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Computation of Lacunarity from Covariance of Spatial Binary Maps

  • Kassel HingeeEmail author
  • Adrian Baddeley
  • Peter Caccetta
  • Gopalan Nair
Article
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Abstract

We consider a spatial binary coverage map (binary pixel image) which might represent the spatial pattern of the presence and absence of vegetation in a landscape. ‘Lacunarity’ is a generic term for the nature of gaps in the pattern: a popular choice of summary statistic is the ‘gliding-box lacunarity’ (GBL) curve. GBL is potentially useful for quantifying changes in vegetation patterns, but its application is hampered by a lack of interpretability and practical difficulties with missing data. In this paper we find a mathematical relationship between GBL and spatial covariance. This leads to new estimators of GBL that tolerate irregular spatial domains and missing data, thus overcoming major weaknesses of the traditional estimator. The relationship gives an explicit formula for GBL of models with known spatial covariance and enables us to predict the effect of changes in the pattern on GBL. Using variance reduction methods for spatial data, we obtain statistically efficient estimators of GBL. The techniques are demonstrated on simulated binary coverage maps and remotely sensed maps of local-scale disturbance and meso-scale fragmentation in Australian forests. Results show in some cases a fourfold reduction in mean integrated squared error and a twentyfold reduction in sensitivity to missing data.

Supplementary materials accompanying the paper appear online and include a software implementation in the R language.

Keywords

Forest disturbance Fractal Gliding box Image analysis Random set Spatial statistics 

Notes

Acknowledgements

Our thanks to Michael Small and his research group for generous permission to use their computing resources.

Funding

This research was supported by the Australian Research Council (Grant No. DP130104470), Australia’s Commonwealth Scientific and Industrial Research Organisation, and an Australian Government Research Training Program Scholarship.

Supplementary material

13253_2019_351_MOESM1_ESM.pdf (19.8 mb)
Supplementary material 1 (pdf 20316 KB)
13253_2019_351_MOESM2_ESM.zip (36.8 mb)
Supplementary material 2 (zip 37662 KB)

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Copyright information

© International Biometric Society 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of Western AustraliaPerthAustralia
  2. 2.Data61CSIROPerthAustralia
  3. 3.Department of Mathematics and StatisticsCurtin UniversityPerthAustralia

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