scholarly journals Issues in clustering algorithm consistency in fixed dimensional spaces. Some solutions for k-means

2021 ◽  
Author(s):  
Mieczysław A. Kłopotek ◽  
Robert A. Kłopotek
Keyword(s):  
Clustering Algorithm ◽  
Dimensional Space ◽  
Clustering Algorithms ◽  
Practical Importance ◽  
Axiomatic System ◽  
One Dimension ◽  
Data Set ◽  
Data Transformations ◽  
Higher Dimensional ◽  
Consistency Property

AbstractKleinberg introduced an axiomatic system for clustering functions. Out of three axioms, he proposed, two (scale invariance and consistency) are concerned with data transformations that should produce the same clustering under the same clustering function. The so-called consistency axiom provides the broadest range of transformations of the data set. Kleinberg claims that one of the most popular clustering algorithms, k-means does not have the property of consistency. We challenge this claim by pointing at invalid assumptions of his proof (infinite dimensionality) and show that in one dimension in Euclidean space the k-means algorithm has the consistency property. We also prove that in higher dimensional space, k-means is, in fact, inconsistent. This result is of practical importance when choosing testbeds for implementation of clustering algorithms while it tells under which circumstances clustering after consistency transformation shall return the same clusters. Two types of remedy are proposed: gravitational consistency property and dataset consistency property which both hold for k-means and hence are suitable when developing the mentioned testbeds.

scholarly journals Subspace Clustering of High-Dimensional Data: An Evolutionary Approach

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Singh Vijendra ◽  
Sahoo Laxman
Keyword(s):  
Clustering Algorithm ◽  
Dimensional Space ◽  
Clustering Algorithms ◽  
High Dimensional Data ◽  
Subspace Clustering ◽  
High Dimensional ◽  
Data Sets ◽  
Real World Data ◽  
Data Set ◽  
Data Points

Clustering high-dimensional data has been a major challenge due to the inherent sparsity of the points. Most existing clustering algorithms become substantially inefficient if the required similarity measure is computed between data points in the full-dimensional space. In this paper, we have presented a robust multi objective subspace clustering (MOSCL) algorithm for the challenging problem of high-dimensional clustering. The first phase of MOSCL performs subspace relevance analysis by detecting dense and sparse regions with their locations in data set. After detection of dense regions it eliminates outliers. MOSCL discovers subspaces in dense regions of data set and produces subspace clusters. In thorough experiments on synthetic and real-world data sets, we demonstrate that MOSCL for subspace clustering is superior to PROCLUS clustering algorithm. Additionally we investigate the effects of first phase for detecting dense regions on the results of subspace clustering. Our results indicate that removing outliers improves the accuracy of subspace clustering. The clustering results are validated by clustering error (CE) distance on various data sets. MOSCL can discover the clusters in all subspaces with high quality, and the efficiency of MOSCL outperforms PROCLUS.

scholarly journals A SELF-ORGANIZING MAP FOR MIXED CONTINUOUS AND CATEGORICAL DATA

2011 ◽  
pp. 24-32 ◽  
Author(s):  
Nicoleta Rogovschi ◽  
Mustapha Lebbah ◽  
Younès Bennani
Keyword(s):  
Clustering Algorithm ◽  
Clustering Algorithms ◽  
Mixed Data ◽  
Categorical Variables ◽  
Data Sets ◽  
Self Organizing Map ◽  
Data Set ◽  
Public Data ◽  
Self Organizing

Most traditional clustering algorithms are limited to handle data sets that contain either continuous or categorical variables. However data sets with mixed types of variables are commonly used in data mining field. In this paper we introduce a weighted self-organizing map for clustering, analysis and visualization mixed data (continuous/binary). The learning of weights and prototypes is done in a simultaneous manner assuring an optimized data clustering. More variables has a high weight, more the clustering algorithm will take into account the informations transmitted by these variables. The learning of these topological maps is combined with a weighting process of different variables by computing weights which influence the quality of clustering. We illustrate the power of this method with data sets taken from a public data set repository: a handwritten digit data set, Zoo data set and other three mixed data sets. The results show a good quality of the topological ordering and homogenous clustering.

