Chebychev Distance Research Articles

The Application of Numerical Measure Variations in K-Means Clustering for Grouping Data

The K-Means Clustering algorithm is commonly used by researchers in grouping data. The main problem in this study was that it has yet to be discovered how optimal the grouping with variations in distance calculations is in K-Means Clustering. The purpose of this research was to compare distance calculation methods with K-Means such as Euclidean Distance, Canberra Distance, Chebychev Distance, Cosine Similarity, Dynamic TimeWarping Distance, Jaccard Similarity, and Manhattan Distance to find out how optimal the distance calculation is in the K-Means method. The best distancecalculation was determined from the smallest Davies Bouldin Index value. This research aimed to find optimal clusters using the K-Means Clustering algorithm with seven distance calculations based on types of numerical measures. This research method compared distance calculation methods in the K-Means algorithm, such as Euclidean Distance, Canberra Distance, Chebychev Distance, Cosine Smilirity, Dynamic Time Warping Distance, Jaccard Smilirity and Manhattan Distance to find out how optimal the distance calculation is in the K-Means method. Determining the best distance calculation can be seen from the smallest Davies Bouldin Index value. The data used in this study was on cosmetic sales at Devi Cosmetics, consisting of cosmetics sales from January to April 2022 with 56 product items. The result of this study was a comparison of numerical measures in the K-Means Clustering algorithm. The optimal cluster was calculating the Euclidean distance with a total of 9 clusters with a DBI value of 0.224. In comparison, the best average DBI value was the calculation of the Euclidean Distance with an average DBI value of 0.265.

Implementasi Metode Learning Vector Quantization (LVQ) Untuk Klasifikasi Keluarga Beresiko Stunting

Stunting is a condition where a child's height is too short compared to children of the same age. This condition affects the health of toddlers in the short and long term, such as suboptimal body posture in adulthood, decreased reproductive health, and decreased learning capacity, resulting in suboptimal performance in school. One of the causes of stunting is a lack of nutrition, basic health facilities, and poor parenting practices. However, the current data collection and classification of families at risk of stunting still use Microsoft Excel, which is ineffective in processing large data. Therefore, the LVQ method, which is an improvement of the Vector Quantization method, is used to accelerate the classification process. In this study, 5 parameters were tested, and the optimal result was achieved by using 7 input neurons, Chebychev distance as the distance measure, a learning rate of 0.1, 7 epochs, and 30% of training data. With these parameters, an accuracy of 99.38% was obtained. Based on these results, the LVQ method can help improve accuracy in classifying families at risk of stunting

Evaluation of Different Clustering Techniques in Classifying the Vegetable Growing Panchayats of Ernakulam District, Kerala

The goal of this study was to evaluate different clustering techniques in classifying the vegetable growing locations of Ernakulam (EKM) district of Kerala so that same nutrient recommendation can be prescribed for panchayats coming under the same cluster. Hierarchical clustering (HC) and K –means clustering were performed to group the panchayats based on soil fertility status and thereafter comparison of various clustering procedures was done using Davies – Bouldin (DB) index. Different dissimilarity measures- Euclidean, squared Euclidean, Chebychev distance and Mahalanobis D2 were determined and single linkage, complete linkage and average linkage methods were adopted under these measures. The results revealed that Mahalanobis D2 was the better clustering procedure with seven clusters (DB index: 0.120) followed by average linkage method under Euclidean distance (DB index: 0.306) with seven clusters. Manjapra and Keerampara panchayats remained as individual clusters. Keerampara had strongly acidic soils (pH -5.17) with high available Mg (158 mg kg-1) while Manjapra soils had low Mg availability (19 mg kg-1) and high S content (57 mg kg-1). Kakkad, Kalady and Vengoor came under cluster I which possessed approximately same EC (0.15-0.19 dS m-1), OC (2-2.4%) and Mg (71-73 mg kg-1) content. Chengamanadu and Vengola came under cluster III while Ayyampuzha and Mudakkuzha belong to cluster IV.

