Introduction to nullcat

Introduction

nullcat provides null model algorithms for categorical and quantitative community ecology data. It extends classic binary null models (e.g., curveball, swap) to work with categorical data, and introduces a stratified randomization framework for continuous data.

Categorical null models generalize binary null models to matrices where cells contain integers representing discrete categories. Whereas binary models preserve totals (e.g. row or column sums), categorical models preserve multisets (e.g. category frequencies in each row or column). For example, if a row contains [1, 1, 2, 3, 3, 3], a row-preserving categorical algorithm would preserve this exact multiset—two 1s, one 2, and three 3s—while randomizing where these categories appear within the row.

Traditional binary models are simply the specific two-category cases of their corresponding general categorical models (e.g. curveball is equivalent to curvecat with only two categories).

Categorical null models

# Create a categorical matrix
set.seed(123)
x <- matrix(sample(1:4, 100, replace = TRUE), nrow = 10)

# Randomize using curvecat (preserves row & column category multisets)
x_rand <- curvecat(x, n_iter = 1000)

# Verify margins are preserved
all.equal(sort(x[1, ]), sort(x_rand[1, ]))  # Row preserved
#> [1] TRUE
all.equal(sort(x[, 1]), sort(x_rand[, 1]))  # Column preserved
#> [1] TRUE

set.seed(456)
x <- matrix(sample(1:3, 60, replace = TRUE), nrow = 10)

# Fixed-fixed: both row and column margins preserved
x_fixed <- curvecat(x, n_iter = 1000)

# Row-constrained: only row margins preserved
x_row <- r0cat(x)

# Column-constrained: only column margins preserved
x_col <- c0cat(x)

# Check row sums by category
table(x[1, ])           # Original row 1
#> 
#> 1 2 3 
#> 2 2 2
table(x_fixed[1, ])     # Preserved in curvecat
#> 
#> 1 2 3 
#> 2 2 2
table(x_row[1, ])       # Preserved in r0cat
#> 
#> 1 2 3 
#> 2 2 2
table(x_col[1, ])       # Not preserved in c0cat
#> 
#> 1 2 3 
#> 1 3 2

Quantitative null models

The quantize() function provides stratified randomization for continuous community data (abundance, biomass, cover, occurrence probability). It works by applying the following steps:

1. Stratify: use a flexible binning scheme to convert continuous values into discrete strata
2. Randomize: apply one of the categorical null models described above to the stratified matrix
3. Reassign: map randomized strata back to original values, subject to a chosen constraint

Why use quantize? Existing quantitative null models lack flexibility in what margins they preserve. While binary curveball elegantly preserves row and column totals, and vegan’s quasiswap models do this for count data, there’s no framework for preserving row and/or column value distributions for continuous data (abundance, biomass, cover). The stratification approach fills this gap: when using fixed = "row" or fixed = "col", one dimension’s value multisets are preserved exactly while the other is preserved at the resolution of strata. This is the closest available approximation to a quantitative fixed-fixed null model. Additionally, stratification provides explicit control over the preservation-randomization tradeoff through the number of strata used.

# Create a quantitative community matrix
set.seed(789)
comm <- matrix(runif(200), nrow = 20)

# Default: curvecat-backed stratified randomization with 5 strata
rand1 <- quantize(comm, n_strata = 5, n_iter = 2000, fixed = "row")

# Values are similar but rearranged
cor(as.vector(comm), as.vector(rand1))
#> [1] 0.1757922
plot(rowSums(comm), rowSums(rand1))

plot(colSums(comm), colSums(rand1))

set.seed(200)
x <- rexp(100, .1)

# More strata = less mixing, higher fidelity to original distribution
s3 <- stratify(x, n_strata = 3)
s10 <- stratify(x, n_strata = 10)

table(s3)   # Coarser bins
#> s3
#>  1  2  3 
#> 89  9  2
table(s10)  # Finer bins
#> s10
#>  1  2  3  4  5  6 10 
#> 49 22 17  4  4  2  2

# Transform before stratifying (e.g., log-transform for skewed data)
s_log <- stratify(x, n_strata = 5, transform = log1p)
table(s_log)
#> s_log
#>  1  2  3  4  5 
#> 14 26 27 27  6

