lacunarity

lacunarity estimates the lacunarity and the generalized lacunarity of unidimensional binary time series. Lacunarity is a scale-dependent measure of translational heterogeneity: it describes how the gaps of a pattern are distributed, and — unlike density alone — it tells apart series that are clustered from series that are evenly spread.

Methodology

For a sliding box of size \(s\), the box mass \(m\) is the number of ones it covers. From the distribution of box masses, the lacunarity index is

\[\Lambda(s) = \frac{\langle m^2 \rangle}{\langle m \rangle^2} = 1 + \frac{\mathrm{Var}(m)}{\langle m \rangle^2} \ge 1,\]

so \(\Lambda(s) = 1\) means a perfectly homogeneous pattern and larger values mean gappier, more heterogeneous textures. The generalized lacunarity \(\Lambda_q(s) = \left(Z_{2q}(s)/Z_q(s)^2\right)^{1/q}\) extends this to an arbitrary moment order \(q\), yielding a spectrum of scaling exponents \(\gamma(q)\). See vignette("lacunarity") for the full theory and references (Allain & Cloitre, 1991; Vernon-Carter et al., 2009).

Installation

You can install the released version from CRAN with:

install.packages("lacunarity")

and the development version from GitHub with:

# install.packages("devtools")
devtools::install_github("Ikarobarreto/lacunarity")

Example

lac() returns the lacunarity scaling exponent y, the lacunarities Ds at each scale and the box scales s:

library(lacunarity)

x <- rbinom(1200, size = 1, prob = 0.8)
lac(x)
#> $y
#> [1] 0.01408786
#> 
#> $Ds
#> [1] 1.129157 1.068018 1.040669 1.024022 1.013987
#> 
#> $s
#>      [,1]
#> [1,]    2
#> [2,]    4
#> [3,]    8
#> [4,]   16
#> [5,]   32

Lacunarity captures texture, not just density. The series z (blocks of ones and zeros) and w (a random shuffle of z) have the same number of ones, yet very different lacunarities:

z <- c(rep(c(rep(1, 8), rep(0, 8)), 25), rep(c(rep(1, 16), rep(0, 16)), 25))
w <- sample(z)

c(density_z = mean(z), density_w = mean(w))      # identical density
#> density_z density_w 
#>       0.5       0.5
c(lac_z = lac(z)$y, lac_w = lac(w)$y)            # different lacunarity
#>     lac_z     lac_w 
#> 0.1908516 0.1142697

genlac() returns the generalized lacunarity; its scaling spectrum yq over the moment orders q summarises how small and large gaps scale:

g <- genlac(x)
round(g$yq, 3)
#>  [1] -0.148 -0.157 -0.166 -0.176 -0.185 -0.192 -0.191 -0.162 -0.092 -0.029
#> [11]  0.014  0.020  0.023  0.025  0.027  0.027  0.028  0.028  0.028  0.028

See vignette("lacunarity") for plots and a full walkthrough.

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