This paper presents a new clustering algorithm called Robust Fuzzy n-Means (RFNM) that can determine the optimal number of clusters in a dataset and is robust to outliers. RFNM is a modification of existing Robust Fuzzy c-Means Clustering (RFCM) and Fuzzy c-Means Clustering (FCM) algorithms. RFCM improves on FCM by making it more resistant to outliers, but requires the user to specify the number of clusters. RFNM retains RFCM's robustness to outliers and does not require the user to specify the number of clusters in advance, allowing it to determine the optimal number of clusters automatically.
A PSO-Based Subtractive Data Clustering AlgorithmIJORCS
There is a tremendous proliferation in the amount of information available on the largest shared information source, the World Wide Web. Fast and high-quality clustering algorithms play an important role in helping users to effectively navigate, summarize, and organize the information. Recent studies have shown that partitional clustering algorithms such as the k-means algorithm are the most popular algorithms for clustering large datasets. The major problem with partitional clustering algorithms is that they are sensitive to the selection of the initial partitions and are prone to premature converge to local optima. Subtractive clustering is a fast, one-pass algorithm for estimating the number of clusters and cluster centers for any given set of data. The cluster estimates can be used to initialize iterative optimization-based clustering methods and model identification methods. In this paper, we present a hybrid Particle Swarm Optimization, Subtractive + (PSO) clustering algorithm that performs fast clustering. For comparison purpose, we applied the Subtractive + (PSO) clustering algorithm, PSO, and the Subtractive clustering algorithms on three different datasets. The results illustrate that the Subtractive + (PSO) clustering algorithm can generate the most compact clustering results as compared to other algorithms.
The document proposes a novel Spatial Fuzzy C-Means (PET-SFCM) clustering algorithm to segment PET scan images of patients with neurodegenerative disorders like Alzheimer's disease. The algorithm incorporates spatial neighborhood information into the traditional Fuzzy C-Means algorithm. It was tested on real patient data sets and showed satisfactory results compared to conventional FCM and K-Means clustering algorithms. The PET-SFCM algorithm provides an effective way to segment PET images and analyze brain changes related to neurological conditions.
CLIQUE is an algorithm for subspace clustering of high-dimensional data. It works in two steps: (1) It partitions each dimension of the data space into intervals of equal length to form a grid, (2) It identifies dense units within this grid and finds clusters as maximal sets of connected dense units. CLIQUE efficiently discovers clusters by identifying dense units in subspaces and intersecting them to obtain candidate dense units in higher dimensions. It automatically determines relevant subspaces for clustering and scales well with large, high-dimensional datasets.
This document summarizes recent convergence results for the fuzzy c-means clustering algorithm (FCM). It discusses both numerical convergence, referring to how well the algorithm attains the minima of an objective function, and stochastic convergence, referring to how accurately the minima represent the actual cluster structure in data. For numerical convergence, the document outlines global and local convergence theorems, showing FCM converges to minima or saddle points globally and linearly to local minima. For stochastic convergence, it discusses a consistency result showing the minima accurately represent cluster structure under certain statistical assumptions.
This document summarizes a research paper that proposes a new density-based clustering technique called Triangle-Density Based Clustering Technique (TDCT) to efficiently cluster large spatial datasets. TDCT uses a polygon approach where the number of data points inside each triangle of a polygon is calculated to determine triangle densities. Triangle densities are used to identify clusters based on a density confidence threshold. The technique aims to identify clusters of arbitrary shapes and densities while minimizing computational costs. Experimental results demonstrate the technique's superiority in terms of cluster quality and complexity compared to other density-based clustering algorithms.
CLIQUE is a grid-based clustering algorithm that identifies dense units in subspaces of high-dimensional data to provide efficient clustering. It works by first partitioning each attribute dimension into equal intervals and then the data space into rectangular grid cells. It finds dense units in subspaces like planes and intersections them to identify dense units in higher dimensions. These dense units are grouped into clusters. CLIQUE scales linearly with size of data and number of dimensions and automatically identifies relevant subspaces for clustering. However, the clustering accuracy may be reduced for simplicity.
Bat-Cluster: A Bat Algorithm-based Automated Graph Clustering Approach IJECEIAES
The document presents a new approach called Bat-Cluster (BC) for automated graph clustering. BC combines the Fast Fourier Domain Positioning (FFDP) algorithm and the Bat Algorithm. FFDP positions graph nodes, then Bat Algorithm optimizes clustering by finding configurations that minimize the Davies-Bouldin Index. BC is tested on four benchmark graphs and outperforms Particle Swarm Optimization, Ant Colony Optimization, and Differential Evolution in providing higher clustering precision.
This document compares the k-means and grid density clustering algorithms. K-means partitions data into k clusters based on minimizing distances between points and cluster centroids. It works well with numerical data but can be affected by outliers. Grid density determines dense grids based on neighbor densities and can handle different shaped and multi-density clusters without knowing the number of clusters beforehand. It has advantages over k-means in that it can handle categorical data, noise and arbitrary shaped clusters.
A PSO-Based Subtractive Data Clustering AlgorithmIJORCS
There is a tremendous proliferation in the amount of information available on the largest shared information source, the World Wide Web. Fast and high-quality clustering algorithms play an important role in helping users to effectively navigate, summarize, and organize the information. Recent studies have shown that partitional clustering algorithms such as the k-means algorithm are the most popular algorithms for clustering large datasets. The major problem with partitional clustering algorithms is that they are sensitive to the selection of the initial partitions and are prone to premature converge to local optima. Subtractive clustering is a fast, one-pass algorithm for estimating the number of clusters and cluster centers for any given set of data. The cluster estimates can be used to initialize iterative optimization-based clustering methods and model identification methods. In this paper, we present a hybrid Particle Swarm Optimization, Subtractive + (PSO) clustering algorithm that performs fast clustering. For comparison purpose, we applied the Subtractive + (PSO) clustering algorithm, PSO, and the Subtractive clustering algorithms on three different datasets. The results illustrate that the Subtractive + (PSO) clustering algorithm can generate the most compact clustering results as compared to other algorithms.
The document proposes a novel Spatial Fuzzy C-Means (PET-SFCM) clustering algorithm to segment PET scan images of patients with neurodegenerative disorders like Alzheimer's disease. The algorithm incorporates spatial neighborhood information into the traditional Fuzzy C-Means algorithm. It was tested on real patient data sets and showed satisfactory results compared to conventional FCM and K-Means clustering algorithms. The PET-SFCM algorithm provides an effective way to segment PET images and analyze brain changes related to neurological conditions.
CLIQUE is an algorithm for subspace clustering of high-dimensional data. It works in two steps: (1) It partitions each dimension of the data space into intervals of equal length to form a grid, (2) It identifies dense units within this grid and finds clusters as maximal sets of connected dense units. CLIQUE efficiently discovers clusters by identifying dense units in subspaces and intersecting them to obtain candidate dense units in higher dimensions. It automatically determines relevant subspaces for clustering and scales well with large, high-dimensional datasets.
This document summarizes recent convergence results for the fuzzy c-means clustering algorithm (FCM). It discusses both numerical convergence, referring to how well the algorithm attains the minima of an objective function, and stochastic convergence, referring to how accurately the minima represent the actual cluster structure in data. For numerical convergence, the document outlines global and local convergence theorems, showing FCM converges to minima or saddle points globally and linearly to local minima. For stochastic convergence, it discusses a consistency result showing the minima accurately represent cluster structure under certain statistical assumptions.
This document summarizes a research paper that proposes a new density-based clustering technique called Triangle-Density Based Clustering Technique (TDCT) to efficiently cluster large spatial datasets. TDCT uses a polygon approach where the number of data points inside each triangle of a polygon is calculated to determine triangle densities. Triangle densities are used to identify clusters based on a density confidence threshold. The technique aims to identify clusters of arbitrary shapes and densities while minimizing computational costs. Experimental results demonstrate the technique's superiority in terms of cluster quality and complexity compared to other density-based clustering algorithms.
