Categorical matrix randomization

Randomize binary or categorical community matrices using categorical generalizations of binary community null model algorithms. Optionally constrain randomization to occur within groups of rows (e.g. regions) and/or columns (e.g. clades).

Usage

nullcat(
  x,
  method = nullcat_methods(),
  n_iter = 1000L,
  output = c("category", "index"),
  swaps = c("auto", "vertical", "horizontal", "alternating"),
  row_groups = NULL,
  col_groups = NULL,
  wt_row = NULL,
  wt_col = NULL,
  seed = NULL
)

Arguments

x

A matrix of categorical data, encoded as integers. Values should represent category or stratum membership for each cell.

method

Character specifying the randomization algorithm to use. Options include the following; see details and linked functions for more info.

"curvecat": categorical analog to curveball; see curvecat() for details.
"swapcat": categorical analog to swap; see swapcat() for details.
"tswapcat": categorical analog to tswap; see tswapcat() for details.
"r0cat": categorical analog to r0; see r0cat() for details.
"c0cat": categorical analog to c0; see c0cat() for details.

n_iter

Number of iterations. Default is 1000. Larger values yield more thorough mixing. Ignored for non-sequential methods. Minimum burn-in times can be estimated with suggest_n_iter(). When row_groups or col_groups is used, n_iter iterations are run separately on every block. The number needed can be more or fewer than for the full matrix, so estimate it by passing the same groups to suggest_n_iter().

output

Character indicating type of result to return:

"category" (default) returns randomized matrix
"index" returns an index matrix describing where original entries (a.k.a. "tokens") moved. Useful mainly for testing, and for applications like quantize() that care about token tracking in addition to generic integer categories.

swaps

Character string controlling the direction of token movement. Only used when method is "curvecat", "swapcat", or "tswapcat". Affects the result only when output = "index", otherwise it only affects computation speed. Options include:

"vertical": Tokens move between rows (stay within columns).
"horizontal": Tokens move between columns (stay within rows).
"alternating": Tokens move in both dimensions, alternating between vertical and horizontal swaps. Provides full 2D mixing without preserving either row or column token sets.
"auto" (default): For output = "category", automatically selects the fastest option based on matrix dimensions. For output = "index", defaults to "alternating" for full mixing.

row_groups, col_groups

Optional vectors assigning each row (for row_groups) or column (for col_groups) of x to a group. Labels can be integers, characters, or factor levels. When supplied, the matrix is split into blocks (every combination of a row group and a column group) and each block is randomized independently with method, so values are only ever exchanged within a block. For algorithms that fix row and column margins, this additionally fixes the category counts of every row group within each column (and every column group within each row): for example, grouping sites into regions gives a null model in which each region keeps its own species pool. Rows or columns that are alone in their group cannot be changed by fixed-margin algorithms. To randomize within a different grouping in each replicate (e.g. random spatial tilings from tile_groups()), pass a groups matrix to nullcat_batch() or quantize_batch().

wt_row, wt_col

Deprecated. Experimental pairwise weight matrices previously used to bias which pairs of rows or columns exchange values. But weighting pair selection changes only how quickly a randomization mixes, not the null distribution it converges to; the only lasting effect of a weight matrix is to prevent exchanges between rows (or columns) that are not connected, directly or through intermediaries, by nonzero weights. Weight matrices are therefore now converted to row_groups / col_groups given by the connected components of their nonzero entries, with a warning. Use row_groups / col_groups instead.

seed

Integer used to seed random number generator, for reproducibility.

Value

A matrix of the same dimensions as x, either randomized categorical values (when output = "category") or an integer index matrix describing the permutation of entries (when output = "index").

Details

curvecat, swapcat, and tswapcat are sequential algorithms that hold category multisets fixed in every row and column. curvecat and swapcat propose symmetric moves, so with enough iterations they converge to the same null distribution: a uniform draw from all matrices with the observed row and column category multisets. curvecat is the most efficient of the two (i.e. fewest steps to become fully mixed). tswapcat repeatedly redraws until it finds a valid swap; this makes each iteration productive, but biases the sampling toward matrices with many available swaps, so its stationary distribution is not uniform (cf. Miklós & Podani 2004). swapcat and tswapcat are thus useful mainly for methodological comparison.

The r0cat algorithm holds category multisets fixed in rows but not columns, while c0cat does the opposite.

Note that categorical null models are for cell-level categorical data. Site-level attributes (e.g., land cover) or species-level attributes (e.g., functional traits) should be analyzed using different approaches. See vignette for details.

References

Miklós, I., & Podani, J. (2004). Randomization of presence–absence matrices: comments and new algorithms. Ecology, 85(1), 86–92.

Examples

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

# Randomize using curvecat method (preserves row & column margins)
x_rand <- nullcat(x, method = "curvecat", n_iter = 1000)

# Check that row multisets are preserved
all.equal(sort(x[1, ]), sort(x_rand[1, ]))
#> [1] TRUE

# Randomize within two regions of five sites each
region <- rep(c("north", "south"), each = 5)
x_reg <- nullcat(x, method = "curvecat", n_iter = 1000, row_groups = region)

# Each region keeps its own category counts in every column
all.equal(rowsum((x == 1) * 1, region), rowsum((x_reg == 1) * 1, region))
#> [1] TRUE