Euclidean Distance Matrix Research Articles

Gait Evaluation Using Procrustes and Euclidean Distance Matrix Analysis.

Objective assessment of gait is important in the treatment and rehabilitation of patients with different diseases. In this paper, we propose a gait evaluation system using the Procrustes and Euclidean distance matrix analysis. We design and develop an android app to collect real time synchronous accelerometer and gyroscope data from two inertial measurement unit sensors through Bluetooth connectivity. The data is collected from 12 young (ten for modeling and two for validation) and 20 older subjects. We analyze the data collected from real world for stride, step, stance, and swing gait features. We validate our method with the measurements of gait features. The generalized Procrustes analysis is used to estimate a standard normal mean gait shape (NMGS) for ten young subjects. Each gait feature of both young and older subjects is then converted to find the best match with the NMGS using the ordinary Procrustes analysis. The shape distance between the NMGS and each gait shape is estimated using Riemannian shape distance, Riemannian size-and-shape distance, Procrustes size-and-shape distance, and root-mean-square deviation. A t-test is performed to provide statistical evidence of gait shape differences between young and older gaits. A mean form, which is considered as a standard normal mean gait form (NMGF), and inter-feature distances are estimated from the set of ten young subjects. The form difference is estimated between the NMGF and individual gaits of young and older. The degree of abnormality is then estimated for individual features and the result is plotted to visualize the feature in a gait. Experimental results demonstrate the performance of the proposed method.

Spatial perspective of hexavalent chromium concentration in superficial waters of the Ciénaga de las Quintas Mangrove Swamp, Cartagena de Indias, Colombia

AbstractA spatial characterization was undertaken to demonstrate the presence of hexavalent chromium (Cr (VI)) in surface waters (maximum depth of 0.5 m) in the Ciénaga de las Quintas mangrove swamp, located in the city of Cartagena de Indias, Colombia. This characterization was geostatistical in nature, an approach that is little used in such studies, although it offers two‐dimensional profiles of the concentration of relevant contaminants (hexavalent chromium in this case) with concentrations between 3 and 9.2 μg/L. Sampling was carried out systematically on a square grid with a cell size of one ha (100 m × 100 m). To corroborate the relationship between the sample positions and concentrations, the Euclidean distance matrices of each variable were calculated previous step for conducting the Mantel test, which indicated a positive correlation between the distance and concentration of hexavalent chromium (R = 0,1674: p value = 0.1275). The Moran Index was also calculated for Cr (VI). The experimental semivariogram (γ)was also calculated and fit to the exponential theoretical model, with a correlation coefficient of 99.4%. Using this analytical model of variance, predictions of Cr (VI) concentrations were made for unsampled locations, using the Kriging method with a new mesh side of only 10 m. This approach indicated the new mesh points exhibiting the higher concentrations coincided with locations in the swamp that receiving sewage effluents from the Bazurto public grocery market and the Chinatown neighbourhood. The Cr (VI) contamination was of anthropogenic origin because its concentration displays a gradient from the zone of human activities to the Cartagena de Indias bay.

Disruption of symmetry: A quantitative assessment of facial skeleton anatomy in children born with unilateral cleft lip and palate.

Unilateral cleft lip and palate (UCLP) is a craniofacial deformity characterized by lip and palate clefting on one side of the face. UCLP originates from failures in neural crest migration and differentiation during embryological development, impairing facial primordia fusion (medial nasal, maxillary, and lateral processes) resulting in clefting. Persons with UCLP experience issues in nasal breathing, speaking, and mastication. Facial directional asymmetry (DA), consisting of left-right side differences biased toward one particular side, can arise from environmental or hereditary factors. This retrospective, cross-sectional study, quantifies DA in the facial skeleton of children with surgically repaired UCLP. We tested the hypothesis that DA is significantly increased in persons with UCLP. Twenty-three anatomical landmark coordinates were measured from cone beam computed tomography images of two age- and sex-matched samples: (1) persons with UCLP post-surgery (n = 26, 7-17 years); (2) typical children (n = 26, 7-17 years). From these coordinates, 81 bilateral paired measurements were calculated and statistically assessed for DA differences using Principal Components Analysis and Euclidean Distance Matrix Analysis. Samples separate in high-dimensional space and 35.8% of bilateral measures are statistically significant. Patterns of significant DA differences between samples were explored based on magnitude. Compared with the control group, in all but five cases, significant DA was greater in persons with UCLP. Regions derived from the maxillary and nasal prominences demonstrated the most DA. These results are important for medical practitioners for identifying the extent to which patients with UCLP deviate from the norm. Clin. Anat. 31:1129-1136, 2018. © 2018 Wiley Periodicals, Inc.

