Ultrametric Distance Research Articles

Diversity and dissimilarity in lines and hierarchies

Within the multi-attribute framework of Nehring and Puppe [Econometrica, 70 (2002) 1155], hierarchies and lines represent the simplest and most fundamental models of diversity. In both cases, the diversity of any set can be recursively determined from the pairwise dissimilarities between its elements. The present paper characterizes the restrictions on the dissimilarity metric entailed by the two models. In the hierarchical case, this generalizes a classical result on the representation of ultrametric distance functions.

Marker-assisted selection for complex trait in common bean (Phaseolus vulgaris L.) using QTL-based index

A procedure was developed for marker-assisted selection of complex traits for common bean (Phaseolus vulgarisL.) using an index based on QTL-linked markers and ultrametric genetic distances between lines and a target parent. A comparison of the mean seed yields of the top five lines selected by different schemes demonstrated that the highest yielding group was selected on the basis of a combination of phenotypic performance and a high QTL-based index,followed by groups identified by a high QTL-based-index, conventional selection,and a low QTL-based-index. This study demonstrated a simple way to use information obtained from QTL studies to make selection decisions. The study also showed that the use of the QTL-based-index in conjunction with the ultrametric genetic distance to the target parent would enablea plant breeder to select lines that retain important QTL in a desirable genetic background. Therefore, this type of MAS would be expected to be superior to the phenotypic selection.

The hierarchy of the cocoons of a graph and its application to image segmentation

A set of particular subgraphs of a valued graph, called cocoons, is introduced. Within the image segmentation framework, the cocoons represent a model of contrasted regions. It is shown that the cocoons are organized into a hierarchy which is a sub-hierarchy of the one produced by the standard clustering algorithm of complete linkage. This result thus offers a new point of view on what the complete linkage algorithm achieves when it is applied on image data. For segmentation purposes, the hierarchy is built on a region adjacency graph valued with a dissimilarity function. It’s construction is efficient, parameter free, and robust towards monotone transformations of the dissimilarities. It is illustrated that the simplest cut criterion in these hierarchies, based on thresholding an associated ultrametric distance, already produces meaningful segmentations.

Zur Konstruktion von Hahn-Ternärkörpern als Ternärkörper formaler Potenzreihen

We endow the set of all formal power series with coefficients in a ternary field and exponents in a totally ordered loop with a ternary field operation such that a uniform valuation is given by the natural ultrametric distance. Any ordering of the coefficient ternary field can be extended to an ordering of the Hahn ternary field which is compatible with the given valuation.

Applying genetic algorithms to search for the best hierarchical clustering of a dataset

We apply genetic algorithms to get the optimal, in a least squares sense, hierarchical clustering of a dataset. We base this on the bijection between the set of hierarchical classifications of a dataset and the set of ultrametric distances. This bijection makes it possible to measure how good a hierarchical classification is, by calculating the L 2 norm between the ultrametric distance matrix associated with the hierarchical classification and the proximity matrix of the dataset. Our results are shown to improve on other methods which have been proposed, based on the ultrametric.

Information and hierarchical structure in financial markets

I investigate the information content present in the time series of stock prices of a portfolio of stocks traded in a financial market. By investigating the correlation coefficient between pairs of stocks I provide a working definition of a generalized distance between the stocks of the portfolio. This generalized distance is used to obtain an ultrametric distance matrix between the stocks. The ultrametric structure of the portfolio investigated has associated a taxonomy which is meaningful from an economic point of view.

Hydrodynamic interactions in self similar structures, such as polymers

We study the hydrodynamic properties of polymers and more generally self-similar structures using a new recursion model. The hydrodynamic interaction between monomers is modeled by the standard Green's function of Stokes flow in which an ultrametric distance is substituted for the usual Euclidean distance. This leads to a model where the long-range hydrodynamic interactions and the long-range correlations of the polymer conformation can both be accounted for and yet allow for analytical solutions. We explore the asymptotic as well as the finite size corrections to the scaling behavior with this model. In order to compare the results of the present scheme with more conventional techniques a generalized version of the existing mean field results by Kirkwood and Riseman for the hydrodynamic drag is introduced.

