Getting started

library(sapo)
library(sf)
#> Linking to GEOS 3.12.1, GDAL 3.8.4, PROJ 9.4.0; sf_use_s2() is TRUE

set.seed(2024)

The sapo package provides a Monte Carlo test for evaluating spatial association between polygons. For reproducibility, set a seed before using the functions.

This vignette illustrates the functionality of the sapo package. sapo, which is Portuguese for “toad,” stands for “Spatial Association Between Polygons”. This package enables you to test whether two types of polygons are spatially associated. This raises two key questions: 1) What do these “polygon types” represent? and 2) What constitutes spatial association?

Polygons can represent various spatial entities. In ecology, for instance, polygons might represent ecological patches (McGarigal 2013), animal home ranges, or forest canopy gaps. It’s often important to determine if two types of spatial phenomena (like those mentioned above) tend to attract or repel each other. Conditional Monte Carlo (MC) hypothesis tests (Godoy, Assunção, and Butler 2022) offer a way to test these hypotheses. The cmc_psat function (conditional MC polygon association test) implements these methods. All methods are based on conditional MC simulation, generating spatial patterns that assume spatial independence (using an adaptation of the toroidal shift¹) while preserving the observed marginal spatial structures.

To illustrate the package’s functionality, consider two simulated polygon types, poly1 and poly2, loaded below as sf objects. These will represent two distinct spatial features. Our goal is to test whether they are spatially independent.

poly1 <- system.file("extdata", "poly1.rds", package = "sapo") |>
  readRDS()
poly2 <- system.file("extdata", "poly2.rds", package = "sapo") |>
  readRDS()

str(poly1)
#> Classes 'sf' and 'data.frame':   20 obs. of  2 variables:
#>  $ id      : chr  "1" "2" "3" "4" ...
#>  $ geometry:sfc_POLYGON of length 20; first list element: List of 1
#>   ..$ : num [1:7, 1:2] 0.00727 0.04896 0.07805 0.07173 0 ...
#>   ..- attr(*, "class")= chr [1:3] "XY" "POLYGON" "sfg"
#>  - attr(*, "sf_column")= chr "geometry"
#>  - attr(*, "agr")= Factor w/ 3 levels "constant","aggregate",..: 
#>   ..- attr(*, "names")= chr(0)
str(poly2)
#> Classes 'sf' and 'data.frame':   25 obs. of  2 variables:
#>  $ id      : chr  "1" "2" "3" "4" ...
#>  $ geometry:sfc_POLYGON of length 25; first list element: List of 1
#>   ..$ : num [1:5, 1:2] 0.6 0.7 0.7 0.6 0.6 0.3 0.3 0.4 0.4 0.3
#>   ..- attr(*, "class")= chr [1:3] "XY" "POLYGON" "sfg"
#>  - attr(*, "sf_column")= chr "geometry"
#>  - attr(*, "agr")= Factor w/ 3 levels "constant","aggregate",..: 
#>   ..- attr(*, "names")= chr(0)

Below, we plot these polygon types from this toy example.

plot(sf::st_geometry(poly1), col = "gray70")
plot(sf::st_geometry(poly2), add = TRUE)

The cmc_psat function requires that its first two arguments, p1 and p2, are sf objects. Other important arguments include:

n_sim (default = 499): An integer specifying the number of conditional MC simulations under the null hypothesis of spatial independence.
alpha (default = 0.05): A real number (between 0 and 1) indicating the significance level for the hypothesis test.

For a deeper understanding of the remaining arguments, refer to Godoy, Assunção, and Butler (2022), which provides a comprehensive explanation of the method. The default argument values generally optimize the test’s power and type I error rate.

The code below tests whether poly1 and poly2 are spatially independent. In essence, it assesses whether the presence of a type 2 polygon affects the likelihood of observing a type 1 polygon.

my_ht <- cmc_psat(poly1, poly2)
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#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
#> all geometries
#> Warning: attribute variables are assumed to be spatially constant throughout
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The test’s p-value is 0.0720, indicating a lack of spatial dependence. If the analysis suggests that the polygons are not spatially independent, a descriptive analysis can be performed using a generalization of the H₁₂ function (typically K₁₂ or L₁₂, with the latter being the package default) used for analyzing spatial point patterns.

We’ll now create point-wise confidence bands based on the L₁₂ null hypothesis. These bands will be plotted against the L₁₂ function calculated from the real data. A significant result with the black curve (representing the real data) below the envelope suggests that the two polygon types repel each other. Conversely, if the black curve is above the envelope, it indicates attraction. Remember that these envelopes are for descriptive analysis, not inference. The p-value determines whether to reject the hypothesis of spatial independence.

sig_level <- my_ht$alpha
sig_plot <-
  data.frame(r = my_ht$distances,
             h = my_ht$mc_funct[NROW(my_ht$mc_funct), ],
             l = apply(my_ht$mc_funct, 2, quantile,
                       prob = .5 * sig_level),
             u = apply(my_ht$mc_funct, 2, quantile,
                       prob = 1 - .5 * sig_level))
## par(mfrow = c(1, 2))
## ## plot density part
## den <- density(my_ht$mc_sample)
## plot(den, main = sprintf("Test statistic (p-value = %.4f)",
##                          my_ht$p_value),
##      xlab = "u", ylab = "density")
## value <- my_ht$mc_sample[length(my_ht$mc_sample)]
## polygon(c(den$x[den$x >= value], value),
##         c(den$y[den$x >= value], 0),
##         col = "coral1",
##         border = "transparent")
## lines(den)
## H function
with(sig_plot, plot(x = r, y = h, type = "l",
                    col = 2,
                    main = "Local envelope",
                    xlab = "r",
                    ylab = expression(H[12](r)),
                    ylim = range(c(h, l, u))))
with(sig_plot, lines(x = r, y = l, lty = 2, col = 2))
with(sig_plot, lines(x = r, y = u, lty = 2, col = 2))
with(sig_plot,
     polygon(c(r, rev(r)), c(l, rev(u)),
             col = "coral1", lty = 0))
with(sig_plot, lines(x = r, y = h,
                     col = 1))

Godoy, Lucas da Cunha, Renato Martins Assunção, and Kevin A. Butler. 2022. “Testing the Spatial Association of Different Types of Polygons.” Spatial Statistics 51: 100695. https://doi.org/https://doi.org/10.1016/j.spasta.2022.100695.

McGarigal, Kevin. 2013. “Landscape Pattern Metrics.” In Encyclopedia of Environmetrics. John Wiley & Sons, Ltd. https://doi.org/https://doi.org/10.1002/9780470057339.val006.pub2.

a technique that shifts the polygons to simulate different spatial arrangements↩︎