Applications of Genetic Distances on Blood-group Gene Frequencies and their Statistical Genetic Similarities

This paper studied the applications of genetic distances on Blood-group gene frequencies and their Statistical genetic similarities characterizing four populations, for data from Eakimo, Bantu, English and Korea. The study compared the following distances: Euclidean, squared Euclidean, Minkowski, Chebychev and City Block on the above mentioned data for the outlined countries. Correlation analysis was applied to evaluate the relationships between these countries on their blood group gene frequencies. Similarity check was also conducted to know the countries that have similar blood-group gene frequency. It was observed that Euclidean distance and Minkowski distance had equal distances. This means that the two distances are more similar in this particular data set than the other studied distances. The study revealed that Chebychev distance had the smallest neighbor distance as compared to other distances studied while City Block had the highest distance. It has been stated in literature that Chebychev and Minkowski distances are concentric circle shape, this suggests the reason behind their equality of distance. It is therefore proposed that the data may be a concentric circle data.

Weighted consensus clustering and its application to Big data

The aim of this study is the development of a weighted consensus clustering that assigns weights to single clustering methods using the purity utility function. In the case of Big data that does not contain labels, the utility function based on the Davies-Bouldin index is proposed in this paper. The Banknote authentication, Phishing, Diabetic, Magic04, Credit card clients, Covertype, Phone accelerometer, and NSL-KDD datasets are used to assess the efficiency of the proposed consensus approach. The proposed approach is evaluated using the Euclidean, Minkowski, squared Euclidean, cosine, and Chebychev distance metrics. It is compared with single clustering algorithms (DBSCAN, OPTICS, CLARANS, k-means, and shared nearby neighbor clustering). The experimental results show the effectiveness of the proposed approach to the Big data clustering in comparison to single clustering methods. The proposed weighted consensus clustering using the squared Euclidean distance metric achieves the highest accuracy, which is a very promising result for Big data clustering. It can be applied to expert systems to help experts make group decisions based on several alternatives. The paper also provides directions for future research on consensus clustering in this area.

Pengenalan Alfabet American Sign Language Menggunakan K-Nearest Neighbors Dengan Ekstraksi Fitur Histogram Of Oriented Gradients

Sign Language use to communicate to people with dissabilities. American Sign Language (ASL) one of popular sign language. Histogram of Oriented Gradient (HOG) can be use as feature extraction. Then feature stored in database. K-Nearest Neighbor use to measure distance between feature train and feature test. There are three distance use in this paper consist of Euclidean Distance, Manhattan Distance and Chebychev Distance. The best result are 0,99 when using Euclidean Distance and Manhattan Distance with k=3 dan k=5

Modeling River Ice Breakup Dates by k-Nearest Neighbor Ensemble

Forecasting of river ice breakup timing is directly related to the local ice-caused flooding management. However, river ice forecasting using k-nearest neighbor (kNN) algorithms is limited. Thus, a kNN stacking ensemble learning (KSEL) method was developed and applied to forecasting breakup dates (BDs) for the Athabasca River at Fort McMurray in Canada. The kNN base models with diverse inputs and distance functions were developed and their outputs were further combined. The performance of these models was examined using the leave-one-out cross validation method based on the historical BDs and corresponding climate and river conditions in 1980–2015. The results indicated that the kNN with the Chebychev distance functions generally outperformed other kNN base models. Through the simple average methods, the ensemble kNN models using multiple-type (Mahalanobis and Chebychev) distance functions had the overall optimal performance among all models. The improved performance indicates that the kNN ensemble is a promising tool for river ice forecasting. The structure of optimal models also implies that the breakup timing is mainly linked with temperature and water flow conditions before breakup as well as during and just after freeze up.

