Create an Object for Null Model Algorithms
commsim.RdThe commsim function can be used to feed Null Model algorithms into
nullmodel analysis.
The make.commsim function returns various predefined algorithm types
(see Details).
These functions represent low level interface for community null model
infrastructure in vegan with the intent of extensibility,
and less emphasis on direct use by users.
Usage
commsim(method, fun, binary, isSeq, mode)
make.commsim(method)
# S3 method for class 'commsim'
print(x, ...)Arguments
- method
Character, name of the algorithm.
- fun
A function. For possible formal arguments of this function see Details.
- binary
Logical, if the algorithm applies to presence-absence or count matrices.
- isSeq
Logical, if the algorithm is sequential (needs burnin and thinning) or not.
- mode
Character, storage mode of the community matrix, either
"integer"or"double".- x
An object of class
commsim.- ...
Additional arguments.
Details
The function fun must return an array of dim(nr, nc, n),
and must take some of the following arguments:
x:input matrix,
n:number of permuted matrices in output,
nr:number of rows,
nc:number of columns,
rs:vector of row sums,
cs:vector of column sums,
rf:vector of row frequencies (non-zero cells),
cf:vector of column frequencies (non-zero cells),
s:total sum of
x,fill:matrix fill (non-zero cells),
thin:thinning value for sequential algorithms,
...:additional arguments.
You can define your own null model, but
several null model algorithm are pre-defined and can be called by
their name. The predefined algorithms are described in detail in the
following chapters. The binary null models produce matrices of zeros
(absences) and ones (presences) also when input matrix is
quantitative. There are two types of quantitative data: Counts are
integers with a natural unit so that individuals can be shuffled, but
abundances can have real (floating point) values and do not have a
natural subunit for shuffling. All quantitative models can handle
counts, but only some are able to handle real values. Some of the null
models are sequential so that the next matrix is derived from the
current one. This makes models dependent from previous models, and usually
you must thin these matrices and study the sequences for stability:
see oecosimu for details and instructions.
See Examples for structural constraints imposed by each algorithm and defining your own null model.
Binary null models
All binary null models preserve fill: number of presences or
conversely the number of absences. The classic models may also
preserve column (species) frequencies (c0) or row frequencies
or species richness of each site (r0) and take into account
commonness and rarity of species (r1, r2). Algorithms
swap, tswap, curveball, quasiswap and
backtracking preserve both row and column frequencies. Three
first ones are sequential but the two latter are non-sequential
and produce independent matrices. Basic algorithms are reviewed by
Wright et al. (1998).
"r00":non-sequential algorithm for binary matrices that only preserves the number of presences (fill).
"r0":non-sequential algorithm for binary matrices that preserves the site (row) frequencies.
"r1":non-sequential algorithm for binary matrices that preserves the site (row) frequencies, but uses column marginal frequencies as probabilities of selecting species.
"r2":non-sequential algorithm for binary matrices that preserves the site (row) frequencies, and uses squared column marginal frequencies as as probabilities of selecting species.
"c0":non-sequential algorithm for binary matrices that preserves species frequencies (Jonsson 2001).
"swap":sequential algorithm for binary matrices that changes the matrix structure, but does not influence marginal sums (Gotelli & Entsminger 2003). This inspects \(2 \times 2\) submatrices so long that a swap can be done.
"tswap":sequential algorithm for binary matrices. Same as the
"swap"algorithm, but it tries a fixed number of times and performs zero to many swaps at one step (according to the thin argument in the call). This approach was suggested by Miklós & Podani (2004) because they found that ordinary swap may lead to biased sequences, since some columns or rows are more easily swapped."curveball":sequential method for binary matrices that implements the ‘Curveball’ algorithm of Strona et al. (2014). The algorithm selects two random rows and finds the set of unique species that occur only in one of these rows. The algorithm distributes the set of unique species to rows preserving the original row frequencies. Zero to several species are swapped in one step, and usually the matrix is perturbed more strongly than in other sequential methods.
