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The analysis of clustering and correlation between points on a linear network, such as traffic accident locations on a street network, depends crucially on how we measure the distance between points. Standard practice is to measure distance by the length of the shortest path. However, this may be inappropriate and even fallacious in some applications. Alternative distance metrics include Euclidean, least-cost, and resistance distances. This paper develops a general framework for the second-order analysis of point patterns on a linear network, using a broad class of distance metrics on the network. We examine the model assumptions that are implicit in choosing a particular distance metric; define appropriate analogues of the K -function and pair correlation function; develop estimators of these characteristics; and study their statistical performance. The methods are tested on several datasets, including a demonstration that different conclusions can be reached using different choices of metric.

Distance measure plays an important role in clustering uncertain data. However, existing distance measures for clustering uncertain data suffer from some issues. Geometric distance measure can not identify the difference between uncertain objects with different distributions heavily overlapping in locations. Probability distribution distance measure can not distinguish the difference between different pairs of completely separated uncertain objects. In this paper, we propose a self-adapted mixture distance measure for clustering uncertain data which considers the geometric distance and the probability distribution distance simultaneously, thus overcoming the issues in previous distance measures. The proposed distance measure consists of three parts: (1) The induced kernel distance: it can be used to measure the geometric distance between uncertain objects. (2) The JensenShannon divergence: it can be used to measure the probability distribution distance between uncertain objects. (3) The self-adapted weight parameter: it can be used to adjust the importance degree of the induced kernel distance and the JensenShannon divergence according to the location overlapping information of the dataset. The proposed distance measure is symmetric, finite and parameter adaptive. Furthermore, we integrate the self-adapted mixture distance measure into the partition-based and density-based algorithms for clustering uncertain data. Extensive experimental results on synthetic datasets, real benchmark datasets and real world uncertain datasets show that our proposed distance measure outperforms the existing distance measures for clustering uncertain data.

Nearest neighbor search is a core process in many data mining algorithms. Finding reliable closest matches of a test instance is still a challenging task as the effectiveness of many general-purpose distance measures such as $$\ell _p$$ -norm decreases as the number of dimensions increases. Their performances vary significantly in different data distributions. This is mainly because they compute the distance between two instances solely based on their geometric positions in the feature space, and data distribution has no influence on the distance measure. This paper presents a simple data-dependent general-purpose dissimilarity measure called ‘ $$m_p$$ -dissimilarity’. Rather than relying on geometric distance, it measures the dissimilarity between two instances as a probability mass in a region that encloses the two instances in every dimension. It deems two instances in a sparse region to be more similar than two instances of equal inter-point geometric distance in a dense region. Our empirical results in k-NN classification and content-based multimedia information retrieval tasks show that the proposed $$m_p$$ -dissimilarity measure produces better task-specific performance than existing widely used general-purpose distance measures such as $$\ell _p$$ -norm and cosine distance across a wide range of moderate- to high-dimensional data sets with continuous only, discrete only, and mixed attributes.

This paper studies properties of parameter estimators obtained by minimizing a distance between the empirical probability generating function and the probability generating function of a model for count data. Specifically, it is shown that, under certain not restrictive conditions, the resulting estimators are consistent and, suitably normalized, asymptotically normal. These properties hold even if the model is misspecified. Three applications of the obtained results are considered. First, we revisit the goodness-of-fit problem for count data and propose a weighted bootstrap estimator of the null distribution of test statistics based on the above cited distance. Second, we give a probability generating function version of the model selection test problem for separate, overlapping and nested families of distributions. Finally, we provide an application to the problem of testing for separate families of distributions. All applications are illustrated with numerical examples.

Vine copulas are a flexible class of dependence models consisting of bivariate building blocks and have proven to be particularly useful in high dimensions. Classical model distance measures require multivariate integration and thus suffer from the curse of dimensionality. In this paper, we provide numerically tractable methods to measure the distance between two vine copulas even in high dimensions. For this purpose, we consecutively develop three new distance measures based on the Kullback–Leibler distance, using the result that it can be expressed as the sum over expectations of KL distances between univariate conditional densities, which can be easily obtained for vine copulas. To reduce numerical calculations, we approximate these expectations on adequately designed grids, outperforming Monte Carlo integration with respect to computational time. For the sake of interpretability, we provide a baseline calibration for the proposed distance measures. We further develop similar substitutes for the Jeffreys distance, a symmetrized version of the Kullback–Leibler distance. In numerous examples and applications, we illustrate the strengths and weaknesses of the developed distance measures.

