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 $\mathbb{R}^d$ has topological and fractal dimension $d$. If the surface is nondifferentiable and rough, the fractal dimension takes...
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ftdatacite:10.48550/arxiv.1101.1444 2023-05-15T15:10:14+02:00 Estimators of Fractal Dimension: Assessing the Roughness of Time Series and Spatial Data Gneiting, Tilmann Ševčíková, Hana Percival, Donald B. 2011 https://dx.doi.org/10.48550/arxiv.1101.1444 https://arxiv.org/abs/1101.1444 unknown arXiv https://dx.doi.org/10.1214/11-sts370 arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ Methodology stat.ME FOS Computer and information sciences article-journal Article ScholarlyArticle Text 2011 ftdatacite https://doi.org/10.48550/arxiv.1101.1444 https://doi.org/10.1214/11-sts370 2022-04-01T14:34:50Z 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 $\mathbb{R}^d$ has topological and fractal dimension $d$. If the surface is nondifferentiable and rough, the fractal dimension takes values between the topological dimension, $d$, and $d+1$. 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 $p>0$ for stochastic processes, and the Hausdorff dimension of their sample paths, appears to be particularly robust and inclusive when $p=1$. : 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) Text Arctic Sea ice DataCite Metadata Store (German National Library of Science and Technology) Arctic |
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Methodology stat.ME FOS Computer and information sciences |
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Methodology stat.ME FOS Computer and information sciences Gneiting, Tilmann Ševčíková, Hana Percival, Donald B. Estimators of Fractal Dimension: Assessing the Roughness of Time Series and Spatial Data |
topic_facet |
Methodology stat.ME FOS Computer and information sciences |
description |
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 $\mathbb{R}^d$ has topological and fractal dimension $d$. If the surface is nondifferentiable and rough, the fractal dimension takes values between the topological dimension, $d$, and $d+1$. 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 $p>0$ for stochastic processes, and the Hausdorff dimension of their sample paths, appears to be particularly robust and inclusive when $p=1$. : 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) |
format |
Text |
author |
Gneiting, Tilmann Ševčíková, Hana Percival, Donald B. |
author_facet |
Gneiting, Tilmann Ševčíková, Hana Percival, Donald B. |
author_sort |
Gneiting, Tilmann |
title |
Estimators of Fractal Dimension: Assessing the Roughness of Time Series and Spatial Data |
title_short |
Estimators of Fractal Dimension: Assessing the Roughness of Time Series and Spatial Data |
title_full |
Estimators of Fractal Dimension: Assessing the Roughness of Time Series and Spatial Data |
title_fullStr |
Estimators of Fractal Dimension: Assessing the Roughness of Time Series and Spatial Data |
title_full_unstemmed |
Estimators of Fractal Dimension: Assessing the Roughness of Time Series and Spatial Data |
title_sort |
estimators of fractal dimension: assessing the roughness of time series and spatial data |
publisher |
arXiv |
publishDate |
2011 |
url |
https://dx.doi.org/10.48550/arxiv.1101.1444 https://arxiv.org/abs/1101.1444 |
geographic |
Arctic |
geographic_facet |
Arctic |
genre |
Arctic Sea ice |
genre_facet |
Arctic Sea ice |
op_relation |
https://dx.doi.org/10.1214/11-sts370 |
op_rights |
arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
op_doi |
https://doi.org/10.48550/arxiv.1101.1444 https://doi.org/10.1214/11-sts370 |
_version_ |
1766341262344978432 |