Euclidean Distance Analysis Enables Nucleotide Skew Analysis in Viral Genomes.

Nucleotide skew analysis is a versatile method to study the nucleotide composition of RNA/DNA molecules, in particular to reveal characteristic sequence signatures. For instance, skew analysis of the nucleotide bias of several viral RNA genomes indicated that it is enriched in the unpaired, single-stranded genome regions, thus creating an even more striking virus-specific signature. The comparison of skew graphs for many virus isolates or families is difficult, time-consuming, and nonquantitative. Here, we present a procedure for a more simple identification of similarities and dissimilarities between nucleotide skew data of coronavirus, flavivirus, picornavirus, and HIV-1 RNA genomes. Window and step sizes were normalized to correct for differences in length of the viral genome. Cumulative skew data are converted into pairwise Euclidean distance matrices, which can be presented as neighbor-joining trees. We present skew value trees for the four virus families and show that closely related viruses are placed in small clusters. Importantly, the skew value trees are similar to the trees constructed by a “classical” model of evolutionary nucleotide substitution. Thus, we conclude that the simple calculation of Euclidean distances between nucleotide skew data allows an easy and quantitative comparison of characteristic sequence signatures of virus genomes. These results indicate that the Euclidean distance analysis of nucleotide skew data forms a nice addition to the virology toolbox.

ADMM-Based Sensor Network Localization Using Low-Rank Approximation

In this paper, we present an efficient localization algorithm for sensor networks using range information. Since each sensor can only communicate with its neighbors, the Euclidean distance matrix (EDM), composed by squared distances between each pair of sensors, is incomplete. The first step of the proposed algorithm is to fulfill the EDM completion and relative maps of sensor networks are then achieved by the multidimensional scaling technique. Besides the EDM, the centralized Gram matrix of sensors’ coordinates is also used to model the localization problem. Compared with other EDM-based localization algorithms, mathematical properties of both the EDM and the Gram matrix are appropriately exploited so as to improve the estimation accuracy. The resulting localization model is formulated as a semidefinite programming problem. An alternating direction method of multipliers is further developed to enhance the scalability of the proposed algorithm. The numerical experiments demonstrate that the proposed algorithm can effectively improve the estimation accuracy of both the EDM completion and the final localization.

A morphometric analysis of developmental instability in children born with unilateral cleft lip and palate.

Unilateral cleft lip and palate (UCLP) is a congenital deformity that occurs due to inadequate merging of the nasal and maxillary prominences during fetal development. Randomly distributed bilateral asymmetries known as fluctuating asymmetry (FA) occur in normally symmetric organisms when evolved mechanisms of developmental stability or equilibrium are disturbed by genetic, environmental, or unknown factors. Here, we hypothesize that facial skeleton FA will be increased in a sample of individuals born with UCLP (n = 24) relative to sex- and age-matched controls (n = 24). To test this hypothesis, 23 anatomical landmarks were measured on individual anonymized cone-beam computerized tomography (CBCT) images in children and adolescents (7-17 years). For each individual, 81 pairs of linear distances were used to estimate FAs across the face. To explore sample variation and statistical differences, a principal components analysis and Euclidean Distance Matrix Analysis multivariate bootstrap approach were carried out. Samples show some separation in multivariate space with 44.44% of FA differences being significantly different. The magnitude of FA was larger in the UCLP sample for every significant measurement. The magnitude of significant FA is highest near regions derived from the maxillary and nasal prominences, such as the nasal aperture. These results are useful for medical and dental practitioners when developing treatment options for children and adolescents with UCLP. Clin. Anat. 32:206-211, 2019. © 2018 Wiley Periodicals, Inc.