Let X be a discrete random variable with a given probability distribution. For any α, 0 ≤ α ≤ 1, we obtain precise values for both the maximum and minimum variational distance between X and another random variable Y under which an α-coupling of these random variables is possible. We also give the maximum and minimum values for couplings of X and Y provided that the variational distance between these random variables is fixed. As a consequence, we obtain a new lower bound on the divergence through variational distance.

The f-divergence evaluates the dissimilarity between two probability distributions defined in terms of the Radon-Nikodym derivative of these two probabilities. The f-divergence generalizes the Hellinger distance and the Kullback-Leibler divergence among other divergence functions. In this paper we define an analogous function for non-additive measures. We discuss them for distorted Lebesgue measures and give examples. Examples focus on the Hellinger distance., Partial support by the Spanish MEC projects ARES (CONSOLIDER INGENIO 2010 CSD2007-00004), eAEGIS (TSI2007-65406-C03-02), and COPRIVACY (TIN2011-27076-C03-03) is acknowledged

We prove asymptotic results and concentration inequalities for a large class of discrete associated kernel estimators, under the total variation distance. We also propose a data driven bandwidth selection procedure aiming to minimize the total variation. Simulations are conducted.

The Monge–Kantorovich distance gives a metric between probability distributions on a metric space $${\mathbb X}$$ and the MK distance is tied to the underlying metric on $${\mathbb X}$$ . The MK distance (or a closely related metric) has been used in many areas of image comparison and retrieval and thus it is of significant interest to compute it efficiently. In the context of finite discrete measures on a graph, we give a linear time algorithm for trees and reduce the case of a general graph to that of a tree. Next, we give a linear time algorithm which computes an approximation to the MK distance between two finite discrete distributions on any graph. Finally, we extend our results to continuous distributions on graphs and give some very general theoretical results in this context.

The problem of characterizing global sensitivity indices of structural response when system uncertainties are represented using probabilistic and (or) non-probabilistic modeling frameworks (which include intervals, convex functions, and fuzzy variables) is considered. These indices are characterized in terms of distance measures between a fiducial model in which uncertainties in all the pertinent variables are taken into account and a family of hypothetical models in which uncertainty in one or more selected variables are suppressed. The distance measures considered include various probability distance measures (Hellinger,l(2), and the Kantorovich metrics, and the Kullback-Leibler divergence) and Hausdorff distance measure as applied to intervals and fuzzy variables. Illustrations include studies on an uncertainly parametered building frame carrying uncertain loads. (C) 2015 Elsevier Ltd. All rights reserved.

This metric distance satisfies on some of the properties and we used this metric for similarity measure.

In this study, the geographically weighted regression (GWR) model is adapted to benefit from a broad range of distance metrics, where it is demonstrated that a well-chosen distance metric can improve model performance. How to choose or define such a distance metric is key, and in this respect, a ‘Minkowski approach’ is proposed that enables the selection of an optimum distance metric for a given GWR model. This approach is evaluated within a simulation experiment consisting of three scenarios. The results are twofold: (1) a well-chosen distance metric can significantly improve the predictive accuracy of a GWR model; and (2) the approach allows a good approximation of the underlying ‘optimal distance metric’, which is considered useful when the ‘true’ distance metric is unknown.

We propose a new distance measure between 2-dimensional points to provide a total order for an entire point set and to reflect the correct geometric meaning of the naturalness of the point ordering. In general, there is no total order for 2-dimensional point sets, so curve reconstruction algorithms do not solve the self-intersection problem because the distance used in the previous methods is the Euclidean distance. A natural distance based on Brownian motion was previously proposed to solve the self-intersection problem. However, the distance reflects the wrong geometric meaning of the naturalness. In this paper, we correct the disadvantage of the natural distance by introducing a polar-natural distance, and we also propose a new curve reconstruction algorithm that is based on the polar-natural distance. Our experiments show that the new distance adequately reflects the correct geometric meaning, so nonsimple curve reconstruction can be solved.

