength(x=traj$steplength, parametric=FALSE) plot4 <- plot_joint_scat(traj, periodic = TRUE, plot_margins=list(ang_dens, step_dens))
This function plots the locations of a trajectory or multiple trajectories.
plot_track(traj = NULL, x_coord = NULL, y_coord = NULL)plot_track(traj = NULL, x_coord = NULL, y_coord = NULL)
traj |
data.frame containing the trajectory produced by e.g.
|
x_coord |
(alternatively) numeric vector of x-coordinates or a list of x-coordinate vectors of multiple trajectories. |
y_coord |
(alternatively) numeric vector of y-coordinates or a list of y-coordinate vectors of multiple trajectories. |
A 'ggplot' object.
Hodel FH, Fieberg JR (2022). “Circular-Linear Copulae for Animal Movement Data.” Methods in Ecology and Evolution. doi:10.1111/2041-210X.13821.
Hodel FH, Fieberg JR (2021). “Cylcop: An R Package for Circular-Linear Copulae with Angular Symmetry.” bioRxiv. doi:10.1101/2021.07.14.452253, https://www.biorxiv.org/content/10.1101/2021.07.14.452253v3/.
plot_cop_scat(),
plot_joint_circ(), plot_cop_surf(), plot_joint_scat().
set.seed(123) traj <- traj_sim(50, copula = cyl_quadsec(0.1), marginal_circ = list(name = "vonmises", coef = list(0, 1)), marginal_lin = list(name = "weibull", coef = list(shape = 3)) ) plot1 <- plot_track(traj=traj) x_coord <- list(runif(10),runif(20),runif(3)) y_coord <- list(runif(10),runif(20),runif(3)) plot2 <- plot_track(x_coord=x_coord, y_coord=y_coord)set.seed(123) traj <- traj_sim(50, copula = cyl_quadsec(0.1), marginal_circ = list(name = "vonmises", coef = list(0, 1)), marginal_lin = list(name = "weibull", coef = list(shape = 3)) ) plot1 <- plot_track(traj=traj) x_coord <- list(runif(10),runif(20),runif(3)) y_coord <- list(runif(10),runif(20),runif(3)) plot2 <- plot_track(x_coord=x_coord, y_coord=y_coord)
cyl_copula' ObjectsMethod for plot() to draw a scatter plot of a
random sample from a circular-linear copula.
## S4 method for signature 'cyl_copula,missing' plot(x, n = 1000, ...)## S4 method for signature 'cyl_copula,missing' plot(x, n = 1000, ...)
x |
R object of class ' |
n |
sample size of the random sample drawn from |
... |
additional arguments passed to |
An invisible NULL. As side effect, a plot is produced.
set.seed(123) plot(cyl_quadsec(0.1)) plot(cyl_vonmises(0,2), n = 100) plot(cyl_quadsec(0.1), xlab = "something", ylab = "something else", main = "clever title", col = "red", fg = "blue", asp= 1)set.seed(123) plot(cyl_quadsec(0.1)) plot(cyl_vonmises(0,2), n = 100) plot(cyl_quadsec(0.1), xlab = "something", ylab = "something else", main = "clever title", col = "red", fg = "blue", asp= 1)
cyl_copula' CopulaThis is a method corresponding to the generic prob() in the
'copula' package.
## S4 method for signature 'cyl_copula' prob(x, l, u)## S4 method for signature 'cyl_copula' prob(x, l, u)
x |
R object of class ' |
l |
numeric vector of length 2 holding the coordinates of the
lower left corner in |
u |
numeric vector of length 2 holding the coordinates of the
upper right corner in |
A numeric in , the probability that a draw from the
2-dimensional copula x falls in the rectangle defined by l and
u.
copula::prob
cop <- cyl_quadsec(0.1) prob(cop, l = c(0.1, 0.3), u = c(0.3, 0.9))cop <- cyl_quadsec(0.1) prob(cop, l = c(0.1, 0.3), u = c(0.3, 0.9))
cyl_copula' ObjectsThese methods can be used, e.g. in other functions, to give users limited access to the parameters of a copula.