CLIQUE is a grid-based clustering algorithm that identifies dense units in subspaces of high-dimensional data to provide efficient clustering. It works by first partitioning each attribute dimension into equal intervals and then the data space into rectangular grid cells. It finds dense units in subspaces like planes and intersections them to identify dense units in higher dimensions. These dense units are grouped into clusters. CLIQUE scales linearly with size of data and number of dimensions and automatically identifies relevant subspaces for clustering. However, the clustering accuracy may be reduced for simplicity.
Bat-Cluster: A Bat Algorithm-based Automated Graph Clustering Approach IJECEIAES
The document presents a new approach called Bat-Cluster (BC) for automated graph clustering. BC combines the Fast Fourier Domain Positioning (FFDP) algorithm and the Bat Algorithm. FFDP positions graph nodes, then Bat Algorithm optimizes clustering by finding configurations that minimize the Davies-Bouldin Index. BC is tested on four benchmark graphs and outperforms Particle Swarm Optimization, Ant Colony Optimization, and Differential Evolution in providing higher clustering precision.
This document compares the k-means and grid density clustering algorithms. K-means partitions data into k clusters based on minimizing distances between points and cluster centroids. It works well with numerical data but can be affected by outliers. Grid density determines dense grids based on neighbor densities and can handle different shaped and multi-density clusters without knowing the number of clusters beforehand. It has advantages over k-means in that it can handle categorical data, noise and arbitrary shaped clusters.
Chapter 11 cluster advanced : web and text miningHouw Liong The
This document provides an overview of advanced clustering analysis techniques discussed in Chapter 11 of the textbook "Data Mining: Concepts and Techniques". It begins with an introduction to probability model-based clustering and fuzzy clustering. It then discusses using the EM algorithm for fuzzy clustering and fitting univariate Gaussian mixture models. Next, it covers challenges with clustering high-dimensional data and methods for subspace clustering. It also briefly introduces clustering graphs and network data as well as clustering with constraints. The document concludes with an outline of the chapter.
The document summarizes the CURE clustering algorithm, which uses a hierarchical approach that selects a constant number of representative points from each cluster to address limitations of centroid-based and all-points clustering methods. It employs random sampling and partitioning to speed up processing of large datasets. Experimental results show CURE detects non-spherical and variably-sized clusters better than compared methods, and it has faster execution times on large databases due to its sampling approach.
Dear Students
Ingenious techno Solution offers an expertise guidance on you Final Year IEEE & Non- IEEE Projects on the following domain
JAVA
.NET
EMBEDDED SYSTEMS
ROBOTICS
MECHANICAL
MATLAB etc
For further details contact us:
enquiry@ingenioustech.in
044-42046028 or 8428302179.
Ingenious Techno Solution
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This document summarizes Chapter 10 of the book "Data Mining: Concepts and Techniques (3rd ed.)" which covers cluster analysis. The chapter introduces different types of clustering methods including partitioning methods like k-means and k-medoids, hierarchical methods, density-based methods, and grid-based methods. It discusses how to evaluate the quality of clustering results and highlights considerations for cluster analysis such as similarity measures, clustering space, and challenges like scalability and high dimensionality.
Machine Learning Algorithms for Image Classification of Hand Digits and Face ...IRJET Journal
This document discusses machine learning algorithms for image classification using five different classification schemes. It summarizes the mathematical models behind each classification algorithm, including Nearest Class Centroid classifier, Nearest Sub-Class Centroid classifier, k-Nearest Neighbor classifier, Perceptron trained using Backpropagation, and Perceptron trained using Mean Squared Error. It also describes two datasets used in the experiments - the MNIST dataset of handwritten digits and the ORL face recognition dataset. The performance of the five classification schemes are compared on these datasets.
A Novel Dencos Model For High Dimensional Data Using Genetic Algorithms ijcseit
Subspace clustering is an emerging task that aims at detecting clusters in entrenched in
subspaces. Recent approaches fail to reduce results to relevant subspace clusters. Their results are
typically highly redundant and lack the fact of considering the critical problem, “the density divergence
problem,” in discovering the clusters, where they utilize an absolute density value as the density threshold
to identify the dense regions in all subspaces. Considering the varying region densities in different
subspace cardinalities, we note that a more appropriate way to determine whether a region in a subspace
should be identified as dense is by comparing its density with the region densities in that subspace. Based
on this idea and due to the infeasibility of applying previous techniques in this novel clustering model, we
devise an innovative algorithm, referred to as DENCOS(DENsity Conscious Subspace clustering), to adopt
a divide-and-conquer scheme to efficiently discover clusters satisfying different density thresholds in
different subspace cardinalities. DENCOS can discover the clusters in all subspaces with high quality, and
the efficiency significantly outperforms previous works, thus demonstrating its practicability for subspace
clustering. As validated by our extensive experiments on retail dataset, it outperforms previous works. We
extend our work with a clustering technique based on genetic algorithms which is capable of optimizing the
number of clusters for tasks with well formed and separated clusters.
This document describes a new distance-based clustering algorithm (DBCA) that aims to improve upon K-means clustering. DBCA selects initial cluster centroids based on the total distance of each data point to all other points, rather than random selection. It calculates distances between all points, identifies points with maximum total distances, and sets initial centroids as the averages of groups of these maximally distant points. The algorithm is compared to K-means, hierarchical clustering, and hierarchical partitioning clustering on synthetic and real data. Experimental results show DBCA produces better quality clusters than these other algorithms.
New Approach for K-mean and K-medoids AlgorithmEditor IJCATR
K-means and K-medoids clustering algorithms are widely used for many practical applications. Original k
medoids algorithms select initial centroids and medoids randomly that affect the quality of the resulting clusters and sometimes it
generates unstable and empty clusters which are meaningless.
expensive and requires time proportional to the product of the number of data items, number of clusters and the number of iterations.
The new approach for the k mean algorithm eliminates the deficiency of exiting k mean. It first calculates the initial centro
requirements of users and then gives better, effective and stable cluster. It also takes less execution time because it eliminates
unnecessary distance computation by using previous iteration. The new approach for k
systematically based on initial centroids. It generates stable clusters to improve accuracy.
Comparison Between Clustering Algorithms for Microarray Data AnalysisIOSR Journals
Currently, there are two techniques used for large-scale gene-expression profiling; microarray and
RNA-Sequence (RNA-Seq).This paper is intended to study and compare different clustering algorithms that used
in microarray data analysis. Microarray is a DNA molecules array which allows multiple hybridization
experiments to be carried out simultaneously and trace expression levels of thousands of genes. It is a highthroughput
technology for gene expression analysis and becomes an effective tool for biomedical research.
Microarray analysis aims to interpret the data produced from experiments on DNA, RNA, and protein
microarrays, which enable researchers to investigate the expression state of a large number of genes. Data
clustering represents the first and main process in microarray data analysis. The k-means, fuzzy c-mean, selforganizing
map, and hierarchical clustering algorithms are under investigation in this paper. These algorithms
are compared based on their clustering model.
NBDT : Neural-backed Decision Tree 2021 ICLRtaeseon ryu
안녕하세요 딥러닝 논문읽기 모임 입니다.
오늘 소개 드릴 논문은 2021년 ICLR 에 억셉된 NBDT : Neural-backed Decision Tree 라는 논문 입니다
초록 :
Machine learning applications such as finance and medicine demand accurate and justifiable predictions, barring most deep learning methods from use. In response, previous work combines decision trees with deep learning, yielding models that (1) sacrifice interpretability for accuracy or (2) sacrifice accuracy for interpretability. We forgo this dilemma by jointly improving accuracy and interpretability using Neural-Backed Decision Trees (NBDTs). NBDTs replace a neural network's final linear layer with a differentiable sequence of decisions and a surrogate loss. This forces the model to learn high-level concepts and lessens reliance on highly-uncertain decisions, yielding (1) accuracy: NBDTs match or outperform modern neural networks on CIFAR, ImageNet and better generalize to unseen classes by up to 16%. Furthermore, our surrogate loss improves the original model's accuracy by up to 2%. NBDTs also afford (2) interpretability: improving human trustby clearly identifying model mistakes and assisting in dataset debugging. Code and pretrained NBDTs are at this https URL.
