In order to develop statistical methods for shapes with a tree-structure, we construct a shape space framework for tree-like shapes and study metrics on the shape space. This shape space has singularities, corresponding to topological transitions in the represented trees. We study two closely related metrics on the shape space, TED and QED. QED is a quotient Euclidean distance arising naturally from the shape space formulation, while TED is the classical tree edit distance. Using Gromov's metric geometry we gain new insight into the geometries defined by TED and QED. We show that the new metric QED has nice geometric properties which facilitate statistical analysis, such as existence and local uniqueness of geodesics and averages. TED, on the other hand, does not share the geometric advantages of QED, but has nice algorithmic properties. We provide a theoretical framework and experimental results on synthetic data trees as well as airway trees from pulmonary CT scans. This way, we effectively illustrate that our framework has both the theoretical and qualitative properties necessary to build a theory of statistical tree-shape analysis.
Statistics on tree-shaped objects receive growing interest in the statistical community. Wang and Marron @cite_18 study metric spaces of trees and define a notion of average tree called the median-mean as well as a version of PCA, which finds modes of variation in terms of , encoding the maximum amount of structural and attributal variation. @cite_6 extend this work by finding efficient algorithms for PCA. This is applied to analysis of brain blood vessels. The metric defined by Wang and Marron does not give a natural geodesic structure on the space of trees, as it places a large emphasis on the tree-topological structure of the trees. The metric has discontinuities in the sense that a sequence of trees with a shrinking branch will not converge to a tree that does not have that branch. Such a metric is not suitable for studying trees with continuous topological variations and noise, such as anatomical tree-structures extracted from medical images, since the emphasis on topology makes the metric sensitive to structural noise.
@cite_18 @cite_6
2076423950 2963432779
In order to develop statistical methods for shapes with a tree-structure, we construct a shape space framework for treelike shapes and study metrics on the shape space. The shape space has singularities, which correspond to topological transitions in the represented trees. We study two closely related metrics, TED and QED. The QED is a quotient euclidean distance arising from the new shape space formulation, while TED is essentially the classical tree edit distance. Using Gromov's metric geometry we gain new insight into the geometries defined by TED and QED. In particular, we show that the new metric QED has nice geometric properties which facilitate statistical analysis, such as existence and local uniqueness of geodesics and averages. TED, on the other hand, has algorithmic advantages, while it does not share the geometric strongpoints of QED. We provide a theoretical framework as well as computational results such as matching of airway trees from pulmonary CT scans and geodesics between synthetic data trees illustrating the dynamic and geometric properties of the QED metric. The mean is often the most important statistic of a dataset as it provides a single point that summarizes the entire set. While the mean is readily defined and computed in Euclidean spaces, no commonly accepted solutions are currently available in more complicated spaces, such as spaces of tree-structured data. In this paper we study the notion of means, both generally in Gromov's CAT(0)-spaces (metric spaces of non-positive curvature), but also specifically in the space of tree-like shapes. We prove local existence and uniqueness of means in such spaces and discuss three different algorithms for computing means. We make an experimental evaluation of the three algorithms through experiments on three different sets of data with tree-like structure: a synthetic dataset, a leaf morphology dataset from images, and a set of human airway subtrees from medical CT scans. This experimental study provides great insight into the behavior of the different methods and how they relate to each other. More importantly, it also provides mathematically well-founded, tractable and robust “average trees”. This statistic is of utmost importance due to the ever-presence of tree-like structures in human anatomy, e.g., airways and vascularization systems. Phylogenetic analysis of DNA or other data commonly gives rise to a collection or sample of inferred evolutionary trees. Principal Components Analysis (PCA) cannot be applied directly to collections of trees since the space of evolutionary trees on a fixed set of taxa is not a vector space. This paper describes a novel geometrical approach to PCA in tree-space that constructs the first principal path in an analogous way to standard linear Euclidean PCA. Given a data set of phylogenetic trees, a geodesic principal path is sought that maximizes the variance of the data under a form of projection onto the path. Due to the high dimensionality of tree-space and the nonlinear nature of this problem, the computational complexity is potentially very high, so approximate optimization algorithms are used to search for the optimal path. Principal paths identified in this way reveal and quantify the main sources of variation in the original collection of trees in terms of both topology and branch lengths. The approach is illustrated by application to simulated sets of trees and to a set of gene trees