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PubMed1993

Normalization of the minimum spanning tree.

A problem of considerable interest in pattern recognition and data analysis is that of describing the spatial structure of a data set. In the field of biology this could be based on graph construction. Although the minimum spanning tree (MST), contains less information than the Relative Neighbourhood, Gabriel and Delaunay graphs [16], this graph has been frequently used [3-9]. The MST is a subgraph of all the preceding graphs. Two main types of parameters can be derived from a graph. Some of the parameters are derived from the structure of the graph (topological parameters), whereas others are based on the Euclidean metrics of the graph (edge lengths). Since these parameters are used to characterize the spatial structure of data sets, they have to be normalized so that different biological structures may be compared. A model for the normalization of the most common parameters derived from the MST is thus presented here. Two aspects of the problem are considered: (i) omission of the metrics associated dimension of the Euclidean parameters in order to compare biological structures at different scale factors and (ii) elimination of border effects to avoid border artefacts.

Data Interpretation, Statistical↗
PubMedDate not supplied

Prospective risk in reciprocal translocation heterozygotes at amniocentesis as determined by potential chromosome imbalance sizes. Data of the European Collaborative Prenatal Diagnosis Centres.

The Date of the European Cooperative Prenatal Diagnosis Laboratories (Boué and Gallano, 1984) of 596 prenatal (amniocyte) diagnoses of familial rcp was examined as to relationships between balanced/unbalanced result and ascertainment, carrier parent and chromosome imbalance size (percentage haploid autosome length). Each rearrangement was graphed once with actual (unbalanced result) or potential (normal or balanced result) imbalances plotted with trisomy as the ordinate and monosomy as the abscissa. The graphed data was divided into 15 regions, each of 2.0 per cent trisomy and 0.75 per cent monosomy and the rate of unbalanced pregnancies determined for each region. The highest rates of chromosomally unbalanced progeny (excluding regions with inadequate data) were found closest to the origin (i.e. associated with the smallest imbalances) and these were for ascertainment category 1 (previous rcp unbalanced child) 22.3 per cent for maternal carriers and 39 per cent for paternal carriers. Overall in pooled data for this ascertainment category (without reference to the imbalance graphs) there were for paternal carriers 28.6 per cent unbalanced pregnancies and for maternal carriers 18.1 per cent. The graphed data, therefore, revealed the higher rates associated with some of the rcp with small potential (combined duplication/deficiency) imbalances. Lesser rates were observed for ascertainment category 2 (carrier parent with a history of recurrent miscarriage) with overall percentages of imbalanced progeny ranging from 2.7 (paternal carriers) to 4.7 (maternal carriers). Again, higher rates were revealed in graphed data for small potential imbalances. All unbalanced results for this group (ascertainment category 2) plotted in the region closest to the origin with rates of 16 per cent (maternal carriers) and 9.5 per cent (paternal carriers) in this region. Remarkably in both ascertainment groups 1 and 2 there was no significant difference in the size of the imbalanced segments for unbalanced progeny. In ascertainment group 1 this was (dup/def; mean +/- S.D.): 1.09 +/- 0.77/0.47 +/- 0.45 and in ascertainment group 2: 1.09 +/- 0.80/0.66 +/- 0.71. From the graphed data which arguably denote viability relationships, a trisomy was approximately 2.7 times as likely to survive until amniocentesis as a monosomy of equivalent size. It is proposed that given further data, risk estimates could be determined for rcp heterozygotes using the present approach where empiric data (from the family history or an analysed series of similar rcp) is not available.

Chromosome Aberrations↗
PubMed1997

Ancestral Processes with Selection

In this paper, we show how to construct the genealogy of a sample of genes for a large class of models with selection and mutation. Each gene corresponds to a single locus at which there is no recombination. The genealogy of the sample is embedded in a graph which we call the ancestral selection graph. This graph contains all the information about the ancestry; it is the analogue of Kingman's coalescent process which arises in the case with no selection. The ancestral selection graph can be easily simulated and we outline an algorithm for simulating samples. The main goal is to analyze the ancestral selection graph and to compare it to Kingman's coalescent process. In the case of no mutation, we find that the distribution of the time to the most recent common ancestor does not depend on the selection coefficient and hence is the same as in the neutral case. When the mutation rate is positive, we give a procedure for computing the probability that two individuals in a sample are identical by descent and the Laplace transform of the time to the most recent common ancestor of a sample of two individuals; we evaluate the first two terms of their respective power series in terms of the selection coefficient. The probability of identity by descent depends on both the selection coefficient and the mutation rate and is different from the analogous expression in the neutral case. The Laplace transform does not have a linear correction term in the selection coefficient. We also provide a recursion formula that can be used to approximate the probability of a given sample by simulating backwards along the sample paths of the ancestral selection graph, a technique developed by Griffiths and Tavare (1994).