"quasiswap":non-sequential algorithm for binary matrices that implements a method where matrix is first filled honouring row and column totals, but with integers that may be larger than one. Then the method inspects random \(2 \times 2\) matrices and performs a quasiswap on them. In addition to ordinary swaps, quasiswap can reduce numbers above one to ones preserving marginal totals (Miklós & Podani 2004). The method is non-sequential, but it accepts
thinargument: the convergence is checked at everythinsteps. This allows performing several ordinary swaps in addition to fill changing swaps which helps in reducing or removing the bias."greedyqswap":A greedy variant of quasiswap. In greedy step, one element of the \(2 \times 2\) matrix is taken from \(> 1\) elements. The greedy steps are biased, but the method can be thinned, and only the first of
thinsteps is greedy. Even modest thinning (saythin = 20) removes or reduces the bias, andthin = 100(1% greedy steps) looks completely safe and still speeds up simulation. The code is experimental and it is provided here for further scrutiny, and should be tested for bias before use."backtracking":non-sequential algorithm for binary matrices that implements a filling method with constraints both for row and column frequencies (Gotelli & Entsminger 2001). The matrix is first filled randomly, but typically row and column sums are reached before all incidences are filled in. After this begins "backtracking", where some of the incidences are removed, and filling is started again, and this backtracking is done so many times that all incidences will be filled into matrix. The results may be biased and should be inspected carefully before use.
Quantitative Models for Counts with Fixed Marginal Sums
These models shuffle individuals of counts and keep marginal sums
fixed, but marginal frequencies are not preserved. Algorithm
r2dtable uses standard R function r2dtable also
used for simulated \(P\)-values in chisq.test.
Algorithm quasiswap_count uses the same, but preserves the
original fill. Typically this means increasing numbers of zero cells
and the result is zero-inflated with respect to r2dtable.
"r2dtable":non-sequential algorithm for count matrices. This algorithm keeps matrix sum and row/column sums constant. Based on
r2dtable."quasiswap_count":non-sequential algorithm for count matrices. This algorithm is similar as Carsten Dormann's
swap.webfunction in the package bipartite. First, a random matrix is generated by ther2dtablefunction preserving row and column sums. Then the original matrix fill is reconstructed by sequential steps to increase or decrease matrix fill in the random matrix. These steps are based on swapping \(2 \times 2\) submatrices (see"swap_count"algorithm for details) to maintain row and column totals.
Quantitative Swap Models
Quantitative swap models are similar to binary swap, but they
swap the largest permissible value. The models in this section all
maintain the fill and perform a quantitative swap only if this can
be done without changing the fill. Single step of swap often changes
the matrix very little. In particular, if cell counts are variable,
high values change very slowly. Checking the chain stability and
independence is even more crucial than in binary swap, and very
strong thinning is often needed. These models should never be
used without inspecting their properties for the current data. These
null models can also be defined using permatswap
function.
"swap_count":sequential algorithm for count matrices. This algorithm find \(2 \times 2\) submatrices that can be swapped leaving column and row totals and fill unchanged. The algorithm finds the largest value in the submatrix that can be swapped (\(d\)). Swap means that the values in diagonal or antidiagonal positions are decreased by \(d\), while remaining cells are increased by \(d\). A swap is made only if fill does not change.
"abuswap_r":sequential algorithm for count or nonnegative real valued matrices with fixed row frequencies (see also
permatswap). The algorithm is similar toswap_count, but uses different swap value for each row of the \(2 \times 2\) submatrix. Each step changes the the corresponding column sums, but honours matrix fill, row sums, and row/column frequencies (Hardy 2008; randomization scheme 2x)."abuswap_c":sequential algorithm for count or nonnegative real valued matrices with fixed column frequencies (see also
permatswap). The algorithm is similar as the previous one, but operates on columns. Each step changes the the corresponding row sums, but honours matrix fill, column sums, and row/column frequencies (Hardy 2008; randomization scheme 3x).
Quantitative Swap and Shuffle Models
Quantitative Swap and Shuffle methods (swsh methods) preserve
fill and column and row frequencies, and also either row or column
sums. The methods first perform a binary quasiswap and then
shuffle original quantitative data to non-zero cells. The
samp methods shuffle original non-zero cell values and can be
used also with non-integer data. The both methods
redistribute individuals randomly among non-zero cells and can only
be used with integer data. The shuffling is either free over the
whole matrix, or within rows (r methods) or within columns
(c methods). Shuffling within a row preserves row sums, and
shuffling within a column preserves column sums. These models can
also be defined with permatswap.
"swsh_samp":non-sequential algorithm for quantitative data (either integer counts or non-integer values). Original non-zero values values are shuffled.
"swsh_both":non-sequential algorithm for count data. Individuals are shuffled freely over non-zero cells.
