Symmetric Dissimilarity Research Articles

The conditional p-dispersion problem

We introduce the conditional p-dispersion problem (c-pDP), an incremental variant of the p-dispersion problem (pDP). In the c-pDP, one is given a set N of n points, a symmetric dissimilarity matrix D of dimensions $$n\times n$$ , an integer $$p\ge 1$$ and a set $$Q\subseteq N$$ of cardinality $$q\ge 1$$ . The objective is to select a set $$P\subset N\setminus Q$$ of cardinality p that maximizes the minimal dissimilarity between every pair of selected vertices, i.e., $$z(P\cup Q) {:}{=}\min \{D(i, j), i, j\in P\cup Q\}$$ . The set Q may model a predefined subset of preferences or hard location constraints in incremental network design. We adapt the state-of-the-art algorithm for the pDP to the c-pDP and include an ad-hoc acceleration mechanism designed to leverage the information provided by the set Q to further reduce the size of the problem instance. We perform exhaustive computational experiments and show that the proposed acceleration mechanism helps reduce the total computational time by a factor of five on average. We also assess the scalability of the algorithm and derive sensitivity analyses.

Decremental Clustering for the Solution of p-Dispersion Problems to Proven Optimality

Given n points, a symmetric dissimilarity matrix D of dimensions n × n, and an integer p ? 2, the p-dispersion problem (pDP) consists of selecting a subset of exactly p points in such a way that the minimum dissimilarity between any pair of selected points is maximum. The pDP is [Formula: see text] when p is an input of the problem. We propose a decremental clustering method to reduce the problem to the solution of a series of smaller pDPs until reaching proven optimality. A k-means algorithm is used to construct and refine the clusterings along the algorithm’s execution. The proposed method can handle problems orders of magnitude larger than the limits of the state-of-the-art solver for the pDP for small values of p.

Embedding road networks and travel time into distance metrics for urban modelling

ABSTRACTUrban environments are restricted by various physical, regulatory and customary barriers such as buildings, one-way systems and pedestrian crossings. These features create challenges for predictive modelling in urban space, as most proximity-based models rely on Euclidean (straight line) distance metrics which, given restrictions within the urban landscape, do not fully capture spatial urban processes. Here, we argue that road distance and travel time provide effective alternatives, and we develop a new low-dimensional Euclidean distance metric based on these distances using an isomap approach. The purpose of this is to produce a valid covariance matrix for Kriging. Our primary methodological contribution is the derivation of two symmetric dissimilarity matrices ( and ), with which it is possible to compute low-dimensional Euclidean metrics for the production of a positive definite covariance matrix with commonly utilised kernels. This new method is implemented into a Kriging predictor to estimate house prices on 3,669 properties in Coventry, UK. We find that a metric estimating a combination of road distance and travel time, in both and , produces a superior house price predictor compared with alternative state-of-the-art methods, that is, a standard Euclidean metric in and a non-restricted road distance metric in and . F

Visual Assessment of Clustering Tendency for Incomplete Data

The iVAT (asiVAT) algorithms reorder symmetric (asymmetric) dissimilarity data so that an image of the data may reveal cluster substructure. Images formed from incomplete data don't offer a very rich interpretation of cluster structure. In this paper, we examine four methods for completing the input data with imputed values before imaging. We choose a best method using contaminated versions of the complete Iris data, for which the desired results are known. Then, we analyze two real world data sets from social networks that are incomplete using the best imputation method chosen in the juried trials with Iris: (i) Sampson's monastery data, an incomplete, asymmetric relation matrix; and (ii) the karate club data, comprising a symmetric similarity matrix that is about 86 percent incomplete.

Spherical and Hyperbolic Embeddings of Data.