from metazoan (animal) species. New representations of tree-structured data objects, using ideas from topological data analysis, enable improved statistical analyses of a population of brain artery trees. A number of representations of each data tree arise from persistence diagrams that quantify branching and looping of vessels at multiple scales. Novel approaches to the statistical analysis, through various summaries of the persistence diagrams, lead to heightened correlations with covariates such as age and sex, relative to earlier analyses of this data set. The correlation with age continues to be signicant even after controlling for correlations from earlier signicant summaries. The active field of Functional Data Analysis (about understanding the variation in a set of curves) has been recently extended to Object Oriented Data Analysis, which considers populations of more general objects. A particularly challenging extension of this set of ideas is to populations of tree-structured objects. We develop an analog of Principal Component Analysis for trees, based on the notion of tree-lines, and propose numerically fast (linear time) algorithms to solve the resulting problems to proven optimality. The solutions we obtain are used in the analysis of a data set of 73 individuals, where each data object is a tree of blood vessels in one person's brain. Our analysis revealed a significant relation between the age of the individuals and their brain vessel structure. There are many metrics available to compare phylogenetic trees since this is a fundamental task in computational biology. In this paper, we focus on one such metric, the l∞-cophenetic metric introduced by This metric works by representing a phylogenetic tree with n labeled leaves as a point in ( R ^ n(n+1) 2 ) known as the cophenetic vector, then comparing the two resulting Euclidean points using the l∞ distance. Meanwhile, the interleaving distance is a formal categorical construction generalized from the definition of , originally introduced to compare persistence modules arising from the field of topological data analysis. We show that the l∞-cophenetic metric is an example of an interleaving distance. To do this, we define phylogenetic trees as a category of merge trees with some additional structure, namely, labelings on the leaves plus a requirement that morphisms respect these labels. Then we can use the definition of a flow on this category to give an interleaving distance. Finally, we show that, because of the additional structure given by the categories defined, the map sending a labeled merge tree to the cophenetic vector is, in fact, an isometric embedding, thus proving that the l∞-cophenetic metric is an interleaving distance. We propose a measure of shape which is appropriate for the study of a complicated geometric structure, defined using the topology of neighborhoods of the structure. One aspect of this measure gives a new notion of fractal dimension. We demonstrate the utility and computability of this measure by applying it to branched polymers, Brownian trees, and self-avoiding random walks. This paper presents a new geometric framework for analysis of planar treelike shapes for applications such as shape matching, recognition and morphology, using the geometry of the space of treelike shapes. Mathematically, the shape space is given the structure of a stratified set which is a quotient of a normed vector space with a metric inherited from the vector space norm. We give examples of geodesic paths in tree-space corresponding to fundamental deformations of small trees, and discuss how these deformations are key building blocks for understanding deformations between larger trees. Several popular methods for phylogenetic inference (or hierarchical clustering) are based on a matrix of pairwise distances between taxa (or any kind of objects): The objective is to construct a tree with branch lengths so that the distances between the leaves in that tree are as close as possible to the input distances. If we hold the structure (topology) of the tree fixed, in some relevant cases (e.g., ordinary least squares) the optimal values for the branch lengths can be expressed using simple combinatorial formulae. Here we define a general form for these formulae and show that they all have two desirable properties: First, the common tree reconstruction approaches (least squares, minimum evolution), when used in combination with these formulae, are guaranteed to infer the correct tree when given enough data (consistency); second, the branch lengths of all the simple (nearest neighbor interchange) rearrangements of a tree can be calculated, optimally, in quadratic time in the size of the tree, thus allowing the efficient application of hill climbing heuristics. The study presented here is a continuation of that by Mihaescu and Pachter on branch length estimation [Mihaescu R, Pachter L (2008) Proc Natl Acad Sci USA 105:13206–13211]. The focus here is on the inference of the tree itself and on providing a basis for novel algorithms to reconstruct trees from distances. Comparing and computing distances between phylogenetic trees are important biological problems, especially for models where edge lengths play an important role. The geodesic distance measure between two phylogenetic trees with edge lengths is the length of the shortest path between them in the continuous tree space introduced by Billera, Holmes, and Vogtmann. This tree space provides a powerful tool for studying and comparing phylogenetic trees, both in exhibiting a natural distance measure and in providing a euclidean-like structure for solving optimization problems