entropy_kde2d()

Introduction

The entropy_kde2d() function estimates the Shannon entropy for a two-dimensional dataset using kernel density estimation (KDE). This function provides a non-parametric measure of entropy, useful for analyzing the uncertainty or randomness in bivariate distributions, such as spatial data of animal trajectories (e.g, Maei et al., 2009).

The parameters for entropy_kde2d() are as follows:

• x: A numeric vector representing the random variable in the first dimension.

• y: A numeric vector representing the random variable in the second dimension.

• n_grid: A numeric value specifying the number of grid cells used to evaluate the density; defaults to 150. Higher values result in finer density grids but increase computational complexity.

The function outputs a single numeric value representing the entropy of the given data.

Example

set.seed(123)
# Generate a 2D normal distribution with a correlation of 0.6
n <- 1000
mean <- c(0, 0)
sd_x <- 1
sd_y <- 5
correlation <- 0.6
sigma <- matrix(
  c(
    sd_x^2,
    correlation * sd_x * sd_y,
    correlation * sd_x * sd_y,
    sd_y^2
  ),
  ncol = 2
)
library(MASS)
simulated_data <- mvrnorm(n, mu = mean, Sigma = sigma)
x <- simulated_data[, 1]
y <- simulated_data[, 2]
# Plot the data
plot(simulated_data)

# Compute entropy using normal entropy formula
cov_matr <- cov(cbind(x, y))
sigmas <- diag(cov_matr)
det_sig <- prod(sigmas)

normal_entropy is a function that computes the entropy of a bivariate normal distribution given the number of dimensions k, the value of \(\pi\), and the determinant of the covariance matrix det_sig. This is used, for example, in Maei et al. (2009) to compute the entropy of mice trajectories in a Morris water maze.

normal_entropy <- function(k, pi, det_sig) {
  (k / 2) * (1 + log(2 * pi)) + (1 / 2) * log(det_sig)
}

entropia <- normal_entropy(k = 2, pi = pi, det_sig)
print(entropia) # Expected value close to 4.3997

## [1] 4.399979

# Compute entropy using entropy_kde2d
result <- entropy_kde2d(x, y, n_grid = 50)
print(result) # Expected value close to 4.2177

## [1] 4.217723