Distribution Of Interpoint Distances Research Articles

Способи побудови розподілу максимальної відстані між випадковими нормальними точками на площині

Many practical problems call for constructing the maximum interpoint distance distribution for random pints in a plane. In the literature, the case of a great number of points is considered, for which an asymptotic distribution is determined. This paper addresses the problem of constructing the maximum interpoint distance distribution for a small number of random points in a plane whose coordinates are independent random quantities that obey the standard normal distribution. The special case of three random points in a plane is considered as the basic one, for which three ways to construct the maximum interpoint distance distribution are studied. The first way is to construct the distribution function from geometrical considerations. To do this, the loci of three points are considered from the condition that the maximum distance between them shall not exceed a certain value. The position of the third point in the plane is determined relative to the two other points: the leftmost and the lowermost one. In this case, the construction of the distribution function involves the successive evaluation of several integrals using numerical methods. The obtained results are in good agreement with those of statistical simulation. The second way is based on studying the distance between pairs of random normal points in a plane. Taken separately, the distances between each pair of random normal points obey one-dimensional Rayleigh distributions, but in the aggregate they prove to be correlated because they are determined from the same pint coordinates. A joint distribution of the squared distances between three points is constructed using the three-dimensional Moran-Downton distribution. Using it, a distribution function of the squared maximum distance between three random normal points, which is identical with the maximum interpoint distance distribution, is obtained. It is found that for small values it underestimates the actual probability of the maximum distance not exceeding a certain value. For great distance values, the above probabilities coincide. The third way uses the Rice distribution (a generalization of the Rayleigh distribution) to approximate the unknown maximum interpoint distance distribution for three random normal points in a plane. The Rice distribution parameters found by the least-squares method are in good agreement with those obtained by statistical simulation. The results for three random normal points are generalized to a greater number of points (up to 30). It is shown that in this case the third way is most efficient.

Characterization of balls by generalized Riesz energy

AbstractWe show that balls, circles and 2‐spheres can be identified by generalized Riesz energy among compact submanifolds of the Euclidean space that are either closed or with codimension 0, where the Riesz energy is defined as the double integral of some power of the distance between pairs of points. As a consequence, we obtain the identification by the interpoint distance distribution.

Asymptotic normality of interpoint distances for high-dimensional data with applications to the two-sample problem

SummaryInterpoint distances have applications in many areas of probability and statistics. Thanks to their simplicity of computation, interpoint distance-based procedures are particularly appealing for analysing small samples of high-dimensional data. In this paper, we first study the asymptotic distribution of interpoint distances in the high-dimension, low-sample-size setting and show that it is normal under regularity conditions. We then construct a powerful test for the two-sample problem, which is consistent for detecting location and scale differences. Simulations show that the test compares favourably with existing distance-based tests.

Shape classification based on interpoint distance distributions

According to Kendall (1989), in shape theory, The idea is to filter out effects resulting from translations, changes of scale and rotations and to declare that shape is what is left. While this statement applies in principle to classical shape theory based on landmarks, the basic idea remains also when other approaches are used. For example, we might consider, for every shape, a suitable associated function which, to a large extent, could be used to characterize the shape. This finally leads to identify the shapes with the elements of a quotient space of sets in such a way that all the sets in the same equivalence class share the same identifying function. In this paper, we explore the use of the interpoint distance distribution (i.e.?the distribution of the distance between two independent uniform points) for this purpose. This idea has been previously proposed by other authors e.g.,?Osada et?al. (2002), Bonetti and Pagano (2005). We aim at providing some additional mathematical support for the use of interpoint distances in this context. In particular, we show the Lipschitz continuity of the transformation taking every shape to its corresponding interpoint distance distribution. Also, we obtain a partial identifiability result showing that, under some geometrical restrictions, shapes with different planar area must have different interpoint distance distributions. Finally, we address practical aspects including a real data example on shape classification in marine biology.