On Cartesian product of Euclidean distance matrices

If A∈Rm×m and B∈Rn×n, we define the product A⊘B as A⊘B=A⊗Jn+Jm⊗B, where ⊗ denotes the Kronecker product and Jn is the n×n matrix of all ones. We refer to this product as the Cartesian product of A and B since if D1 and D2 are the distance matrices of graphs G1 and G2 respectively, then D1⊘D2 is the distance matrix of the Cartesian product G1□G2. We study Cartesian products of Euclidean distance matrices (EDMs). We prove that if A and B are EDMs, then so is the product A⊘B. We show that if A is an EDM and U is symmetric, then A⊗U is an EDM if and only if U=cJn for some c. It is shown that for EDMs A and B, A⊘B is spherical if and only if both A and B are spherical. If A and B are EDMs, then we derive expressions for the rank and the Moore–Penrose inverse of A⊘B. In the final section we consider the product A⊘B for arbitrary matrices. For A∈Rm×m,B∈Rn×n, we show that all nonzero minors of A⊘B of order m+n−1 are equal. An explicit formula for a nonzero minor of order m+n−1 is proved. The result is shown to generalize the familiar fact that the determinant of the distance matrix of a tree on n vertices does not depend on the tree and is a function only of n.

A hierarchical clustering approach to identify repeated enrollments in web survey data.

IntroductionOnline surveys are a valuable tool for social science research, but the perceived anonymity provided by online administration may lead to problematic behaviors from study participants. Particularly, if a study offers incentives, some participants may attempt to enroll multiple times. We propose a method to identify clusters of non-independent enrollments in a web-based study, motivated by an analysis of survey data which tests the effectiveness of an online skin-cancer risk reduction program.MethodsTo identify groups of enrollments, we used a hierarchical clustering algorithm based on the Euclidean distance matrix formed by participant responses to a series of Likert-type eligibility questions. We then systematically identified clusters that are unusual in terms of both size and similarity, by repeatedly simulating datasets from the empirical distribution of responses under the assumption of independent enrollments. By performing the clustering algorithm on the simulated datasets, we determined the distribution of cluster size and similarity under independence, which is then used to identify groups of outliers in the observed data. Next, we assessed 12 other quality indicators, including previously proposed and study-specific measures. We summarized the quality measures by cluster membership, and compared the cluster groupings to those found when using the quality indicators with latent class modeling.Results and conclusionsWhen we excluded the clustered enrollments and/or lower-quality latent classes from the analysis of study outcomes, the estimates of the intervention effect were larger. This demonstrates how including repeat or low quality participants can introduce bias into a web-based study. As much as is possible, web-based surveys should be designed to verify participant quality. Our method can be used to verify survey quality and identify problematic groups of enrollments when necessary.

Wireless Sensor Network Localization via Matrix Completion Based on Bregman Divergence

One of the main challenges faced by wireless sensor network (WSN) localization is the positioning accuracy of the WSN node. The existing algorithms are arduous to use for dealing with the pulse noise that is universal and ineluctable in practical considerations, resulting in lower positioning accuracy. Aimed at this problem and introducing Bregman divergence, we propose in this paper a novel WSN localization algorithm via matrix completion (LBDMC). Based on the natural low-rank character of the Euclidean Distance Matrix (EDM), the problem of EDM recovery is formulated as an issue of matrix completion in a noisy environment. A regularized matrix completion model is established, smoothing the pulse noise by leveraging -norm and the multivariate function Bregman divergence is defined to solve the model to obtain the EDM estimator. Furthermore, node localization is available based on the multi-dimensional scaling (MDS) method. Multi-faceted comparison experiments with existing algorithms, under a variety of noise conditions, demonstrate the superiority of LBDMC to other algorithms regarding positioning accuracy and robustness, while ensuring high efficiency. Notably, the mean localization error of LBDMC is about ten times smaller than that of other algorithms when the sampling rate reaches a certain level, such as >30%.

