24,539 research outputs found
Exact sampling and counting for fixed-margin matrices
The uniform distribution on matrices with specified row and column sums is
often a natural choice of null model when testing for structure in two-way
tables (binary or nonnegative integer). Due to the difficulty of sampling from
this distribution, many approximate methods have been developed. We will show
that by exploiting certain symmetries, exact sampling and counting is in fact
possible in many nontrivial real-world cases. We illustrate with real datasets
including ecological co-occurrence matrices and contingency tables.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1131 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org). arXiv admin note: text overlap with
arXiv:1104.032
On the number of matrices and a random matrix with prescribed row and column sums and 0-1 entries
We consider the set Sigma(R,C) of all mxn matrices having 0-1 entries and
prescribed row sums R=(r_1, ..., r_m) and column sums C=(c_1, ..., c_n). We
prove an asymptotic estimate for the cardinality |Sigma(R, C)| via the solution
to a convex optimization problem. We show that if Sigma(R, C) is sufficiently
large, then a random matrix D in Sigma(R, C) sampled from the uniform
probability measure in Sigma(R,C) with high probability is close to a
particular matrix Z=Z(R,C) that maximizes the sum of entropies of entries among
all matrices with row sums R, column sums C and entries between 0 and 1.
Similar results are obtained for 0-1 matrices with prescribed row and column
sums and assigned zeros in some positions.Comment: 26 pages, proofs simplified, results strengthene
Exact Enumeration and Sampling of Matrices with Specified Margins
We describe a dynamic programming algorithm for exact counting and exact
uniform sampling of matrices with specified row and column sums. The algorithm
runs in polynomial time when the column sums are bounded. Binary or
non-negative integer matrices are handled. The method is distinguished by
applicability to non-regular margins, tractability on large matrices, and the
capacity for exact sampling
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