on trees. An important open problem is to find a polynomial time algorithm for finding geodesics in tree space. This paper gives such an algorithm, which starts with a simple initial path and moves through a series of successively shorter paths until the geodesic is attained. Statistical phylogenetic inference methods use tree rearrangement operations such as subtree–prune–regraft (SPR) to perform Markov chain Monte Carlo (MCMC) across tree topologies. The structure of the graph induced by tree rearrangement operations is an important determinant of the mixing properties of MCMC, motivating the study of the underlying SPR graph in greater detail. In this paper, we investigate the SPR graph of rooted trees (rSPR graph) in a new way: by calculating the Ricci–Ollivier curvature with respect to uniform and Metropolis–Hastings random walks. This value quantifies the degree to which a pair of random walkers from specified points move towards each other; negative curvature means that they move away from one another on average, while positive curvature means that they move towards each other. In order to calculate this curvature, we develop fast new algorithms for rSPR graph computation. We then develop formulas characterizing how the number of rSPR neighbors of a tree changes after an rSPR operation is applied to that tree. These give bounds on the curvature, as well as a flatness-in-the-limit theorem indicating that paths of small topology changes are easy to traverse. However, we find that large topology changes (i.e. moving a large subtree) give pairs of trees with negative curvature. We show using simulation that mean access time distributions depend on distance, degree, and curvature, demonstrating the relevance of these results to stochastic tree search. We study languages of nested trees—structures obtained by augmenting trees with sets of nested jump-edges. These graphs can naturally model branching behaviors of pushdown programs, so that the problem of branching-time software model checking may be phrased as a membership question for such languages. We define finite-state automata accepting such languages—these automata can pass states along jump-edges as well as tree edges. We find that the model-checking problem for these automata on pushdown systems is EXPTIME-complete, and that their alternating versions are expressively equivalent to NT-μ, a recently proposed temporal logic for nested trees that can express a variety of branching-time, “context-free” requirements. We also show that monadic second order logic (MSO) cannot exploit the structure: MSO on nested trees is too strong in the sense that it has an undecidable model checking problem, and seems too weak to capture NT-μ. Background 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. We propose a compact and efficient index structure for massive data sets. Several indexing techniques are widely-used and well-known such as binary trees and B+trees. Unfortunately, we find that these techniques suffer major two shortcomings when applied to massive sets; first, their indices are so large they could overflow regular main memory, and, second, they suffer from a variety of penalties (e.g., conditional branches, low cache hits, and TLB misses), which restricts the number of instructions executed per processor cycle. Our state-of-the-art index structure, called VAST-Tree, classifies branch nodes into multiple layers. It applies existing techniques such as cache-conscious, aligned, and branch-free structures to the top layers of branch nodes in trees. Next, it applies the adaptive compression technique to save space and harness data parallelism with SIMD instructions to the middle and bottom layers of branch nodes. Moreover, a processor-friendly compression technique is applied to leaf nodes. The end result is that trees are much more compact and traversal efficiency is high. We implement a prototype and show its resulting index size and performance as compared to binary trees, and the hardware-conscious technique called FAST which currently offers the highest performance. Compared to current alternatives, VAST-Tree compacts the branch nodes by more than 95 , and the overall index size by 47-84 given that there are 230 keys. With 228 keys, it has roughly 6.0-times and 1.24-times throughput and saves the memory consumption by more than 94.7 and 40.5 as compared to binary trees and FAST, respectively. This paper provides a novel technique for the analysis of randomized algorithms for optimization problems on metric spaces, by relating the randomized performance ratio for any, metric space to the randomized performance ratio for a set of "simple" metric spaces. We define a notion of a set of metric spaces that probabilistically-approximates another metric space. We prove that any metric space can be probabilistically-approximated by hierarchically well-separated trees (HST) with a polylogarithmic distortion. These metric spaces are "simple" as being: (1) tree metrics; (2) natural for applying a divide-and-conquer algorithmic approach. The technique presented is of particular interest in the context of on-line computation. A large number of on-line algorithmic problems, including metrical task systems, server problems, distributed paging, and dynamic storage rearrangement are defined in terms of some metric space. Typically for these problems, there are linear lower bounds on the competitive ratio of deterministic algorithms. Although randomization against an oblivious adversary has the potential of overcoming these high ratios, very little progress has been made in the analysis. We demonstrate the use of our technique by obtaining