Journal Article↗
PubMed1981

Towards a theory of cell assemblies.

The term cell assembly, first introduced by D. O. Hebb, is defined in the framework of graph theory. This definition leads to some beautiful problems concerning the number and size of cell assemblies in large graphs. Some approaches to solve these problems are presented. In particular, the graphs Kn X Km are constructed that have n . m points, n + m - 2 connections per point, and at least 2n + 2m - 4 assemblies. Several new notions of connectivity in directed graphs are introduced and their relationships are investigated. The insight into these notions and their relationships will be helpful for further construction of graphs with many assemblies and/or high connectivity. The resulting graphs are not only important for the idea of cell assemblies in the content of neurodynamics, they may also find applications in the construction of communication networks and associative memories.

Brain↗
PubMed2005

A fast algorithm for the construction of universal footprinting templates in DNA.

We introduce and give a complete description of a new graph to be used for DNA sequencing questions. This graph has the advantage over the classical de Bruijn graph that it fully accounts for the double stranded nature of DNA, rather than dealing with single strands. Technically, our graph may be thought of as the quotient of the de Bruijn graph under the natural involution of sending a DNA strand to its complementary strand. However, this involution has fixed points, and this complicates the structure of the quotient graph which we have therefore modified herein. As an application and motivating example, we give an efficient algorithm for constructing universal footprinting templates for n-mers. This problem may be formulated as the task of finding a shortest possible segment of DNA which contains every possible sequence of base pairs of some fixed length n. Previous work by Kwan et al has attacked this problem from a numerical point of view and generated minimal length universal footprinting templates for n = 2, 3, 5, 7, together with unsubstantiated candidates for the case n = 4. We show that their candidates for n = 4 are indeed minimal length universal footprinting templates.

Algorithms↗
PubMed2005

A comparison of graphical and textual presentations of time series data to support medical decision making in the neonatal intensive care unit.

OBJECTIVE: To compare expert-generated textual summaries of physiological data with trend graphs, in terms of their ability to support neonatal Intensive Care Unit (ICU) staff in making decisions when presented with medical scenarios. METHODS: Forty neonatal ICU staff were recruited for the experiment, eight from each of five groups--junior, intermediate and senior nurses, junior and senior doctors. The participants were presented with medical scenarios on a computer screen, and asked to choose from a list of 18 possible actions those they thought were appropriate. Half of the scenarios were presented as trend graphs, while the other half were presented as passages of text. The textual summaries had been generated by two human experts and were intended to describe the physiological state of the patient over a short period of time (around 40 minutes) but not to interpret it. RESULTS: In terms of the content of responses there was a clear advantage for the Text condition, with participants tending to choose more of the appropriate actions when the information was presented as text rather than as graphs. In terms of the speed of response there was no difference between the Graphs and Text conditions. There was no significant difference between the staff groups in terms of speed or content of responses. In contrast to the objective measures of performance, the majority of participants reported a subjective preference for the Graphs condition. CONCLUSIONS: In this experimental task, participants performed better when presented with a textual summary of the medical scenario than when it was presented as a set of trend graphs. If the necessary algorithms could be developed that would allow computers automatically to generate descriptive summaries of physiological data, this could potentially be a useful feature of decision support tools in the intensive care unit.

Computer Graphics↗
PubMed1988

Automata with hierarchical control and evolutionary learning.

We propose an automata-theoretical framework for structured hierarchical control, in terms of rules and meta-rules, for sequences of moves on a graph. This leads to a notion of a "universal" hierarchically structured automaton mu which can move on a given graph in such a way as to emulate any automaton which moves on that graph in response to inputs. This emulation is achieved via a mapping of the inputs in the given automaton to those of mu, and we think of such a mapping as an encoding of the given automaton. We see in several examples that efficient encodings of graph-search algorithms correspond to their natural hierarchical structure (in terms of rules and meta-rules), and this leads one to a precise notion of the "depth" of an automaton which moves on a given graph. By way of application, we discuss a proposed structure of a series of stochastic neural networks which can learn, by example, to encode a given sequence of moves on a graph, so that the encoding obtained is structurally the "natural" one for the given sequence of moves. Thus, such a learning system would perform both structural pattern recognition (in terms of "patterns" of moves), and encoding based on a desired outcome.

Algorithms↗
PubMed1998

Hemolytic disease of the fetus: a comparison of the Queenan and extended Liley methods.

