3,167 research outputs found
Estimators of Fractal Dimension: Assessing the Roughness of Time Series and Spatial Data
The fractal or Hausdorff dimension is a measure of roughness (or smoothness)
for time series and spatial data. The graph of a smooth, differentiable surface
indexed in has topological and fractal dimension . If the
surface is nondifferentiable and rough, the fractal dimension takes values
between the topological dimension, , and . We review and assess
estimators of fractal dimension by their large sample behavior under infill
asymptotics, in extensive finite sample simulation studies, and in a data
example on arctic sea-ice profiles. For time series or line transect data,
box-count, Hall--Wood, semi-periodogram, discrete cosine transform and wavelet
estimators are studied along with variation estimators with power indices 2
(variogram) and 1 (madogram), all implemented in the R package fractaldim.
Considering both efficiency and robustness, we recommend the use of the
madogram estimator, which can be interpreted as a statistically more efficient
version of the Hall--Wood estimator. For two-dimensional lattice data, we
propose robust transect estimators that use the median of variation estimates
along rows and columns. Generally, the link between power variations of index
for stochastic processes, and the Hausdorff dimension of their sample
paths, appears to be particularly robust and inclusive when .Comment: Published in at http://dx.doi.org/10.1214/11-STS370 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
A Transfer Matrix for the Backbone Exponent of Two-Dimensional Percolation
Rephrasing the backbone of two-dimensional percolation as a monochromatic
path crossing problem, we investigate the latter by a transfer matrix approach.
Conformal invariance links the backbone dimension D_b to the highest eigenvalue
of the transfer matrix T, and we obtain the result D_b=1.6431 \pm 0.0006. For a
strip of width L, T is roughly of size 2^{3^L}, but we manage to reduce it to
\sim L!. We find that the value of D_b is stable with respect to inclusion of
additional ``blobs'' tangent to the backbone in a finite number of points.Comment: 19 page
On Simulation of Manifold Indexed Fractional Gaussian Fields
To simulate fractional Brownian motion indexed by a manifold poses serious numerical problems: storage, computing time and choice of an appropriate grid. We propose an effective and fast method, valid not only for fractional Brownian fields indexed by a manifold, but for any Gaussian fields indexed by a manifold. The performance of our method is illustrated with different manifolds (sphere, hyperboloid).
Random fractal strings: their zeta functions, complex dimensions and spectral asymptotics
In this paper a string is a sequence of positive non-increasing real numbers which sums to one. For our purposes a fractal string is a string formed from the lengths of removed sub-intervals created by a recursive decomposition of the unit interval. By using the so called complex dimensions of the string, the poles of an associated zeta function, it is possible to obtain detailed information about the behaviour of the asymptotic properties of the string. We consider random versions of fractal strings. We show that using a random recursive self-similar construction it is possible to obtain similar results to those for deterministic self-similar strings. In the case of strings generated by the excursions of stable subordinators, we show that the complex dimensions can only lie on the real line. The results allow us to discuss the geometric and spectral asymptotics of one-dimensional domains with random fractal boundary
Dirac operators and spectral triples for some fractal sets built on curves
We construct spectral triples and, in particular, Dirac operators, for the
algebra of continuous functions on certain compact metric spaces. The triples
are countable sums of triples where each summand is based on a curve in the
space. Several fractals, like a finitely summable infinite tree and the
Sierpinski gasket, fit naturally within our framework. In these cases, we show
that our spectral triples do describe the geodesic distance and the Minkowski
dimension as well as, more generally, the complex fractal dimensions of the
space. Furthermore, in the case of the Sierpinski gasket, the associated
Dixmier-type trace coincides with the normalized Hausdorff measure of dimension
.Comment: 48 pages, 4 figures. Elementary proofs omitted. To appear in Adv.
Mat
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