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Programita, a software to perform point pattern analysis with Ripley’s L and the O-ring statistic
I developed Programita for my 1999, 2001, 2003, 2005, and 2007 graduate course “ Patrones espaciales en ecología: modelos y análisis”, at the Escuela para Graduados, Facultad de Agronomia, University Buenos Aires, Argentina.
The Programita software allows you to perform univariate and bivariate point-pattern analysis with Ripley's L-function L(r), the pair-correlation function g(r), the O-ring statistic O(r), and the distribution G(y) of nearest neighbor distances y . Programita contains standard and non-standard procedures for most practical applications. Procedures for non-standard situations include the possibility to perform point-pattern analyses for arbitrarily shaped study regions and Programita offers a range of non-standard null models such as heterogeneous Poisson null models or cluster null models.
The calculation of L(r), g(r), G(y), and O(r) is done within a grid-based framework which greatly simplifies the computation of L, g, and O for non-standard situations. The second-order statistics are based on the distance between all pairs of points of a pattern and count the number of points within (or at) a certain distance, r, of each point, with r taking a range of scales. While the L-function is basically related to the mean number of neighbors in a circle of radius r, the pair-correlation function g(r) and the O-ring statistic are related to the density of neighbors in an annulus of radius r.
Programita tests for significance of a given null model by comparing the observed data with Monte Carlo envelopes from multiple simulations of the null model. However, to aid interpretation of the simulation envelopes and to avoid underestimation of type I error due to simultaneous inference, Programita includes a goodness-of-fit (GoF) test which allows you to determine the accurate type I error rate for a distance interval of interest.
You can request Programita and a draft version of the user manual (with extensive examples) by contacting me via email. At a later step I will make Programita freely available from this site.
Programita has the following features:
The univariate null models include
The bivariate null models include
Combine results from replicate plots
For statistical analysis it is common to map several replicate plots of a larger point pattern under identical conditions. In this case the resulting second-order statistics of the individual replicate plots can be combined into average second-order statistics. This is of particular interest if the number of points in each replicate plot is relatively low. In this case the confidence limits of individual analyses would become wide, but combining the data of several replicate plots into average second-order statistics increases the sample size and thus narrows the confidence limits. Average second-order statistics are also an effective way of summarizing the results of several replicate plots.
You can download background information and instructions for combining the results of replicate plots into one mean, weighted function: PDF (134K).
Goodness-of-fit (GoF) test
The traditional approach in point pattern analysis of using simulation envelopes [e.g., Kmin(r) and Kmax(r)] to judge departure of the data from a given null model involves simultaneous inference because several tests are performed simultaneously, one at each different distance r. Simultaneous inference yields an underestimation of type I error and therefore the upper and lower bounds of the simulation envelope cannot be interpreted as (1 – α)% confidence intervals [Stoyan and Stoyan (1994, Chapter 15.8), Diggle (2003: Chapter 2), Loosmore and Ford (2006, Ecology 87: 1925). The inaccurate type I error rate of a simulation envelope based statistical test is e.g., α = 1/s where s – 1 is the number of patterns used to construct the envelope, and the additional 1 in the denominator accounts for the observed pattern being tested.
Underestimation of type I error is an especial issue when using cumulative statistics such as Ripley’s K-function. In this case a high proportion of all point-point pairs used to evaluate K(r+1) are also used for evaluating K(r), thus K(r+1) and K(r) are not statistically independent. Although Kmin(r) ≤ K(r) ≤ Kmax(r) for each fixed r, the probability for the joined occurrence of Kmin(r1) ≤ K(1) ≤ Kmax(r1) and Kmin(r2) ≤ K(2) ≤ Kmax(r2) cannot be computed easily.
However, underestimation of type I error is less an issue when using non accumulative statistics such as the pair-correlation function g(r) (or the O-ring statistic) which use for each distance r a different set of point-point pairs.
To aid the interpretation of the simulation envelopes, Programita includes the algorithm of the goodness-of-fit test proposed in Diggle (2003: Chapter 2). The single test statistic used in this test represents the total squared deviation between the observed pattern and the theoretical result across the distances of interest.
The Programita software allows you to perform analysis analogous to point-pattern analysis which considers the finite size and irregular shape of objects (e.g., plants) and to compare the results with that of the conventional point approximation. The plants are approximated by using an underlying grid and may occupy several adjacent grid cells depending on their size and shape. Null models correspond to that of point pattern analysis but need to be modified to account for the finite size and irregular shape of plants.
Here you can download my recent article describing the analysis of objects of finite size and irregular shape:
Ripley's K- function
To determine the bivariate K-function of two patterns occurring in a study region R, a circle of radius r is centered i each point of pattern 1 and the number of points of pattern 2 inside the circle is counted. For n2 points of pattern 2 distributed in a study region R with area A, the density l = n2/A gives the mean number of points per unit area, assumed approximately constant through R (i.e., a homogeneous pattern). The function λ K12(r) gives the expected number of points of pattern 2 within radius r of an arbitrary point of pattern 1:
λK12(r) = E(#(points of pattern ≤ r from an arbitrary point of pattern 1))
where # means “the number of”, and E() is the expectation operator. Note that this definition applies also for univariate patterns, in this case K(r) = K11(r). If the points are independent (random), the expected value of K12(r) equals π r2, i.e., the area of a circle of radius r. Thus, this null model depends on the spatial scale r. To remove this scale dependence of K12(r) and to stabilize the variance, a transformation called L-function, is used instead:
It follows L12(r) > 0 [L(r) > 0] for attraction [clustering], whereas L12(r) < 0 [L(r) < 0] indicates repulsion [regularity].