M Statistic Commands: Interpoint Distance Distribution Analysis

We implement the commands mstat and mtest to perform inference based on the M statistic, a statistic that can be used to compare the interpoint distance distribution across groups of observations. The analyses are based on the study of the interpoint distances between n points in a k-dimensional setting to produce a one-dimensional real-valued test statistic. The locations are distributed in a region of the plane. When we consider all ( n2) interpoint distances, the dependencies among them are difficult to express analytically, but their distribution is informative, and the M statistic can be built to summarize one aspect of this information. The two commands can be used on a wide class of data sets to test the null hypothesis that two groups have the same (spatial) distribution. mstat and mtest return the exact M test statistic. Moreover, mtest executes a Monte Carlo–type permutation test, which returns the empirical p-value together with its confidence interval. This is the command to use in most situations, because the convergence of M to its asymptotic chi-squared distribution is slow. Both commands can be used to obtain graphical output of the empirical density function of the interpoint distance distributions in the two groups and the two-dimensional map of the n observations in the plane. The descriptions of the commands are accompanied by examples of applications with real and simulated data. We run the test on the Alt and Vach grave site dataset (Manjourides and Pagano, forthcoming, Statistics in Medicine) and reject the null hypothesis, in contradiction to other published analyses. We also show how to adapt the techniques to discrete datasets with more than one unit in each location. Finally, we report an extensive application on breast cancer data in Massachusetts; in the application, we show the compatibility of the M commands with Pisati's spmap package.

The choice of the number of bins for the [formula omitted] statistic

Methods to monitor spatial patterns of disease in populations are of interest in public health practice. The M statistic uses interpoint distances between cases to detect abnormalities in the spatial patterns of diseases. This statistic compares the observed distribution of interpoint distances with that which is expected when no unusual spatial patterns exist. We show the relationship of M to Pearson’s Chi Square statistic, X n 2 . Both statistics require the discretization of continuous data into bins and then are formed by creating a quadratic form, scaled by an appropriate variance covariance matrix. We seek to choose the number and type of these bins for the M statistic so as to maximize the power to detect spatial anomalies. By showing the relationship between M to X n 2 , we argue for the extension of the theory that has been developed for the selection of the number and type of bins for X n 2 to M . We further show that spatial data provides a unique insight into the problem through examples with simulated data and spatial data from a health care provider. In the spatial setting, these indicate that the optimal number of bins depends on the size of the cluster. For large clusters, a smaller number of bins appears to be preferable, however for small clusters having many bins increases the power. Further, results indicate that the number of bins does not appear to vary with m , the number of spatial locations. We discuss the implications of this result for further work.

Data Clustering Based on Maximization of Outlier Factor

There exist many data clustering algorithms, but they can not adequately handle the number of clusters or cluster shapes. Their performance mainly depends on a choice of algorithm parameters. Our approach to data clustering and algorithm does not require the parameter choice; it can be treated as a natural adaptation to the existing structure of distances between data points. The outlier factor introduced by the author specifies a degree of being an outlier for each data point. The outlier factor notion is based on the difference between the frequency distribution of interpoint distances in a given dataset and the corresponding distribution of uniformly distributed points. Then data clusters can be determined by maximizing the outlier factor function. The data points in dataset are divided into clusters according to the attractor regions of local optima. An experimental evaluation of the proposed algorithm shows that the proposed method can identify complex cluster shapes. Key advantages of the approach are: good clustering properties for datasets with comparatively large amount of noise (an additional data points), and an absence of important parameters which adequate choice determines the quality of results.

Bioterrorism. Mathematical modeling applications in homeland security.

Bioterrorism. Mathematical modeling applications in homeland security.

Real time spatial cluster detection using interpoint distances among precise patient locations

BackgroundPublic health departments in the United States are beginning to gain timely access to health data, often as soon as one day after a visit to a health care facility. Consequently, new approaches to outbreak surveillance are being developed. When cases cluster geographically, an analysis of their spatial distribution can facilitate outbreak detection. Our method focuses on detecting perturbations in the distribution of pair-wise distances among all patients in a geographical region. Barring outbreaks, this distribution can be quite stable over time. We sought to exemplify the method by measuring its cluster detection performance, and to determine factors affecting sensitivity to spatial clustering among patients presenting to hospital emergency departments with respiratory syndromes.MethodsThe approach was to (1) define a baseline spatial distribution of home addresses for a population of patients visiting an emergency department with respiratory syndromes using historical data; (2) develop a controlled feature set simulation by inserting simulated outbreak data with varied parameters into authentic background noise, thereby creating semisynthetic data; (3) compare the observed with the expected spatial distribution; (4) establish the relative value of different alarm strategies so as to maximize sensitivity for the detection of clustering; and (5) measure factors which have an impact on sensitivity.ResultsOverall sensitivity to detect spatial clustering was 62%. This contrasts with an overall alarm rate of less than 5% for the same number of extra visits when the extra visits were not characterized by geographic clustering. Clusters that produced the least number of alarms were those that were small in size (10 extra visits in a week, where visits per week ranged from 120 to 472), diffusely distributed over an area with a 3 km radius, and located close to the hospital (5 km) in a region most densely populated with patients to this hospital. Near perfect alarm rates were found for clusters that varied on the opposite extremes of these parameters (40 extra visits, within a 250 meter radius, 50 km from the hospital).ConclusionMeasuring perturbations in the interpoint distance distribution is a sensitive method for detecting spatial clustering. When cases are clustered geographically, there is clearly power to detect clustering when the spatial distribution is represented by the M statistic, even when clusters are small in size. By varying independent parameters of simulated outbreaks, we have demonstrated empirically the limits of detection of different types of outbreaks.

