287,857 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
Number Counting among Students with Mild Intellectual Disability in Penang: A Case Study
AbstractThis study examined number counting among students with mild intellectual disability. This study also investigates their skills as well as the difficulty they are facing on number counting. This is a quantitative research and had involved thirty participants as respondents. All respondents were selected through purposive sampling technique. Results of this study showed that students with mild intellectual disability have understanding of number counting using counting instructional models. Results of this study also showed that female students with mild intellectual disability have the higher mean scores compared to male students on number counting using n+1>n rule counting instructional model. On the other hand, male students with mild intellectual disability have higher mean score achievement on number counting using enumeration instructional model than female. It is shown from the result of the study that there is no significant difference in the skills of n + 1 > n rule counting instructional model among male and female students with mild intellectual disability. Also there is no significant difference among male and female in the skills of enumeration counting instructional model
Efficient Sampling Algorithms for Approximate Motif Counting in Temporal Graph Streams
A great variety of complex systems, from user interactions in communication
networks to transactions in financial markets, can be modeled as temporal
graphs consisting of a set of vertices and a series of timestamped and directed
edges. Temporal motifs are generalized from subgraph patterns in static graphs
which consider edge orderings and durations in addition to topologies. Counting
the number of occurrences of temporal motifs is a fundamental problem for
temporal network analysis. However, existing methods either cannot support
temporal motifs or suffer from performance issues. Moreover, they cannot work
in the streaming model where edges are observed incrementally over time. In
this paper, we focus on approximate temporal motif counting via random
sampling. We first propose two sampling algorithms for temporal motif counting
in the offline setting. The first is an edge sampling (ES) algorithm for
estimating the number of instances of any temporal motif. The second is an
improved edge-wedge sampling (EWS) algorithm that hybridizes edge sampling with
wedge sampling for counting temporal motifs with vertices and edges.
Furthermore, we propose two algorithms to count temporal motifs incrementally
in temporal graph streams by extending the ES and EWS algorithms referred to as
SES and SEWS. We provide comprehensive analyses of the theoretical bounds and
complexities of our proposed algorithms. Finally, we perform extensive
experimental evaluations of our proposed algorithms on several real-world
temporal graphs. The results show that ES and EWS have higher efficiency,
better accuracy, and greater scalability than state-of-the-art sampling methods
for temporal motif counting in the offline setting. Moreover, SES and SEWS
achieve up to three orders of magnitude speedups over ES and EWS while having
comparable estimation errors for temporal motif counting in the streaming
setting.Comment: 27 pages, 11 figures; overlapped with arXiv:2007.1402
Counting and Sampling from Markov Equivalent DAGs Using Clique Trees
A directed acyclic graph (DAG) is the most common graphical model for
representing causal relationships among a set of variables. When restricted to
using only observational data, the structure of the ground truth DAG is
identifiable only up to Markov equivalence, based on conditional independence
relations among the variables. Therefore, the number of DAGs equivalent to the
ground truth DAG is an indicator of the causal complexity of the underlying
structure--roughly speaking, it shows how many interventions or how much
additional information is further needed to recover the underlying DAG. In this
paper, we propose a new technique for counting the number of DAGs in a Markov
equivalence class. Our approach is based on the clique tree representation of
chordal graphs. We show that in the case of bounded degree graphs, the proposed
algorithm is polynomial time. We further demonstrate that this technique can be
utilized for uniform sampling from a Markov equivalence class, which provides a
stochastic way to enumerate DAGs in the equivalence class and may be needed for
finding the best DAG or for causal inference given the equivalence class as
input. We also extend our counting and sampling method to the case where prior
knowledge about the underlying DAG is available, and present applications of
this extension in causal experiment design and estimating the causal effect of
joint interventions
Algorithmic Pirogov-Sinai theory
We develop an efficient algorithmic approach for approximate counting and
sampling in the low-temperature regime of a broad class of statistical physics
models on finite subsets of the lattice and on the torus
. Our approach is based on combining contour
representations from Pirogov-Sinai theory with Barvinok's approach to
approximate counting using truncated Taylor series. Some consequences of our
main results include an FPTAS for approximating the partition function of the
hard-core model at sufficiently high fugacity on subsets of with
appropriate boundary conditions and an efficient sampling algorithm for the
ferromagnetic Potts model on the discrete torus at
sufficiently low temperature
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