We consider the problem of choosing the best subset of sensors that results in a prescribed error probability P e in Bayesian setting. Since minimizing the error probability is often difficult to evaluate and manipulate, conventional methods adopt Bhattacharyya distance instead of it. In fact, Chernoff distance is the best achievable exponent in the Bayesian error probability and it is more accurate than Bhattacharyya distance. In this paper, we design Chernoff Cyclical Iteration Algorithm (CCIA) to choose the best subset of sensors, provided the conditional distributions of sensor measurements. By leveraging on the convexity of Chernoff distance, we are able to derive the closed form of optimal parameters for the conditionally independent Gaussian measurements, exponential measurements and gamma measurements. Specifically, the optimal parameter is searched by using gold-cut algorithm. Furthermore, we prove that Chernoff distance is equivalent to Bhattacharyya distance for Gaussian measurements with uncommon means and common covariances structure under both hypotheses. In numerical examples, we supplement the comparisons between our proposed CCIA and the approach based on Bhattacharyya distance for exponential measurements and gamma measurements, which show that the detection performance based on Chernoff distance is superior to Bhattacharyya distance in the prescribed threshold.

In this work, we analyze the distance statistics corresponding to points in a Matérn cluster (offspring points) and points that do not belong to that cluster (non-offspring points). We first derive the probability density function (PDF) of the distance between an offspring point and a non-offspring point of a Matérn cluster. We then formulate the probability generating functional based on this PDF. Since many wireless networks (e.g., device-to-device (D2D) networks and cognitive radio systems) exhibit device clustering, this formalism enables us to efficiently formulate and evaluate expressions that describe the interference statistics and connection probability in clustered networks. We validate our theoretical analysis with numerical simulations, and illustrate that traditional methods of evaluating similar performance metrics (based on point process statistics instead of distance statistics) are unsuitable for use in such complex scenarios.

In this paper, we identify a class of absolutely continuous probability distributions, and show that the differential entropy is uniformly convergent over this space under the metric of total variation distance. One of the advantages of this class is that the requirements could be readily verified for a given distribution., 4 pages

Least-cost modelling and circuit theory are common analogs used in ecology and evolution to model gene flow or animal movement across landscapes. Least-cost modelling estimates the least-cost distance, whereas circuit theory estimates resistance distance. The bias added in choosing one method over the other has not been well documented. We designed an experiment to test whether both methods were linearly related. We also tested the sensitivity of these metrics to variation in Euclidean distance, spatial autocorrelation, the number of pixels representing the landscape, and data aggregation. We found that least-cost and resistance distance were not linearly related unless a transformation was applied. Resistance distance was less sensitive to the number of pixels representing a landscape and was also less sensitive than least-cost distance to the Euclidean distance between nodes. Spatial autocorrelation did not affect either method or the relationship between methods. Resistance distance was more sensitive to aggregation in any form compared to least-cost distance. Therefore, the metric used to infer movement or gene flow and the manipulations applied to the data used to calculate these metrics may govern findings.

We propose the L1 distance between the distribution of a binned data sample and a probability distribution from which it is hypothetically drawn as a statistic for testing agreement between the data and a model. We study the distribution of this distance for N-element samples drawn from k bins of equal probability and derive asymptotic formulae for the mean and dispersion of L1 in the large-N limit. We argue that the L1 distance is asymptotically normally distributed, with the mean and dispersion being accurately reproduced by asymptotic formulae even for moderately large values of N and k.

Recently, it has been shown that under a broad set of conditions, the commonly used distance functions will become unstable in high-dimensional data space; i.e., the distance to the farthest data point approaches the distance to the nearest data point of a given query point with increasing dimensionality. It has been shown that if dimensions are independently distributed, and normalized to have zero mean and unit variance, instability happens. In this paper, it is shown that the normalization condition is not necessary, but all appropriate moments must be finite. Furthermore, a new distance function, namely multiplicative distance, is introduced. It is theoretically proved that this function is stable for data with independent dimensions (with identical or nonidentical distribution). In contrast to usual distance functions which are based on the summation of distances over all dimensions (distance components), the multiplicative distance is based on the multiplication of distance components. Experimental results show the stability of the multiplicative distance for data with independent and correlated dimensions in the high-dimensional space and the superiority of the multiplicative distance over the norm distances for the high-dimensional data.