OBJECTIVE: To compare the clinical utility of the Liley and Queenan methods to monitor the severity of fetal hemolytic disease. METHODS: Amniotic fluid bilirubin was measured in specimens from 73 women sensitized to red blood cell antigens. Chloroform-extracted amniotic fluid was evaluated spectrophotometrically for bilirubin content by using the change-from-expected value of the optical density at 450 nm. Values in the four Queenan zones were compared with those of the four zones of the Liley graph (middle zone subdivided). Clinical utility and accuracy of the two methods were compared. RESULTS: Treatment was based on interpretation of bilirubin values plotted on the Liley graph. Hydrops fetalis was not observed. The highest value for each patient was significantly more likely to be plotted in the highest zone using the Queenan method (23 of 73 compared with eight of 73 patients; P < .001). Overestimation of risk occurred with greater frequency when using the Queenan method (13 of 67 compared with seven of 67 patients; P = .031). Overestimation of risk by the Queenan method also was more likely at or before 28 weeks' gestation (10 of 49 compared with four of 49 patients; P = .031). In nine cases (13%), the Queenan graph and method would have prompted unnecessary or premature umbilical vein sampling that was withheld using the Liley graph. CONCLUSION: The performance of the linearly extended Liley graph was superior to that of the Queenan graph, because the Queenan method frequently overestimated risk.

Amniotic Fluid↗
PubMed2005

A combinatorial approach to the electron correlation problem.

Starting from a path-integral formulation of quantum statistical mechanics expressed in a space of Slater determinants, we develop a method for the Monte Carlo evaluation of the energy of a correlated electronic system. The path-integral expression for the partition function is written as a contracted sum over graphs. A graph is a set of distinct connected determinants on which paths can be represented. The weight of a graph is given by the sum over exponentially large numbers of paths which visit the vertices of the graph. We show that these weights are analytically computable using combinatorial techniques, and they turn out to be sufficiently well behaved to allow stable Monte Carlo simulations in which graphs are stochastically sampled according to a Metropolis algorithm. In the present formulation, graphs of up to four vertices have been included. In a Hartree-Fock basis, this allows for paths which include up to sixfold excitations relative to the Hartree-Fock determinant. As an illustration, we have studied the dissociation curve of the N(2) molecule in a VDZ basis, which allows comparison with full configuration-interaction calculations.

Journal Article↗
PubMed2026

ARISE: RNA-anchored shared-edge topology and hierarchical fusion for spatial multi-omics integration.

MOTIVATION: Spatial multi-omics technologies jointly profile transcriptomes, proteins and chromatin accessibility in situ, enabling integrative analysis of tissue organization across molecular layers. However, most existing graph-based integration methods rely on independently constructed modality-specific k-nearest-neighbor graphs. When auxiliary modalities are sparse or noisy, these graphs can become topologically discordant, propagate spurious edges, weaken cross-modal alignment, and reduce spatial domain resolution. RESULTS: We present Anchored RNA for Integrated Spatial Embedding (ARISE), an RNA expression anchored framework for spatial multi-omics integration. ARISE defines a shared-edge topology by intersecting RNA feature-similarity and spatial-proximity graphs, encodes auxiliary modalities on this common scaffold, and integrates them through inside-out hierarchical fusion. We further show theoretically that graph intersection minimizes false-positive edges within a broad class of k-of-r graph fusion rules, providing a principled basis for topology anchoring. Across various spatial multi-omics benchmarks spanning simulated and real datasets in bi-modal and tri-modal settings, ARISE improves spatial domain identification, cross-modal consistency, and preservation of tissue structure relative to existing methods. Furthermore, the learned representation supports biologically meaningful downstream analyses, including marker-based domain annotation, pathway enrichment, and cis-regulatory inference, indicating that ARISE yields a robust and interpretable framework for spatial multi-omics integration. AVAILABILITY AND IMPLEMENTATION: The source code is available at https://github.com/XiangxiangWang-code/ARISE. The archived version used in this study is available at https://doi.org/10.6084/m9.figshare.32686137.v2.

Multiomics↗
PubMed2005

An efficient randomized algorithm for contact-based NMR backbone resonance assignment.

