134,796 research outputs found
Computing the Distribution of a Tree Metric
The Robinson-Foulds (RF) distance is by far the most widely used measure of
dissimilarity between trees. Although the distribution of these distances has
been investigated for twenty years, an algorithm that is explicitly polynomial
time has yet to be described for computing this distribution (which is also the
distribution of trees around a given tree under the popular Robinson-Foulds
metric). In this paper we derive a polynomial-time algorithm for this
distribution. We show how the distribution can be approximated by a Poisson
distribution determined by the proportion of leaves that lie in `cherries' of
the given tree. We also describe how our results can be used to derive
normalization constants that are required in a recently-proposed maximum
likelihood approach to supertree construction.Comment: 16 pages, 3 figure
Computing the distribution of a tree metric
The Robinson-Foulds (RF) distance is by far the most widely used measure of dissimilarity between trees. Although the distribution of these distances has been investigated for twenty years, an algorithm that is explicitly polynomial time has yet to be described for computing this distribution (which is also the dis- tribution of trees around a given tree under the popular Robinson-Foulds metric). In this paper we derive a polynomial-time algorithm for this distribution. We show how the distribution can be approximated by a Poisson distribution determined by the proportion of leaves that lie in ācherriesā of the given tree. We also describe how our results can be used to derive normalization constants that are required in a recently-proposed maximum likelihood approach to supertree construction
Empirical geodesic graphs and CAT(k) metrics for data analysis
A methodology is developed for data analysis based on empirically constructed
geodesic metric spaces. For a probability distribution, the length along a path
between two points can be defined as the amount of probability mass accumulated
along the path. The geodesic, then, is the shortest such path and defines a
geodesic metric. Such metrics are transformed in a number of ways to produce
parametrised families of geodesic metric spaces, empirical versions of which
allow computation of intrinsic means and associated measures of dispersion.
These reveal properties of the data, based on geometry, such as those that are
difficult to see from the raw Euclidean distances. Examples of application
include clustering and classification. For certain parameter ranges, the spaces
become CAT(0) spaces and the intrinsic means are unique. In one case, a minimal
spanning tree of a graph based on the data becomes CAT(0). In another, a
so-called "metric cone" construction allows extension to CAT() spaces. It is
shown how to empirically tune the parameters of the metrics, making it possible
to apply them to a number of real cases.Comment: Statistics and Computing, 201
Efficient Construction of Probabilistic Tree Embeddings
In this paper we describe an algorithm that embeds a graph metric
on an undirected weighted graph into a distribution of tree metrics
such that for every pair , and
. Such embeddings have
proved highly useful in designing fast approximation algorithms, as many hard
problems on graphs are easy to solve on tree instances. For a graph with
vertices and edges, our algorithm runs in time with high
probability, which improves the previous upper bound of shown by
Mendel et al.\,in 2009.
The key component of our algorithm is a new approximate single-source
shortest-path algorithm, which implements the priority queue with a new data
structure, the "bucket-tree structure". The algorithm has three properties: it
only requires linear time in the number of edges in the input graph; the
computed distances have a distance preserving property; and when computing the
shortest-paths to the -nearest vertices from the source, it only requires to
visit these vertices and their edge lists. These properties are essential to
guarantee the correctness and the stated time bound.
Using this shortest-path algorithm, we show how to generate an intermediate
structure, the approximate dominance sequences of the input graph, in time, and further propose a simple yet efficient algorithm to converted
this sequence to a tree embedding in time, both with high
probability. Combining the three subroutines gives the stated time bound of the
algorithm.
Then we show that this efficient construction can facilitate some
applications. We proved that FRT trees (the generated tree embedding) are
Ramsey partitions with asymptotically tight bound, so the construction of a
series of distance oracles can be accelerated
Cophenetic metrics for phylogenetic trees, after Sokal and Rohlf
Phylogenetic tree comparison metrics are an important tool in the study of
evolution, and hence the definition of such metrics is an interesting problem
in phylogenetics. In a paper in Taxon fifty years ago, Sokal and Rohlf proposed
to measure quantitatively the difference between a pair of phylogenetic trees
by first encoding them by means of their half-matrices of cophenetic values,
and then comparing these matrices. This idea has been used several times since
then to define dissimilarity measures between phylogenetic trees but, to our
knowledge, no proper metric on weighted phylogenetic trees with nested taxa
based on this idea has been formally defined and studied yet. Actually, the
cophenetic values of pairs of different taxa alone are not enough to single out
phylogenetic trees with weighted arcs or nested taxa. In this paper we define a
family of cophenetic metrics that compare phylogenetic trees on a same set of
taxa by encoding them by means of their vectors of cophenetic values of pairs
of taxa and depths of single taxa, and then computing the norm of the
difference of the corresponding vectors. Then, we study, either analytically or
numerically, some of their basic properties: neighbors, diameter, distribution,
and their rank correlation with each other and with other metrics.Comment: The "authors' cut" of a paper published in BMC Bioinformatics 14:3
(2013). 46 page
The space of ultrametric phylogenetic trees
The reliability of a phylogenetic inference method from genomic sequence data
is ensured by its statistical consistency. Bayesian inference methods produce a
sample of phylogenetic trees from the posterior distribution given sequence
data. Hence the question of statistical consistency of such methods is
equivalent to the consistency of the summary of the sample. More generally,
statistical consistency is ensured by the tree space used to analyse the
sample.
In this paper, we consider two standard parameterisations of phylogenetic
time-trees used in evolutionary models: inter-coalescent interval lengths and
absolute times of divergence events. For each of these parameterisations we
introduce a natural metric space on ultrametric phylogenetic trees. We compare
the introduced spaces with existing models of tree space and formulate several
formal requirements that a metric space on phylogenetic trees must possess in
order to be a satisfactory space for statistical analysis, and justify them. We
show that only a few known constructions of the space of phylogenetic trees
satisfy these requirements. However, our results suggest that these basic
requirements are not enough to distinguish between the two metric spaces we
introduce and that the choice between metric spaces requires additional
properties to be considered. Particularly, that the summary tree minimising the
square distance to the trees from the sample might be different for different
parameterisations. This suggests that further fundamental insight is needed
into the problem of statistical consistency of phylogenetic inference methods.Comment: Minor changes. This version has been published in JTB. 27 pages, 9
figure
Indexing Metric Spaces for Exact Similarity Search
With the continued digitalization of societal processes, we are seeing an
explosion in available data. This is referred to as big data. In a research
setting, three aspects of the data are often viewed as the main sources of
challenges when attempting to enable value creation from big data: volume,
velocity and variety. Many studies address volume or velocity, while much fewer
studies concern the variety. Metric space is ideal for addressing variety
because it can accommodate any type of data as long as its associated distance
notion satisfies the triangle inequality. To accelerate search in metric space,
a collection of indexing techniques for metric data have been proposed.
However, existing surveys each offers only a narrow coverage, and no
comprehensive empirical study of those techniques exists. We offer a survey of
all the existing metric indexes that can support exact similarity search, by i)
summarizing all the existing partitioning, pruning and validation techniques
used for metric indexes, ii) providing the time and storage complexity analysis
on the index construction, and iii) report on a comprehensive empirical
comparison of their similarity query processing performance. Here, empirical
comparisons are used to evaluate the index performance during search as it is
hard to see the complexity analysis differences on the similarity query
processing and the query performance depends on the pruning and validation
abilities related to the data distribution. This article aims at revealing
different strengths and weaknesses of different indexing techniques in order to
offer guidance on selecting an appropriate indexing technique for a given
setting, and directing the future research for metric indexes
- ā¦