set_cop_param(copula, param_val, param_name, ...) ## S4 method for signature 'cyl_cubsec' set_cop_param(copula, param_val, param_name) ## S4 method for signature 'cyl_quadsec' set_cop_param(copula, param_val, param_name) ## S4 method for signature 'cyl_rect_combine' set_cop_param(copula, param_val, param_name) ## S4 method for signature 'cyl_rot_combine' set_cop_param(copula, param_val, param_name) ## S4 method for signature 'cyl_vonmises' set_cop_param(copula, param_val, param_name)set_cop_param(copula, param_val, param_name, ...) ## S4 method for signature 'cyl_cubsec' set_cop_param(copula, param_val, param_name) ## S4 method for signature 'cyl_quadsec' set_cop_param(copula, param_val, param_name) ## S4 method for signature 'cyl_rect_combine' set_cop_param(copula, param_val, param_name) ## S4 method for signature 'cyl_rot_combine' set_cop_param(copula, param_val, param_name) ## S4 method for signature 'cyl_vonmises' set_cop_param(copula, param_val, param_name)
copula |
R object of class ' |
param_val |
numeric vector holding the values to which the
parameters given in |
param_name |
vector of character strings holding the names of the parameters to be changed. |
... |
additional arguments. |
Note that for a rectangular patchwork copula
('cyl_rect_combine')
the attribute rectangles_symmetric cannot be changed by set_cop_param(),
since rectangular patchwork copulas with symmetric rectangles are treated as
distinct from rectangular patchwork copulas with potentially asymmetric rectangles.
Therefore, when changing one of the bounds of the lower rectangle of such a copula,
the corresponding bound of the upper rectangle is automatically changed as well
(see examples).
A 'cyl_copula' object with the changed parameters.
cop <- cyl_rect_combine(copula::normalCopula(0.2),low_rect = c(0.1,0.4), up_rect="symmetric") cop cop <- set_cop_param(cop, param_val = c(0.1, 0.3), param_name = c("rho.1", "low_rect2")) cop <- cyl_rect_combine(copula::normalCopula(0.2),low_rect = c(0.1,0.4), up_rect=c(0.6,0.9)) cop cop <- set_cop_param(cop, param_val = 0.3, param_name = "low_rect2") copcop <- cyl_rect_combine(copula::normalCopula(0.2),low_rect = c(0.1,0.4), up_rect="symmetric") cop cop <- set_cop_param(cop, param_val = c(0.1, 0.3), param_name = c("rho.1", "low_rect2")) cop <- cyl_rect_combine(copula::normalCopula(0.2),low_rect = c(0.1,0.4), up_rect=c(0.6,0.9)) cop cop <- set_cop_param(cop, param_val = 0.3, param_name = "low_rect2") cop
cyl_copula' ObjectsMethods for function show() in package cylcop
## S4 method for signature 'cyl_copula' show(object) ## S4 method for signature 'cyl_rect_combine' show(object) ## S4 method for signature 'cyl_rot_combine' show(object)## S4 method for signature 'cyl_copula' show(object) ## S4 method for signature 'cyl_rect_combine' show(object) ## S4 method for signature 'cyl_rot_combine' show(object)
object |
R object of class ' |
An invisible NULL. As side effect, information on object is
printed.