Clustering Bathymetric Data for Electronic Navigational Charts

2016 ◽  
Vol 69 (5) ◽  
pp. 1143-1153 ◽  
Author(s):  
Marta Wlodarczyk–Sielicka ◽  
Andrzej Stateczny
Keyword(s):  
Clustering Algorithm ◽  
Search Algorithm ◽  
Clustering Algorithms ◽  
Data Set ◽  
Bathymetric Data ◽  
Large Sets ◽  
Analysis Of Results ◽  
Comparison And Analysis ◽  
Self Organising Map ◽  
Source Of Information

An electronic navigational chart is a major source of information for the navigator. The component that contributes most significantly to the safety of navigation on water is the information on the depth of an area. For the purposes of this article, the authors use data obtained by the interferometric sonar GeoSwath Plus. The data were collected in the area of the Port of Szczecin. The samples constitute large sets of data. Data reduction is a procedure to reduce the size of a data set to make it easier and more effective to analyse. The main objective of the authors is the compilation of a new reduction algorithm for bathymetric data. The clustering of data is the first part of the search algorithm. The next step consists of generalisation of bathymetric data. This article presents a comparison and analysis of results of clustering bathymetric data using the following selected methods:K-means clustering algorithm, traditional hierarchical clustering algorithms and self-organising map (using artificial neural networks).

scholarly journals Data Fusion Using a Multi-Sensor Sparse-Based Clustering Algorithm

2020 ◽  
Vol 12 (23) ◽  
pp. 4007
Author(s):  
Kasra Rafiezadeh Shahi ◽  
Pedram Ghamisi ◽  
Behnood Rasti ◽  
Robert Jackisch ◽  
Paul Scheunders ◽  
...  
Keyword(s):  
Clustering Algorithm ◽  
Spatial Information ◽  
Clustering Algorithms ◽  
Hyperspectral Data ◽  
Sensor Data ◽  
Data Sets ◽  
Data Types ◽  
Data Set ◽  
Multiple Data Sets ◽  
Imaging Sensors

The increasing amount of information acquired by imaging sensors in Earth Sciences results in the availability of a multitude of complementary data (e.g., spectral, spatial, elevation) for monitoring of the Earth’s surface. Many studies were devoted to investigating the usage of multi-sensor data sets in the performance of supervised learning-based approaches at various tasks (i.e., classification and regression) while unsupervised learning-based approaches have received less attention. In this paper, we propose a new approach to fuse multiple data sets from imaging sensors using a multi-sensor sparse-based clustering algorithm (Multi-SSC). A technique for the extraction of spatial features (i.e., morphological profiles (MPs) and invariant attribute profiles (IAPs)) is applied to high spatial-resolution data to derive the spatial and contextual information. This information is then fused with spectrally rich data such as multi- or hyperspectral data. In order to fuse multi-sensor data sets a hierarchical sparse subspace clustering approach is employed. More specifically, a lasso-based binary algorithm is used to fuse the spectral and spatial information prior to automatic clustering. The proposed framework ensures that the generated clustering map is smooth and preserves the spatial structures of the scene. In order to evaluate the generalization capability of the proposed approach, we investigate its performance not only on diverse scenes but also on different sensors and data types. The first two data sets are geological data sets, which consist of hyperspectral and RGB data. The third data set is the well-known benchmark Trento data set, including hyperspectral and LiDAR data. Experimental results indicate that this novel multi-sensor clustering algorithm can provide an accurate clustering map compared to the state-of-the-art sparse subspace-based clustering algorithms.

scholarly journals Structure determinations for random-tiling quasicrystals

2000 ◽  
Vol 215 (10) ◽  
Author(s):  
C.L. Henley ◽  
V. Elser ◽  
M. Mihalkovic
Keyword(s):  
Structural Information ◽  
Direct Method ◽  
Dimensional Space ◽  
Waller Factor ◽  
Data Set ◽  
3 Dimensional ◽  
Random Tiling ◽  
Debye Waller Factor ◽  
Higher Dimensional ◽  
Structure Determinations

How, in principle, could one solve the atomic structure of a quasicrystal, modeled as a random tiling decorated by atoms, and what techniques are available to do it? One path is to solve the phase problem first, obtaining the density in a higher dimensional space which yields the averaged scattering density in 3-dimensional space by the usual construction of an incommensurate cut. A novel direct method for this is summarized and applied to an i(AlPdMn) data set. This averaged density falls short of a true structure determination (which would reveal the typical unaveraged atomic patterns.) We discuss the problematic validity of inferring an ideal structure by simply factoring out a "perp-space" Debye-Waller factor, and we test this using simulations of rhombohedral tilings. A second, "unified" path is to relate the measured and modeled intensities directliy, by adjusting parameters in a simulation to optimize the fit. This approach is well suited for unifying structural information from diffraction and from minimizing total energies derived ultimately from ab-initio calculations. Finally, we discuss the special pitfalls of fitting random-tiling decagonal phases.