"swsh_samp_r":non-sequential algorithm for quantitative data. Non-zero values (samples) are shuffled separately for each row.
"swsh_samp_c":non-sequential algorithm for quantitative data. Non-zero values (samples) are shuffled separately for each column.
"swsh_both_r":non-sequential algorithm for count matrices. Individuals are shuffled freely for non-zero values within each row.
"swsh_both_c":non-sequential algorithm for count matrices. Individuals are shuffled freely for non-zero values with each column.
Quantitative Shuffle Methods
Quantitative shuffle methods are generalizations of binary models
r00, r0 and c0. The _ind methods
shuffle individuals so that the grand sum, row sum or column sums
are preserved. These methods are similar as r2dtable but
with still slacker constraints on marginal sums. The _samp
and _both methods first apply the corresponding binary model
with similar restriction on marginal frequencies and then distribute
quantitative values over non-zero cells. The _samp models
shuffle original cell values and can therefore handle also non-count
real values. The _both models shuffle individuals among
non-zero values. The shuffling is over the whole matrix in
r00_, and within row in r0_ and within column in
c0_ in all cases.
"r00_ind":non-sequential algorithm for count matrices. This algorithm preserves grand sum and individuals are shuffled among cells of the matrix.
"r0_ind":non-sequential algorithm for count matrices. This algorithm preserves row sums and individuals are shuffled among cells of each row of the matrix.
"c0_ind":non-sequential algorithm for count matrices. This algorithm preserves column sums and individuals are shuffled among cells of each column of the matrix.
"r00_samp":non-sequential algorithm for count or nonnegative real valued (
mode = "double") matrices. This algorithm preserves grand sum and cells of the matrix are shuffled."r0_samp":non-sequential algorithm for count or nonnegative real valued (
mode = "double") matrices. This algorithm preserves row sums and cells within each row are shuffled."c0_samp":non-sequential algorithm for count or nonnegative real valued (
mode = "double") matrices. This algorithm preserves column sums constant and cells within each column are shuffled."r00_both":non-sequential algorithm for count matrices. This algorithm preserves grand sum and cells and individuals among cells of the matrix are shuffled.
"r0_both":non-sequential algorithm for count matrices. This algorithm preserves grand sum and cells and individuals among cells of each row are shuffled.
"c0_both":non-sequential algorithm for count matrices. This algorithm preserves grand sum and cells and individuals among cells of each column are shuffled.
Value
An object of class commsim with elements
corresponding to the arguments (method, binary,
isSeq, mode, fun).
If the input of make.comsimm is a commsim object,
it is returned without further evaluation. If this is not the case,
the character method argument is matched against
predefined algorithm names. An error message is issued
if none such is found. If the method argument is missing,
the function returns names of all currently available
null model algorithms as a character vector.
References
Gotelli, N.J. & Entsminger, N.J. (2001). Swap and fill algorithms in null model analysis: rethinking the knight's tour. Oecologia 129, 281–291.
Gotelli, N.J. & Entsminger, N.J. (2003). Swap algorithms in null model analysis. Ecology 84, 532–535.
Hardy, O. J. (2008) Testing the spatial phylogenetic structure of local communities: statistical performances of different null models and test statistics on a locally neutral community. Journal of Ecology 96, 914–926.
Jonsson, B.G. (2001) A null model for randomization tests of nestedness in species assemblages. Oecologia 127, 309–313.
Miklós, I. & Podani, J. (2004). Randomization of presence-absence matrices: comments and new algorithms. Ecology 85, 86–92.
Patefield, W. M. (1981) Algorithm AS159. An efficient method of generating r x c tables with given row and column totals. Applied Statistics 30, 91–97.
Strona, G., Nappo, D., Boccacci, F., Fattorini, S. & San-Miguel-Ayanz, J. (2014). A fast and unbiased procedure to randomize ecological binary matrices with fixed row and column totals. Nature Communications 5:4114 doi:10.1038/ncomms5114 .
Wright, D.H., Patterson, B.D., Mikkelson, G.M., Cutler, A. & Atmar, W. (1998). A comparative analysis of nested subset patterns of species composition. Oecologia 113, 1–20.
See also
See permatfull, permatswap for
alternative specification of quantitative null models. Function
oecosimu gives a higher-level interface for applying null
models in hypothesis testing and analysis of models. Function
nullmodel and simulate.nullmodel are used to
generate arrays of simulated null model matrices.
