Traditional Euclidean Distance Research Articles

A new type of distance metric and its use for clustering

In order to address high dimensional problems, a new ‘direction-aware’ metric is introduced in this paper. This new distance is a combination of two components: (1) the traditional Euclidean distance and (2) an angular/directional divergence, derived from the cosine similarity. The newly introduced metric combines the advantages of the Euclidean metric and cosine similarity, and is defined over the Euclidean space domain. Thus, it is able to take the advantage from both spaces, while preserving the Euclidean space domain. The direction-aware distance has wide range of applicability and can be used as an alternative distance measure for various traditional clustering approaches to enhance their ability of handling high dimensional problems. A new evolving clustering algorithm using the proposed distance is also proposed in this paper. Numerical examples with benchmark datasets reveal that the direction-aware distance can effectively improve the clustering quality of the k-means algorithm for high dimensional problems and demonstrate the proposed evolving clustering algorithm to be an effective tool for high dimensional data streams processing.

A novel pathway-based distance score enhances assessment of disease heterogeneity in gene expression

BackgroundDistance based unsupervised clustering of gene expression data is commonly used to identify heterogeneity in biologic samples. However, high noise levels in gene expression data and relatively high correlation between genes are often encountered, so traditional distances such as Euclidean distance may not be effective at discriminating the biological differences between samples. An alternative method to examine disease phenotypes is to use pre-defined biological pathways. These pathways have been shown to be perturbed in different ways in different subjects who have similar clinical features. We hypothesize that differences in the expressions of genes in a given pathway are more predictive of differences in biological differences compared to standard approaches and if integrated into clustering analysis will enhance the robustness and accuracy of the clustering method. To examine this hypothesis, we developed a novel computational method to assess the biological differences between samples using gene expression data by assuming that ontologically defined biological pathways in biologically similar samples have similar behavior.ResultsPre-defined biological pathways were downloaded and genes in each pathway were used to cluster samples using the Gaussian mixture model. The clustering results across different pathways were then summarized to calculate the pathway-based distance score between samples. This method was applied to both simulated and real data sets and compared to the traditional Euclidean distance and another pathway-based clustering method, Pathifier. The results show that the pathway-based distance score performs significantly better than the Euclidean distance, especially when the heterogeneity is low and genes in the same pathways are correlated. Compared to Pathifier, we demonstrated that our approach achieves higher accuracy and robustness for small pathways. When the pathway size is large, by downsampling the pathways into smaller pathways, our approach was able to achieve comparable performance.ConclusionsWe have developed a novel distance score that represents the biological differences between samples using gene expression data and pre-defined biological pathway information. Application of this distance score results in more accurate, robust, and biologically meaningful clustering results in both simulated data and real data when compared to traditional methods. It also has comparable or better performance compared to Pathifier.

The Dynamic-Time-Warping-based k-means++ clustering and its application in phenoregion delineation

ABSTRACTThe phenoregion delineation facilitates more effective monitoring and more accurate forecasting of land-surface phenology (LSP), and thereby can greatly improve natural resources management. This article delineated a series of phenoregion maps by applying the Dynamic-Time-Warping (DTW)-based k-means++ clustering on normalized difference vegetation index (NDVI) time series. The DTW distance, a well-known shape-based similarity measure for time series data, was used as the distance measure instead of the traditional Euclidean distance in k-means++ clustering. These phenoregion maps were compared with the ones clustered based on the similarity of phenological forcing variables. The results demonstrated that the DTW-based k-means++ clustering can capture much more homogeneous phenological cycles within each phenoregion; the two types of phenoregion maps have a medium degree of spatial concordance, and their representativeness of vegetation types is comparable. The phenocycle-based phenoregion map with 15 phenoregions was selected as the optimal one, based on the criteria of cluster cohesion and separation.

A new method of fast distance transform 3D image based on “neighborhood between voxels in space” theory