Iterative echo labeling technique for room geometry estimation

Room geometry estimation using acoustic room impulse responses (RIRs) has been tackled in various ways. Most of them detect wall locations using the time of arrival (TOA) of echoes between multiple microphones and loudspeakers. To combine TOA information from multiple room impulse responses, echoes from the same walls should be collected and processed together. When multiple microphones are widely distributed in a room, however, it is hard to identify walls responsible for the generation of each echo. In this work, an iterative update algorithm is proposed for resolving the echo labeling problem with low computational complexity. Unlike conventional algorithms utilizing the rank property of Euclidean distance matrix, the proposed algorithm utilizes the property of convex hull formed by ellipses representing individual TOAs. The location of the reflectors are sequentially identified from common tangent lines constituting the convex hull, and the search result is iteratively updated by inspecting later echoes in time. In addition, a termination criterion is also developed in order to enable the geometry estimation without the prior knowledge of the number of the reflectors.

A Fast Matrix Majorization-Projection Method for Penalized Stress Minimization With Box Constraints

Kruskal's stress minimization, though nonconvex and nonsmooth, has been a major computational model for dissimilarity data in multidimensional scaling. Semidefinite programming (SDP) relaxation (by dropping the rank constraint) would lead to a high number of SDP cone constraints. This has rendered the SDP approach computationally challenging even for problems of small size. In this paper, we reformulate the stress optimization as an Euclidean distance matrix (EDM) optimization with box constraints. A key element in our approach is the conditional positive-semidefinite cone with rank cut. Although nonconvex, this geometric object allows a fast computation of the projection onto it, and it naturally leads to a majorization-minimization algorithm with the minimization step having a closed-form solution. Moreover, we prove that our EDM optimization follows a continuously differentiable path, which greatly facilitated the analysis of the convergence to a stationary point. The superior performance of the proposed algorithm is demonstrated against some of the state-of-the-art solvers in the field of sensor network localization and molecular conformation.

Параллельный алгоритм вычисления матрицы Евклидовых расстояний для многоядерного процессора Intel Xeon Phi поколения Knights Landing

Computation of a Euclidean distance matrix (EDM) is a typical task in a wide spectrum of problems connected with data mining. Currently, many parallel algorithms for this task have been developed for graphical processors. These developments, however, cannot be directly applied to the Intel Many Integrated Core systems. In this paper, we suggest a parallel algorithm for EDM computation on Intel Xeon Phi Knights Landing processor in the case when the input data fit into the main memory. The algorithm exploits block-oriented scheme of computations that allows for the efficient utilization of Intel Xeon Phi vectorization abilities. In the algorithm, we also apply apply a sophisticated data layout to store data points in main memory so as to reduce the number of processor cache misses during EDM computations. Experimental evaluation of the algorithm on real-world and synthetic datasets shows that t is highly scalable and outruns analogues in the case of rectangular matrices with low-dimensional data points.

Noise Tolerant Localization for Sensor Networks

Most range-based localization approaches for wireless sensor networks WSNs rely on accurate and sufficient range measurements, yet noise and data missing are inevitable in distance ranging. Existing localization approaches often suffer from unsatisfied accuracy in the co-existence of incomplete and corrupted range measurements. In this paper, we propose LoMaC, a noise-tolerant localization scheme, to address this problem. Specifically, we first employ Frobenius-norm and $L_{1}$ -norm to formulate the reconstruction of noisy and missing Euclidean distance matrix EDM as a norm-regularized matrix completion NRMC problem. Second, we design an efficient algorithm based on alternating direction method of multiplier to solve the NRMC problem. Third, based on the completed EDM, we further employ a multi-dimension scaling method to localize unknown nodes. Meanwhile, to accelerate our algorithm, we also adopt some acceleration techniques to reduce the computation cost. Finally, extensive experimental results show that our algorithm not only achieves significantly better localization performance than prior algorithms but also provides an accurate position prediction of outlier, which is useful for malfunction diagnosis in WSNs.

