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Programita, a software to perform point pattern analysis in ecology

I developed Programita for my 1999, 2001, 2003,  2005,  2007, and 2008 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 the pair-correlation function g(r), the O-ring statistic  O(r), and the distribution function D(y) of nearest neighbor distances y, and the L-function L(r) which is based on Ripley’s K. Programita contains standard and non-standard procedures for most practical applications in ecology. 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 (Wiegand et al. 2007a) and you can fit cluster processes that incorporate one or two critical scales of clustering to your data (Wiegand et al. 2007b).

Additionally to standard point pattern analysis, Programita allows you to perform analysis analogous to point pattern analysis, but for objects of  finite size and irregular shape of objects. Programita contains a feature that allows the uses direct comparison of the results with that of the conventional point approximation. This technique is explained in detail in Wiegand et al. (2006)..

In the spirit of exploratory analysis, 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.

Programita allows for a variety of specific null models for univariate and bivariate point-patterns. The procedures used by Programita are described in detail in Wiegand and Moloney 2004, Wiegand et al. 2006,  Wiegand et al. (2007a), and Wiegand et al. (2007b).. An updated user manual fill follow soon.

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:

• calculates the pair-correlation function g(r) and the O-ring statistic for uni- and bivariate point-patterns

• calculates L-function for uni- and bivariate point-patterns

• allows calculation of second-order statistics for arbitrarily shaped study areas

• allows you to combine the results from several replicate plots into a mean, weigthed second-order statsitsic

• allows you to perform a goodness-of-fit (GoF) test, based on the approach of Diggle (2003: Chapter 2), to avoid underestimation of typ I error when using simulation envelopes.

• includes inhomogeneous K- and g-functions for heterogeneous point patterns proposed by Baddeley et al. (2000, Statistica Neerlandica 54: 329). A detailed description of this will follow  in an updated manual.

The univariate null models include

• CSR: the points of the pattern are randomized over the entire study region (which can be of any arbitrary shape).

• Heterogeneous Poisson process:  the points of the pattern are randomized in accordance with the first-order intensity of the pattern.

• Random labeling: Univariate random labeling can be used to correct for underlying environmental heterogeneity if a "control" pattern is available to act as surrogate for the varying environmental factor. The null model distributes the points of the pattern randomly over the joined locations of pattern and control.

• Thomas cluster process: if the univariate pattern shows clustering, a Thomas process can be used as null model. Programita  fits the two parameters of the Thomas process to the L and the g-functions calculated from the data and simulates the fitted Thomas process for construction of confidence envelopes. Programita also contains null modles for cluster processes with two critical scales of clustering.

• Hard (or soft) core: simple hard (or soft core) null model can be used to describe small-scale regularity which can be caused by the finite size of the points (e.g, the canopy of a tree). Programita enables you to combine a hard (or soft) core null model with the most other null models.

The bivariate null models include

• independence: a toroidal shift null model is used that conserves the spatial structure of both patterns, but it breaks the connection between them.

• random labeling: bivariate random labeling assumes that one process created a pattern and a subsequent process a binary label such as surviving-dead. Programita offers a variety of test statistics to test departures from random labeling. Each test  statistic evaluates a different biological effects. 

• Antecedent conditions: the location of pattern 1 is kept fixed (e.g., adult trees) while pattern 2 (e.g., seedlings) is distributed in accordance to a specific null model.

• Cluster processes: Programita includes several bivariate cluster processes which are based on the Thomas process. Programita fits the parameters of the bivariate cluster processes to the data and simulates  the fitted process for construction of confidence envelopes. The  bivariate cluster processes range between the extreme cases where (1) the points of pattern 1 are the cluster centres (= parents) of the clusters of pattern 2, and where (2) the clusters of pattern 1 and pattern 2 share the same parents.

• Hard (or soft) core: a hard (or soft core) null model can be used to describe small-scale regularity which can be caused by the finite size of the points (e.g, the canopy of a tree). Programita enables you to combine a hard (or soft) core null model with the most other null models.

Example of the analysis of a heterogeneous point-pattern with the O-ring function. The O-ring function describes the local neighborhood density of the points of the pattern (= probability to find a point in distance r away from an arbitrary point of the pattern). The grey horizontals give the average density of the pattern in the rectangular study region. The standard null model delivers aggregation at al scales r > 4 while the null model that accounts for the large-scale heterogeneity correctly reveals randomness at scales r > 4.

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., K_min(r) and K_max(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.

 

Extending point pattern analysis for objects of finite size and irregular shape

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.

Left: categorical map showing individuals of two shrub species.  Right: conventional point approximation where each shrub is approximated by its centre of mass.

 

 

Here you can download my recent article describing the analysis of objects of finite size and irregular shape:

• Wiegand, T., Kissling, W.D., Cipriotti, P.A., and Aguiar, M.R. 2006. Extending point pattern analysis to objects of finite size and irregular shape. Journal of Ecology 94: 825-837 Abstract and pdf

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

The bivariate pair-correlation function g₁₂(r) is the analogy to Ripley’s K₁₂(r) when replacing the circles of radius r by (infinitesimal) rings with radius r. The function λ₂ g₁₂(r) gives the expected density of points of pattern 2 at distance r of an arbitrary point of pattern 1:

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.

2007

Birkhofer, K. S. Scheu, and D. H. Wise. 2007. Small-scale spatial pattern of web-building ppiders (Araneae) in alfalfa: relationship to disturbance from cutting, prey availability, and intraguild interactions. Environmental Entomology 36: 801–810.

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

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.

Watson, D.M., D. A. Roshier, and T. Wiegand. 2007 Spatial ecology of a parasitic shrub: patterns and predictions. Austral Ecology 32: 359-369.

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. 

2008

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.

Chung, M. Y., and C.-W. Park. 2008. Fixation of alleles and depleted levels of genetic variation within populations of the endangered lithophytic orchid Amitostigma gracile (Orchidaceae) in South Korea: implications for conservation. Plant Systematics and Evolution 272:119-130.

De Luis, M., J. Raentós, T. Wiegand, and J. C. González-Hidalgo. in press. Temporal and spatial differentiation in seedling emergence may promote species coexistence in Mediterranean fire-prone ecosystems.Ecography 31: 620-629

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

Getzin, S., T. Wiegand, K. Wiegand, and F. He.  Heterogeneity influences spatial patterns and demographics in forest stands. Journal of Ecology 96: 807-820.

Giesselmann, U.C., T. Wiegand, J. Meyer, R. Brandl, and M. Vogel. 2008. Spatial patterns in the sociable weaver (Philetairus socius). Austral Ecology 32:359 - 369.

Lawes, M.J.; Griffiths, M.E.; Midgley, J.J.; Boudreau, S.; Eeley, H.A.C. and C. A. Chapman. 2008. Tree spacing and area of competitive influence do not scale with tree size in an African rainforest. Journal of Vegetation Science 19: 729-738

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

Moustakasa, A, K. Wiegand, S. Getzin, D. Ward, K. M. Meyer, M. Guenther, and K.-H. Mueller. 2008. Spacing patterns of an Acacia tree in the Kalahari over a 61-year period: How clumped becomes regular and vice versa. Acta Oecologica 33:355-364

Rosental, G., and D. Lederbogen. in press. Response of the clonal plant Apium repens (Jacq.) Lag. to extensive grazing. FLORA 203:141-151.

Schmidt, J. P. 2008. Sex ratio and spatial pattern of males and females in the dioecious sandhill shrub, Ceratiola ericoides ericoides (Empetraceae) Michx. Plant Ecology: 196: 281-288.

in press