Many computer vision and pattern recognition problems may be posed as the analysis of a set of dissimilarities between objects. For many types of data, these dissimilarities are not euclidean (i.e., they do not represent the distances between points in a euclidean space), and therefore cannot be isometrically embedded in a euclidean space. Examples include shape-dissimilarities, graph distances and mesh geodesic distances. In this paper, we provide a means of embedding such non-euclidean data onto surfaces of constant curvature. We aim to embed the data on a space whose radius of curvature is determined by the dissimilarity data. The space can be either of positive curvature (spherical) or of negative curvature (hyperbolic). We give an efficient method for solving the spherical and hyperbolic embedding problems on symmetric dissimilarity data. Our approach gives the radius of curvature and a method for approximating the objects as points on a hyperspherical manifold without optimisation. For objects which do not reside exactly on the manifold, we develop a optimisation-based procedure for approximate embedding on a hyperspherical manifold. We use the exponential map between the manifold and its local tangent space to solve the optimisation problem locally in the euclidean tangent space. This process is efficient enough to allow us to embed data sets of several thousand objects. We apply our method to a variety of data including time warping functions, shape similarities, graph similarity and gesture similarity data. In each case the embedding maintains the local structure of the data while placing the points in a metric space.

Adaptive conformal semi-supervised vector quantization for dissimilarity data

Existing semi-supervised learning algorithms focus on vectorial data given in Euclidean space. But many real life data are non-metric, given as (dis-)similarities which are not widely addressed. We propose a conformal prototype-based classifier for dissimilarity data to semi-supervised tasks. A ‘secure region’ of unlabeled data is identified to improve the trained model based on labeled data and to adapt the model complexity. The new approach (i) can directly deal with arbitrary symmetric dissimilarity matrices, (ii) offers intuitive classification by sparse prototypes, (iii) adapts the model complexity. Experiments confirm the effectiveness of our approach in comparison to state-of-the-art methods.

Sparse conformal prediction for dissimilarity data

Existing classification algorithms focus on vectorial data given in Euclidean space or representations by means of positive semi-definite kernel matrices. Many real world data, like biological sequences are not vectorial, often non-euclidean and given only in the form of (dis-)similarities between examples, requesting for efficient and interpretable models. Vectorial embeddings or transformations to get a valid kernel are limited and current dissimilarity classifiers often lead to dense complex models which are hard to interpret by domain experts. They also fail to provide additional information about the confidence of the classification. In this paper we propose a prototype-based conformal classifier for dissimilarity data. It is based on a prototype dissimilarity learner and extended by the conformal prediction methodology. It (i) can deal with dissimilarity data characterized by an arbitrary symmetric dissimilarity matrix, (ii) offers intuitive classification in terms of sparse prototypical class representatives, (iii) leads to state-of-the-art classification results supported by a confidence measure and (iv) the model complexity is automatically adjusted. In experiments on dissimilarity data we investigate the effectiveness with respect to accuracy and model complexity in comparison to different state of the art classifiers.

A Structural Equation Multidimensional Scaling Model for One-Mode Asymmetric Dissimilarity Data

A multidimensional scaling (MDS) model is proposed for 2-way 1-mode asymmetric dissimilarity data, to estimate the unknown symmetric subjacent dissimilarity matrix while the objects are represented in a low-dimensional space. In the context of least squares MDS allowing transformations, and considering both triangular parts of the asymmetric dissimilarity matrix as effects of the unobserved symmetric dissimilarities, an alternating estimation procedure is proposed in which the unknown symmetric dissimilarity matrix is estimated in a covariance structure framework. Real and artificial data are analyzed to illustrate the proposed procedure.

An Improved Exact Algorithm for Least-Squares Unidimensional Scaling

Given n objects and an symmetric dissimilarity matrix D with zero main diagonal and nonnegative off-diagonal entries, the least-squares unidimensional scaling problem asks to find an arrangement of objects along a straight line such that the pairwise distances between them reflect dissimilarities represented by the matrix D. In this paper, we propose an improved branch-and-bound algorithm for solving this problem. The main ingredients of the algorithm include an innovative upper bounding technique relying on the linear assignment model and a dominance test which allows considerably reducing the redundancy in the enumeration process. An initial lower bound for the algorithm is provided by an iterated tabu search heuristic. To enhance the performance of this heuristic we develop an efficient method for exploring the pairwise interchange neighborhood of a solution in the search space. The basic principle and formulas of the method are also used in the implementation of the dominance test. We report computational results for both randomly generated and real-life based problem instances. In particular, we were able to solve to guaranteed optimality the problem defined by a Morse code dissimilarity matrix.

