Package {lacunarity}

lacunarity: lacunarity and generalized lacunarity for binary time series

Description

Estimators of the gliding-box lacunarity index and of the generalized (multifractal-like) lacunarity for one-dimensional binary series. See the package vignette, vignette("lacunarity"), for the underlying theory and worked examples.

Main functions

lac

ordinary lacunarity index \Lambda(s) and its scaling exponent.

genlac

generalized lacunarity \Lambda_q(s) and the spectrum of exponents \gamma(q).

Author(s)

Maintainer: Ikaro Barreto daniel.carvalho.ib@gmail.com

See Also

Useful links:

• https://github.com/Ikarobarreto/lacunarity

• Report bugs at https://github.com/Ikarobarreto/lacunarity/issues

Generalized lacunarity of a binary series

Description

Computes the generalized (multifractal-like) lacunarity \Lambda_q(s) of a binary time series and the spectrum of scaling exponents \gamma(q).

Usage

genlac(x)

Arguments

Details

The ordinary lacunarity is extended to an arbitrary moment order q by

\Lambda_q(s) = \left[ \frac{Z(2q,s)}{Z(q,s)^2} \right]^{1/q},

where Z(q,s) is the q-th moment of the gliding-box mass distribution at scale s. Large positive q emphasises dense boxes and negative q emphasises sparse boxes, so the curve q \mapsto \gamma(q), with \gamma(q) the slope of \log_{10}\Lambda_q(s) on \log_{10} s, describes how gaps of different magnitudes scale. Orders q range over \{-10, \dots, 10\} \setminus \{0\}.

Value

A list with components:

s

the dyadic box scales s = 2^i.

q

the moment orders.

yq

the generalized scaling exponents \gamma(q).

Dqs

the matrix of generalized lacunarities \Lambda_q(s) (rows index q, columns index s).

References

Vernon-Carter, J., Lobato-Calleros, C., Escarela-Perez, R., Rodriguez, E. and Alvarez-Ramirez, J. (2009). A suggested generalization for the lacunarity index. Physica A, 388(20), 4305-4314.

Allain, C. and Cloitre, M. (1991). Characterizing the lacunarity of random and deterministic fractal sets. Physical Review A, 44(6), 3552-3558.

See Also

lac for the ordinary lacunarity index.

Examples

x <- rbinom(1000, 1, 0.85)
genlac(x)

Lacunarity index of a binary series

Description

Computes the gliding-box lacunarity index \Lambda(s) of a binary time series across dyadic scales, together with its scaling exponent.

Usage

lac(x)

Arguments

Details

A box of size s is slid one observation at a time along the series and its mass m (the number of ones it covers) is recorded. Writing Z(q,s) for the q-th moment of the resulting box-mass distribution, the lacunarity index is

\Lambda(s) = \frac{Z(2,s)}{Z(1,s)^2} = 1 + \frac{\mathrm{Var}(m)}{\mathrm{mean}(m)^2},

so that \Lambda(s) \ge 1, with equality only for a translationally homogeneous pattern. Larger values indicate gappier, more heterogeneous textures. The scaling exponent y is the slope of \log_2 \Lambda(s) regressed on \log_2 s. Scales are dyadic, s = 2^i, and capped by the longest run of ones.

Value

A list with components:

y

the lacunarity scaling exponent \gamma.

Ds

the lacunarity \Lambda(s) at each scale.

s

the dyadic box scales s = 2^i.

References

Allain, C. and Cloitre, M. (1991). Characterizing the lacunarity of random and deterministic fractal sets. Physical Review A, 44(6), 3552-3558.

Plotnick, R. E., Gardner, R. H., Hargrove, W. W., Prestegaard, K. and Perlmutter, M. (1996). Lacunarity analysis: a general technique for the analysis of spatial patterns. Physical Review E, 53(5), 5461-5468.

See Also

genlac for the generalized lacunarity spectrum.

Examples

x <- rbinom(1000, 1, 0.85)
lac(x)

Moment generating function of the box masses

Description

Internal helper that computes the q-th moment Z(q) = \sum_m m^q\, Q(m) of a box-mass frequency distribution, where Q(m) is the relative frequency of boxes carrying mass m.

Usage

zqs(mat, q)

Arguments

Value

The q-th moment of the box-mass distribution.