Matrix Permutation Algorithms for PresenceAbsence and Count Data
Description
Individual (for count data) or incidence (for presenceabsence data) based null models can be generated for community level simulations. Options for preserving characteristics of the original matrix (rows/columns sums, matrix fill) and restricted permutations (based on strata) are discussed in the Details section.
Usage
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20  permatfull(m, fixedmar = "both", shuffle = "both", strata = NULL,
mtype = "count", times = 99, ...)
permatswap(m, method = "quasiswap", fixedmar="both", shuffle = "both",
strata = NULL, mtype = "count", times = 99,
burnin = 0, thin = 1, ...)
## S3 method for class 'permat'
print(x, digits = 3, ...)
## S3 method for class 'permat'
summary(object, ...)
## S3 method for class 'summary.permat'
print(x, digits = 2, ...)
## S3 method for class 'permat'
plot(x, type = "bray", ylab, xlab, col, lty,
lowess = TRUE, plot = TRUE, text = TRUE, ...)
## S3 method for class 'permat'
lines(x, type = "bray", ...)
## S3 method for class 'permat'
as.ts(x, type = "bray", ...)
## S3 method for class 'permat'
as.mcmc(x)

Arguments
m 
A community data matrix with plots (samples) as rows and species (taxa) as columns. 
fixedmar 
character, stating which of the row/column sums should
be preserved ( 
strata 
Numeric vector or factor with length same as