MOTIVATION: Backbone resonance assignment is a critical bottleneck in studies of protein structure, dynamics and interactions by nuclear magnetic resonance (NMR) spectroscopy. A minimalist approach to assignment, which we call 'contact-based', seeks to dramatically reduce experimental time and expense by replacing the standard suite of through-bond experiments with the through-space (nuclear Overhauser enhancement spectroscopy, NOESY) experiment. In the contact-based approach, spectral data are represented in a graph with vertices for putative residues (of unknown relation to the primary sequence) and edges for hypothesized NOESY interactions, such that observed spectral peaks could be explained if the residues were 'close enough'. Due to experimental ambiguity, several incorrect edges can be hypothesized for each spectral peak. An assignment is derived by identifying consistent patterns of edges (e.g. for alpha-helices and beta-sheets) within a graph and by mapping the vertices to the primary sequence. The key algorithmic challenge is to be able to uncover these patterns even when they are obscured by significant noise. RESULTS: This paper develops, analyzes and applies a novel algorithm for the identification of polytopes representing consistent patterns of edges in a corrupted NOESY graph. Our randomized algorithm aggregates simplices into polytopes and fixes inconsistencies with simple local modifications, called rotations, that maintain most of the structure already uncovered. In characterizing the effects of experimental noise, we employ an NMR-specific random graph model in proving that our algorithm gives optimal performance in expected polynomial time, even when the input graph is significantly corrupted. We confirm this analysis in simulation studies with graphs corrupted by up to 500% noise. Finally, we demonstrate the practical application of the algorithm on several experimental beta-sheet datasets. Our approach is able to eliminate a large majority of noise edges and to uncover large consistent sets of interactions. AVAILABILITY: Our algorithm has been implemented in the platform-independent Python code. The software can be freely obtained for academic use by request from the authors.

Algorithms↗
PubMedDate not supplied

Learning the topological properties of brain tumors.

This work presents a graph-based representation (a.k.a., cell-graph) of histopathological images for automated cancer diagnosis by probabilistically assigning a link between a pair of cells (or cell clusters). Since the node set of a cell-graph can include a cluster of cells as well as individual ones, it enables working with low-cost, low-magnification photomicrographs. The contributions of this work are twofold. First, it is shown that without establishing a pairwise spatial relation between the cells (i.e., the edges of a cell-graph), neither the spatial distribution of the cells nor the texture analysis of the images yields accurate results for tissue level diagnosis of brain cancer called malignant glioma. Second, this work defines a set of global metrics by processing the entire cell-graph to capture tissue level information coded into the histopathological images. In this work, the results are obtained on the photomicrographs of 646 archival brain biopsy samples of 60 different patients. It is shown that the global metrics of cell-graphs distinguish cancerous tissues from noncancerous ones with high accuracy (at least 99 percent accuracy for healthy tissues with lower cellular density level, and at least 92 percent accuracy for benign tissues with similar high cellular density level such as nonneoplastic reactive/inflammatory conditions).

Algorithms↗
PubMed2001

Correctness of belief propagation in Gaussian graphical models of arbitrary topology.

Graphical models, such as Bayesian networks and Markov random fields, represent statistical dependencies of variables by a graph. Local "belief propagation" rules of the sort proposed by Pearl (1988) are guaranteed to converge to the correct posterior probabilities in singly connected graphs. Recently, good performance has been obtained by using these same rules on graphs with loops, a method we refer to as loopy belief propagation. Perhaps the most dramatic instance is the near Shannon-limit performance of "Turbo codes," whose decoding algorithm is equivalent to loopy propagation. Except for the case of graphs with a single loop, there has been little theoretical understanding of loopy propagation. Here we analyze belief propagation in networks with arbitrary topologies when the nodes in the graph describe jointly gaussian random variables. We give an analytical formula relating the true posterior probabilities with those calculated using loopy propagation. We give sufficient conditions for convergence and show that when belief propagation converges, it gives the correct posterior means for all graph topologies, not just networks with a single loop. These results motivate using the powerful belief propagation algorithm in a broader class of networks and help clarify the empirical performance results.

Journal Article↗
PubMed2005

Estimation and marginalization using the Kikuchi approximation methods.

In this letter, we examine a general method of approximation, known as the Kikuchi approximation method, for finding the marginals of a product distribution, as well as the corresponding partition function. The Kikuchi approximation method defines a certain constrained optimization problem, called the Kikuchi problem, and treats its stationary points as approximations to the desired marginals. We show how to associate a graph to any Kikuchi problem and describe a class of local message-passing algorithms along the edges of any such graph, which attempt to find the solutions to the problem. Implementation of these algorithms on graphs with fewer edges requires fewer operations in each iteration. We therefore characterize minimal graphs for a Kikuchi problem, which are those with the minimum number of edges. We show with empirical results that these simpler algorithms often offer significant savings in computational complexity, without suffering a loss in the convergence rate. We give conditions for the convexity of a given Kikuchi problem and the exactness of the approximations in terms of the loops of the minimal graph. More precisely, we show that if the minimal graph is cycle free, then the Kikuchi approximation method is exact, and the converse is also true generically. Together with the fact that in the cycle-free case, the iterative algorithms are equivalent to the well-known belief propagation algorithm, our results imply that, generically, the Kikuchi approximation method can be exact if and only if traditional junction tree methods could also solve the problem exactly.