The function calculates step lengths and turn angles from x- and y-coordinates and calculates pseudo-observations from those step lengths and turn angles.
traj_get(x_coords, y_coords)traj_get(x_coords, y_coords)
x_coords |
vector of numeric values containing the x-coordinates of a trajectory. |
y_coords |
vector of numeric values containing the y-coordinates of a trajectory. |
A data.frame containing the trajectory. It has 6 columns containing the x and y coordinates, the turn angles, the step lengths, and the pseudo-observations.
traj_sim().
set.seed(123) traj <- traj_sim(n = 5, copula = cyl_quadsec(0.1), marginal_circ = list(name="vonmises",coef=list(0, 1)), marginal_lin = list(name="weibull",coef=list(shape=3)) ) traj_from_coords <- traj_get(traj[,1], traj[,2])set.seed(123) traj <- traj_sim(n = 5, copula = cyl_quadsec(0.1), marginal_circ = list(name="vonmises",coef=list(0, 1)), marginal_lin = list(name="weibull",coef=list(shape=3)) ) traj_from_coords <- traj_get(traj[,1], traj[,2])
The function draws values from a circular-linear bivariate distribution of turn angles and step lengths specified by the marginal distributions and a circular-linear copula. From the start point (0,0) and the second (potentially user specified) point, a trajectory is then built with these turn angles and step lengths.
traj_sim( n, copula, marginal_circ, marginal_lin, ignore_first = TRUE, pos_2 = NULL )traj_sim( n, copula, marginal_circ, marginal_lin, ignore_first = TRUE, pos_2 = NULL )
n |
integer, number of trajectory steps to generate. |
copula |
' |
marginal_circ |
named list (for parametric estimates) or
a ' |
marginal_lin |
named list (for parametric estimates) or
a ' |
ignore_first |
logical value. If |
pos_2 |
(optional) numeric vector of length 2 containing the coordinates of the second point in the trajectory. The first point is always at (0,0). If no value is specified, the second point is obtained by going in a random direction from the first point for a distance drawn from the marginal step length distribution. |
Samples are drawn from the circular-linear copula and then transformed
using the quantile functions of the marginal circular and the marginal linear
distribution. To generate draws from any bivariate joint distribution (not
necessarily a circular-linear one) without also producing a trajectory,
the function rjoint() can be used.
If entered "by hand", the named lists describing the parametric distributions
(marginal_circ and marginal_lin) must contain 2 entries:
name:
a character string denoting the name of the distribution.
For the circular distribution, it can be "vonmises", "vonmisesmix", or
"wrappedcauchy". For the linear distribution, it must be a
string denoting the name of a linear distribution in the environment, i.e. the name of its
distribution function without the "p",
e.g. "norm" for normal distribution
coef: For the circular distribution coef is a (named) list of
parameters of the circular
marginal distribution as taken by the functions
qvonmises(), qvonmisesmix(),
or qwrappedcauchy(). For the linear distribution, coef is
a named list containing the parameters of the distribution given in "name".
A data.frame containing the trajectory. It has 6 columns containing the x and y coordinates, the turn angles, the step lengths, and the values sampled from the copula.
traj_get(),
fit_steplength(), fit_angle(),
plot_track(), plot_cop_scat(),
plot_joint_scat(), plot_joint_circ().
require(circular) set.seed(123) traj <- traj_sim(n = 5, copula = cyl_quadsec(0.1), marginal_circ = list(name="vonmises",coef=list(0, 1)), marginal_lin = list(name="weibull",coef=list(shape=3)) ) traj angles <- rvonmisesmix(100, mu = c(0, pi), kappa = c(2, 3), prop = c(0.4, 0.6) ) angles <- full2half_circ(angles) bw <- opt_circ_bw(theta = angles, method = "nrd", kappa.est = "trigmoments") marg_ang <- fit_angle(theta = angles, parametric = FALSE, bandwidth = bw) steplengths <- rlnorm(100, 0, 0.3) marg_stepl <- fit_steplength(x = steplengths, parametric = "lnorm") traj_sim(n = 5, copula = cyl_quadsec(0.1), marginal_circ = marg_ang, marginal_lin = marg_stepl, ignore_first = FALSE, pos_2 = c(5,5) )require(circular) set.seed(123) traj <- traj_sim(n = 5, copula = cyl_quadsec(0.1), marginal_circ = list(name="vonmises",coef=list(0, 1)), marginal_lin = list(name="weibull",coef=list(shape=3)) ) traj angles <- rvonmisesmix(100, mu = c(0, pi), kappa = c(2, 3), prop = c(0.4, 0.6) ) angles <- full2half_circ(angles) bw <- opt_circ_bw(theta = angles, method = "nrd", kappa.est = "trigmoments") marg_ang <- fit_angle(theta = angles, parametric = FALSE, bandwidth = bw) steplengths <- rlnorm(100, 0, 0.3) marg_stepl <- fit_steplength(x = steplengths, parametric = "lnorm") traj_sim(n = 5, copula = cyl_quadsec(0.1), marginal_circ = marg_ang, marginal_lin = marg_stepl, ignore_first = FALSE, pos_2 = c(5,5) )
The number of components in the mixed von Mises distribution is specified by the length of the parameter vectors. The quantiles are numerically obtained from the distribution function using monotone cubic splines.