A Dynamic Genetic Algorithm for Clustering Problems

2013 ◽  
Vol 411-414 ◽  
pp. 1884-1893
Author(s):  
Yong Chun Cao ◽  
Ya Bin Shao ◽  
Shuang Liang Tian ◽  
Zheng Qi Cai
Keyword(s):  
Genetic Algorithm ◽  
Clustering Algorithm ◽  
Clustering Algorithms ◽  
Real Life ◽  
Search Space ◽  
Adaptive Mutation ◽  
Data Sets ◽  
Data Set ◽  
Local Optima ◽  
Clustering Problems

Due to many of the clustering algorithms based on GAs suffer from degeneracy and are easy to fall in local optima, a novel dynamic genetic algorithm for clustering problems (DGA) is proposed. The algorithm adopted the variable length coding to represent individuals and processed the parallel crossover operation in the subpopulation with individuals of the same length, which allows the DGA algorithm clustering to explore the search space more effectively and can automatically obtain the proper number of clusters and the proper partition from a given data set; the algorithm used the dynamic crossover probability and adaptive mutation probability, which prevented the dynamic clustering algorithm from getting stuck at a local optimal solution. The clustering results in the experiments on three artificial data sets and two real-life data sets show that the DGA algorithm derives better performance and higher accuracy on clustering problems.

SPSM: A NEW HYBRID DATA CLUSTERING ALGORITHM FOR NONLINEAR DATA ANALYSIS

2009 ◽  
Vol 23 (08) ◽  
pp. 1701-1737 ◽  
Author(s):  
UREERAT WATTANACHON ◽  
CHIDCHANOK LURSINSAP
Keyword(s):  
Clustering Algorithm ◽  
Color Image ◽  
Clustering Algorithms ◽  
Noisy Data ◽  
Second Phase ◽  
Data Sets ◽  
Data Set ◽  
Cluster Distance ◽  
Data Points ◽  
Hybrid Data

Existing clustering algorithms, such as single-link clustering, k-means, CURE, and CSM are designed to find clusters based on predefined parameters specified by users. These algorithms may be unsuccessful if the choice of parameters is inappropriate with respect to the data set being clustered. Most of these algorithms work very well for compact and hyper-spherical clusters. In this paper, a new hybrid clustering algorithm called Self-Partition and Self-Merging (SPSM) is proposed. The SPSM algorithm partitions the input data set into several subclusters in the first phase and, then, removes the noisy data in the second phase. In the third phase, the normal subclusters are continuously merged to form the larger clusters based on the inter-cluster distance and intra-cluster distance criteria. From the experimental results, the SPSM algorithm is very efficient to handle the noisy data set, and to cluster the data sets of arbitrary shapes of different density. Several examples for color image show the versatility of the proposed method and compare with results described in the literature for the same images. The computational complexity of the SPSM algorithm is O(N2), where N is the number of data points.

scholarly journals Adaptive Initialization Method for K-Means Algorithm

2021 ◽  
Vol 4 ◽  
Author(s):  
Jie Yang ◽  
Yu-Kai Wang ◽  
Xin Yao ◽  
Chin-Teng Lin
Keyword(s):  
Time Complexity ◽  
Clustering Algorithm ◽  
Clustering Algorithms ◽  
Real Life ◽  
Superior Performance ◽  
Local Optima ◽  
Initial Cluster ◽  
Higher Dimensional ◽  
Real World Datasets ◽  
Random Method

