Landscape Ecology - Lecture Notes

Quantifying Landscape Pattern: Methods and Applications

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Regionalized variable theory
• Regionalized variable theory (Matheron 1963) provides an important theoretical basis for spatial statistics and geostatistics, which recognizes that observations taken at short distance from each other are more alike, i.e., have lower variances, than those taken further apart.

• Regionalized variable theory has implications for field sampling and experimentation.  Indeed, geostatistical principles are generating a rethinking of conventional field experimentation.

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Spatial Statistics
• The objectives of spatial statistics are: to describe how things are distributed in space (e.g., random, clustered, uniform), and to determine if spatial proximity (spatial autocorrelation) is playing a causal role in the observed distribution.

• It is important to distinguish pattern caused by environmental or spatial heterogeneity from that caused by spatial dependence or autocorrelation.

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Geostatistics
• Geostatistics is a set of statistical tools that detect, model and estimate spatial dependence or pattern of variables.

• Geostatistics allows for optimized estimation of such "regionalized variables" at unsampled locations through quantification of its spatial dependence structure.

 How does geostatistics differ from traditional statistics?

• Commonly used statistical tests (t-test, ANOVA, etc.) assume independence and normal distribution of sample values. Geostatistical tests specifically look for and quantify spatial dependence.
• The difference between geostatistical and conventionally used assumptions is that the former recognizes the existence of a spatial relationship for (co)variances which depends only on the separation vector h.  Thus, if observations separated by different distances are compared, their expected (co)variances may be unequal.

 Why is this beneficial to ecological studies?

• Space and time dependence and continuity are inherent in ecological phenomena. Organisms are not distributed randomly across their environment. Environmental variables like temperature and soil moisture do not change from one location to the next in an unrelated way.
• Spatial dependence or temporal dependence in one direction or another may indicate an underlying ecological process and direct further research.
• The application of geostatistical tools may cause erroneous results if: (i) if the sampling domain does not adhere to the intrinsic hypothesis, or (ii) geostatistical tools are inappropriately used.

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Autocorrelation indices:
 
• Moran's I

• Geary's c

where xi and xj are the values of grid cells i and j, and all other parameters are defined as in the equation for Moran's I.

• It seems desirable to use the two commonly used indices together in that Moran's I is intended to measure the degree of correlation between neighboring sites, whereas Geary's c measures how different they are. The degree of autocorrelation may be indicative of spatial distribution of landcover types (aggregated, regular, or random) or patchiness in landscape pattern (e.g., Legendre and Fortin 1989, Qi and Wu 1996).

An example:




Spatial autocorrelation indices, Moran's I and Geary's c

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• Semivariance analysis provides another means for examining autocorrelation in spatial data, which does not require second-order stationarity (while autocorrelation measures such as Moran's I do).
• Semivariogram can define the range or distance over which spatial dependence exists.
  

The semivariance statistic for one-dimension datasets is:







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Exmple (Robertson, P. 1987. Geostatistics in ecology: Interpolating with known variance. Ecology 68: 744-748):

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Relationship between fractal and semivariance analysis:
 
Fractal dimension D can be related to semivariance through the following formula (a function of the slope of a log-log semivariogram plot; Burrough 1981, 1986 ):

       D = 2 - m / 2

       where D = the Hausdorff-Besicovitch statistic,  m = the slope of a log-log semivariogram.

References:

• Wu, J. G., D. E. Jelinski, M. Luck, and P. T. Tueller. 2000. Multiscale analysis of landscape heterogeneity: Scale variance and pattern metrics. Geographic Information Sciences 6:6-19.
• Fortin, M.-J., M. R. T. Dale, and J. ver Hoef. 2002. Spatial analysis in ecology. Encyclopedia of Environmetrics 4:2051-2058.
  
• Fortin, M.-J. and M. R. T. Dale. 2005. Spatial Analysis:  A Guide for Ecologists. Cambridge University Press, Cambridge.
  
• Griffith, D. A. 2012. Spatial statistics: A quantitative geographer's perspective. Spatial Statistics 1:3-15.