rvonmisesmix(n, mu, kappa, prop) dvonmisesmix(theta, mu, kappa, prop) pvonmisesmix(theta, mu, kappa, prop) qvonmisesmix(p, mu, kappa, prop)rvonmisesmix(n, mu, kappa, prop) dvonmisesmix(theta, mu, kappa, prop) pvonmisesmix(theta, mu, kappa, prop) qvonmisesmix(p, mu, kappa, prop)
n |
integer value, the number of random samples to be
generated with |
mu |
|
kappa |
|
prop |
numeric vector, holding the mixing proportions of the components. |
theta |
numeric vector giving the angles where the density or distribution function is evaluated. |
p |
numeric vector giving the probabilities where the quantile function is evaluated. |
dvonmisesmix() gives a vector of length length(theta)
containing the density at theta.
pvonmisesmix() gives a
vector of length length(theta) containing
the distribution function at the corresponding values of theta.
qvonmisesmix() gives a vector of length length(p)
containing the quantiles at the corresponding values of p.
rvonmisesmix() generates a vector of length n
containing the random samples, i.e. angles in .
rvonmisesmix(10, mu = c(0, pi, pi/2), kappa = c(2, 2, 4), prop = c(0.6, 0.3, 0.1)) dvonmisesmix(c(0, 2, pi, 1), mu = c(0, pi), kappa = c(2, 2), prop = c(0.6, 0.4)) prob <- pvonmisesmix(c(0.1, pi), mu = c(0, pi, pi/2), kappa = c(2, 2, 4), prop = c(0.6, 0.3, 0.1)) prob qvonmisesmix(prob, mu = c(0, pi, pi/2), kappa = c(2, 2, 4), prop = c(0.6, 0.3, 0.1))rvonmisesmix(10, mu = c(0, pi, pi/2), kappa = c(2, 2, 4), prop = c(0.6, 0.3, 0.1)) dvonmisesmix(c(0, 2, pi, 1), mu = c(0, pi), kappa = c(2, 2), prop = c(0.6, 0.4)) prob <- pvonmisesmix(c(0.1, pi), mu = c(0, pi, pi/2), kappa = c(2, 2, 4), prop = c(0.6, 0.3, 0.1)) prob qvonmisesmix(prob, mu = c(0, pi, pi/2), kappa = c(2, 2, 4), prop = c(0.6, 0.3, 0.1))
The Wasserstein distance is calculated based on the Euclidean distance between two copula PDFs on a grid, or between a copula PDF and pseudo-observations.
wasserstein( copula, copula2 = NULL, theta = NULL, x = NULL, n_grid = 2500, p = 2 )wasserstein( copula, copula2 = NULL, theta = NULL, x = NULL, n_grid = 2500, p = 2 )
copula |
R object of class ' |
copula2 |
R object of class ' |
theta |
(alternatively) numeric vector of angles (measurements of a circular variable) or "circular" component of pseudo-observations. |
x |
(alternatively) numeric vector of step lengths (measurements of a linear variable) or "linear" component of pseudo-observations. |
n_grid |
integer number of grid cells at which the PDF of the copula(s) is calculated Default is 2500 |
p |
integer power (1 or 2) to which the Euclidean distance between points is taken in order to compute transportation costs. |
Note that when comparing 2 copula PDFs (i.e. theta = NULL and x = NULL),
the calculated Wasserstein distance will depend on the number of grid cells
(n_grid) used to approximate the PDFs. The distance will converge to a certain
value with a higher number of grid cells, but the computational time will also increase.