Digital core technology is a new type of tool for analyzing and explaining the flow characteristics and fluid distribution of reservoir. Digital core technology has been widely used in recent years to describe features of pose space and simulate the process of fluid flow. As the basement of segmentation of pore space and reconstruction of pore network, the improvement of distance transform method has great impact on the development of digital core analysis technology. The accuracy and computation speed of distance transform method can directly affect the size of digital data and the detailedness of pore network model. Euclidean distance transform is the most precise one among all the distance transform methods, which means it is suitable for processing digital core data and calculating distance map. For traditional Euclidean distance transform method application in three-dimensional space data, there exist problems, such as too many search directions, large amount of data, and time-consuming. Large-scale data of digital core is hard to be transformed by this method. Therefore, a new theory of space based geometric topology neighbor relationship distance search algorithm was proposed in this paper. By introducing theory of neighborhood in 3D space, the relationship between 3 ´ 3 ´ 3 neighborhood with whole core data can be constructed, the computational area is greatly narrowed so that computation speed can be improved markedly. Then, instead of calculating every distance between pore voxels and skeleton voxels, the Euclidean distance of a pore voxel can be obtained by scanning the distance value of its 3 ´ 3 ´ 3 neighborhood. Exact Euclidean distance map of digital core data includes large-scale data showed after only two-scans. Noteworthily, due to the disturbing of boundary points which out the range of data size, special treatment is needed to process the pore voxels which near boundary of digital core data. Compared to existing methods, according to the interior of the rock pore structure characteristics, we simplified the comparison rules of the neighbor domain Euclidean distance value so that we can significantly improve computing capacity and computation speed of the Euclidean distance transform method. By this way, a large number of operations by the complex Euclidean distance structure can be avoided. And the complexity of the algorithm is better understood and applied. This article describes the process of the algorithm in detail and the method is extended to characterize pore space segmentation work of digital cores. Fractured-cave digital core, fractured digital core and kinds of digital core data were transformed by the new method, the results show that the method is more accurate and efficient for the segmentation and reconstruction of pore space model. On this basis, pore structure characteristics of pore space can be extracted and analyzed by digital core technology, and more parameters like permeability, formation factor, and so on, can be simulated. This paper created a new distance transform method on digital rock identification and extraction, which laid a theoretical foundation for the efficient development of microscopic description of oil and gas reservoirs, especially for fractured and vuggy reservoirs.

Acoustic source localization in ocean waveguides using principles of Riemannian geometry

Source localization in underwater acoustics entails a comparison between acoustic fields involving some measure of correlation, looking for similarity between the acoustic field propagated from the true source location and replica fields propagated from different locations in the waveguide. The uniqueness of the deterministic Green function between pairs of source-receiver positions forms the basis for the solution to this inverse problem. We consider a novel approach to source localization based on non-Euclidean geometry, where the “distance” between cross-spectral density matrices (CSDMs) is used to estimate the spatial location of the source. The traditional Euclidean distance is not necessarily appropriate because CSDMs are not arbitrary points in space; rather, they form a manifold constrained by the facts that CSDMs are both Hermitian and positive definite. These properties naturally lead to the interpretation of geodesic distance between CSDMs as a measure of similarity between acoustic fields with this minimum distance, parametrized by replica source location, establishing an estimate of the source position. We discuss the underlying concepts and present simulation results for a waveguide with internal wave-induced ocean variability. Several Riemannian metrics are considered and compared to more traditional approaches to matched-field localization. [Work supported by the Office of Naval Research.]

Local Indicator of Colocation Quotient with a Statistical Significance Test: Examining Spatial Association of Crime and Facilities

Most existing point-based colocation methods are global measures (e.g., join count statistic, cross K function, and global colocation quotient). Most recently, a local indicator such as the local colocation quotient has been proposed to capture the variability of colocation across areas. Our research advances this line of work by developing a simulation-based statistical test for the local indicator of colocation quotient (LCLQ). The study applies the indicator to examine the association of land use facilities with crime patterns. Moreover, we use the street network distance in addition to the traditional Euclidean distance in defining neighbors because human activities (including facilities and crimes) usually occur along a street network. The method is applied to analyze the colocation of three types of crimes and three categories of facilities in a city in Jiangsu Province, China. The findings demonstrate the value of the proposed method in colocation analysis of crime and facilities and, in general, colocation analysis of point data.

A Novel Method for Detecting Spam Email using KNN Classification with Spearman Correlation as Distance Measure

E-mail is the most prevalent methods for correspondence because of its availability, quick message exchange and low sending cost. Spam mail appears as a serious issue influencing this application today's internet. Spam may contain suspicious URL’s, or may ask for financial information as money exchange information or credit card details. Here comes the scope of filtering spam from legitimate emails. Classification is a way to get rid of those spam messages. Various researches are proposed for spam filtering by classifying them into labels of spam and business messages. Bayesian classification based spam filtering technique is a popular method. Also SVM based classifications are also used. K-nearest neighbour classification is simple, straightforward and easy to implement and has high Fmeasure compare to Bayesian and SVM classification. But accuracy of traditional KNN is lower than Bayesian classification. In this work a detection of spam mail is proposed by using Knearest neighbour classification method by combining Spearman’s correlation coefficient as distance measure rather than traditional Euclidean distance. Experimental results present a significant improvement in accuracy with higher Fmeasure compare to traditional algorithms.