Image quality assessment using natural image statistics in gradient domain

It is a challenging work to find an efficient metric of image quality assessment, which is an important approach for many image processing tasks. In this paper, we propose a novel reduced reference image quality assessment based on natural image statistical prior in gradient domain. The derivatives of image describe the detailed geometric features of images which are appealing for visual prediction of human beings. Resistor average distance as a symmetric dissimilarity measure between distributions of images is utilized in the definition of distortion measure. Our method is general purpose for different distortion types and has relatively low complexity. The experimental results upon image database show that the proposed method is consistent with the subjective assessment of human beings and has a good performance for all distortion types.

Heuristic Implementation of Dynamic Programming for Matrix Permutation Problems in Combinatorial Data Analysis

Dynamic programming methods for matrix permutation problems in combinatorial data analysis can produce globally-optimal solutions for matrices up to size 30×30, but are computationally infeasible for larger matrices because of enormous computer memory requirements. Branch-and-bound methods also guarantee globally-optimal solutions, but computation time considerations generally limit their applicability to matrix sizes no greater than 35×35. Accordingly, a variety of heuristic methods have been proposed for larger matrices, including iterative quadratic assignment, tabu search, simulated annealing, and variable neighborhood search. Although these heuristics can produce exceptional results, they are prone to converge to local optima where the permutation is difficult to dislodge via traditional neighborhood moves (e.g., pairwise interchanges, object-block relocations, object-block reversals, etc.). We show that a heuristic implementation of dynamic programming yields an efficient procedure for escaping local optima. Specifically, we propose applying dynamic programming to reasonably-sized subsequences of consecutive objects in the locally-optimal permutation, identified by simulated annealing, to further improve the value of the objective function. Experimental results are provided for three classic matrix permutation problems in the combinatorial data analysis literature: (a) maximizing a dominance index for an asymmetric proximity matrix; (b) least-squares unidimensional scaling of a symmetric dissimilarity matrix; and (c) approximating an anti-Robinson structure for a symmetric dissimilarity matrix.

Optimal Least-Squares Unidimensional Scaling: Improved Branch-and-Bound Procedures and Comparison to Dynamic Programming

There are two well-known methods for obtaining a guaranteed globally optimal solution to the problem of least-squares unidimensional scaling of a symmetric dissimilarity matrix: (a) dynamic programming, and (b) branch-and-bound. Dynamic programming is generally more efficient than branch-and-bound, but the former is limited to matrices with approximately 26 or fewer objects because of computer memory limitations. We present some new branch-and-bound procedures that improve computational efficiency, and enable guaranteed globally optimal solutions to be obtained for matrices with up to 35 objects. Experimental tests were conducted to compare the relative performances of the new procedures, a previously published branch-and-bound algorithm, and a dynamic programming solution strategy. These experiments, which included both synthetic and empirical dissimilarity matrices, yielded the following findings: (a) the new branch-and-bound procedures were often drastically more efficient than the previously published branch-and-bound algorithm, (b) when computationally feasible, the dynamic programming approach was more efficient than each of the branch-and-bound procedures, and (c) the new branch-and-bound procedures require minimal computer memory and can provide optimal solutions for matrices that are too large for dynamic programming implementation.

Graph coloring, minimum-diameter partitioning, and the analysis of confusion matrices

It is well known that minimum-diameter partitioning of symmetric dissimilarity matrices can be framed within the context of coloring the vertices of a graph. Although confusion data are typically represented in the form of asymmetric similarity matrices, they are also amenable to a graph-coloring perspective. In this paper, we propose the integration of the minimum-diameter partitioning method with a neighborhood-based coloring approach for analyzing digraphs corresponding to confusion data. This procedure is capable of producing minimum-diameter partitions with the added desirable property that vertices with the same color have similar in-neighborhoods (i.e., directed edges entering the vertex) and out-neighborhoods (i.e., directed edges exiting the vertex) for the digraph corresponding to the minimum partition diameter.