Directed polymers on a Cayley tree with spatially correlated disorder

In this paper we consider directed walks on a tree with a fixed branching ratio K at a finite temperature T. We consider the case where each site (or link) is assigned a random energy uncorrelated in time, but correlated in the transverse direction i.e. within the shell. In this paper we take the transverse distance to be the hierarchical ultrametric distance, but other possibilities are discussed. We compute the free energy for the case of quenched disorder and show that there is a fundamental difference between the case of short range spatial correlations of the disorder which behaves similarly to the non-correlated case considered previously by Derrida and Spohn and the case of long range correlations which has a totally different overlap distribution which approaches a single delta function about q=1 for large L, where L is the length of the walk. In the latter case the free energy is not extensive in L for the intermediate and also relevant range of L values, although in the true thermodynamic limit extensivity is restored. We identify a crossover temperature which grows with L, and whenever T<T_c(L) the system is always in the low temperature phase. Thus in the case of long-ranged correlation as opposed to the short-ranged case a phase transition is absent.

A variational approach to relaxation in ultrametric spaces

We investigate the relaxation behavior of complex systems endowed with a rugged free energy landscape from a variational perspective. We focus first on the dynamics in generic ultrametric spaces and then we specialize our generic results to a coarse-grained description of RNA folding, where ultrametricity holds as a limit description of conformation space. Using variational calculus in the generic context, we obtain the brachistochrone or fastest relaxation pathway for the ultrametric distance as a function of the barrier size and conclude that the brachistochrone may only be realized by those systems that follow the exponential or Debye law. Our approach not only reproduces the phenomenological generic relaxation law in the ultrametric limit, but is also justified a posteriori in specialized contexts: It reproduces meaningful folding pathways for the search in conformational space performed by renaturing RNA molecules which are targets of natural selection, reflecting a maximization in the efficiency of the folding process.

Homogeneous Ultrametric Spaces

This is a continuation of our papers ‘‘Generalized Ultrametric Spaces,’’ Ž w x. I, II see 9, 10 , where we studied spaces X, endowed with an ultrametric distance d with values in a partially ordered set G. In those papers, we developed a full theory, which concerned equivalence relations compatible with the distance, the skeleton, the associated spherically complete space, and the Hahn space. A main topic was the consideration of immediate and of dense extensions as well as the question of embedding into the Hahn space. It should be pointed out that the ultrametric space X was not assumed to carry any further structure. The present paper is devoted to homogeneous ultrametric spaces}the main class of such spaces is obtained when a group G is acting transitively on the ultrametric space X. Accordingly, the results in the preceding papers give rise, in the situation of homogeneous spaces, to more specific statements, which reflect the homogeneity.

A topological property of rational ω-languages

We prove that any rational subset K of A ω is the set of cluster points of A ω equipped with some ultrametric distance having some specific properties.

Hierarchical structures and asymmetric stochastic processes on p-adics and adeles

The space of states of some phenomena, in physics and other sciences, displays a hierarchical structure. When that is the case, it is natural to label the states by a p-adic number field. Both the classification of the states and their relationships are then based on a notion of distance with ultrametric properties. The dynamics of the phenomena, that is, the transition between different states, is also a function of the p-adic distance dp. However, because the distance is a symmetric function, probabilistic processes which depend only on dp have a uniform invariant probability measure, that is, all states are equally probable at large times. This being a severe limitation for cases of physical interest, processes with asymmetric transition functions have been studied. In addition to the dependence on the ultrametric distance, the asymmetric transition functions are allowed to depend also on the probability of the target state, leading to any desired invariant probability measure. When each state of a physical system is associated to several distinct hierarchical structures or parametrizations, an appropriate labeling set is the ring of adeles. Stochastic processes on the adeles are also constructed.

Hierarchical dynamics in large assemblies of interacting oscillators

We study a collection of phase-coupled oscillators possessing a hierarchical coupling structure. We establish a necessary condition for the existence of a phase transition to collective synchrony for finite values of the coupling strength in terms of an inequality involving the connectivity between clusters of oscillators, the rate at which coupling strengths decrease with ultrametric distance, and the dispersion of intrinsic frequencies. When the inequality is not satisfied, there is a cascade of discrete transitions to intracluster synchrony as the coupling strength is increased, but no global synchronization is possible in the infinite size limit.