This paper presents an econometric analysis that compares the performance of different measures of distance in a gravity model using state data for Mexico. The estimation shows that at this geographic scale, the definition of distance does not affect the explanatory power of the model significantly. However, time-based definitions of distance have a marginal improvement on the model fit in comparison to length-based measures. When geographic specific fixed effects are unknown, the model shows that distance measured as road network distance is a better predictor. The paper concludes that time-based definitions of distance present several advantages in comparison to traditional length-based definitions. However, at large geographic scales, where relative distances between every geographic unit are long, the use of length-based distance instead of time-based distance to approximate travel costs generates similar results.

Spatial point pattern is an important tool for describing the spatial distribution of species in ecology. Negative binomial distribution (NBD) is widely used to model spatial aggregation. In this paper, we derive the probability distribution model of event-to-event nearest neighbor distance (distance from a focal individual to its n-th nearest individual). Compared with the probability distribution model of point-to-event nearest neighbor distance (distance from a randomly distributed sampling point to the n-th nearest individual), the new probability distribution model is more flexible. We propose that spatial aggregation can be detected by fitting this probability distribution model to event-to-event nearest neighbor distances. The performance is evaluated using both simulated and empirical spatial point patterns.

David Hume, the renowned philosopher in his Treatise of Human Nature, describes probability as the amount of evidence that accompanies uncertainty, a reasoning from conjecture. Probability theory is the branch of mathematics dealing with the study of uncertainty.

Stochastic optimization problems, which arise in different areas, such as simple asset allocation problems or problems in insurance, often involve coherent risk measures. In these real-world problems the considered risk measure is frequently built on empirical distributions. It is therefore of interest to understand the potential deviation, which occurs when evaluating a risk measure for a perturbed, or slightly changed, probability measure. This paper addresses the potential deviation. It turns out that the Wasserstein distance, a well-known distance for probability measures, provides a valuable notion of distance in the present context: Many risk measures allow a precise quantification in terms of the Wasserstein distance, and important risk measures are continuous with respect to this distance. For specific random variables, which often occur in concrete, real-world problems, it is moreover demonstrated that the derived constants, describing the continuity properties, cannot be improved. The associated...

In information geometry a Riemannian manifold of probability distributions is considered with the metric given by Fisher information matrix. In this paper we approach explicit forms for the Fisher-Rao distance in spaces composed by multivariate normal distributions of particular forms. Upper bounds for the distance between two general normal distributions are presented and their tightness are discussed in specific cases.

This paper uses a decision theoretic approach for updating a probability measure representing beliefs about an unknown parameter. A cumulative loss function is considered, which is the sum of two terms: one depends on the prior belief and the other one on further information obtained about the parameter. Such information is thus converted to a probability measure and the key to this process is shown to be the Kullback-Leibler divergence. The Bayesian approach can be derived as a natural special case. Some illustrations are presented. © The Institute of Statistical Mathematics, Tokyo 2012.

Probability and time are integral dimensions of virtually any decision. To treat them together, we consider the prospect of receiving outcome x with a probability p at time t. We define risk and time distance, and show that if these two distances are traded off linearly, then preferences are characterized by three functions: a value function, a probability discount rate function, and a psychological distance function. The concavity of the psychological distance function explains the common ratio and common difference effects. A decreasing probability discount rate accounts for the magnitude effect. The discount rate and the risk premium depend on the shape of these three functions. This paper was accepted by Peter Wakker, decision analysis.

Through social network analysis using portal site information, we study the relation of the probability distributions that appear in statistics textbooks with probability distributions that appear in daily life. Based on daily life, we discuss probability distributions that must be emphasized in frequent use.

This paper deals with the deviation distance to visit a facility from pre-planned routes. Facilities are approximated by both points and lines on a continuous plane. To see the relationship between the deviation distance and the availability of facilities, we derive the distributions of the rectilinear deviation distance for regular and random patterns of facilities. These distributions demonstrate how the shortest distance and the relative position of origin and destination affect the deviation distance. We also show that the deviation distance is a generalization of the nearest neighbour distance.

Hidden Markov model (HMM) has been found useful in modeling complex time series in various applications. An appropriate distance measure between two HMMs is of theoretical interests and it is also important in HMM-based applications. Kullback-Leibler (KL) and modified KL are usually used as distance measures between two HMMs. However, these measures do not satisfy the necessary properties of a distance measure, such as the triangle inequality. A novel distance measure, which is based on the HMM stationary cumulative distribution function, is proposed to discriminate two HMMs. It is proved that the measure can fulfill the properties requirements. The distance measure is evaluated by making comparisons to KL distance in experiments on a series of models. Also clustering on both synthesized data and real world data is performed with the new distance and KL distance, respectively. The results show that the proposed distance is more effective and reasonable in discriminating HMMs.