A branch-and-bound algorithm for fitting anti-robinson structures to symmetric dissimilarity matrices

The seriation of proximity matrices is an important problem in combinatorial data analysis and can be conducted using a variety of objective criteria. Some of the most popular criteria for evaluating an ordering of objects are based on (anti-) Robinson forms, which reflect the pattern of elements within each row and/or column of the reordered matrix when moving away from the main diagonal. This paper presents a branch-and-bound algorithm that can be used to seriate a symmetric dissimilarity matrix by identifying a reordering of rows and columns of the matrix optimizing an anti-Robinson criterion. Computational results are provided for several proximity matrices from the literature using four different anti-Robinson criteria. The results suggest that with respect to computational efficiency, the branch-and-bound algorithm is generally competitive with dynamic programming. Further, because it requires much less storage than dynamic programming, the branch-and-bound algorithm can provide guaranteed optimal solutions for matrices that are too large for dynamic programming implementations.

RAPD-PCR analysis of genetic variation among isolates of Alternaria solani and Alternaria alternata from potato and tomato

Random amplified polymorphic DNA-PCR (RAPD-PCR) analysis was used to investigate genetic variation among 35 isolates of Alternaria solani, the causal agent of early blight on solanaceous crops, and 30 isolates of A. alternata, an opportunistic pathogen. A symmetric dissimilarity matrix, based on Euclidean metric distances, was used to calculate interpopulation genetic distances (Φst) between the fungal species, geographic origin, and host plant. Of the total genetic variation observed, 34% (Φst = 0.340, P < 0.0001) resided between the two fungal species. A very highly significant interpopulation genetic distance (Φst = 0.2275, P < 0.0001) was observed between isolates of A. solani collected from potato and tomato, suggesting the possibility of pathogenic specialization. There was a significant difference between isolates of A. solani (Φst = 0.1569, P 0.017) and A. alternata (Φst = 0.1096, P = 0.02) from foreign and domestic sources, but no difference between isolates from Pennsylvania and other states. One primer, P248, generated polymorphisms that could be used to distinguish between the two fungal species.

Fitting a Restricted Asymmetric Dissimilarity Model to a Symmetric Dissimilarity Matrix

This paper proposes a model to analyze the symmetric dissimilarity matrix by an Euclidean distance model with additional terms. The proposed model is a restricted model of Weeks and Bentler’s model. The symmetric dissimilarity is assumed to be induced from asymmetric dissimilarities. A Monte Carlo simulation study conducted demonstrated that the proposed least squares algorithm was effective in recovering the true number of dimensions and true configuration. This algorithm was applied to the real data set. The local minimum problem is discussed.

A new approach to the species classification problem in floristic analysis

AbstractA new method of species (inverse) classification of vegetation data, i.e. classification of species into groups with similar ecological tolerances, is presented which overcomes the problems of species abundance distorting the results. The algorithm TWO‐STEP is based on the use of an asymmetric measure of dissimilarity: image where i, j are species, h is the stand, n is the total number of stands, and xih is the amount of species i in stand h.The algorithm uses the rows of the asymmetric dissimilarity matrix generated as above to form a second symmetric dissimilarity matrix using the measure: image where m is the number of species and k the species.Flexible sorting is applied to generate a species classification. Comparison of results after applying the TWO‐STEP algorithm and a standard alternative to an artificial data set demonstrates its efficacy. TWO‐STEP also shows considerable advantages over previous analyses for a Queensland rainforest data set (quantitative) and an English heath (qualitative) data set. Normalization of species data appears advantageous for quantitative data only.

Perceptual features of normally hearing and hearing impaired subjects revisited

Perceptual strategies in rating dissimilarities among 24 prevocalic consonants paired with /a/ were investigated for normally hearing and two groups of hearing impaired subjects (i.e., one with flat, and one with sloping sensorineural losses). A 576-item equal appearing interval scaling data collection method was employed. The diadic-paired stimuli were presented bilaterally at MCL. Subjects' ratings on a 1–7 scale were converted to 24 × 24 full symmetric dissimilarity matrices and submitted to INDSCAL analysis. Individual group and combined analyses were performed. Results revealed features (e.g., sibilant, sonorant, and nasal) common to all groups, as well as group-specific features showing interactions among manner, place, and voicing categories. Results were compared to earlier investigations and data showed that subjects used features in their perception of the speech stimuli, that both common and group-specific features were used, and that individual differences in subjects' weightings of features were present.