HEALTH ASSESSMENT FOR HYDRAULIC SERVO SYSTEM USING MANIFOLD LEARNING BASED ON EMD

This study proposes a health assessment method for the hydraulic servo system using manifold learning based on empirical mode decomposition (EMD). An RBF neural network is adopted as a fault observer for the hydraulic servo system to generate a residual error signal. Then, the residual error signal is decomposed by EMD to form the initial feature matrix. To extract more sensitive features and reduce time consumption, isometric mapping algorithm is introduced to reduce the dimensionality of the initial feature matrix. Furthermore, the singular values of the reduced feature matrix are extracted for the subsequent health assessment. Considering the traditional Euclidean distance metric can only reflect local consistency, this study utilizes manifold distance (ManiD) to measure the health condition of the hydraulic servo system. Finally, the ManiD is converted into a confidence value, which directly represents the health status. Experiment results demonstrate the effectiveness of the proposed method.

Generalized Distance Between Compact Convex Sets: Algorithms and Applications

This paper presents algorithms to compute the generalized distance between two separated or penetrating compact convex sets, which is defined as the minimum or maximum scale factor of a given gauge set such that the scaled gauge set intersects or is contained in the Minkowski difference of the two sets. The traditional Euclidean distance is a special case where the origin-centered unit ball is used as the gauge set. While the generalized distance was proposed almost a decade ago, the only practical method for its computation has been general-purpose numerical optimization, which is computationally expensive. In contrast, our geometry-based algorithms are efficient and guarantee globally optimal solutions. Important applications of the algorithms in robotics include collision detection and grasp planning. The algorithm for computing the penetration distance also provides an accurate and efficient approach to flatness error evaluation, which is a fundamental problem in manufacturing. We demonstrate that our algorithms possess superior efficiency and accuracy in these applications.

Classification of coastal wetlands in eastern China using polarimetric SAR data

This study was initiated to classify Jiangsu coastal wetlands, which are situated on the north bank of the Yangtze River in eastern China, using fully polarimetric synthetic aperture radar (PolSAR) data with an improved classification scheme. First, Cloude-Pottier decomposition was completed to obtain polarimetric parameters. Then, the data were classified into 24 clusters using the decomposed parameters. Third, the agglomerative hierarchical-clustering algorithm was applied to merge the clusters into seven classes; the outcomes were regarded as initial values in the subsequent processing. Finally, the improved fuzzy C-means (FCM) algorithm, in which a fuzzy factor was introduced and the traditional Euclidean distance was replaced by the Wishart distance, was used to adjust the land cover boundaries. The experiment results revealed that for similar scattering mechanisms with different scattering intensities, the proposed method presented a satisfactory performance, with an overall accuracy of 86.93 %. The accuracies of seven land cover types were much higher and the boundaries were more clear when using the proposed method compared with two other methods. The study demonstrated the potential of using L-band fully polarimetric SAR data in coastal wetland classification.

Image Classification Based on New Matrix Distance

Image classiflcation problem has been the highlight in the fleld of machine learning. Most of the traditional learning algorithms are based on the vector space model, but in computer vision research, gray images are naturally represented by matrices. However, learning machine based on matrix often sufiers from the problem that the Traditional Euclidean Distance (TED) for images without preserving the natural matrix structure. In this paper, we present a novel Linear Support Matrix Machine (LSMM) based on New Matrix Distance (NMD), which integrates the merits of Classical Support Vector Machine (C-SVM) and the NMD of matrix data. Difierent with the TED, which is constrained by orthogonality assumption, NMD measures the distance between matrix points by considering the spatial relationships of pixels of matrix. Therefore, NMD is robust to small perturbation of images. Using some benchmark face recognition databases, we demonstrate a consistent performance improvement of LSMM embedded with the NMD over their original versions.