Actuarial Risk Matrices: The Nearest Positive Semidefinite Matrix Problem

The manner in which a group of insurance risks are interrelated is commonly presented via a correlation matrix. Actuarial risk correlation matrices are often constructed using output from disparate modeling sources and can be subjectively adjusted, for example, increasing the estimated correlation between two risk sources to confer reserving prudence. Hence, while individual elements still obey the assumptions of correlation values, the overall matrix is often not mathematically valid (not positive semidefinite). This can prove problematic in using the matrix in statistical models. The first objective of this article is to review existing techniques that address the nearest positive semidefinite matrix problem in a very general setting. The chief approaches studied are Semidefinite Programming (SDP) and the Alternating Projections Method (APM). The second objective is to finesse the original problem specification to consider imposition of a block structure on the initial risk correlation matrix. This commonly employed technique identifies off-diagonal subsets of the matrix where values can or should be set equal to some constant. This may be due to similarity of the underlying risks and/or with the goal of increasing computational efficiency for processes involving large matrices. Implementation of further linear constraints of this nature requires adaptation of the standard SDP and APM algorithms. In addition, a new Shrinking Method is proposed to provide an alternative solution in the context of this increased complexity. “Nearness” is primarily considered in terms of two summary measures for differences between matrices: the Chebychev Norm (maximum element distance) and the Frobenius Norm (sum of squared element distances). Among the existing methods, adapted to function appropriately for actuarial risk matrices, APM is extremely efficient in producing solutions that are optimal in the Frobenius norm. An efficient algorithm that would return a positive semidefinite matrix that is optimal in Chebychev norm is currently unknown. However, APM is used to highlight the existence of matrices close to such an optimum and exploited, via the Shrinking Method, to find high-quality solutions. All methods are shown to work well both on artificial and real actuarial risk matrices provided under collaboration with Tokio Marine Kiln (TMK). Convergence speeds are calculated and compared and sample data and MATLAB code is provided. Ultimately the APM is identified as being superior in Frobenius distance and convergence speed. The Shrinking Method, building on the output of the APM algorithm, is demonstrated to provide excellent results at low computational cost for minimizing Chebychev distance.

Proof of a conjecture of Kl\xf8ve on permutation codes under the Chebychev distance

Let d be a positive integer and x a real number. Let A_{d, x} be a dtimes 2d matrix with its entries ai,j=xfor1⩽j⩽d+1-i,1ford+2-i⩽j⩽d+i,0ford+1+i⩽j⩽2d.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} a_{i,j}=\\left\\{ \\begin{array}{ll} x\\ \\ &{} \\quad \\text{ for } \\ 1\\leqslant j\\leqslant d+1-i,\\\\ 1\\ \\ &{} \\quad \\text{ for } \\ d+2-i\\leqslant j\\leqslant d+i,\\\\ 0\\ \\ &{} \\quad \\text{ for } \\ d+1+i\\leqslant j\\leqslant 2d. \\end{array} \\right. \\end{aligned}$$\\end{document}Further, let R_d be a set of sequences of integers as follows: Rd=(ρ1,ρ2,…,ρd)|1⩽ρi⩽d+i,1⩽i⩽d,andρr≠ρsforr≠s.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} R_d=\\left\\{ (\\rho _1, \\rho _2,\\ldots , \\rho _d)|1\\leqslant \\rho _i\\leqslant d+i, 1\\leqslant i \\leqslant d,\\ \\text{ and }\\ \\rho _r\\ne \\rho _s\\ \\quad \\text{ for }\\ r\\ne s\\right\\} . \\end{aligned}$$\\end{document}and define Ωd(x)=∑ρ∈Rda1,ρ1a2,ρ2…ad,ρd.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\Omega _d(x)=\\sum _{\\rho \\in R_d}a_{1,\\rho _1}a_{2, \\rho _2}\\ldots a_{d,\\rho _d}. \\end{aligned}$$\\end{document}In order to give a better bound on the size of spheres of permutation codes under the Chebychev distance, Kløve introduced the above function and conjectured that Ωd(x)=∑m=0ddm(m+1)d(x-1)d-m.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\Omega _d(x)=\\sum _{m=0}^d{d\\atopwithdelims ()m}(m+1)^d(x-1)^{d-m}. \\end{aligned}$$\\end{document}In this paper, we settle down this conjecture positively.