The K-means algorithm is a widely used clustering algorithm that offers simplicity and efficiency. However, the traditional K-means algorithm uses a random method to determine the initial cluster centers, which make clustering results prone to local optima and then result in worse clustering performance. In this research, we propose an adaptive initialization method for the K-means algorithm (AIMK) which can adapt to the various characteristics in different datasets and obtain better clustering performance with stable results. For larger or higher-dimensional datasets, we even leverage random sampling in AIMK (name as AIMK-RS) to reduce the time complexity. 22 real-world datasets were applied for performance comparisons. The experimental results show AIMK and AIMK-RS outperform the current initialization methods and several well-known clustering algorithms. Specifically, AIMK-RS can significantly reduce the time complexity to O (n). Moreover, we exploit AIMK to initialize K-medoids and spectral clustering, and better performance is also explored. The above results demonstrate superior performance and good scalability by AIMK or AIMK-RS. In the future, we would like to apply AIMK to more partition-based clustering algorithms to solve real-life practical problems.

scholarly journals GRAPH BASED CLUSTERING WITH CONSTRAINTS AND ACTIVE LEARNING

2021 ◽  
Vol 37 (1) ◽  
pp. 71-89
Author(s):  
Vu-Tuan Dang ◽  
Viet-Vu Vu ◽  
Hong-Quan Do ◽  
Thi Kieu Oanh Le
Keyword(s):  
Active Learning ◽  
Clustering Algorithm ◽  
Side Information ◽  
Clustering Algorithms ◽  
Real Data ◽  
Data Sets ◽  
Data Set ◽  
Supervised Clustering ◽  
Class Labels ◽  
Graph Based Clustering

During the past few years, semi-supervised clustering has emerged as a new interesting direction in machine learning research. In a semi-supervised clustering algorithm, the clustering results can be significantly improved by using side information, which is available or collected from users. There are two main kinds of side information that can be learned in semi-supervised clustering algorithms: the class labels - called seeds or the pairwise constraints. The first semi-supervised clustering was introduced in 2000, and since that, many algorithms have been presented in literature. However, it is not easy to use both types of side information in the same algorithm. To address the problem, this paper proposes a semi-supervised graph based clustering algorithm that tries to use seeds and constraints in the clustering process, called MCSSGC. Moreover, we introduces a simple but efficient active learning method to collect the constraints that can boost the performance of MCSSGC, named KMMFFQS. In order to verify effectiveness of the proposed algorithm, we conducted a series of experiments not only on real data sets from UCI, but also on a document data set applied in an Information Extraction of Vietnamese documents. These obtained results show that the proposed algorithm can significantly improve the clustering process compared to some recent algorithms.

scholarly journals Data Analysis Using Representation Theory and Clustering Algorithms

2021 ◽  
Vol 19 ◽  
pp. 310-320
Author(s):  
Suboh Alkhushayni ◽  
Taeyoung Choi ◽  
Du’a Alzaleq
Keyword(s):  
Data Analysis ◽  
Random Forest ◽  
Hierarchical Clustering ◽  
Clustering Algorithm ◽  
Clustering Algorithms ◽  
Optimal Number ◽  
Categorical Variables ◽  
Common Disease ◽  
Agglomerative Hierarchical Clustering ◽  
Data Set

This work aims to expand the knowledge of the area of data analysis through both persistence homology, as well as representations of directed graphs. To be specific, we looked for how we can analyze homology cluster groups using agglomerative Hierarchical Clustering algorithms and methods. Additionally, the Wine data, which is offered in R studio, was analyzed using various cluster algorithms such as Hierarchical Clustering, K-Means Clustering, and PAM Clustering. The goal of the analysis was to find out which cluster's method is proper for a given numerical data set. By testing the data, we tried to find the agglomerative hierarchical clustering method that will be the optimal clustering algorithm among these three; K-Means, PAM, and Random Forest methods. By comparing each model's accuracy value with cultivar coefficients, we came with a conclusion that K-Means methods are the most helpful when working with numerical variables. On the other hand, PAM clustering and Gower with random forest are the most beneficial approaches when working with categorical variables. All these tests can determine the optimal number of clustering groups, given the data set, and by doing the proper analysis. Using those the project, we can apply our method to several industrial areas such that clinical, business, and others. For example, people can make different groups based on each patient who has a common disease, required therapy, and other things in the clinical society. Additionally, for the business area, people can expect to get several clustered groups based on the marginal profit, marginal cost, or other economic indicators.

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