Fault Diagnosis of Supervision and Homogenization Distance Based on Local Linear Embedding Algorithm

In view of the problems of uneven distribution of reality fault samples and dimension reduction effect of locally linear embedding (LLE) algorithm which is easily affected by neighboring points, an improved local linear embedding algorithm of homogenization distance (HLLE) is developed. The method makes the overall distribution of sample points tend to be homogenization and reduces the influence of neighboring points using homogenization distance instead of the traditional Euclidean distance. It is helpful to choose effective neighboring points to construct weight matrix for dimension reduction. Because the fault recognition performance improvement of HLLE is limited and unstable, the paper further proposes a new local linear embedding algorithm of supervision and homogenization distance (SHLLE) by adding the supervised learning mechanism. On the basis of homogenization distance, supervised learning increases the category information of sample points so that the same category of sample points will be gathered and the heterogeneous category of sample points will be scattered. It effectively improves the performance of fault diagnosis and maintains stability at the same time. A comparison of the methods mentioned above was made by simulation experiment with rotor system fault diagnosis, and the results show that SHLLE algorithm has superior fault recognition performance.

An efficient implementation of k-means clustering for time series data with DTW distance

Time series clustering is one of the crucial tasks in time series data mining. The most popular method in time series clustering is k-means algorithm due to its simplicity and flexibility. So far, k-means for time series clustering has been most used with Euclidean distance. Dynamic time warping DTW distance measure has increasingly been used as a similarity measurement for various data mining tasks in place of traditional Euclidean distance due to its superiority in sequence-alignment flexibility. However, there exist some difficulties in clustering with DTW distance, for example, the problem of shape averaging in DTW or the problem of speeding up DTW distance calculation. In this paper, we compare the performance of the three shape averaging methods in DTW: nonlinear alignment and averaging filter NLAAF, prioritised shape averaging PSA and DTW barycenter averaging DBA and propose an efficient method to implement k-means clustering for time series data with DTW distance. In our method, we choose to use DBA method for shape-based time series averaging, apply early abandoning method for speeding up DTW distance calculation and median-based method for determining initial centroids for k-means clustering. The experimental results on benchmark datasets validate our proposed implementation method for time series k-means clustering with DTW.

Tensor Distance Based Least Square Twin Support Tensor Machine

Nowadays, there have been many data which are represented by tensor, that how to deal with these tensor data directly remains a significant challenge. In this paper, we propose a new tensor distance (TD) based least square twin support tensor machine (called TDLS-TSTM). Unlike the traditional Euclidean distance, TD considers the relationship information of various coordinates. TDLS-TSTM works directly on tensor data and aims to find two nonparallel hyperplanes for classification based on TD which can make full of structural information of data, solves two systems of linear equations rather than two quadratic programming problems. Compared with other classifiers, our method has the advantages of higher precision and accepted time consumption. The numerical experiments show the valid and efficient of TDLS-TSTM.

Supplier Selection and Evaluation Based on Grey Correlation Degree and TOPSIS

As logistic entity providers of supply chain, suppliers play a crucial role in assuring the smooth running of production and logistics, and realizing on time delivery of client orders. Proper selection of suppliers can not only reduce operations cost but also determine client order response speed, corporate flexibility, and product delivery, etc., hence having great impact on corporate competitiveness. This paper proposes supplier selection decision-making method based on grey correlation degree and Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS). By introducing grey correlation degree to traditional TOPSIS and combining grey correlation degree and traditional Euclidean distance, a new relative closeness is built, which reflects position relation between ideal solution and negative ideal solution, and similarity difference of data curve. With all the information above serving as the basis of solution order preference, evaluation on supplier optimal solution is thus realized. Simulation results indicate that the decision-making method by combination of grey correlation degree and TOPSIS can get rid of errors caused by subjective factors using one single method, improve reliability of selection solution evaluation, and have a great value of application.

Empirical analysis of city contact in Zhujiang (Pearl) River Delta, China

In traditional urban geography, city contact research is a classic study element in city research. In general, researchers use the traditional gravity model to characterize the contacts that exist between two cities. The traditional gravity model assumes ideal conditions, but these preconditions and their results often do not exist in realistic conditions. Thus, we used a modified gravity model to characterize the city contacts within a specific region. This model considers factors such as intercity complementarities, government intervention, and the diversity of the transportation infrastructure which is characterized as the transportation distance instead of the traditional Euclidean distance. We applied this model to an empirical study of city contact in the Zhujiang (Pearl) River Delta (PRD) of China. The regression results indicated that the modified gravity model could measure city contact more accurately and comprehensively than the traditional gravity model, i.e., it yielded a higher adjusted R 2 value (0.379) than the traditional gravity model result (0.259). Our study also suggests that, in addition to urban-regional and metropolitan development, the complementarities of the basic functions of cities at the administrative and market levels, as well as the corporeal and immaterial levels, play very significant roles in the characterization of city contact. Given the complexity of city contact, it will be necessary to consider more relevant influential factors in the modified gravity model to characterize the features of city contact in the future.