mtype 
Matrix data type, either 
times 
Number of permuted matrices. 
method 
Character for method used for the swap algorithm
( 
shuffle 
Character, indicating whether individuals
( 
burnin 
Number of null communities discarded before proper
analysis in sequential ( 
thin 
Number of discarded permuted matrices between two
evaluations in sequential ( 
x, object 
Object of class 
digits 
Number of digits used for rounding. 
ylab, xlab, col, lty 
graphical parameters for the 
type 
Character, type of plot to be displayed: 
lowess, plot, text 
Logical arguments for the 
... 
Other arguments passed to 
Details
The function permatfull
is useful when matrix fill is
allowed to vary, and matrix type is count
. The fixedmar
argument is used to set constraints for permutation. If none
of the margins are fixed, cells are randomised within the matrix. If
rows
or columns
are fixed, cells within rows or columns
are randomised, respectively. If both
margins are fixed, the
r2dtable
function is used that is based on Patefield's
(1981) algorithm. For presence absence data, matrix fill should be
necessarily fixed, and permatfull
is a wrapper for the function
make.commsim
. The r00, r0, c0, quasiswap
algorithms of make.commsim
are used for "none",
"rows", "columns", "both"
values of the fixedmar
argument,
respectively
The shuffle
argument only have effect if the mtype =
"count"
and permatfull
function is used with "none",
"rows", "columns"
values of fixedmar
. All other cases for
count data are individual based randomisations. The "samp"
and
"both"
options result fixed matrix fill. The "both"
option means that individuals are shuffled among non zero cells
ensuring that there are no cell with zeros as a result, then cell
(zero and new valued cells) are shuffled.
The function permatswap
is useful when with matrix fill
(i.e. the proportion of empty cells) and row/columns sums should be
kept constant. permatswap
uses different kinds of swap
algorithms, and row and columns sums are fixed in all cases. For
presenceabsence data, the swap
and tswap
methods of
make.commsim
can be used. For count data, a special
swap algorithm ('swapcount') is implemented that results in permuted
matrices with fixed marginals and matrix fill at the same time.
The 'quasiswapcount' algorithm (method="quasiswap"
and
mtype="count"
) uses the same trick as Carsten Dormann's
swap.web
function in the package
bipartite. First, a random matrix is generated by the
r2dtable
function retaining 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 2x2 submatrices (see 'swapcount' algorithm for
details) to maintain row and column totals. This algorithm generates
independent matrices in each step, so burnin
and thin
arguments are not considered. This is the default method, because this
is not sequential (as swapcount
is) so independence of subsequent
matrices does not have to be checked.
The swapcount
algorithm (method="swap"
and
mtype="count"
) tries to find 2x2 submatrices (identified by 2
random row and 2 random column indices), that can be swapped in order
to leave column and row totals and fill unchanged. First, the
algorithm finds the largest value in the submatrix that can be swapped
(d) and whether in diagonal or antidiagonal way. Submatrices
that contain values larger than zero in either diagonal or
antidiagonal position can be swapped. 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
doesn't change. This algorithm is sequential, subsequent matrices are
not independent, because swaps modify little if the matrix is
large. In these cases many burnin steps and thinning is needed to get
independent random matrices. Although this algorithm is implemented in
C, large burnin and thin values can slow it down
considerably. WARNING: according to simulations, this algorithm seems
to be biased and non random, thus its use should be avoided!
The algorithm "swsh"
in the function permatswap
is a
hybrid algorithm. First, it makes binary quasiswaps to keep row and
column incidences constant, then nonzero values are modified
according to the shuffle
argument (only "samp"
and
"both"
are available in this case, because it is applied only
on nonzero values). It also recognizes the fixedmar
argument which cannot be "both"
(vegan versions <= 2.0
had this algorithm with fixedmar = "none"
).
The algorithm "abuswap"
produces two kinds of null models
(based on fixedmar="columns"
or fixedmar="rows"
) as
described in Hardy (2008; randomization scheme 2x and 3x,
respectively). These preserve column and row occurrences, and column
or row sums at the same time. (Note that similar constraints
can be achieved by the non sequential "swsh"
algorithm
with fixedmar
argument set to "columns"
or
"rows"
, respectively.)
Constraints on row/column sums, matrix fill, total sum and sums within
strata can be checked by the summary
method. plot
method
is for visually testing the randomness of the permuted matrices,
especially for the sequential swap algorithms. If there are any
tendency in the graph, higher burnin
and thin
values can
help for sequential methods. New lines can be added to existing plot
with the lines
method.
Unrestricted and restricted permutations: if strata
is
NULL
, functions perform unrestricted permutations. Otherwise,
it is used for restricted permutations. Each strata should contain at
least 2 rows in order to perform randomization (in case of low row
numbers, swap algorithms can be rather slow). If the design is not
well balanced (i.e. same number of observations within each stratum),
permuted matrices may be biased because same constraints are forced on
submatrices of different dimensions. This often means, that the number
of potential permutations will decrease with their dimensions. So the
more constraints we put, the less randomness can be expected.
The plot
method is useful for graphically testing for trend and
independence of permuted matrices. This is especially important when
using sequential algorithms ("swap", "tswap", "abuswap"
).
The as.ts
method can be used to extract BrayCurtis
dissimilarities or Chisquared values as time series. This can further
used in testing independence (see Examples). The method as.mcmc
is useful for accessing diagnostic tools available in the coda
package.
Value
Functions permatfull
and permatswap
return an
object of class "permat"
containing the the function call
(call
), the original data matrix used for permutations
(orig
) and a list of permuted matrices with length times
(perm
).
The summary
method returns various statistics as a list
(including mean BrayCurtis dissimilarities calculated pairwise among
original and permuted matrices, Chisquare statistics, and check
results of the constraints; see Examples). Note that when
strata
is used in the original call, summary calculation may
take longer.
The plot
creates a plot as a side effect.
The as.ts
method returns an object of class "ts"
.
Author(s)
Péter Sólymos, solymos@ualberta.ca and Jari Oksanen
References
Original references for presenceabsence algorithms are
given on help page of make.commsim
.
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.
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.
See Also
For other functions to permute matrices:
make.commsim
, r2dtable
,
sample
, swap.web
.
For the use of these permutation algorithms: oecosimu
,
adipart
, hiersimu
.
For timeseries diagnostics: Box.test
,
lag.plot
, tsdiag
, ar
,
arima
For underlying ‘low level’ implementation:
commsim
and nullmodel
.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53  ## A simple artificial community data matrix.
m < matrix(c(
1,3,2,0,3,1,
0,2,1,0,2,1,
0,0,1,2,0,3,
0,0,0,1,4,3
), 4, 6, byrow=TRUE)
## Using the quasiswap algorithm to create a
## list of permuted matrices, where
## row/columns sums and matrix fill are preserved:
x1 < permatswap(m, "quasiswap")
summary(x1)
## Unrestricted permutation retaining
## row/columns sums but not matrix fill:
x2 < permatfull(m)
summary(x2)
## Unrestricted permutation of presenceabsence type
## not retaining row/columns sums:
x3 < permatfull(m, "none", mtype="prab")
x3$orig ## note: original matrix is binarized!
summary(x3)
## Restricted permutation,
## check sums within strata:
x4 < permatfull(m, strata=c(1,1,2,2))
summary(x4)
## NOTE: 'times' argument usually needs to be >= 99
## here much lower value is used for demonstration
## Not sequential algorithm
data(BCI)
a < permatswap(BCI, "quasiswap", times=19)
## Sequential algorithm
b < permatswap(BCI, "abuswap", fixedmar="col",
burnin=0, thin=100, times=19)
opar < par(mfrow=c(2,2))
plot(a, main="Not sequential")
plot(b, main="Sequential")
plot(a, "chisq")
plot(b, "chisq")
par(opar)
## Extract BrayCurtis dissimilarities
## as time series
bc < as.ts(b)
## Lag plot
lag.plot(bc)
## First order autoregressive model
mar < arima(bc, c(1,0,0))
mar
## LjungBox test of residuals
Box.test(residuals(mar))
## Graphical diagnostics
tsdiag(mar)

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