The O-ring statistics and the pair-correlation function
We obtain g12(r) = 1 [g(r)=1] for independent [random] patterns g12(r) > 1 [g(r)>1] for attraction [clustering], whereas g12(r) < 0 [g(r) <0] indicates repulsion (regularity). Wiegand et al. (1999) modified the pair-correlation function g12(r) to obtain a measure with the direct interpretation of a neighborhood density, the O-ring statistics
O12(r) = λ2g12(r)
For a univariate pattern, O(r) = O11(r). If pattern 2 is independent from pattern 1 we obtain O12(r)= λ2 and O12(r) < λ2 for repulsion, whereas O12(r) > λ2 for attraction.
Publications that use Programita
Birkhofer, K., J.R. Henschel, and S. Scheu. 2006. Spatial-pattern analysis in a
territorial spider: evidence for multi-scale effects. Ecography 29:
Chung, M.Y, Nash, J.N., and M.G. Chung. 2007. Effects of population succession on demographic and genetic processes: predictions and tests in the daylily Hemerocallis thunbergii (Liliaceae) . Molecular Ecology 16: 2816–2829.
Chung, M.Y, and J. D. Nason. 2007. Spatial demographic and genetic consequences of harvesting within populations of the terrestrial orchid Cymbidium goeringii. Biological Conservation 137: 125-137
Feagin, R.A., and X.B. Wu. 2007. The spatial patterns of functional groups and sand dune plant community succession. Journal of Rangeland Ecology and Management 60: 417-425
Getzin, S., and K. Wiegand. 2007. Asymmetric tree growth at the stand level: Random crown patterns and the response to slope. Forest Ecology and Management 242: 165-174
Hao, Z, J. Zhang, B. Song, J. Ye and B. Li. 2007.
Vertical structure and spatial associations of dominant tree species in an
old-growth temperate forest. Forest Ecology and Management 252: 1-11
Jacquemyn, H., R. Brys, K. Vandepitte, O. Honnay, I. Roldán-Ruiz and T. Wiegand. 2007 A spatially-explicit analysis of seedling recruitment in the terrestrial orchid Orchis purpurea. New Phytologist 176: 448–459
Lee, A. C, R. M. Lucas. 2007. A LiDAR-derived canopy density model for tree stem and crown mapping in Australian forests. Remote Sensing of Environment 111: 493–518
Ramsay, P. M., and R M. Fotherby. 2007. Implications of the spatial pattern of Vigur's Eyebright (Euphrasia vigursii) for heathland management. Basic and Applied Ecology 8: 242-251
Schmidt, J. P. 2007. Sex ratio and spatial pattern of males and females in the dioecious sandhill shrub, Ceratiola ericoides ericoides (Empetraceae) Michx. Plant Ecology, online first
Stockholm, D., R. Benchaouir, J. Picot, P. Rameau, T. M. A. Neildez, G. Landini, C. Laplace-Builhé, and A. Paldi. 2007. The origin of phenotypic heterogeneity in a clonal cell population in vitro. PLoS ONE. 2007; 2(4): e394.
Wiegand, T, C.V.S. Gunatilleke, and I.A.U.N. Gunatilleke. 2007. Species associations in a heterogeneous Sri Lankan Dipterocarp forest. The American Naturalist 170 E77–E95.
Wiegand, T, C.V.S. Gunatilleke, I.A.U.N. Gunatilleke, and T. Okuda. 2007. Analyzing the spatial structure of a Sri Lankan tree species with multiple scales of clustering. Ecology 88: 3088–3102.
Blanco, P.D., C. M. Rostagno, H. F. del Valle, A. M. Beeskow, and T. Wiegand. 2008. Grazing impacts in vegetated dune fields: predictions from spatial pattern analysis. Rangeland Ecology and Management 61: 194-203
Chung, M. Y. 2008. Variation in demographic and fine-scale genetic structure with population-history stage of Hemerocallis taeanensis (Liliaceae) across the landscape. Ecological Research 23: 83-90
Djossa, B.A., Fahr, J., Wiegand, T., Ayihouénou, B.E., Kalko, E.K.V., and B. A. Sinsin. 2008 Land use impact on Vitellaria paradoxa C.F. Gaerten. stand structure and distribution patterns: a comparison of Biosphere Reserve of Pendjari in Atacora district in Benin. Agroforestry Systems 72: 205-220.
Felinks, B. and T. Wiegand. 2008. Analysis of spatial pattern in early stages of primary succession on former lignite mining sites. Journal of Vegetation Science 19:267-276
Meyer, K.M., Ward, D., Wiegand, K. & Moustakas, A. 2008. Multi-proxy evidence for competition between savanna woody species. Perspectives in Ecology, Evolution and Systematic 10: 63-72
Getzin, S., T. Wiegand, K. Wiegand, and F. He. Heterogeneity influences spatial patterns and demographics in forest stands. Journal of Ecology
Giesselmann, U.C., T. Wiegand, J. Meyer, R. Brandl, and M. Vogel. in press. Spatial patterns in the sociable weaver (Philetairus socius). Austral Ecology.
Rosental, G., and D. Lederbogen. in press. Response of the clonal plant Apium repens (Jacq.) Lag. to extensive grazing. FLORA.
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