Kolmogorov–Smirnov test for spatially correlated data |
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Authors: | Ricardo A Olea Vera Pawlowsky-Glahn |
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Institution: | (1) Eastern Energy Resources Team, US Geological Survey, 12201 Sunrise Valley Dr., MS 956, Reston, VA 20192, USA;(2) Department d’Informàtica i Matemàtica Aplicada, Universitat de Girona, Campus Montilivi, Edifici P4, 17071 Girona, Spain |
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Abstract: | The Kolmogorov–Smirnov test is a convenient method for investigating whether two underlying univariate probability distributions
can be regarded as undistinguishable from each other or whether an underlying probability distribution differs from a hypothesized
distribution. Application of the test requires that the sample be unbiased and the outcomes be independent and identically
distributed, conditions that are violated in several degrees by spatially continuous attributes, such as topographical elevation.
A generalized form of the bootstrap method is used here for the purpose of modeling the distribution of the statistic D of the Kolmogorov–Smirnov test. The innovation is in the resampling, which in the traditional formulation of bootstrap is
done by drawing from the empirical sample with replacement presuming independence. The generalization consists of preparing
resamplings with the same spatial correlation as the empirical sample. This is accomplished by reading the value of unconditional
stochastic realizations at the sampling locations, realizations that are generated by simulated annealing. The new approach
was tested by two empirical samples taken from an exhaustive sample closely following a lognormal distribution. One sample
was a regular, unbiased sample while the other one was a clustered, preferential sample that had to be preprocessed. Our results
show that the p-value for the spatially correlated case is always larger that the p-value of the statistic in the absence of spatial correlation, which is in agreement with the fact that the information content
of an uncorrelated sample is larger than the one for a spatially correlated sample of the same size. |
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Keywords: | Probability distribution Statistic Test Bootstrap Simulated annealing |
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