Abstract
We carry out a simulation study of the estimation of fractal dimension in a grid-based setting typical of ecological species distributions, using null landscape models. We calculate the box-counting dimension for samples taken in various types of sampling geometry. Sampler geometries include simple blocks,Cantor grids and line transects. This method may be used to measure fractal dimension of a species distribution, but the accuracy depends on a number of criteria. The most important is sampling effort: any estimate will be inaccurate if the sampling effort is low. We also find the geometry of the sampler to be important. For a given sampling effort, schemes based on the Cantor grids performed better than either line transects or simple blocks. Sampling effort can be improved either by using a bigger sampler over a larger area or by repeated sampling of a smaller area: optimum performance is often a trade-off between these two mechanisms. However, performance is also highly sensitive to the type of fractal object being sampled, with certain types of object requiring a much greater effort for an accurate estimate of fractal dimension. These results raise the possibilities of using novel sampling techniques to estimate fractal dimension, when confronted with limited resources and time, but underline also the need for an understanding of the “type” of fractality expected in ecological situations.
Similar content being viewed by others
References
Azovsky A.I. 2000. Concept of scale in marine ecology: linking the words or the worlds? Web Ecology 1: 28-34.
Bellehumeur C. and Legendre P. 1998. Multiscale sources of variation in ecological variables: modeling spatial dispersion, elaborating sampling designs. Landsc. Ecol. 13: 15-25.
Berntson G.M. and Stoll P. 1997. Correcting for finite spatial scales of self-similarity when calculating the fractal dimensions of real world structures. In: Proceedings of the Royal Society London, series B 264., pp. 1531-1537.
Borgani S., Murante G., Provenzale A. and Valdarini R. 1993. Multifractal analysis of the galaxy distribution: Reliability of results from finite data sets. Phys. Rev. E47: 3879-3888.
Buczkowski S., Hilden P. and Cartilier L. 1998. Measurements of fractal dimension be box counting: a critical analysis of data scatter. Physica A252: 23-34.
Colasanti R.L. and Grime J.P. 1993. Resource Dynamics and Vegetation Process: a Deterministic Model Using Two-Dimensional Cellular Automata. Funct. Ecol. 7: 169-176.
Cutler C.D. 1993. A Review of the Theory and Estimation of Fractal Dimension. In: Tong H. (ed.), Nonlinear Time Series and Chaos, Dimension Estimation and Models. Vol. I. World Scientific, Singapore, pp. 1-107.
Dale M.R. 1998. Spatial Pattern Analysis in Plant Ecology. Cambridge University Press, New York, New York, USA.
Daley D.J. and Vere-Jones D. 1988. An Introduction to the Theory of Point Processes. Springer Verlag, New York, New York, USA.
Drake J.B. and Weishampel J.F. 2000. Multifractal analysis of canopy height measures in a longleaf pine savanna. Forest Ecol. Management 128: 121-127.
Falconer K. 1990. Fractal Geometry. J. Wiley and sons, London, UK.
Feder J. 1988. Fractals. Plenum Press, New York, New York, USA.
Gardner R.H. 1999. RULE: Map generation and a spatial pattern analysis program. In: Klopatek J.M. and Gardner R.H. (eds), Landscape Ecological Analysis Issues and Applications. Springer, New York, New York, USA, pp. 280-303.
Gunnarsson B. 1992. Fractal dimension of plants and body size distribution in spiders. Funct. Ecol. 6: 636-641.
Hall P. and Wood A. 1993. On the performance of box counting estimators of fractal dimension. Biometrika 80: 246-252.
Halley J.M., Comins H.N., Lawton J.H. and Hassell M.P. 1994. Competition, Succession and Pattern in Fungal Communities-Towards a Cellular Automation Model. Oikos 70: 435-442.
Hamburger D., Biham O. and Avnir D. 1996. Apparent Fractality Emerging from Models of Random Distributions. Phys. Rev. E53: 3342-3358.
Hastings H.M. and Sugihara G. 1993. Fractals a User's Guide for the Natural Sciences. Oxford University Press, New York, New York, USA.
He F.L. and Gaston K.J. 2000. Estimating species abundance from occurrence. Amer. Naturalist 156: 553-559.
Hentschel H.G.E. and Procaccia I. 1983. The infinite number of generalized dimensions of fractals and strange attractors. Physica D8: 435-444.
Keitt T.H. 1997. Stability and complexity on a lattice: coexistence of species in an individual-based food web model. Ecol. Model. 102: 243-258.
Krummel J.R., Gardner R.H., Sugihara G., O'Neill R.V. and Coleman P.R. 1987. Landscape patterns in a disturbed environment. Oikos 48: 321-324.
Kunin W.E. 1998. Extrapolating Species Abundance Across Spatial Scales. Science 281: 1513-1515.
Kunin W.E., Hartley S. and Lennon J.J. 2000. Scaling down: on the challenge of estimating abundance from occurrence patterns. Amer. Naturalist 156: 560-566.