오늘 논문 리뷰를 이미지 처리팀 안종식님이 자세하고 친절한 리뷰 도와주셨습니다.
감사합니다
문의 : tfkeras@kakao.com
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Puducherry.
Email Id: praveen@nexgenproject.com
Mobile: 9791938249
Telephone: 0413-2211159
www.nexgenproject.com
CLIQUE Automatic subspace clustering of high dimensional data for data mining...Raed Aldahdooh
The document describes the CLIQUE subspace clustering algorithm. CLIQUE identifies subspaces that contain dense clusters in high-dimensional data in a bottom-up approach. It then identifies clusters within these subspaces and generates a minimal description of each cluster in disjunctive normal form. Empirical tests showed CLIQUE can accurately recover known clusters, scales linearly with data size and dimensionality, and is insensitive to input order.
K-MEDOIDS CLUSTERING USING PARTITIONING AROUND MEDOIDS FOR PERFORMING FACE R...ijscmc
Face recognition is one of the most unobtrusive biometric techniques that can be used for access control as well as surveillance purposes. Various methods for implementing face recognition have been proposed with varying degrees of performance in different scenarios. The most common issue with effective facial biometric systems is high susceptibility of variations in the face owing to different factors like changes in pose, varying illumination, different expression, presence of outliers, noise etc. This paper explores a novel technique for face recognition by performing classification of the face images using unsupervised learning approach through K-Medoids clustering. Partitioning Around Medoids algorithm (PAM) has been used for performing K-Medoids clustering of the data. The results are suggestive of increased robustness to noise and outliers in comparison to other clustering methods. Therefore the technique can also be used to increase the overall robustness of a face recognition system and thereby increase its invariance and make it a reliably usable biometric modality
This is very simple introduction to Clustering with some real world example. At the end of lecture I use stackOverflow API to test some clustering. I also wants to try facebook but it has some problem with it's API
A COMPREHENSIVE ANALYSIS OF QUANTUM CLUSTERING : FINDING ALL THE POTENTIAL MI...IJDKP
Quantum clustering (QC), is a data clustering algorithm based on quantum mechanics which is
accomplished by substituting each point in a given dataset with a Gaussian. The width of the Gaussian is a
σ value, a hyper-parameter which can be manually defined and manipulated to suit the application.
Numerical methods are used to find all the minima of the quantum potential as they correspond to cluster
centers. Herein, we investigate the mathematical task of expressing and finding all the roots of the
exponential polynomial corresponding to the minima of a two-dimensional quantum potential. This is an
outstanding task because normally such expressions are impossible to solve analytically. However, we
prove that if the points are all included in a square region of size σ, there is only one minimum. This bound
is not only useful in the number of solutions to look for, by numerical means, it allows to to propose a new
numerical approach “per block”. This technique decreases the number of particles by approximating some
groups of particles to weighted particles. These findings are not only useful to the quantum clustering
problem but also for the exponential polynomials encountered in quantum chemistry, Solid-state Physics
and other applications.
A Novel Approach to Mathematical Concepts in Data Miningijdmtaiir
-This paper describes three different fundamental
mathematical programming approaches that are relevant to
data mining. They are: Feature Selection, Clustering and
Robust Representation. This paper comprises of two clustering
algorithms such as K-mean algorithm and K-median
algorithms. Clustering is illustrated by the unsupervised
learning of patterns and clusters that may exist in a given
databases and useful tool for Knowledge Discovery in
Database (KDD). The results of k-median algorithm are used
to collecting the blood cancer patient from a medical database.
K-mean clustering is a data mining/machine learning algorithm
used to cluster observations into groups of related observations
without any prior knowledge of those relationships. The kmean algorithm is one of the simplest clustering techniques
and it is commonly used in medical imaging, biometrics and
related fields.
This document discusses various clustering analysis methods including k-means, k-medoids (PAM), and CLARA. It explains that clustering involves grouping similar objects together without predefined classes. Partitioning methods like k-means and k-medoids (PAM) assign objects to clusters to optimize a criterion function. K-means uses cluster centroids while k-medoids uses actual data points as cluster representatives. PAM is more robust to outliers than k-means but does not scale well to large datasets, so CLARA applies PAM to samples of the data. Examples of clustering applications include market segmentation, land use analysis, and earthquake studies.
1. Clustering high-dimensional data presents unique challenges as traditional distance measures become less meaningful and clusters may only exist in subspaces of the data. 2. Subspace clustering methods aim to find clusters that exist in subspaces of the feature space rather than the entire space. 3. Popular subspace clustering methods include subspace search approaches that examine various subspaces, bi-clustering methods, and dimensionality reduction techniques.
This document provides guidance on creating an enterprise mobile strategy. It discusses:
1) Setting objectives like brand extension, customer service, revenue generation, and process optimization.
2) Understanding the target audience by identifying users and segmenting based on needs.
3) Defining and prioritizing mobile offerings by identifying unique capabilities, comparing to competitors, and developing a roadmap.
4) Coordinating delivery by engaging stakeholders like IT, marketing, and ensuring governance, training, and support.
The document recommends assessing the current environment, defining a strategy, creating recommendations, reviewing with sponsors, and socializing with stakeholders.
This document discusses different categories of artifacts and their meanings. It begins by asking how information is embedded in material objects and how this may differ from fine art objects. It defines artifacts as objects that can reveal cultural information about their creators. Vernacular objects are everyday objects with wide popularity but obscure origins. Designed objects are those carefully crafted with both function and aesthetics in mind, differing from mere styling changes. The document explores examples from art, furniture, packaging and vehicle design to illustrate these concepts. It emphasizes that studying humble everyday objects can provide historical insights, and good design can improve people's lives by creating user-friendly products and environments.
- In the 1990s and 2000s, iconic fashion designers and brands experienced major successes and changes in leadership. Gianni Versace launched new lines and gained celebrity clients for his bold designs until his tragic death in 1997, after which his sister Donatella took over as head designer. Jean Paul Gaultier designed Madonna's iconic cone bra outfit in 1990. Vera Wang established her bridal business in 1990 known for sleek modern gowns. Anna Sui debuted her first collection in 1991. Daymond John launched the urban streetwear brand FUBU in 1992. Dolce & Gabbana also designed costumes for Madonna in 1993. Gucci hired Tom Ford in 1994 to revamp the brand's image. John
1) Several notable historical events have occurred on or around Christmas, including the first test run of the internet in 1990, George Washington's victory over Hessian mercenaries in 1776, and temporary ceasefires and soccer games between British and German soldiers during WWI.
2) Other events include the Soviet invasion of Afghanistan in 1979, Sir Isaac Newton's birth in 1642, the Apollo 8 mission reaching the moon's orbit in 1968, and Mikhail Gorbachev's resignation in 1991 ending the Soviet Union.
3) On Christmas Day 1868, U.S. President Andrew Johnson pardoned all Confederate soldiers from the Civil War.
Chapter 11 cluster advanced : web and text miningHouw Liong The
This document provides an overview of advanced clustering analysis techniques discussed in Chapter 11 of the textbook "Data Mining: Concepts and Techniques". It begins with an introduction to probability model-based clustering and fuzzy clustering. It then discusses using the EM algorithm for fuzzy clustering and fitting univariate Gaussian mixture models. Next, it covers challenges with clustering high-dimensional data and methods for subspace clustering. It also briefly introduces clustering graphs and network data as well as clustering with constraints. The document concludes with an outline of the chapter.