A Novel Ranking-based Optimal Guides Selection Strategy in MOPSO

A challenging issue with multi-objective particle swarm optimization (MOPSO) is the mechanism to select the optimal guides. This paper presents a new strategy based on ranking dominance and integrates into MOPSO. By using the ranking information and incorporating the chebychev distance of particle in objective space, we implement the selection of gbest and pbest simply and elegantly. On the basis of ranking, we propose a new maintenance strategy for updating the external archive which can obtain a more diverse and uniform distribution. Furthermore, a qualitative and quantitative analysis in terms of convergence analysis over some benchmarks is presented, providing a basis for conclusions about the proposed method. showing that the proposed method performs better than the adopted algorithms.

Comparison of Different Face Recognition Method Based On PCA

This paper is about human face recognition in image files. Face recognition involves matching a given image with the database of images and identifying the image that it resembles the most. Here, face recognition is done using:
 (a) Eigen faces and
 (b) applying Principal Component Analysis (PCA) on image.
 The aim is to successfully demonstrate the human face recognition using Principal component analysis & comparison of Manhattan distance, Eucleadian distance & Chebychev distance for face matching.

Suitability analysis for siting MSW landfills and its multicriteria spatial decision support system: Method, implementation and case study

Multicriteria spatial decision support systems (MC-SDSS) have emerged as an integration of geographical information systems (GIS) and multiple criteria decision analysis (MCDA) methods for incorporating conflicting objectives and decision makers’ (DMs’) preferences into spatial decision models. This article presents a raster-based MC-SDSS that combines the analytic hierarchy process (AHP) and compromise programming methods, such as TOPSIS (technique for order preference by similarity to the ideal solution) and Ideal Point Methods. To the best of our knowledge it is the first time that a synergy of AHP and compromise programming methods is implemented in raster-driven GIS-based landfill suitability analysis. This procedure is supported by a spatial decision support system (SDSS) that was developed within a widely used commercial GIS software package. A real case study in the Thrace region in northeast Greece serves as a guide on how to conduct a suitability analysis for a MSW landfill site with the proposed MC-SDSS. Moreover, the procedure for identifying MSW disposal sites is accomplished by performing four computational models for synthesizing the DMs per criterion preferential system. Based on the case study results, a comparison analysis is performed according to suitability index estimations. According to them Euclidean distance metric and TOPSIS present strong similarities. When compared with Euclidean distance metric, TOPSIS seems to generate results closer to that derived by Manhattan distance metric. The comparison of Chebychev distance metric with all the other approaches revealed the greatest deviations.

A visualization metric for dimensionality reduction

Data visualization of high-dimensional data is possible through the use of dimensionality reduction techniques. However, in deciding which dimensionality reduction techniques to use in practice, quantitative metrics are necessary for evaluating the results of the transformation and visualization of the lower dimensional embedding. In this paper, we propose a manifold visualization metric based on the pairwise correlation of the geodesic distance in a data manifold. This metric is compared with other metrics based on the Euclidean distance, Mahalanobis distance, City Block metric, Minkowski metric, cosine distance, Chebychev distance, and Spearman distance. The results of applying different dimensionality reduction techniques on various types of nonlinear manifolds are compared and discussed. Our experiments show that our proposed metric is suitable for quantitatively evaluating the results of the dimensionality reduction techniques if the data lies on an open planar nonlinear manifold. This has practical significance in the implementation of knowledge-based visualization systems and the application of knowledge-based dimensionality reduction methods.

Lower bounds on the size of spheres of permutations under the Chebychev distance

Lower bounds on the number of permutations p of {1, 2, . . . , n} satisfying |p i ? i| ? d for all i are given.

Synthesis of Sparse Planar Arrays Using Modified Real Genetic Algorithm

In array design, the positions of sparse array elements is an important concern for optimal performance in terms of its ability to achieve minimum peak sidelobe level (SLL). This paper proposes a modified real genetic algorithm (MGA) based on resetting of chromosome for the element position optimization of sparse planar arrays with rectangular boundary. And here the multiple optimization constraints include the number of elements, the aperture and the minimum element spacing. By simplifying the space between the elements from the actual distance to Chebychev distance, the MGA searches a smaller solution space by means of indirect description of individual, and it can avoid infeasible solution during the optimization process by two novel genetic operators. Finally, the simulation results confirming the great efficiency and the robustness of the proposed method are shown in this paper