lacunarity {lacunr}

Calculate gliding-box lacunarity

Description

Generates \Lambda(r) lacunarity curves for a specified set of box sizes, using one of two versions of the gliding-box algorithm

Usage

lacunarity(x, box_sizes = "twos", periodic = FALSE)

Arguments

x

A 3-dimensional array of integer values

box_sizes

Which box sizes to use for calculating lacunarity:

• "twos" (the default) returns box sizes for all powers of two less than or equal to the smallest dimension of x.

• "all" calculates every possible box size up to the smallest dimension of x.

• Alternatively, users may supply their own vector of custom box sizes. This vector must be of type "numeric" and can only contain positive values. Values which exceed the dimensions of x are ignored.

periodic

A Boolean. Determines which boundary algorithm to use, the classic fixed boundary by Allain and Cloitre (default) or the periodic boundary algorithm introduced by Feagin et al. 2007. The latter is slightly slower but is more robust to edge effects.

Details

The raw \Lambda(r) values depend on the proportion of occupied voxels within the data space. As a result, it is difficult to compare two spatial patterns with different occupancy proportions because the curves will begin at different y-intercepts. This is rectified by normalizing the curve, typically by log-transforming it and dividing by the lacunarity value at the smallest box size (i.e. \log \Lambda(r)/\log \Lambda(1)). lacunarity() outputs both normalized and non-normalized \Lambda(r) curves for convenience.

The function also computes \mathrm{H}(r), a transformed lacunarity curve introduced by Feagin 2003. \mathrm{H}(r) rescales normalized \Lambda(r) in terms of the Hurst exponent, where values greater than 0.5 indicate heterogeneity and values less than 0.5 indicate homogeneity. Where \Lambda(r) describes a pattern's deviation from translational invariance, \mathrm{H}(r) describes its deviation from standard Brownian motion.

Value

A data.frame containing box sizes and their corresponding \Lambda(r), normalized \Lambda(r), and \mathrm{H}(r) values. Lacunarity is always computed for box size 1, even if the user supplies a custom box_sizes vector that omits it, as this value is required to calculate normalized lacunarity.

References

Allain, C., & Cloitre, M. (1991). Characterizing the lacunarity of random and deterministic fractal sets. Physical Review A, 44(6), 3552–3558. doi:10.1103/PhysRevA.44.3552.

Feagin, R. A. (2003). Relationship of second-order lacunarity, Hurst exponent, Brownian motion, and pattern organization. Physica A: Statistical Mechanics and its Applications, 328(3-4), 315-321. doi:10.1016/S0378-4371(03)00524-7.

Feagin, R. A., Wu, X. B., & Feagin, T. (2007). Edge effects in lacunarity analysis. Ecological Modelling, 201(3–4), 262–268. doi:10.1016/j.ecolmodel.2006.09.019.

Examples

# generate array
a <- array(data = rep(c(1,0), 125), dim = c(5,5,5))
# calculate lacunarity with default options
lacunarity(a)
# supply custom vector of box sizes
lacunarity(a, box_sizes = c(1,3,5))
# calculate lacunarity at all box sizes using the periodic boundary algorithm
lacunarity(a, box_sizes = "all", periodic = TRUE)