# Rank transform creates equal-occupancy strata
s_rank <- stratify(x, n_strata = 5, transform = rank)
table(s_rank)  # Nearly equal counts per stratum
#> s_rank
#>  1  2  3  4  5 
#> 20 20 20 20 20

# Separate zeros into their own stratum
x_with_zeros <- c(0, 0, 0, x)
s_zero <- stratify(x_with_zeros, n_strata = 4, zero_stratum = TRUE)
table(s_zero)
#> s_zero
#>  1  2  3  4 
#>  3 89  9  2

set.seed(100)
comm <- matrix(rexp(100), nrow = 10)

# Preserve row value multisets (quantitative row sums maintained)
rand_row <- quantize(comm, n_strata = 5, fixed = "row", n_iter = 2000)
all.equal(rowSums(comm), rowSums(rand_row))
#> [1] TRUE

# Preserve column value multisets (quantitative column sums maintained)
rand_col <- quantize(comm, n_strata = 5, fixed = "col", n_iter = 2000)
all.equal(colSums(comm), colSums(rand_col))
#> [1] TRUE

# Cell-level preservation: each value moves with its original cell location
# The categorical randomization determines WHERE cells go, but each cell
# carries its original value with it
rand_cell <- quantize(comm, n_strata = 5, fixed = "cell", n_iter = 2000)

# Values shuffled globally within strata, holding none of the above fixed
rand <- quantize(comm, n_strata = 5, fixed = "stratum", n_iter = 2000)

# For non-fixed-fixed methods like r0cat or c0cat, only some fixed options make sense:
# r0cat with fixed="col" or c0cat with fixed="row" would be incompatible

Efficient repeated randomization

The quantize routine includes some setup, including stratification, that doesn’t need to be repeatedly recomputed when generating multiple randomized versions of the same matrix. For generating null distributions, you can pre-compute overhead once using quantize_prep(), or you can use *_batch() functions to efficiently generate multiple replicates (optionally using parallel processing).

set.seed(400)
comm <- matrix(rexp(200), nrow = 20)

# Prepare once
prep <- quantize_prep(comm, n_strata = 5, fixed = "row", n_iter = 2000)

# Generate many randomizations efficiently
rand1 <- quantize(prep = prep)
rand2 <- quantize(prep = prep)
rand3 <- quantize(prep = prep)

# Or use batch functions
nulls <- quantize_batch(comm, n_reps = 99, n_strata = 5, 
                       fixed = "row", n_iter = 2000)
dim(nulls)  # 20 rows × 10 cols × 99 replicates
#> [1] 20 10 99

Convergence diagnostics

Sequential algorithms (curvecat, swapcat, tswapcat) require sufficient iterations to reach stationarity. You can use the trace_cat() function with either nullcat or quantize to assess how many iterations are needed for your particular dataset and algorithm. The trace shows how similarity to the original matrix changes over iterations; when the trace stabilizes around zero, the algorithm has reached stationarity. You can also use suggest_n_iter() to get an automated suggestion for a reasonable number of burn-in iterations to reach stationarity.

set.seed(300)
x <- matrix(sample(1:5, 400, replace = TRUE), nrow = 20)

# Generate trace showing mixing over iterations
trace <- trace_cat(x, fun = "nullcat", method = "curvecat",
                   n_iter = 1000, n_chains = 3, thin = 10)

# Visual inspection
plot(trace)

# Automatic burn-in suggestion
suggested <- suggest_n_iter(trace, tail_frac = 0.3)
print(suggested)
#> suggested_n_iter object
#> -----------------------
#> Converged: TRUE 
#> Suggested n iterations: 410

Weighted pair sampling

Standard null models assume that any row can freely exchange tokens with any other row—by default, all randomization algorithms select pairs of rows (or columns) uniformly at random for token exchange. But in many ecological contexts, this is unrealistic. Species assemblages at sites 3000 km apart shouldn’t contribute equally to a null expectation designed to test local community assembly.

Weighted pair sampling addresses this by making nearby (or similar) sites and/or species exchange tokens more frequently, while preserving the same marginal constraints as the unweighted model. This changes the null hypothesis being tested. An unweighted null model asks whether the observed matrix is structured relative to unconstrained random assembly; a weighted null model asks whether it is structured relative to assembly constrained by the supplied pairwise relationships. There is no correction for asymmetric pair selection (as there would be in, e.g., a Metropolis-Hastings sampler), so the weights directly shape the stationary distribution.