The document summarizes the CURE clustering algorithm, which uses a hierarchical approach that selects a constant number of representative points from each cluster to address limitations of centroid-based and all-points clustering methods. It employs random sampling and partitioning to speed up processing of large datasets. Experimental results show CURE detects non-spherical and variably-sized clusters better than compared methods, and it has faster execution times on large databases due to its sampling approach.
Dear Students
Ingenious techno Solution offers an expertise guidance on you Final Year IEEE & Non- IEEE Projects on the following domain
JAVA
.NET
EMBEDDED SYSTEMS
ROBOTICS
MECHANICAL
MATLAB etc
For further details contact us:
enquiry@ingenioustech.in
044-42046028 or 8428302179.
Ingenious Techno Solution
#241/85, 4th floor
Rangarajapuram main road,
Kodambakkam (Power House)
http://www.ingenioustech.in/
This document summarizes Chapter 10 of the book "Data Mining: Concepts and Techniques (3rd ed.)" which covers cluster analysis. The chapter introduces different types of clustering methods including partitioning methods like k-means and k-medoids, hierarchical methods, density-based methods, and grid-based methods. It discusses how to evaluate the quality of clustering results and highlights considerations for cluster analysis such as similarity measures, clustering space, and challenges like scalability and high dimensionality.
Machine Learning Algorithms for Image Classification of Hand Digits and Face ...IRJET Journal
This document discusses machine learning algorithms for image classification using five different classification schemes. It summarizes the mathematical models behind each classification algorithm, including Nearest Class Centroid classifier, Nearest Sub-Class Centroid classifier, k-Nearest Neighbor classifier, Perceptron trained using Backpropagation, and Perceptron trained using Mean Squared Error. It also describes two datasets used in the experiments - the MNIST dataset of handwritten digits and the ORL face recognition dataset. The performance of the five classification schemes are compared on these datasets.
A Novel Dencos Model For High Dimensional Data Using Genetic Algorithms ijcseit
Subspace clustering is an emerging task that aims at detecting clusters in entrenched in
subspaces. Recent approaches fail to reduce results to relevant subspace clusters. Their results are
typically highly redundant and lack the fact of considering the critical problem, “the density divergence
problem,” in discovering the clusters, where they utilize an absolute density value as the density threshold
to identify the dense regions in all subspaces. Considering the varying region densities in different
subspace cardinalities, we note that a more appropriate way to determine whether a region in a subspace
should be identified as dense is by comparing its density with the region densities in that subspace. Based
on this idea and due to the infeasibility of applying previous techniques in this novel clustering model, we
devise an innovative algorithm, referred to as DENCOS(DENsity Conscious Subspace clustering), to adopt
a divide-and-conquer scheme to efficiently discover clusters satisfying different density thresholds in
different subspace cardinalities. DENCOS can discover the clusters in all subspaces with high quality, and
the efficiency significantly outperforms previous works, thus demonstrating its practicability for subspace
clustering. As validated by our extensive experiments on retail dataset, it outperforms previous works. We
extend our work with a clustering technique based on genetic algorithms which is capable of optimizing the
number of clusters for tasks with well formed and separated clusters.
This document describes a new distance-based clustering algorithm (DBCA) that aims to improve upon K-means clustering. DBCA selects initial cluster centroids based on the total distance of each data point to all other points, rather than random selection. It calculates distances between all points, identifies points with maximum total distances, and sets initial centroids as the averages of groups of these maximally distant points. The algorithm is compared to K-means, hierarchical clustering, and hierarchical partitioning clustering on synthetic and real data. Experimental results show DBCA produces better quality clusters than these other algorithms.
New Approach for K-mean and K-medoids AlgorithmEditor IJCATR
K-means and K-medoids clustering algorithms are widely used for many practical applications. Original k
medoids algorithms select initial centroids and medoids randomly that affect the quality of the resulting clusters and sometimes it
generates unstable and empty clusters which are meaningless.
expensive and requires time proportional to the product of the number of data items, number of clusters and the number of iterations.
The new approach for the k mean algorithm eliminates the deficiency of exiting k mean. It first calculates the initial centro
requirements of users and then gives better, effective and stable cluster. It also takes less execution time because it eliminates
unnecessary distance computation by using previous iteration. The new approach for k
systematically based on initial centroids. It generates stable clusters to improve accuracy.
Comparison Between Clustering Algorithms for Microarray Data AnalysisIOSR Journals
Currently, there are two techniques used for large-scale gene-expression profiling; microarray and
RNA-Sequence (RNA-Seq).This paper is intended to study and compare different clustering algorithms that used
in microarray data analysis. Microarray is a DNA molecules array which allows multiple hybridization
experiments to be carried out simultaneously and trace expression levels of thousands of genes. It is a highthroughput
technology for gene expression analysis and becomes an effective tool for biomedical research.
Microarray analysis aims to interpret the data produced from experiments on DNA, RNA, and protein
microarrays, which enable researchers to investigate the expression state of a large number of genes. Data
clustering represents the first and main process in microarray data analysis. The k-means, fuzzy c-mean, selforganizing
map, and hierarchical clustering algorithms are under investigation in this paper. These algorithms
are compared based on their clustering model.
NBDT : Neural-backed Decision Tree 2021 ICLRtaeseon ryu
안녕하세요 딥러닝 논문읽기 모임 입니다.
오늘 소개 드릴 논문은 2021년 ICLR 에 억셉된 NBDT : Neural-backed Decision Tree 라는 논문 입니다
초록 :
Machine learning applications such as finance and medicine demand accurate and justifiable predictions, barring most deep learning methods from use. In response, previous work combines decision trees with deep learning, yielding models that (1) sacrifice interpretability for accuracy or (2) sacrifice accuracy for interpretability. We forgo this dilemma by jointly improving accuracy and interpretability using Neural-Backed Decision Trees (NBDTs). NBDTs replace a neural network's final linear layer with a differentiable sequence of decisions and a surrogate loss. This forces the model to learn high-level concepts and lessens reliance on highly-uncertain decisions, yielding (1) accuracy: NBDTs match or outperform modern neural networks on CIFAR, ImageNet and better generalize to unseen classes by up to 16%. Furthermore, our surrogate loss improves the original model's accuracy by up to 2%. NBDTs also afford (2) interpretability: improving human trustby clearly identifying model mistakes and assisting in dataset debugging. Code and pretrained NBDTs are at this https URL.
오늘 논문 리뷰를 이미지 처리팀 안종식님이 자세하고 친절한 리뷰 도와주셨습니다.
감사합니다
문의 : tfkeras@kakao.com
GET IEEE BIG DATA,JAVA ,DOTNET,ANDROID ,NS2,MATLAB,EMBEDED AT LOW COST WITH BEST QUALITY PLEASE CONTACT BELOW NUMBER
FOR MORE INFORMATION PLEASE FIND THE BELOW DETAILS:
Nexgen Technology
No :66,4th cross,Venkata nagar,
Near SBI ATM,
Puducherry.
Email Id: praveen@nexgenproject.com
Mobile: 9791938249
Telephone: 0413-2211159
www.nexgenproject.com
CLIQUE Automatic subspace clustering of high dimensional data for data mining...Raed Aldahdooh
The document describes the CLIQUE subspace clustering algorithm. CLIQUE identifies subspaces that contain dense clusters in high-dimensional data in a bottom-up approach. It then identifies clusters within these subspaces and generates a minimal description of each cluster in disjunctive normal form. Empirical tests showed CLIQUE can accurately recover known clusters, scales linearly with data size and dimensionality, and is insensitive to input order.