AC susceptibilities in spin glasses and dynamics in an ultrametric space

The authors show that the temperature and frequency dependence of the AC susceptibilities chi ( omega ) in the 2D and 3D+or-J Ising spin glasses (SGs) can be reproduced by a model based on the picture of motion in phase space as thermally activated hopping in an ultrametric space. The characteristic difference in the frequency dependence of Im chi ( omega ) in the 2D and 3D SGs, previously found by the Monte Carlo (MC) simulation, is reduced to a difference in a scaling of free-energy barriers with ultrametric distance. One assumption is needed to combine dynamics in an ultrametric space with a correlation function of the magnetization in the SGs. chi ( omega )-values that were obtained previously by the MC simulation are reproduced based on the authors' assumption with better results than those of Sibani (1987). Further the relation obtained by the Lundgren et al. (1981) lm chi =-( pi /2)(d(Re chi )/d(In omega )) holds without taking a special limit z to 1, where z is a branching ratio of the hierarchical structure.

Piecewise hierarchical clustering

We consider two or more ultrametric distance matrices defined over different, possibly overlapping, subsets. These matrices are merged into one ultrametric matrix defined over the whole set. Necessary and sufficient conditions for uniqueness of the merging are established. when these conditions are not satisfied, consistent algorithms are given.

A validation study of a variable weighting algorithm for cluster analysis

De Soete (1986, 1988) proposed a variable weighting procedure when Euclidean distance is used as the dissimilarity measure with an ultrametric hierarchical clustering method. The algorithm produces weighted distances which approximate ultrametric distances as closely as possible in a least squares sense. The present simulation study examined the effectiveness of the De Soete procedure for an applications problem for which it was not originally intended. That is, to determine whether or not the algorithm can be used to reduce the influence of variables which are irrelevant to the clustering present in the data. The simulation study examined the ability of the procedure to recover a variety of known underlying cluster structures. The results indicate that the algorithm is effective in identifying extraneous variables which do not contribute information about the true cluster structure. Weights near 0.0 were typically assigned to such extraneous variables. Furthermore, the variable weighting procedure was not adversely effected by the presence of other forms of error in the data. In general, it is recommended that the variable weighting procedure be used for applied analyses when Euclidean distance is employed with ultrametric hierarchical clustering methods.

Display aspects in hierarchical clustering

AbstractUltrametric inter‐object distances found by hierarchical clustering may be used to generate and appropriate constellation of points in a plane. A configuration of points which reproduce the ultrametric distances in a low‐dimensional space as closely as possible can be found by Torgerson's method of multidimensional scaling. The resulting display supports the cognition of clusters very well, and for this purpose it seems to be superior to, for instance, the frequently applied principal components display. In certain situations dedrograms are expected to become simpler to interpret when such complementary ‘cluster display’ is also consulted. An example from analytical chemistry is presented.

Relaxational dynamics for a class of disordered ultrametric models.

The relaxational dynamics for a class of disordered, ultrametric spaces with arbitrary and irregular branchings $K$ is considered. It is shown that, if the transfer rates between sites depend upon their ultrametric distance and obey a certain constraint, the relaxational eigenvalues and eigenmodes can be obtained exactly for an arbitrary tree. The plausibility of the constraint is given on physical grounds. Approximate ensemble averagings are performed for the decay law leading to a power law similar to that found in regular models. This generalizes the exactly soluble regular $K=2$ model of Ogielski and Stein, and makes closer contact with real disordered systems.

Dynamics on ultrametric spaces.

We present an exact solution to the problem of a random walk on an ultrametric space with arbitrary transition probabilities. The solution provides a description of fluctuation dynamics of systems with many degenerate states separated by a hierarchy of energy barriers. Dynamical behavior is found to depend on the scaling of barriers with ultrametric distance. We find a range of interesting behavior, from temperature-dependent power-law decay to the Kohlrausch law.

Euclidean consensus dendrograms and other classification structures

By embedding the ultrametric distances between objects in a classification structure into a Euclidean space, two linear-algebraic procedures can be used to obtain a consensus among any combination of dendrograms, partitions, coverings, or more general arrangements of overlapping groups; the consensus has the property that the sum of the squared distances from the objects in each of the separate representations to their positions in the consensus is minimized in the Euclidean space. A consensus classification structure (usually a dendrogram) is obtained by reversing the embedding procedure. Admissibility criteria for a consensus are briefly considered.