In nullcat, the wt_row and wt_col parameters let you bias pair selection frequencies using pairwise weight matrices, enabling spatially, environmentally, or taxonomically constrained null models. This weighted pair sampling works with the core categorical curvecat(), swapcat(), and tswapcat() functions, as well as with quantize() for continuous data.

Using weights has minimal effect on computation speed (<5% increase per iteration in benchmarks). However, sparse weight matrices slow mixing because fewer pairs are available for exchange, so a larger number of iterations is usually necessary with weights. Use trace_cat() and suggest_n_iter() to calibrate—they automatically account for the weight structure.

set.seed(42)

# 20 sites x 15 species categorical matrix
x <- matrix(sample(1:4, 300, replace = TRUE), nrow = 20)

# Site coordinates
coords <- cbind(runif(20), runif(20))

# Distance-decay weight matrix: nearby sites exchange more often
d <- as.matrix(dist(coords))
W <- exp(-d / 0.3)  # Gaussian kernel with scale = 0.3

# Spatially constrained randomization
x_spatial <- curvecat(x, n_iter = 3000, wt_row = W)

# Margins are still preserved
all.equal(sort(x[1, ]), sort(x_spatial[1, ]))
#> [1] TRUE
all.equal(sort(x[, 1]), sort(x_spatial[, 1]))
#> [1] TRUE

set.seed(42)
x <- matrix(sample(1:3, 80, replace = TRUE), nrow = 8)

# Two island groups: rows 1-4 and 5-8
W <- matrix(0, 8, 8)
W[1:4, 1:4] <- 1
W[5:8, 5:8] <- 1
diag(W) <- 0  # diagonal is ignored, but zero is cleaner

# Tokens only move within islands
x_islands <- curvecat(x, n_iter = 3000, wt_row = W, output = "index")

# Verify: tokens from rows 1-4 stayed in rows 1-4
idx <- matrix(1:length(x), nrow(x))
all(x_islands[1:4, ] %in% idx[1:4, ])
#> [1] TRUE

set.seed(42)
x <- matrix(sample(1:4, 200, replace = TRUE), nrow = 20)

# Trait-based column weights (e.g., species with similar body size exchange more)
traits <- runif(10)
W_col <- exp(-as.matrix(dist(traits)) / 0.3)

x_trait <- curvecat(x, n_iter = 3000, wt_col = W_col)

set.seed(42)
x <- matrix(sample(1:4, 200, replace = TRUE), nrow = 20)

# Spatial weights for rows, trait weights for columns
coords <- cbind(runif(20), runif(20))
W_row <- exp(-as.matrix(dist(coords)) / 0.3)

traits <- runif(10)
W_col <- exp(-as.matrix(dist(traits)) / 0.3)

x_dual <- curvecat(x, n_iter = 3000, wt_row = W_row, wt_col = W_col)

Integration with vegan

nullcat includes several functions to integrate with the vegan package’s null model framework. For both nullcat() and quantize(), there are sequential and non-sequential variants of these functions that create vegan::commsim() objects suitable for use with vegan::nullmodel() and vegan::oecosimu(). Examples include:

library(vegan)

# Categorical data
x_cat <- matrix(sample(1:4, 100, replace = TRUE), nrow = 10)
cs <- nullcat_commsim_seq(method = "curvecat")
nm <- nullmodel(x_cat, cs)
sims <- simulate(nm, nsim = 99, burnin = 1000, thin = 100)

# Quantitative data
x_quant <- matrix(rexp(100), nrow = 10)
cs_quant <- quantize_commsim(n_strata = 5, method = "curvecat", n_iter = 2000)
nm_quant <- nullmodel(x_quant, cs_quant)
sims_quant <- simulate(nm_quant, nsim = 99)

References

• Strona et al. (2014). A fast and unbiased procedure to randomize ecological binary matrices. Nature Communications, 5, 4114.
• Gotelli (2000). Null model analysis of species co-occurrence patterns. Ecology, 81(9), 2606-2621.

Further reading

• Package website: https://matthewkling.github.io/nullcat/
• Report issues: https://github.com/matthewkling/nullcat/issues