Realizing Euclidean distance matrices by sphere intersection

This paper presents the theoretical properties of an algorithm to find a realization of a (full) n×n Euclidean distance matrix in the smallest possible embedding dimension. Our algorithm performs linearly in n, and quadratically in the minimum embedding dimension, which is an improvement w.r.t. other algorithms.

On yielding and jointly yielding entries of Euclidean distance matrices

An n×n matrix D is a Euclidean distance matrix (EDM) if there exist points p1,…,pn in some Euclidean space such that dij=‖pi−pj‖2 for all i,j=1,…,n. Let D be an EDM and let Eij be the n×n symmetric matrix with 1's in the ijth and jith entries and 0's elsewhere. We say that [lij,uij] is the yielding interval of entry dij if it holds that D+tEij is an EDM if and only if lij≤t≤uij. If the yielding interval of entry dij has length 0, i.e., if lij=uij=0, then dij is said to be unyielding. Otherwise, if lij≠uij, then dij is said to be yielding. Let dij and dik be two unyielding entries of D. We say that dij and dik are jointly yielding if D+t1Eij+t2Eik is an EDM for some nonzero scalars t1 and t2.In this paper, we characterize the yielding and the jointly yielding entries of an EDM D in terms of Gale transforms of p1,…,pn. Moreover, for each yielding entry, we present explicit formulae of its yielding interval. Finally, we specialize our results to the case where points p1,…,pn are in general position.

Evaluation of genotypic variation for growth of rice seedlings under optimized hydroponics medium

Although rice (Oryza sativa L.) is a semi-aquatic plant, its growth in solution culture is often problematic. In commonly used rice hydroponics media, plants exhibited mild nitrogen deficiency, leaf tip burn, salt deposition, along with zinc and iron deficiency. Therefore, we aimed to optimize the nutrient media for growing rice plants taking into consideration the nutrient concentration, pH and ratio of NH4 + to NO3 – nitrogen (N) and named it as ‘Pusa Rice Hydroponics’ (PusaRicH). PusaRicH contains higher amounts of macronutrients, significantly lower B, Mn and Cl and higher amounts of Zn than two commonly used hydroponics media, ‘Yoshida’ and ‘Kumura B’. The optimal ratio of NH4 + to NO3 – – N in PusaRicH medium was 0.5 mM NH4 + and 7.0 mM NO3 – with pH 5.0. The PusaRicH medium was validated by growing 100 diverse rice genotypes and it significantly outperformed the widely cited ‘Yoshida’ and ‘Kimura B’. Cluster analysis carried out on the squared Euclidean distance matrix of biomass and leaf area values of genotypes revealed four major clusters in all hydroponics medium. However, only PusaRicH medium resulted in ten genotypes as good performers in comparison to other two widely cited media. Therefore, the optimized PusaRicH medium can be used successfully to grow rice seedlings in hydroponic system which will aid in screening large number of genotypes in breeding and other physiological experiments.

Muscle‐intensive and High‐Impact Exercises Differentially Influence Whole Bone 3D Morphology in Young Outbred Male Mice

