Lacunarity and generalized lacunarity for binary time series

Gliding-box masses

Fix a box (a contiguous window) of size \(s\) and slide it one observation at a time along the series. At each of the \(N - s + 1\) positions the box mass is the number of occupied sites it covers, \[ m \;=\; \sum_{t = j}^{\,j + s - 1} x_t , \] that is, a running sum of the series over a window of width \(s\). Sliding the box produces an empirical distribution of masses \(Q(m, s)\), the relative frequency of boxes carrying mass \(m\) at scale \(s\). The package evaluates the masses on dyadic scales \(s = 2^{i}\), \(i = 1, \dots, p\), with \(p = \lceil \log_2(\text{longest run of ones}) \rceil\).

Moment generating function

All the estimators are built from the \(q\)-th moment of the box-mass distribution, \[ Z(q, s) \;=\; \sum_{m} m^{q}\, Q(m, s) , \] computed by the helper zqs(). The argument mat is a two-column table with the distinct masses x and their frequencies freq; internally the frequencies are normalised to probabilities \(Q = \text{freq}/\sum \text{freq}\).

The lacunarity index

The lacunarity at scale \(s\) is the ratio of the second moment to the square of the first moment of the box masses, \[ \Lambda(s) \;=\; \frac{Z(2, s)}{\big[Z(1, s)\big]^{2}} \;=\; \frac{\langle m^{2} \rangle}{\langle m \rangle^{2}} \;=\; 1 + \frac{\operatorname{Var}(m)}{\langle m \rangle^{2}} . \] The last identity shows that \(\Lambda(s) \ge 1\) always, and that lacunarity is exactly one plus the squared coefficient of variation of the box masses:

• \(\Lambda(s) = 1\) means every box of size \(s\) carries the same mass — the pattern is translationally homogeneous at that scale;
• \(\Lambda(s) > 1\) signals heterogeneity (gappiness), growing with the spread of the masses.

As the box grows, local fluctuations average out and \(\Lambda(s)\) typically decays towards \(1\). For self-similar sets the lacunarity follows a power law (Allain and Cloitre 1991) \[ \Lambda(s) \;=\; \beta\, s^{-\gamma} , \] whose lacunarity scaling exponent \(\gamma\) measures how fast the heterogeneity decays with scale: large \(\gamma\) means a quickly homogenising pattern, while \(\gamma \to 0\) marks a scale-invariant one. lac() estimates the exponent y from the slope of \(\log_2 \Lambda(s)\) on \(\log_2 s\) fitted through the origin, and also returns the vector Ds of lacunarities \(\Lambda(s)\) and the vector s of scales.

Generalized lacunarity

Vernon-Carter et al. (2009) generalised the index by replacing the fixed orders \(1\) and \(2\) with the generalized moments \(Z(q, s) = \sum_m m^{q} Q(m, s)\), giving \[ \Lambda_q(s) \;=\; \left[ \frac{Z(2q, s)}{\big[Z(q, s)\big]^{2}} \right]^{1/q} , \qquad q \in \{-10, \dots, 10\} \setminus \{0\}. \] The exponent \(1/q\) keeps the units of \(\Lambda_q(s)\) those of a ratio of quadratic masses for every \(q\), so all orders are directly comparable. The order \(q\) acts as a magnifying glass: large positive \(q\) emphasises the dense (high-mass) boxes, while negative \(q\) emphasises the sparse (low-mass) boxes. As in the ordinary case, self-similar sets obey a power law \(\Lambda_q(s) = \beta_q\, s^{-\gamma(q)}\), and genlac() estimates the generalized scaling exponent \(\gamma(q)\) from the (origin-constrained) slope of \(\log_{10}\Lambda_q(s)\) on \(\log_{10} s\).

The whole curve \(q \mapsto \gamma(q)\) is a lacunarity spectrum. Following Vernon-Carter et al. (2009), a set is monofractal when \(\gamma(q)\) is essentially constant in \(q\) — small and large gaps organise the same way — and multilacunar when \(\gamma(q)\) varies appreciably with \(q\), revealing a richer arrangement of small and large gaps. genlac() returns the scales s, the orders q, the exponents yq (\(= \gamma(q)\)) and the full matrix Dqs of \(\Lambda_q(s)\) values.

A single series

We simulate a Bernoulli series with \(80\%\) ones and compute its lacunarity.

x <- rbinom(2000, size = 1, prob = 0.8)
rx <- lac(x)
rx
#> $y
#> [1] 0.01311362
#> 
#> $Ds
#> [1] 1.130717 1.067124 1.036955 1.020647 1.011371
#> 
#> $s
#>      [,1]
#> [1,]    2
#> [2,]    4
#> [3,]    8
#> [4,]   16
#> [5,]   32

Ds decreases towards \(1\) as the scale grows, the signature of an essentially homogeneous (gap-poor) pattern. The decay is clearest on a log–log plot:

plot(rx$s, rx$Ds, log = "xy", type = "b", pch = 19,
     xlab = "scale s", ylab = expression(Lambda(s)),
     main = "Lacunarity curve of a random binary series")
abline(h = 1, lty = 2, col = "grey50")

Same density, different texture

The point of lacunarity is that density alone does not determine texture. Below, z is a strongly clustered sequence (blocks of ones and zeros) and w is a random shuffle of z: both have exactly the same number of ones, hence the same density, but a very different spatial organisation.

block <- c(rep(c(rep(1, 8),  rep(0, 8)),  60),
           rep(c(rep(1, 16), rep(0, 16)), 60))
z <- block
w <- sample(block)               # same ones, gaps reshuffled

mean(z) == mean(w)               # identical density
#> [1] TRUE

lz <- lac(z)
lw <- lac(w)
c(clustered = lz$y, shuffled = lw$y)
#>  clustered   shuffled 
#> 0.19072431 0.07468284

The clustered series has a substantially larger lacunarity at every scale:

ylim <- range(lz$Ds, lw$Ds)
plot(lz$s, lz$Ds, log = "xy", type = "b", pch = 19, ylim = ylim,
     xlab = "scale s", ylab = expression(Lambda(s)),
     main = "Same density, different lacunarity")
lines(lw$s, lw$Ds, type = "b", pch = 1, lty = 2)
legend("topright", c("clustered", "shuffled"),
       pch = c(19, 1), lty = c(1, 2), bty = "n")

Although z and w are indistinguishable by their mean, \(\Lambda(s)\) cleanly separates the blocky texture from the random one — exactly the information that density and most second-order summaries miss.

The generalized lacunarity spectrum

Finally we compare the spectra \(\gamma(q)\) of the clustered and the random series with genlac().

gz <- genlac(z)
gw <- genlac(w)

plot(gz$q, gz$yq, type = "b", pch = 19,
     xlab = "moment order q", ylab = expression(gamma(q)),
     main = "Generalized lacunarity spectrum",
     ylim = range(gz$yq, gw$yq))
lines(gw$q, gw$yq, type = "b", pch = 1, lty = 2)
legend("topleft", c("clustered", "shuffled"),
       pch = c(19, 1), lty = c(1, 2), bty = "n")

The clustered series produces a spectrum that varies markedly with \(q\) — a multilacunar texture in the terminology of Vernon-Carter et al. (2009), where small and large gaps scale differently. The shuffled series gives a nearly flat spectrum, the monofractal signature of a structureless, single-scale gap distribution.