A novel artificial immune algorithm for spatial clustering with obstacle constraint and its applications.

An important component of a spatial clustering algorithm is the distance measure between sample points in object space. In this paper, the traditional Euclidean distance measure is replaced with innovative obstacle distance measure for spatial clustering under obstacle constraints. Firstly, we present a path searching algorithm to approximate the obstacle distance between two points for dealing with obstacles and facilitators. Taking obstacle distance as similarity metric, we subsequently propose the artificial immune clustering with obstacle entity (AICOE) algorithm for clustering spatial point data in the presence of obstacles and facilitators. Finally, the paper presents a comparative analysis of AICOE algorithm and the classical clustering algorithms. Our clustering model based on artificial immune system is also applied to the case of public facility location problem in order to establish the practical applicability of our approach. By using the clone selection principle and updating the cluster centers based on the elite antibodies, the AICOE algorithm is able to achieve the global optimum and better clustering effect.

Sparse representation for robust abnormality detection in crowded scenes

In crowded scenes, the extracted low-level features, such as optical flow or spatio-temporal interest point, are inevitably noisy and uncertainty. In this paper, we propose a fully unsupervised non-negative sparse coding based approach for abnormality event detection in crowded scenes, which is specifically tailored to cope with feature noisy and uncertainty. The abnormality of query sample is decided by the sparse reconstruction cost from an atomically learned event dictionary, which forms a sparse coding bases. In our algorithm, we formulate the task of dictionary learning as a non-negative matrix factorization (NMF) problem with a sparsity constraint. We take the robust Earth Mover's Distance (EMD), instead of traditional Euclidean distance, as distance metric reconstruction cost function. To reduce the computation complexity of EMD, an approximate EMD, namely wavelet EMD, is introduced and well combined into our approach, without losing performance. In addition, the combination of wavelet EMD with our approach guarantees the convexity of optimization in dictionary learning. To handle both local abnormality detection (LAD) and global abnormality detection, we adopt two different types of spatio-temporal basis. Experiments conducted on four public available datasets demonstrate the promising performance of our work against the state-of-the-art methods.

Optimized Laplacian SVM With Distance Metric Learning for Hyperspectral Image Classification

Laplacian support vector machine (LapSVM), as a benchmark method, which includes an additional regularization term with graph Laplacian, has been successfully applied to remote sensing image classification. However, using the Euclidean distance to construct weights, the graph in LapSVM may not really represent the inherent distribution of the data. In this paper, optimized LapSVMs are developed for semisupervised hyperspectral image classification, by introducing distance metric learning instead of the traditional Euclidean distance which is used in the existing LapSVM. In the procedure of constructing graph with distance metric learning, equivalence and non-equivalence pairwise constraints are imposed for better capturing similarity of samples from different classes. In this way, two new optimization problems are reformulated for building LapSVM with normalized and unnormalized graph Laplacian respectively. Experiments are conducted on two real hyperspectral datasets. Corresponding results obtained with low number of labeled training samples demonstrate the effectiveness of our proposed methods for hyperspectral image classification.

Tensor distance based multilinear globality preserving embedding: A unified tensor based dimensionality reduction framework for image and video classification

Image and video classification tasks often suffer from the problem of high-dimensional feature space. How to discover the meaningful, low-dimensional representations of such high-order, high-dimensional observations remains a fundamental challenge. In this paper, we present a unified framework for tensor based dimensionality reduction including a new tensor distance (TD) metric and a novel multilinear globality preserving embedding (MGPE) strategy. Different with the traditional Euclidean distance, which is constrained by orthogonality assumption, TD measures the distance between data points by considering the relationships among different coordinates of high-order data. To preserve the natural tensor structure in low-dimensional space, MGPE directly works on the high-order form of input data and employs an iterative strategy to learn the transformation matrices. To provide faithful global representation for datasets, MGPE intends to preserve the distances between all pairs of data points. According to the proposed TD metric and MGPE strategy, we further derive two algorithms dubbed tensor distance based multilinear multidimensional scaling (TD-MMDS) and tensor distance based multilinear isometric embedding (TD-MIE). TD-MMDS finds the transformation matrices by keeping the TDs between all pairs of input data in the embedded space, while TD-MIE intends to preserve all pairwise distances calculated according to TDs along shortest paths in the neighborhood graph. By integrating tensor distance into tensor based embedding, TD-MMDS and TD-MIE perform tensor based dimensionality reduction through the whole learning procedure and achieve obvious performance improvement on various standard datasets.