The default of 2500 seems to be a good (empirically determined) compromise.
The same is true when calculating the Wasserstein distance between a copula
PDF and pseudo-observations. There, it is also important to only compare distances
that use the same number of observations.
The code is based on the functions transport::wasserstein()
and transport::semidiscrete().
numeric, the pth Wasserstein distance
set.seed(1234) copula1 <- cyl_quadsec(0.1) copula2 <- cyl_rect_combine(copula::frankCopula(2)) wasserstein(copula=copula1,copula2 = copula2,p=2,n_grid=20) wasserstein(copula=copula1,copula2 = copula1,p=2,n_grid=20) wasserstein(copula=copula1,copula2 = copula::frankCopula(2),p=2,n_grid=20) sample <- rjoint(10, copula1, marginal_1 = list(name = "vonmises", coef = list(0, 1)), marginal_2 = list(name = "weibull", coef = list(3,4)) ) wasserstein(copula=copula1, theta=sample[,1], x=sample[,2], n_grid=20)set.seed(1234) copula1 <- cyl_quadsec(0.1) copula2 <- cyl_rect_combine(copula::frankCopula(2)) wasserstein(copula=copula1,copula2 = copula2,p=2,n_grid=20) wasserstein(copula=copula1,copula2 = copula1,p=2,n_grid=20) wasserstein(copula=copula1,copula2 = copula::frankCopula(2),p=2,n_grid=20) sample <- rjoint(10, copula1, marginal_1 = list(name = "vonmises", coef = list(0, 1)), marginal_2 = list(name = "weibull", coef = list(3,4)) ) wasserstein(copula=copula1, theta=sample[,1], x=sample[,2], n_grid=20)
The number of components in the mixed Weibull distribution is specified by the length of the parameter vectors. The quantiles are numerically obtained from the distribution function using monotone cubic splines.
rweibullmix(n, shape, scale, prop) dweibullmix(x, shape, scale, prop) pweibullmix(q, shape, scale, prop) qweibullmix(p, shape, scale, prop)rweibullmix(n, shape, scale, prop) dweibullmix(x, shape, scale, prop) pweibullmix(q, shape, scale, prop) qweibullmix(p, shape, scale, prop)
n |
integer value, the number of random samples to be
generated with |
shape |
numeric vector holding the shape parameter of the components. |
scale |
numeric vector holding the scale parameter of the components. |
prop |
numeric vector, holding the mixing proportions of the components. |
x |
numeric vector giving the points where the density function is evaluated. |
q |
numeric vector giving the quantiles where the distribution function is evaluated. |
p |
numeric vector giving the probabilities where the quantile function is evaluated. |
dweibullmix() gives a vector of length length(x)
containing the density at x.
pweibullmix() gives a
vector of length length(q) containing
the distribution function at the corresponding values of q.
qweibullmix() gives a vector of length length(p)
containing the quantiles at the corresponding values of p.
rweibullmix() generates a vector of length n
containing the random samples.