Algorithms↗
PubMed1998

Comparison of serial monitoring of peak expiratory flow and FEV1 in the diagnosis of occupational asthma.

Peak expiratory flow (PEF) monitoring is often used to establish the relationship between occupational exposure and asthma. FEV1 has been found to be a better physiologic index than PEF in the measurement of airflow obstruction. The aim of this study was to compare the accuracy of serial monitoring of PEF and FEV1 in the diagnosis of occupational asthma. Twenty consecutive subjects referred for possible occupational asthma were asked to perform serial monitoring of PEF and FEV1 using a portable ventilometer. Two sets of graphs were plotted for both PEF and FEV1: graphs with the best of all values and graphs with the best of two reproducible values. Three observers interpreted both PEF and FEV1 recordings by the visual method in a blind, randomized manner as either compatible with occupational asthma or not. Eleven of the subjects had a positive inhalation challenge test (high-molecular-weight agents, n = 6; low-molecular-weight agents, n = 5). In the case of analysis of the graphs plotted with the best of all values, the sensitivity of the PEF recording interpreted by the three observers was 82, 73, and 73%, and of the FEV1 recording as 55, 55, and 45%; specificity of PEF recording was 89, 100, and 100%, and of FEV1 was 56, 89, and 100%. When an agreement between two of the three readers was required to define occupational asthma, sensitivity and specificity were 73 and 100% for PEF and 55 and 89% for FEV1. Lower sensitivities were found when the same analyses were performed with the graphs plotted with the best of two reproducible values. It was concluded that unsupervised FEV1 is not more accurate than unsupervised PEF monitoring in the diagnosis of occupational asthma. Plotting graphs using the best value gives better diagnostic accuracy than plotting them with the best of two reproducible values.

Adult↗
PubMed1997

Graphic data representation in anaesthesiological journals: a proposed methodology for assessment of appropriateness.

Few authors have addressed the topic of graphic data presentation. The purpose of our study was to combine several guidelines in order to evaluate three anaesthesiology journals listed in Index Medicus (Australian, American and Italian) in terms of the appropriateness and the quality of presentation of graphs. Our analysis was based on concepts expressed by Cox and Tufte. We calculated the optimization of the amount of information in each graph using two parameters: Data Density Index (DDI) and Data Ink Ratio (DIR). The correctness and clearness of each component of the graph (scale, title, axes, legends and abbreviations) was evaluated on the basis of a binary score. We analysed 300 exploratory plots, quantitative graphs and summaries of statistical analysis. About 50% of papers had more than three graphs. Mean scores were 3.22 for the Italian journal, 3.47 for the American journal and 3.82 for the Australian journal. Tufte parameters were calculated on 42 scatterplots: DDI was 5.4 +/- 13.9 and DIR was 0.7 +/- 0.1. The criteria applied in our study appear sufficiently sensitive to differentiate the quality of graphs.

Anesthesiology↗
PubMed2006

Transformations of mathematical and stimulus functions.

Following a pretest, 8 participants who were unfamiliar with algebraic and trigonometric functions received a brief presentation on the rectangular coordinate system. Next, they participated in a computer-interactive matching-to-sample procedure that trained formula-to-formula and formula-to-graph relations. Then, they were exposed to 40 novel formula-to-graph tests and 10 novel graph-to-formula tests. Seven of the 8 participants showed substantial improvement in identifying formula-to-graph relations; however, in the test of novel graph-to-formula relations, participants tended to select equations in their factored form. Next, we manipulated contextual cues in the form of rules regarding mathematical preferences. First, we informed participants that standard forms of equations were preferred over factored forms. In a subsequent test of 10 additional novel graph-to-formula relations, participants shifted their selections to favor equations in their standard form. This preference reversed during 10 more tests when financial reward was made contingent on correct identification of formulas in factored form. Formula preferences and transformation of novel mathematical and stimulus functions are discussed.

Adult↗
PubMed2004

Scalable distance similarity of chemical structures.

Screening a library of molecular graphs for an exact or approximate match with one particular molecular graph, the query graph, is reduced to list comparisons. The lists contain lengths of shortest paths in graph Voronoi regions. This induces the notion of shortest path similarity. All graphs that are shortest path similar to the query graph are efficiently retrievable. The same applies to approximate or similarity matching. For the retrieval of all superstructures of a query, shortest path lists are modified to distance patterns. This also allows algorithmic support for query construction.

Algorithms↗