The interpoint distance distribution as a descriptor of point patterns, with an application to spatial disease clustering

The topic of this paper is the distribution of the distance between two points distributed independently in space. We illustrate the use of this interpoint distance distribution to describe the characteristics of a set of points within some fixed region. The properties of its sample version, and thus the inference about this function, are discussed both in the discrete and in the continuous setting. We illustrate its use in the detection of spatial clustering by application to a well-known leukaemia data set, and report on the results of a simulation experiment designed to study the power characteristics of the methods within that study region and in an artificial homogenous setting.

Randomness of spatial distributions of two proteins in the cell nucleus involved in mRNA synthesis and their relationship.

Randomness and relation between point positions play an important role in archeology, cosmology, geography, and biology. An often-discarded effect is the edge effect, the effect that points are bound to a certain region. Without an appropriate correction, the outcome will be wrong. We studied the problem of randomness by comparing the distribution of interpoint distances with what can be expected for randomly distributed points (pair correlation function in statistics), and applied this to two sets of nuclear proteins inside the cell nucleus. The technique comprised labelling the proteins with a fluorescent dye, recording the fluorescent distribution with a 3D confocal microscope, and detecting the positions of the individual fluorescent spots. Results showed that, apart from studying randomness, the method is well equipped to quantitatively analyze a spot detection procedure, as the resolving power and the subvoxel accuracy were clearly visible. Given the results of assessing the randomness of the general transcription factor BRG1 and the RNA synthesizing protein RNA polymerase II (polII) in the cell nucleus, we concluded that the high intensity spots of the BRG1 protein are regularly spaced. The low intensity spots of the BRG1 protein and the low- and high-intensity spots of the polII protein showed more random behavior. The BRG1 and polII proteins showed correlation; unexpectedly, the relation was also found for the low-intensity spots, which were expected to have a more random behavior.

Line transects, covariance functions and set convergence

We define the ‘linear scan transform' G of a set in ℝd using information observable on its one-dimensional linear transects. This transform determines the set covariance function, interpoint distance distribution, and (for convex sets) the chord length distribution. Many basic integral-geometric formulae used in stereology can be expressed as identities for G. We modify a construction of Waksman (1987) to construct a metric η for ‘regular' subsets of ℝd defined as the L1 distance between their linear scan transforms. For convex sets only, η is topologically equivalent to the Hausdorff metric. The set covariance function (of a generally non-convex set) depends continuously on its set argument, with respect to η and the uniform metric on covariance functions.

Line transects, covariance functions and set convergence

We define the ‘linear scan transform' G of a set in ℝ d using information observable on its one-dimensional linear transects. This transform determines the set covariance function, interpoint distance distribution, and (for convex sets) the chord length distribution. Many basic integral-geometric formulae used in stereology can be expressed as identities for G. We modify a construction of Waksman (1987) to construct a metric η for ‘regular' subsets of ℝ d defined as the L 1 distance between their linear scan transforms. For convex sets only, η is topologically equivalent to the Hausdorff metric. The set covariance function (of a generally non-convex set) depends continuously on its set argument, with respect to η and the uniform metric on covariance functions.

Investigating Group-Specific Clustering in Univariate Spatial Data with Many Small Groups

In a situation where no assumption of stationarity or distributional form can be made, the problem is to investigate whether there tend to be local clusters within groups of observations. The null hypothesis is that the points are grouped randomly. One method is to compare the distribution of interpoint distances within groups with that in the pooled sample. Another method is to look at the rank distances between neighbouring points within groups, the ranks being obtained from the pooled sample. Many small rank distances indicates group-specific clustering. These methods are applied to a medical problem.

On the distinctness of clusters

A common problem in classification is the assignment of objects to meaningful clusters given their relative positions in reduced ordination space. We examine the distinctiveness of such putative clusters. Clusters are enlarged by simulation and then the relative frequency distributions of interpoint distances in the “enlarged” clusters are compared with the distribution of the original points in the original space. This procedure may be repeated for all reasonable clustering hypotheses until the best fit is found. We suggest that one way of looking at the distinctness of clusters is to look at the distribution of interpoint distances for various hypothetical clusters from which the data might have been sampled. The best clustering, given the data, is the one that best matches the original distribution of interpoint distances.