K-MEDOIDS CLUSTERING USING PARTITIONING AROUND MEDOIDS FOR PERFORMING FACE R...ijscmc
Face recognition is one of the most unobtrusive biometric techniques that can be used for access control as well as surveillance purposes. Various methods for implementing face recognition have been proposed with varying degrees of performance in different scenarios. The most common issue with effective facial biometric systems is high susceptibility of variations in the face owing to different factors like changes in pose, varying illumination, different expression, presence of outliers, noise etc. This paper explores a novel technique for face recognition by performing classification of the face images using unsupervised learning approach through K-Medoids clustering. Partitioning Around Medoids algorithm (PAM) has been used for performing K-Medoids clustering of the data. The results are suggestive of increased robustness to noise and outliers in comparison to other clustering methods. Therefore the technique can also be used to increase the overall robustness of a face recognition system and thereby increase its invariance and make it a reliably usable biometric modality
This is very simple introduction to Clustering with some real world example. At the end of lecture I use stackOverflow API to test some clustering. I also wants to try facebook but it has some problem with it's API
A COMPREHENSIVE ANALYSIS OF QUANTUM CLUSTERING : FINDING ALL THE POTENTIAL MI...IJDKP
Quantum clustering (QC), is a data clustering algorithm based on quantum mechanics which is
accomplished by substituting each point in a given dataset with a Gaussian. The width of the Gaussian is a
σ value, a hyper-parameter which can be manually defined and manipulated to suit the application.
Numerical methods are used to find all the minima of the quantum potential as they correspond to cluster
centers. Herein, we investigate the mathematical task of expressing and finding all the roots of the
exponential polynomial corresponding to the minima of a two-dimensional quantum potential. This is an
outstanding task because normally such expressions are impossible to solve analytically. However, we
prove that if the points are all included in a square region of size σ, there is only one minimum. This bound
is not only useful in the number of solutions to look for, by numerical means, it allows to to propose a new
numerical approach “per block”. This technique decreases the number of particles by approximating some
groups of particles to weighted particles. These findings are not only useful to the quantum clustering
problem but also for the exponential polynomials encountered in quantum chemistry, Solid-state Physics
and other applications.
A Novel Approach to Mathematical Concepts in Data Miningijdmtaiir
-This paper describes three different fundamental
mathematical programming approaches that are relevant to
data mining. They are: Feature Selection, Clustering and
Robust Representation. This paper comprises of two clustering
algorithms such as K-mean algorithm and K-median
algorithms. Clustering is illustrated by the unsupervised
learning of patterns and clusters that may exist in a given
databases and useful tool for Knowledge Discovery in
Database (KDD). The results of k-median algorithm are used
to collecting the blood cancer patient from a medical database.
K-mean clustering is a data mining/machine learning algorithm
used to cluster observations into groups of related observations
without any prior knowledge of those relationships. The kmean algorithm is one of the simplest clustering techniques
and it is commonly used in medical imaging, biometrics and
related fields.
This document discusses various clustering analysis methods including k-means, k-medoids (PAM), and CLARA. It explains that clustering involves grouping similar objects together without predefined classes. Partitioning methods like k-means and k-medoids (PAM) assign objects to clusters to optimize a criterion function. K-means uses cluster centroids while k-medoids uses actual data points as cluster representatives. PAM is more robust to outliers than k-means but does not scale well to large datasets, so CLARA applies PAM to samples of the data. Examples of clustering applications include market segmentation, land use analysis, and earthquake studies.
1. Clustering high-dimensional data presents unique challenges as traditional distance measures become less meaningful and clusters may only exist in subspaces of the data. 2. Subspace clustering methods aim to find clusters that exist in subspaces of the feature space rather than the entire space. 3. Popular subspace clustering methods include subspace search approaches that examine various subspaces, bi-clustering methods, and dimensionality reduction techniques.
This document provides guidance on creating an enterprise mobile strategy. It discusses:
1) Setting objectives like brand extension, customer service, revenue generation, and process optimization.
2) Understanding the target audience by identifying users and segmenting based on needs.
3) Defining and prioritizing mobile offerings by identifying unique capabilities, comparing to competitors, and developing a roadmap.
4) Coordinating delivery by engaging stakeholders like IT, marketing, and ensuring governance, training, and support.
The document recommends assessing the current environment, defining a strategy, creating recommendations, reviewing with sponsors, and socializing with stakeholders.
This document discusses different categories of artifacts and their meanings. It begins by asking how information is embedded in material objects and how this may differ from fine art objects. It defines artifacts as objects that can reveal cultural information about their creators. Vernacular objects are everyday objects with wide popularity but obscure origins. Designed objects are those carefully crafted with both function and aesthetics in mind, differing from mere styling changes. The document explores examples from art, furniture, packaging and vehicle design to illustrate these concepts. It emphasizes that studying humble everyday objects can provide historical insights, and good design can improve people's lives by creating user-friendly products and environments.
- In the 1990s and 2000s, iconic fashion designers and brands experienced major successes and changes in leadership. Gianni Versace launched new lines and gained celebrity clients for his bold designs until his tragic death in 1997, after which his sister Donatella took over as head designer. Jean Paul Gaultier designed Madonna's iconic cone bra outfit in 1990. Vera Wang established her bridal business in 1990 known for sleek modern gowns. Anna Sui debuted her first collection in 1991. Daymond John launched the urban streetwear brand FUBU in 1992. Dolce & Gabbana also designed costumes for Madonna in 1993. Gucci hired Tom Ford in 1994 to revamp the brand's image. John
1) Several notable historical events have occurred on or around Christmas, including the first test run of the internet in 1990, George Washington's victory over Hessian mercenaries in 1776, and temporary ceasefires and soccer games between British and German soldiers during WWI.
2) Other events include the Soviet invasion of Afghanistan in 1979, Sir Isaac Newton's birth in 1642, the Apollo 8 mission reaching the moon's orbit in 1968, and Mikhail Gorbachev's resignation in 1991 ending the Soviet Union.
3) On Christmas Day 1868, U.S. President Andrew Johnson pardoned all Confederate soldiers from the Civil War.
The document provides an overview of major events, people, trends, and developments across various categories from the 1990s decade. Some of the key topics covered include the Gulf War, OJ Simpson trial, Clinton presidency, rise of the internet, boy bands and grunge fashion, blockbuster films like Titanic, and dominance of Chicago Bulls and Michael Jordan in basketball. Major scientific advances included the Hubble telescope and cloning of Dolly the sheep. Popular culture was influenced by toys like Beanie Babies and music from Nirvana, Backstreet Boys, and Britney Spears.
This document summarizes entertainment and popular culture during the 1980s. It provides lists of the top 10 movies, music videos, video games, historical events, and more from the decade. It also profiles some of the most popular movie genres, directors, actors, soundtracks, and themes from 1980s films. Two individuals, Regina and Erin, each share their own personal top 10 favorite movies from the 1980s as well.
Wallpaper is a common interior design product used for decorating. There are many types of wallpaper available with different patterns, colors, materials, and designs. Wallpaper patterns include paintings, drawings, dimensional figures, and other unique designs. In addition to wallpaper, other common interior design products include furniture, decals, paintings, artwork, vases, sculptures, clocks, and more. These items add beauty and style to home interiors. Online interior design firms offer a wide selection of affordable wallpaper and other products for customers' home decorating needs.
Session 1 - Introduction to iOS 7 and SDKVu Tran Lam
This document provides an overview and introduction to iOS application development. It discusses the iOS 7 SDK, Xcode developer tools, Objective-C programming language, and building a simple "Hello World" iOS app. Key topics covered include the iOS architecture and frameworks, a roadmap for becoming an iOS developer, and documentation resources for developing iOS 7 applications.
Interior designers work indoors decorating and designing spaces to meet clients' needs and tastes. They must be creative and knowledgeable about colors, textures, trends and different climates. Interior designers typically need a bachelor's degree with courses in art and design and can expect to earn an average salary of $43,000 annually or $20 per hour, with potential to earn up to $82,800. The field is expected to grow 21% between 2010-2020.