Examples
## write the r00 algorithm
f <- function(x, n, ...)
array(replicate(n, sample(x)), c(dim(x), n))
(cs <- commsim("r00", fun=f, binary=TRUE,
isSeq=FALSE, mode="integer"))
#> An object of class “commsim”
#> ‘r00’ method (binary, non-sequential, integer mode)
#>
## retrieving the sequential swap algorithm
(cs <- make.commsim("swap"))
#> An object of class “commsim”
#> ‘swap’ method (binary, sequential, integer mode)
#>
## feeding a commsim object as argument
make.commsim(cs)
#> An object of class “commsim”
#> ‘swap’ method (binary, sequential, integer mode)
#>
## making the missing c1 model using r1 as a template
## non-sequential algorithm for binary matrices
## that preserves the species (column) frequencies,
## but uses row marginal frequencies
## as probabilities of selecting sites
f <- function (x, n, nr, nc, rs, cs, ...) {
out <- array(0L, c(nr, nc, n))
J <- seq_len(nc)
storage.mode(rs) <- "double"
for (k in seq_len(n))
for (j in J)
out[sample.int(nr, cs[j], prob = rs), j, k] <- 1L
out
}
cs <- make.commsim("r1")
cs$method <- "c1"
cs$fun <- f
## structural constraints
diagfun <- function(x, y) {
c(sum = sum(y) == sum(x),
fill = sum(y > 0) == sum(x > 0),
rowSums = all(rowSums(y) == rowSums(x)),
colSums = all(colSums(y) == colSums(x)),
rowFreq = all(rowSums(y > 0) == rowSums(x > 0)),
colFreq = all(colSums(y > 0) == colSums(x > 0)))
}
evalfun <- function(meth, x, n) {
m <- nullmodel(x, meth)
y <- simulate(m, nsim=n)
out <- rowMeans(sapply(1:dim(y)[3],
function(i) diagfun(attr(y, "data"), y[,,i])))
z <- as.numeric(c(attr(y, "binary"), attr(y, "isSeq"),
attr(y, "mode") == "double"))
names(z) <- c("binary", "isSeq", "double")
c(z, out)
}
x <- matrix(rbinom(10*12, 1, 0.5)*rpois(10*12, 3), 12, 10)
algos <- make.commsim()
a <- t(sapply(algos, evalfun, x=x, n=10))
print(as.table(ifelse(a==1,1,0)), zero.print = ".")
#> binary isSeq double sum fill rowSums colSums rowFreq colFreq
#> r00 1 . . 1 1 . . . .
#> c0 1 . . 1 1 . 1 . 1
#> r0 1 . . 1 1 1 . 1 .
#> r1 1 . . 1 1 1 . 1 .
#> r2 1 . . 1 1 1 . 1 .
#> quasiswap 1 . . 1 1 1 1 1 1
#> greedyqswap 1 . . 1 1 1 1 1 1
#> swap 1 1 . 1 1 1 1 1 1
#> tswap 1 1 . 1 1 1 1 1 1
#> curveball 1 1 . 1 1 1 1 1 1
#> backtrack 1 . . 1 1 1 1 1 1
#> r2dtable . . . 1 . 1 1 . .
#> swap_count . 1 . 1 1 1 1 . .
#> quasiswap_count . . . 1 1 1 1 . .
#> swsh_samp . . 1 1 1 . . 1 1
#> swsh_both . . . 1 1 . . 1 1
#> swsh_samp_r . . 1 1 1 1 . 1 1
#> swsh_samp_c . . 1 1 1 . 1 1 1
#> swsh_both_r . . . 1 1 1 . 1 1
#> swsh_both_c . . . 1 1 . 1 1 1
#> abuswap_r . 1 1 1 1 1 . 1 1
#> abuswap_c . 1 1 1 1 . 1 1 1
#> r00_samp . . 1 1 1 . . . .
#> c0_samp . . 1 1 1 . 1 . 1
#> r0_samp . . 1 1 1 1 . 1 .
#> r00_ind . . . 1 . . . . .
#> c0_ind . . . 1 . . 1 . .
#> r0_ind . . . 1 . 1 . . .
#> r00_both . . . 1 1 . . . .
#> c0_both . . . 1 1 . 1 . 1
#> r0_both . . . 1 1 1 . 1 .