Skeletal morphology is predominantly dictated by the genetic background and the habitual loading environment of individual bones. Mechanical loads placed on bones are derived from intracorporeal contractions of muscles and from extracorporeal ground reaction forces generated by bearing weight and by substrate reaction forces during locomotion. We investigated the relative effects of these two sources of load on the growing skeleton using an outbred mouse model system. The 3D shape of the femur was compared across groups of mice following discrete exercise regimens designed to emphasize specific loading environments. Hsd:ICR (CD‐1) mice (N=80; 20 per group) were randomly assigned to one of four treatment groups at four weeks old: swimming which emphasizes muscle forces, dropping which emphasizes impact forces, wheel running which results in a combination of muscle and impact forces, and non‐exercised controls which were free to move about their cages. Durations of swimming, running, and impact loading were matched so that mice received approximately the same intensity. After 20 days mice were euthanized and right hind limbs were micro‐CT scanned (~20 μm3 voxel size). Femoral shape was defined by digitizing 23 landmarks on each scan. To adjust for size, each individual femur was scaled to the geometric mean of all possible linear distances among landmarks. Group differences were analyzed using Euclidean Distance Matrix Analysis (EDMA), and differences >5% were considered substantial. In all three activity groups, we find decreased robusticity of the femoral diaphysis and increased distal condylar breadth compared to non‐exercised controls. Additionally, each group displays unique femoral morphologies when compared to controls. The swimming treatment group shows increased length of the femoral neck. The wheel running treatment group shows a narrower femoral neck with a larger femoral head. The impact loading group shows a lengthened and reoriented distal diaphysis. Overall, the running and swimming groups show similar patterns of morphology that diverge from the non‐exercised mice, suggesting muscle forces differentially affect the loading environment during running in mice compared to impact forces. Further, these results indicate the regional sensitivity of femoral morphology to muscle‐ and impact‐derived loads, and emphasize how activity patterns during adolescence can lead to distinct morphologies in the limb skeleton.Support or Funding InformationResearch support: University of Missouri School of MedicineThis abstract is from the Experimental Biology 2018 Meeting. There is no full text article associated with this abstract published in The FASEB Journal.

A novel low-rank matrix completion approach to estimate missing entries in Euclidean distance matrix

A Euclidean distance matrix (EDM) is a table of distance-square between points on a k-dimensional Euclidean space, with applications in many fields (e.g., engineering, geodesy, economics, genetics, biochemistry, and psychology). A problem that often arises is the absence (or uncertainty) of some EDM elements. In many situations, only a subset of all pairwise distances is available and it is desired to have some procedure to estimate the missing distances. In this paper, we address the problem of missing data in EDM through low-rank matrix completion techniques. We exploit the fact that the rank of a EDM is at most $$k+2$$ and does not depend on the number of points, which is, in general, much bigger then k. We use a singular value decomposition approach that considers the rank of the matrix to be completed and computes, in each iteration, a parameter that controls the convergence of the method. After performing a number of computational experiments, we could observe that our proposal was able to recover, with high precision, random EDMs with more than 1000 points and up to 98% of missing data in few minutes. In addition, our method required a smaller number of iterations when compared to other competitive state-of-art technique.

Joint localization of multiple sources from incomplete noisy Euclidean distance matrix in wireless networks

One major challenge of massive wireless networks is to identify the locations of source nodes from partially observed and noisy distance information. This is especially important for wireless sensor networks (WSN) and wireless local area networks (WLAN). In this paper, we propose a unified localization framework of multiple sources from an Euclidean distance matrix (EDM) with noise and outliers both in WSN and WLAN scenarios. We first develop a semidefinite programming (SDP) based low rank matrix completion (LRMC) estimator by using the semidefinite embedding lemma to recover EDM. Based on our recovered EDM, two robust localization estimators, namely, semidefinite relaxation localization (SDRL) and weighted semidefinite relaxation localization (WSDRL), are derived to efficiently relax our non-convex localization problem into a convex one, and yield more accuarate location estimates. As compared with existing techniques, our proposed techniques are more robust to noise and outliers with higher accuracies both in EDM recovery and source localization. Simulations and real data experiments are included to evaluate the performance of the proposed algorithms by comparing them with some existing methods.

Euclidean Distance Matrices and Separations in Communication Complexity Theory

A Euclidean distance matrix \(D(\alpha )\) is defined by \(D_{ij}=(\alpha _i-\alpha _j)^2\), where \(\alpha =(\alpha _1,\ldots ,\alpha _n)\) is a real vector. We prove that \(D(\alpha )\) cannot be written as a sum of \(\left[ 2\sqrt{n}-2\right] \) nonnegative rank-one matrices, provided that the coordinates of \(\alpha \) are algebraically independent. As a corollary, we provide an asymptotically optimal separation between the complexities of quantum and classical communication protocols computing a given matrix in expectation.