The document provides an overview of life in the 1980s decade. It describes trends of the time such as movies, music, fashion, and popular culture icons. Lifestyles and trends from the 80s still influence culture today. While fashions from the era like big hair are now seen as outlandish, the 80s helped shape modern American life. Music and stars from that decade became even more popular and helped sculpt the identity of the country.
The 1970s in Britain saw economic decline, labor unrest, and sectarian violence. The UK faced stagflation as well as power cuts and three-day work weeks due to strikes by unions. The British auto and steel industries declined due to outdated practices and foreign competition. Meanwhile, the IRA carried out bombing campaigns and riots broke out in response to police harassment of black communities. Punk music emerged as a form of social protest against this backdrop of economic malaise and social tensions.
The document provides brief descriptions of people, events, inventions and cultural aspects from the 1980s. Some of the topics covered include the first music video channel MTV, the first female black model Vanessa Williams, the blockbuster movie Thriller and its 14 minute music video, the deaths of John Lennon and Len Bias, the introduction of compact discs, the election of the first female Supreme Court Justice Sandra Day O'Connor, the eruption of Mount St. Helens, the emergence of the AIDS epidemic, Olympic hockey victory over the Soviet Union, the rise of video game company Nintendo, supermodel Gia Carangi and her death from drug addiction, and the 1983 bombing of the US embassy in Beirut
Fashion designing is the art of applying design and aesthetics to clothing and accessories. It is influenced by various cultural and social factors and has varied over time and place. Some of the top fashion designers in the world include Valentino Garavani, Tom Ford, Betsey Johnson, Domenico Dolce and Stefano Gabbana, and Stella McCartney. Fashion also has a large industry in India, with top Indian designers being Ritu Beri, Rohit Bal, Rina Dhaka, Manish Malhotra, and Abu Jani and Sandeep Khosla. Fashion is showcased through important fashion shows, represented by famous brands, and brought to the public through magazines
Geometric Correction for Braille Document Images csandit
Image processing is an important research area in computer vision. clustering is an unsupervised
study. clustering can also be used for image segmentation. there exist so many methods for image
segmentation. image segmentation plays an important role in image analysis.it is one of the first
and the most important tasks in image analysis and computer vision. this proposed system
presents a variation of fuzzy c-means algorithm that provides image clustering. the kernel fuzzy
c-means clustering algorithm (kfcm) is derived from the fuzzy c-means clustering
algorithm(fcm).the kfcm algorithm that provides image clustering and improves accuracy
significantly compared with classical fuzzy c-means algorithm. the new algorithm is called
gaussian kernel based fuzzy c-means clustering algorithm (gkfcm)the major characteristic of
gkfcm is the use of a fuzzy clustering approach ,aiming to guarantee noise insensitiveness and
image detail preservation.. the objective of the work is to cluster the low intensity in homogeneity
area from the noisy images, using the clustering method, segmenting that portion separately using
content level set approach. the purpose of designing this system is to produce better segmentation
results for images corrupted by noise, so that it can be useful in various fields like medical image
analysis, such as tumor detection, study of anatomical structure, and treatment planning.
GAUSSIAN KERNEL BASED FUZZY C-MEANS CLUSTERING ALGORITHM FOR IMAGE SEGMENTATIONcscpconf
Image processing is an important research area in computer vision. clustering is an unsupervised study. clustering can also be used for image segmentation. there exist so many methods for image segmentation. image segmentation plays an important role in image analysis.it is one of the first and the most important tasks in image analysis and computer vision. this proposed system presents a variation of fuzzy c-means algorithm that provides image clustering. the kernel fuzzy
c-means clustering algorithm (kfcm) is derived from the fuzzy c-means clustering algorithm(fcm).the kfcm algorithm that provides image clustering and improves accuracy significantly compared with classical fuzzy c-means algorithm. the new algorithm is called gaussian kernel based fuzzy c-means clustering algorithm (gkfcm)the major characteristic of gkfcm is the use of a fuzzy clustering approach ,aiming to guarantee noise insensitiveness and image detail preservation.. the objective of the work is to cluster the low intensity in homogeneity area from the noisy images, using the clustering method, segmenting that portion separately using content level set approach. the purpose of designing this system is to produce better segmentation results for images corrupted by noise, so that it can be useful in various fields like medical image analysis, such as tumor detection, study of anatomical structure, and treatment planning.
GAUSSIAN KERNEL BASED FUZZY C-MEANS CLUSTERING ALGORITHM FOR IMAGE SEGMENTATIONcsandit
Image processing is an important research area in computer vision. clustering is an unsupervised
study. clustering can also be used for image segmentation. there exist so many methods for image
segmentation. image segmentation plays an important role in image analysis.it is one of the first
and the most important tasks in image analysis and computer vision. this proposed system
presents a variation of fuzzy c-means algorithm that provides image clustering. the kernel fuzzy
c-means clustering algorithm (kfcm) is derived from the fuzzy c-means clustering
algorithm(fcm).the kfcm algorithm that provides image clustering and improves accuracy
significantly compared with classical fuzzy c-means algorithm. the new algorithm is called
gaussian kernel based fuzzy c-means clustering algorithm (gkfcm)the major characteristic of
gkfcm is the use of a fuzzy clustering approach ,aiming to guarantee noise insensitiveness and
image detail preservation.. the objective of the work is to cluster the low intensity in homogeneity
area from the noisy images, using the clustering method, segmenting that portion separately using
content level set approach. the purpose of designing this system is to produce better segmentation
results for images corrupted by noise, so that it can be useful in various fields like medical image
analysis, such as tumor detection, study of anatomical structure, and treatment planning.
A Density Based Clustering Technique For Large Spatial Data Using Polygon App...IOSR Journals
This document presents a density-based clustering technique called TDCT (Triangle-density based clustering technique) for efficiently clustering large spatial datasets. The technique uses a polygon approach where the number of data points inside each triangle of a polygon is calculated. If the ratio of point densities between two neighboring triangles exceeds a threshold, the triangles are merged into the same cluster. The technique is capable of identifying clusters of arbitrary shapes and densities. Experimental results demonstrate the technique has superior cluster quality and complexity compared to other methods.
Clustering Using Shared Reference Points Algorithm Based On a Sound Data ModelWaqas Tariq
A novel clustering algorithm CSHARP is presented for the purpose of finding clusters of arbitrary shapes and arbitrary densities in high dimensional feature spaces. It can be considered as a variation of the Shared Nearest Neighbor algorithm (SNN), in which each sample data point votes for the points in its k-nearest neighborhood. Sets of points sharing a common mutual nearest neighbor are considered as dense regions/ blocks. These blocks are the seeds from which clusters may grow up. Therefore, CSHARP is not a point-to-point clustering algorithm. Rather, it is a block-to-block clustering technique. Much of its advantages come from these facts: Noise points and outliers correspond to blocks of small sizes, and homogeneous blocks highly overlap. This technique is not prone to merge clusters of different densities or different homogeneity. The algorithm has been applied to a variety of low and high dimensional data sets with superior results over existing techniques such as DBScan, K-means, Chameleon, Mitosis and Spectral Clustering. The quality of its results as well as its time complexity, rank it at the front of these techniques.
FUAT – A Fuzzy Clustering Analysis ToolSelman Bozkır
This document summarizes fuzzy c-means clustering (FCM) and introduces a software tool called FUAT that aims to address some of the difficulties with FCM. FCM is a soft clustering method that allows data elements to belong to more than one cluster. It is based on fuzzy set theory and combines c-means clustering with handling fuzziness in data. FUAT stands for Fuzzy Unsupervised Analysis Tool and provides features like automatic cluster number detection, interactive viewers for insights into results, and connectivity to R for further analysis. It aims to make fuzzy clustering more transparent and help with challenges like selecting initial centroids and evaluating clusters.
