Standard and Generalized Lacunarity for Binary Time Series.
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.