mi_cyl {cylcop}

Estimate the Mutual Information Between a Circular and a Linear Random Variable

Description

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.

Usage

mi_cyl(theta, x, normalize = TRUE, symmetrize = FALSE)

Arguments

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 [0,1].

symmetrize

logical value whether it should be assumed that right and left turns are equivalent. If theta can take values in [-\pi, \pi), this means that positive and negative angles are equivalent.

Details

First, the two components of the empirical copula, u and v are obtained. Then the mutual information is calculated via discretizing u and v 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 (0,0) to (1,1), 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 [-\pi, \pi) 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.

Value

A numeric value, the mutual information between theta and x in nats.

References

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/.

See Also

cor_cyl(), fit_cylcop_cor().

Examples

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
)