An Efficient Clustering Method for Aggregation on Data FragmentsIJMER
Clustering is an important step in the process of data analysis with applications to numerous fields. Clustering ensembles, has emerged as a powerful technique for combining different clustering results to obtain a quality cluster. Existing clustering aggregation algorithms are applied directly to large number of data points. The algorithms are inefficient if the number of data points is large. This project defines an efficient approach for clustering aggregation based on data fragments. In fragment-based approach, a data fragment is any subset of the data. To increase the efficiency of the proposed approach, the clustering aggregation can be performed directly on data fragments under comparison measure and normalized mutual information measures for clustering aggregation, enhanced clustering aggregation algorithms are described. To show the minimal computational complexity. (Agglomerative, Furthest, and Local Search); nevertheless, which increases the accuracy.
Data Science - Part VII - Cluster AnalysisDerek Kane
This lecture provides an overview of clustering techniques, including K-Means, Hierarchical Clustering, and Gaussian Mixed Models. We will go through some methods of calibration and diagnostics and then apply the technique on a recognizable dataset.
A Novel Penalized and Compensated Constraints Based Modified Fuzzy Possibilis...ijsrd.com
A cluster is a group of objects which are similar to each other within a cluster and are dissimilar to the objects of other clusters. The similarity is typically calculated on the basis of distance between two objects or clusters. Two or more objects present inside a cluster and only if those objects are close to each other based on the distance between them.The major objective of clustering is to discover collection of comparable objects based on similarity metric. Fuzzy Possibilistic C-Means (FPCM) is the effective clustering algorithm available to cluster unlabeled data that produces both membership and typicality values during clustering process. In this approach, the efficiency of the Fuzzy Possibilistic C-means clustering approach is enhanced by using the penalized and compensated constraints based FPCM (PCFPCM). The proposed PCFPCM approach differ from the conventional clustering techniques by imposing the possibilistic reasoning strategy on fuzzy clustering with penalized and compensated constraints for updating the grades of membership and typicality. The performance of the proposed approaches is evaluated on the University of California, Irvine (UCI) machine repository datasets such as Iris, Wine, Lung Cancer and Lymphograma. The parameters used for the evaluation is Clustering accuracy, Mean Squared Error (MSE), Execution Time and Convergence behavior.
This document compares the k-means and grid density clustering algorithms. It summarizes that grid density clustering determines dense grids based on the densities of neighboring grids, and is able to handle different shaped clusters in multi-density environments. The grid density algorithm does not require distance computation and is not dependent on the number of clusters being known in advance like k-means. The document concludes that grid density clustering is better than k-means clustering as it can handle noise and outliers, find arbitrary shaped clusters, and has lower time complexity.
This document provides a summary of different image segmentation techniques through clustering. It discusses exclusive clustering methods like k-means clustering, overlapping clustering methods like fuzzy c-means, and hierarchical clustering. The paper reviews these clustering approaches and their application to image segmentation, which is the process of partitioning a digital image into multiple segments. Image segmentation through clustering has various uses including computer vision, medical imaging, and remote sensing.
This document presents a survey of contemporary research on image segmentation through clustering techniques. It discusses various clustering approaches including exclusive clustering (e.g. k-means), overlapping clustering (e.g. fuzzy c-means), hierarchical clustering, and probabilistic D-clustering. It provides details on the algorithms and steps involved in each technique. The paper analyzes different clustering methods for image segmentation and concludes that fuzzy c-means is superior but has high computational costs, while probabilistic D-clustering can avoid this issue.
Unsupervised Clustering Classify theCancer Data withtheHelp of FCM AlgorithmIOSR Journals
This document discusses using fuzzy C-means (FCM) clustering to classify cancer data. It begins with an introduction to cluster analysis and cluster validity. It then provides details on FCM clustering, including the FCM algorithm and an example of classifying 4 cancer patient data points into 2 clusters based on medicine efficiency and penetration speed. The summary finds that initially 1 patient is classified into the improving cluster and 3 into the collapsing cluster, but after FCM classification 3 patients are classified as improving and 1 as collapsing.
This document discusses using fuzzy clustering to group real estate properties. It presents a case study clustering 46 real estate listings into 3 groups based on price, area, and region attributes. The fuzzy c-means clustering algorithm in MATLAB is used to assign membership levels and cluster centroids. The results identify 3 clusters - one for mid-priced properties in good regions and average areas, one for high-priced properties in excellent regions and large areas, and one for low-priced properties in poor regions and small areas. Graphs and tables show the clustered properties and centroids.
E XTENDED F AST S EARCH C LUSTERING A LGORITHM : W IDELY D ENSITY C LUSTERS ,...csandit
CFSFDP (clustering by fast search and find of densi
ty peaks) is recently developed density-
based clustering algorithm. Compared to DBSCAN, it
needs less parameters and is
computationally cheap for its non-iteration. Alex.
at al have demonstrated its power by many
applications. However, CFSFDP performs not well whe
n there are more than one density peak
for one cluster, what we name as "no density peaks"
. In this paper, inspired by the idea of a
hierarchical clustering algorithm CHAMELEON, we pro
pose an extension of CFSFDP,
E_CFSFDP, to adapt more applications. In particular
, we take use of original CFSFDP to
generating initial clusters first, then merge the s
ub clusters in the second phase. We have
conducted the algorithm to several data sets, of wh
ich, there are "no density peaks". Experiment
results show that our approach outperforms the orig
inal one due to it breaks through the strict
claim of data sets
This document provides an overview of several clustering algorithms. It begins by defining clustering and its importance in data mining. It then categorizes clustering algorithms into four main types: partitional, hierarchical, grid-based, and density-based. For each type, some representative algorithms are described briefly. The document also reviews several popular clustering algorithms like k-means, CLARA, PAM, CLARANS, and BIRCH in more detail. It discusses aspects like the algorithms' time complexity, types of data handled, ability to detect clusters of different shapes, required input parameters, and advantages/disadvantages. Overall, the document aims to guide selection of suitable clustering algorithms for specific applications by surveying their key characteristics.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Literature Survey On Clustering TechniquesIOSR Journals
This document provides a literature review of different clustering techniques. It begins by defining clustering and describing the main categories of clustering methods: hierarchical, partitioning, density-based, grid-based, and model-based. It then summarizes some examples of algorithms for each category in 1-2 sentences. For hierarchical methods, it discusses BIRCH, CURE, and CHAMELEON. For partitioning methods, it mentions k-means clustering and k-medoids. For density-based methods, it lists DBSCAN, OPTICS, DENCLUE. For grid-based methods, it lists CLIQUE, STING, MAFIA, WAVE CLUSTER, O-CLUSTER, ASGC, and
The document provides a literature review of different clustering techniques. It begins by defining clustering and its applications. It then categorizes and describes several clustering methods including hierarchical (BIRCH, CURE, CHAMELEON), partitioning (k-means, k-medoids), density-based (DBSCAN, OPTICS, DENCLUE), grid-based (CLIQUE, STING, MAFIA), and model-based (RBMN, SOM) methods. For each method, it discusses the algorithm, advantages, disadvantages and time complexity. The document aims to provide an overview of various clustering techniques for classification and comparison.
Juha vesanto esa alhoniemi 2000:clustering of the somArchiLab 7
This document summarizes a research paper that proposes clustering the self-organizing map (SOM) as a way to analyze cluster structure in data. The paper discusses:
1) Using a two-stage process where data is first mapped to prototypes using a SOM, then the SOM prototypes are clustered, reducing computational load compared to direct data clustering.
2) Different clustering algorithms that can be applied to the SOM prototypes, including hierarchical and partitive (k-means) methods.
3) The benefits of clustering the SOM include noise reduction and being able to cluster large datasets more efficiently.