substantially improved results for two different on-line problems. We present a novel image-based technique for modeling complex unfoliaged trees. Existing tree modeling tools either require capturing a large number of views for dense 3D reconstruction or rely on user inputs and botanic rules to synthesize natural-looking tree geometry. In this paper, we focus on faithfully recovering real instead of realistically-looking tree geometry from a sparse set of images. Our solution directly integrates 2D 3D tree topology as shape priors into the modeling process. For each input view, we first estimate a 2D skeleton graph from its matte image and then find a 2D skeleton tree from the graph by imposing tree topology. We develop a simple but effective technique for computing the optimal 3D skeleton tree most consistent with the 2D skeletons. For each edge in the 3D skeleton tree, we further apply volumetric reconstruction to recover its corresponding curved branch. Finally, we use piecewise cylinders to approximate each branch from the volumetric results. We demonstrate our framework on a variety of trees to illustrate the robustness and usefulness of our technique. Many phylogenetic algorithms search the space of possible trees using topological rearrangements and some optimality criterion. FastME is such an approach that uses the balanced minimum evolution (BME) principle, which computer studies have demonstrated to have high accuracy. FastME includes two variants: balanced subtree prune and regraft (BSPR) and balanced nearest neighbor interchange (BNNI) . These algorithms take as input a distance matrix and a putative phylogenetic tree. The tree is modified using SPR or NNI operations, respectively, to reduce the BME length relative to the distance matrix, until a tree with (locally) shortest BME length is found. Following computer simulations, it has been conjectured that BSPR and BNNI are consistent, i.e. for an input distance that is a tree-metric, they converge to the corresponding tree. We prove that the BSPR algorithm is consistent. Moreover, even if the input contains small errors relative to a tree-metric, we show that the BSPR algorithm still returns the corresponding tree. Whether BNNI is consistent remains open. Contour trees can represent the topology of large volume data sets in a relatively compact, discrete data structure. However, the resulting trees often contain many thousands of nodes; thus, many graph drawing techniques fail to produce satisfactory results. Therefore, several visualization methods were proposed recently for the visualization of contour trees. Unfortunately, none of these techniques is able to handle uncertain contour trees although any uncertainty of the volume data inevitably results in partially uncertain contour trees. In this work, we visualize uncertain contour trees by combining the contour trees of two morphologically filtered versions of a volume data set, which represent the range of uncertainty. These two contour trees are combined and visualized within a single image such that a range of potential contour trees is represented by the resulting visualization. Thus, potentially erroneous topological structures are visually distinguished from more certain structures. Moreover, topological structures can be revealed that are otherwise obscured by data errors. We present and discuss results obtained with a prototypical implementation using well-known volume data sets. For some positive constant , we give a (3 2- )-approximation algorithm for the following problem: given a graph G_0=(V,E_0), find the shortest tour that visits every vertex at least once. This is a special case of the metric traveling salesman problem when the underlying metric is defined by shortest path distances in G_0. The result improves on the 3 2-approximation algorithm due to Christofides [C76] for this special case. Similar to Christofides, our algorithm finds a spanning tree whose cost is upper bounded by the optimum, then it finds the minimum cost Eulerian augmentation (or T-join) of that tree. The main difference is in the selection of the spanning tree. Except in certain cases where the solution of LP is nearly integral, we select the spanning tree randomly by sampling from a maximum entropy distribution defined by the linear programming relaxation. Despite the simplicity of the algorithm, the analysis builds on a variety of ideas such as properties of strongly Rayleigh measures from probability theory, graph theoretical results on the structure of near minimum cuts, and the integrality of the T-join polytope from polyhedral theory. Also, as a byproduct of our result, we show new properties of the near minimum cuts of any graph, which may be of independent interest. We introduce a novel measure called e-four-pointscondition (e-4PC), which assigns a value e ∈ [0,1] to every metric space quantifying how close the metric is to a tree metric. Data-sets taken from real Internet measurements indicate remarkable closeness of Internet latencies to tree metrics based on this condition. We study embeddings of e-4PC metric spaces into trees and prove tight upper and lower bounds. Specifically, we show that there are constants c1 and c2 such that, (1) every metric (X,d) which satisfies the e-4PC can be embedded into a tree with distortion (1+e)c1log|X|, and (2) for every e ∈: [0,1] and any number of nodes, there is a metric space (X,d) satisfying the e-4PC that does not embed into a tree with distortion less than (1+e)c2log|X|. In addition, we prove a lower bound on approximate distance labelings of e-4PC metrics, and give tight bounds for tree embeddings with additive error guarantees.