A generalized Statistical Complexity Measure (SCM) is a functional that characterizes the probability distribution P associated to the time series generated by a given dynamical system. It quantifies not only randomness but also the presence of correlational structures. We review here several fundamental issues in such a respect, namely, (a) the selection of the information measure I; (b) the choice of the probability metric space and associated distance D; (c) the question of defining the so-called generalized disequilibrium Q;(d) the adequate way of picking up the probability distribution P associated to a dynamical system or time series under study, which is indeed a fundamental problem. In this communication we show (point d) that sensible improvements in the final results can be expected if the underlying probability distribution is "extracted" via appropriate consideration regarding causal effects in the system's dynamics. Fil: Rosso, Osvaldo Aníbal. Universidad de Newcastle; Australia. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Cálculo; Argentina. University Drive; Australia. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina Fil: de Micco, Luciana. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata. Instituto de Investigaciones Científicas y Tecnológicas en Electrónica. Universidad Nacional de Mar del Plata. Facultad de Ingeniería. Instituto de Investigaciones Científicas y Tecnológicas en Electrónica; Argentina Fil: Larrondo, Hilda Angela. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata. Instituto de Investigaciones Científicas y Tecnológicas en Electrónica. Universidad Nacional de Mar del Plata. Facultad de Ingeniería. Instituto de Investigaciones Científicas y Tecnológicas en Electrónica; Argentina Fil: Martín, María Teresa. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Instituto de Física La Plata. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Instituto de Física La Plata; Argentina Fil: Plastino, Ángel Luis. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Instituto de Física La Plata. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Instituto de Física La Plata; Argentina

We provide bounds on the average distance between two points uniformly and independently chosen from a compact convex subset of the s-dimensional Euclidean space.

This paper studies the generalization and normalization issues of information-theoretic distance measures for clustering validation. Along this line, we first introduce a uniform representation of distance measures, defined as quasi-distance, which is induced based on a general form of conditional entropy. The quasi-distance possesses three properties: symmetry, the triangle law, and the minimum reachable. These properties ensure that the quasi-distance naturally lends itself as the external measure for clustering validation. In addition, we observe that the ranges of the distance measures are different when they apply for clustering validation on different data sets. Therefore, when comparing the performances of clustering algorithms on different data sets, distance normalization is required to equalize ranges of the distance measures. A critical challenge for distance normalization is to obtain the ranges of a distance measure when a data set is provided. To that end, we theoretically analyze the computation of the maximum value of a distance measure for a data set. Finally, we compare the performances of the partition clustering algorithm K-means on various real-world data sets. The experiments show that the normalized distance measures have better performance than the original distance measures when comparing clusterings of different data sets. Also, the normalized Shannon distance has the best performance among four distance measures under study.

Distance based reconstruction methods of phylogenetic trees consist of two independent parts: first, inter-species distances are inferred assuming some stochastic model of sequence evolution; then the inferred distances are used to construct a tree. In this paper we concentrate on the task of inter-species distance estimation. Specifically, we characterize the family of valid distance functions for the assumed substitution model and show that deliberate selection of distance function significantly improves the accuracy of distance estimates and, consequently, also improves the accuracy of the reconstructed tree. Our contribution consists of three parts: first, we present a general framework for constructing families of additive distance functions for stochastic evolutionary models. Then, we present a method for selecting (near) optimal distance functions, and we conclude by presenting simulation results which support our theoretical analysis.

We present a mathematical theory of objective, frequentist chance phenomena that uses as a model a set of probability measures. In this work, sets of measures are not viewed as a statistical compound hypothesis or as a tool for modeling imprecise subjective behavior. Instead we use sets of measures to model stable (although not stationary in the traditional stochastic sense) physical sources of finite time series data that have highly irregular behavior. Such models give a coarse-grained picture of the phenomena, keeping track of the range of the possible probabilities of the events. We present methods to simulate finite data sequences coming from a source modeled by a set of probability measures, and to estimate the model from finite time series data. The estimation of the set of probability measures is based on the analysis of a set of relative frequencies of events taken along subsequences selected by a collection of rules. In particular, we provide a universal methodology for finding a family of subsequence selection rules that can estimate any set of probability measures with high probability.