rweibullmix(10, shape = c(1, 3, 7), scale = c(2, 2, 4), prop = c(0.6, 0.3, 0.1)) dweibullmix(c(0, 2, 1), shape = c(1, 3), scale = c(2, 2), prop = c(0.6, 0.4)) prob <- pweibullmix(c(0.1, 7), shape = c(1, 3, 7), scale = c(2, 2, 4), prop = c(0.6, 0.3, 0.1)) prob qweibullmix(prob, shape = c(1, 3, 7), scale = c(2, 2, 4), prop = c(0.6, 0.3, 0.1))rweibullmix(10, shape = c(1, 3, 7), scale = c(2, 2, 4), prop = c(0.6, 0.3, 0.1)) dweibullmix(c(0, 2, 1), shape = c(1, 3), scale = c(2, 2), prop = c(0.6, 0.4)) prob <- pweibullmix(c(0.1, 7), shape = c(1, 3, 7), scale = c(2, 2, 4), prop = c(0.6, 0.3, 0.1)) prob qweibullmix(prob, shape = c(1, 3, 7), scale = c(2, 2, 4), prop = c(0.6, 0.3, 0.1))
The distribution function (pwrappedcauchy()) and quantiles
(qwrappedcauchy()) of the wrapped Cauchy distribution cannot
be obtained analytically. They are therefore missing in the
'circular' package and are obtained here numerically.
Random number generation (rwrappedcauchy()) and density
(dwrappedcauchy()) don't need a numerical
approximation and are provided here for consistency in parametrization
with the other wrapped Cauchy functions.
rwrappedcauchy(n, location = 0, scale = 1) dwrappedcauchy(theta, location = 0, scale = 1) pwrappedcauchy(theta, location = 0, scale = 1, K = 100, check_prec = FALSE) qwrappedcauchy(p, location = 0, scale = 1, K = 100, check_prec = FALSE)rwrappedcauchy(n, location = 0, scale = 1) dwrappedcauchy(theta, location = 0, scale = 1) pwrappedcauchy(theta, location = 0, scale = 1, K = 100, check_prec = FALSE) qwrappedcauchy(p, location = 0, scale = 1, K = 100, check_prec = FALSE)
n |
integer value, the number of random samples to be
generated with |
location |
numeric value, the mean of the distribution. |
scale |
numeric value, the parameter tuning the spread of the density. It must be non-negative. |
theta |
numeric vector giving the angles where the density or distribution function is evaluated. |
K |
integer value, the number of "wraps" used in each direction to approximate the distribution. |
check_prec |
logical, whether to check if the precision of the numerical approximation with the current parameters is higher than 99%. |
p |
numeric vector giving the probabilities where the quantile function is evaluated. |
One could alternatively convert scale to rho via
rho = exp(-scale) and use
circular::rwrappedcauchy(theta, mu=location rho=rho) or
circular::dwrappedcauchy(theta, mu=location rho=rho).
The wrapped Cauchy cdf, for which there is no analytical expression,
is calculated by wrapping the Cauchy distribution times
around the circle in each direction and summing the Cauchy cdfs at each point of
the circle. Let follow a Cauchy distribution and
a wrapped Cauchy distribution, where can take values
.
is approximated as
The quantiles are calculated by numerical inversion.
dwrappedcauchy() gives a vector of length length(theta)
containing the density at theta.
pwrappedcauchy() gives a
vector of length length(theta) containing
the distribution function at the corresponding values of theta.
qwrappedcauchy() gives a vector of length length(p)
containing the quantiles at the corresponding values of p.
rwrappedcauchy() generates a vector of length n
containing the random samples, i.e. angles in .
circular::dwrappedcauchy(),
circular::rwrappedcauchy().
set.seed(123) rwrappedcauchy(10, location = 0, scale =3) dwrappedcauchy(c(0.1, pi), location = pi, scale =2) circular::dwrappedcauchy(circular::circular(c(0.1,pi)), mu = circular::circular(pi), rho =exp(-2)) prob <- pwrappedcauchy(c(0.1, pi), location = pi, scale =2) prob qwrappedcauchy(prob, location = pi, scale =2)set.seed(123) rwrappedcauchy(10, location = 0, scale =3) dwrappedcauchy(c(0.1, pi), location = pi, scale =2) circular::dwrappedcauchy(circular::circular(c(0.1,pi)), mu = circular::circular(pi), rho =exp(-2)) prob <- pwrappedcauchy(c(0.1, pi), location = pi, scale =2) prob qwrappedcauchy(prob, location = pi, scale =2)