Lavorel S., Gardner R.H. and O'Neill R.V. 1993. Analysis of patterns in hierarchically structured landscapes. Oikos 67: 521-528.
Leduc A., Prairie Y.T. and Bergeron Y. 1994. Fractal dimension estimates of a fragmented landscape-sources of variability. Landsc. Ecol. 9: 279-286.
Li B.L. 2000. Fractal geometry applications in description and analysis of patch patterns and patch dynamics. Ecol. Model. 132: 33-50.
Lin Y.X. and McCabe M. 1999. An example of applying the asymptotic quasi likelihood to dimension estimation for random spatial patterns. Journal of statistical planning and inference 80: 197-209.
Lobo A., Moloney K., Chic O. and Chiariello N. 1998. Analysis of fine scale spatial pattern of a grassland from remotely sensed imagery and field collected data. Landsc. Ecol. 13: 111-131.
Loehle C. and Li B.L. 1996. Statistical properties of ecological and geologic fractals. Ecol. Model. 85: 271-284.
Mandelbrot B.B. 1982. The Fractal Geometry of Nature. W.H. Freeman, San Francisco.
Manly B.F.J. 1997. Randomization Bootstrap and Monte Carlo Methods in Biology. 2nd edn. Chapman and Hall, London, UK.
McIntyre N.E. and Wiens J.A. 1999. Interactions between habitat abundance and configuration: experimental validation of some predictions from percolation theory. Oikos 86: 129-137.
Milne B.T. 1992. Spatial aggregation and neutral models in fractal landscapes. Amer. Naturalist 139: 32-57.
Nikora V.I., Pearson C.P. and Shankar U. 1999. Scaling properties in landscape patterns: New Zealand experience. Landsc. Ecol. 14: 17-33.
Ogata Y. and Katsura K. 1991. Maximum-Likelihood-Estimates of the Fractal Dimension for Random Spatial Patterns. Biometrika 78: 463-474.
Oleshko K., Brambila F., Aceff F. and Mora L.P. 1998. From fractal analysis along a line to fractals on the plane. Soil Till. Res. 45: 389-406.
Palmer M.W. 1988. Fractal geometry: a tool for describing spatial patterns of plant communities. Vegetatio 75: 91-102.
Palmer M.W. 1992. The coexistence of species in fractal landscapes. Amer. Naturalist 193: 375-397.
Peitgen H.O., Jurgens H. and Saupe D. 1992. Chaos and Fractals New Frontiers of Science. Springer Verlag, New York, New York, USA.
Pickett S.T.A. and Cadenasso M.L. 1995. Landscape ecology: spatial heterogeneity in ecological systems. Science 269: 331-334.
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. Phys. Rev. E53: 5461-5468.
Ramsey J.B. and Yuan H.J. 1990. The statistical properties of dimension calculations using small data sets. Nonlinearity 3: 155-176.
Ricotta C. 2000. From theoretical ecology to statistical physics and back: self similar landscape metrics as a synthesis of ecological diversity and geometrical complexity. Ecol. Model. 125: 245-253.
Ritchie M.E. 1998. Scale dependent foraging and patch choice in fractal environments. Evol. Ecol. 12: 309-330.
Shorrocks B., Marsters J., Wars J. and Evennett P.J. 1991. The fractal dimension of lichens and the distribution of arthropod body lengths. Funct. Ecol. 5: 457-460.
Theiler J. 1990. Statistical Precision of dimension estimators. Phys. Rev. A41: 3038-3051.
Tischendorf L. 2001. Can landscape indices predict ecological processes constantly? Landsc. Ecol. 16: 235-254.
Turcotte D.L. 1992. Fractals and Chaos in Geology and Geophysics. Cambridge University Press, Cambridge, UK.
Vere-Jones D., Davies R.B., Harte D., Mikosch T. and Wang Q. 1997. Problems and examples in the estimation of fractal dimesion from meteorological and earthquake data. In: Subba Rao T., Priestley M.B. and Lessi O. (eds), Applications of Time Series Analysis in Astronomy and Meteorology. Chapman and Hall, London, UK, pp. 359-375.
Wallis J.R. and Matalas N.C. 1970. Small sample properties of H and K-Estimators of the Hurst Coefficient h. Water Resour. Res. 6: 1583-1594.
With K.A. and King A.W. 1998. Extinction thresholds for species in fractal landscapes. Conserv. Biol. 13: 314-326.
With K.A. and King A.W. 2001. Analysis of landscape sources and sinks: the effect of spatial pattern on avian demography. Biol. Conserv. 100: 75-88.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kallimanis, A.S., Sgardelis, S.P. & Halley, J.M. Accuracy of fractal dimension estimates for small samples of ecological distributions. Landscape Ecology 17, 281–297 (2002). https://doi.org/10.1023/A:1020285932506
Issue Date:
DOI: https://doi.org/10.1023/A:1020285932506