Juha vesanto esa alhoniemi 2000:clustering of the som
RFNM-Aranda-Final.PDF
1. Robust Fuzzy n-Means Clustering
A Research Paper
Presented to
the Faculty of the Division of Mathematical Sciences
Midwestern State University
In Partial Fulfillment
of the Requirements of the Degree
Master of Science
by
Thomas G. Aranda
October 2000
2. Abstract
Clustering is a data segmentation method with a wide range of applications including
pattern recognition, document classification and data mining. This paper focuses on the problem
of unsupervised clustering when the optimal number of clusters is not known. This paper
presents an algorithm that can determine the ideal number of clusters and be robust to the
influence of outliers. A modification of the Robust Fuzzy c-Means Clustering Algorithm
(RFCM) was developed. This modification retains the robustness (ability to ignore outliers) of
RFCM, yet it does not increase the complexity of the algorithm. A Robust Fuzzy n-Means
Clustering Algorithm (RFNM) is presented. This method produces a good partition without a
priori knowledge of the optimal number of clusters.
3. 1
Introduction
This research is motivated by the requirement to segment large images in real time
without prior knowledge of the image’s structure. The ultimate goal is to identify and classify
sections of data into categories. For example, given an aerial photograph, the computer should
be able to distinguish between grass, concrete, water and asphalt. One technique for segmenting
data in this way is called clustering.
Many methods for clustering are in use, including Validity-Guided Clustering [1] and c-
Means Clustering [4]. However, these algorithms have two drawbacks. First, they are very
susceptible to the presence of outliers in some data sets. Consequently, they do not identify the
clusters properly. Some algorithms solve this problem by using robust centering statistics.
Second, these algorithms require the user to input the desired number of clusters. Often times,
the correct number of clusters is not known prior to execution. Therefore, it would be beneficial
to develop an algorithm that does not require such knowledge. This paper presents such an
algorithm.
Data Segmentation via Clustering
Background on Clustering
The classification of objects into categories is the subject of cluster analysis. It plays a
large role in pattern recognition. However, it has many other applications such as the
classification of documents in a database, the development of social demographics, data mining
and the construction of taxonomies in biology.
Ultimately, clustering attempts to identify groups of similar data. Given a set of data X,
the problem of clustering is to find several cluster centers that properly characterize relevant
classes of X [10]. For example, a good clustering of an image by color would identify the
4. 2
various shades of red as one cluster, the blues as another cluster, etc. A clustering of a 3D set of
points over a Euclidean space would find groups (clusters) of points that are close together.
After the cluster centers are identified, the data set X is partitioned by labeling each data element
with the exemplar (cluster center) closest to it.
In 1967 Ball and Hall introduced the ISODATA process [2]. This technique, which is
also called Hard c-Means Clustering (HCM), is one of the most popular clustering methods [4].
However, the user is required to input the desired number N of clusters. It uses an alternating
optimization (AO) technique to minimize an objective function. The definition of the objective
function and the AO technique can be found in [7]. One problem with HCM is that it tends to
get caught in local minima [7]. In other words, it does not find the global minimum of the
objective function and therefore does not properly identify the cluster centers.
Zadeh introduced fuzzy set theory in 1965 as a way to represent the vagueness of
everyday life [3]. In a nutshell, fuzzy set theory allows data elements to belong to a set in
varying degrees. Each element has a membership value [ ]1,0∈u that represents the degree to
which the data element belongs to that set. In other words, data elements can have a partial
membership in a set. This fuzziness allows one to mathematically represent vague concepts such
as “pretty soon” or “very far.”
Dunn applied fuzzy set theory to the ISODATA clustering process in 1973 [7]. His
method, called Fuzzy c-Means Clustering (FCM), allows data elements to belong to several
clusters in varying degrees. For example, a data element can have a 30% membership in one
cluster and a 70% membership in a second cluster, instead of discretely belonging to one cluster
or the other. Consider the clustering by color example: a dark violet could partially belong to the
red cluster and partially belong to the blue cluster.
5. 3
Fuzzy c-Means Clustering (FCM) uses an alternating optimization (AO) technique that is
very similar to HCM. After the algorithm finishes execution and the cluster centers are
identified, the clusters are “defuzzified” by discretely assigning each data element to the cluster
in which it has the highest membership. If a light orange color had a 45% membership in red, a
52% membership in yellow and a 3% membership in blue, then the color would be assigned to
the yellow cluster. Experiments have shown that the fuzzy clustering method is less likely to be
trapped in a local minimum [7] and, therefore, avoids one disadvantage of HCM.
FCM typically produces better results than HCM, but it is susceptible to the influence of
outliers—extraneous data elements that are very far away from the cluster centers. Outliers may
be the result of errors in the data, or they could be real information: such as a highly reflective
piece of aluminum foil appearing in a radar image of a grass field. Regardless of what the
outliers are, their presence often disrupts the clustering process.
Kersten’s Fuzzy c-Medians Clustering Algorithm (FCMED), which uses the fuzzy
median as its centering statistic, is more robust than FCM [8]. In other words, it is more resistant
to the influence of outliers. However, its time complexity of ( )NcpNO lg and space complexity
of ( )NO make it very slow [8]. Conversely, Choi’s and Krishnapuram’s Robust Fuzzy c-Means
Algorithm (RFCM) solves the outlier problem in linear time. [6]. Kersten’s implementation of
RFCM uses Huber’s weighting functions to reduce the influence of outliers [9]. Experiments
have shown RFCM to be very robust.
One disadvantage of RFCM is that it requires the user to input the correct number of
clusters. Often times the user does not know enough about the structure of the data to provide
such information. This is especially true in data mining applications. The research described in
this paper developed a new algorithm, Robust Fuzzy n-Means (RFNM), which is robust to
6. 4
outliers and capable of determining the proper number of clusters. This algorithm is a
modification of FCM and RFCM. In order to provide the reader with a complete understanding
of the new RFNM algorithm, this paper will describe its parent algorithms in detail.
Fuzzy c-Means Clustering
Fuzzy c-means clustering (FCM) is defined well by [4]. Consider N data samples
forming the data set denoted by { }NxxxX ,,, 21 K= . Assume there are c clusters and
( ) [ ]1,0∈= kiik xuu is the membership of the k-th sample kx in the i-th cluster iv , where
{ }cvvvv ,,, 21 K= is the set of exemplars (cluster centers) and U is the membership matrix.
Normally, a cluster center refers to an actual pattern in the data and an exemplar refers to a
pattern identified by the algorithm. However, these terms will be used interchangeably in this
paper. The membership value of each data element kx satisfies the requirement that
∑=
=
c
i
iku
1
1 (1)
for all Nk ℵ∈ . In other words, all of a particular data element’s membership values must add up
to one. In addition, each cluster must contain some, but not all of the data points’ membership.
Defined mathematically, this means that for every ci ℵ∈
Nu
N
k
ik << ∑=1
0 . (2)
The goal of the FCM algorithm is to minimize the objective function
( ) ∑∑= =
=
N
k
c
i
ik
m
ik duvUJ c
1 1
2
, (3)
where 2kiik xvd −= (the Euclidean distance between the exemplar and the data element). The
power cm of the membership function is called the weighting exponent. It expresses the
7. 5
“fuzziness” of the algorithm. Setting 1=cm and only allowing discrete membership values will
convert the fuzzy algorithm into traditional HCM [9].
The objective function (3) is the weighted square error of the exemplars. The closer data
elements are to their respective cluster centers, the lower the value of the function will be.
Furthermore, the number of exemplars c will have an effect on the value of ( )vUJ , . Increasing
the number of exemplars will lower the value the objective function. In an extreme case, when
the number of clusters equals the number of data elements ( )Nc = , the objective function will go
to zero. Although using a large number of clusters will reduce the value of ( )vUJ , , it is more
important to choose a value of c that represents the actual number of clusters in the data.
Fuzzy c-Means Clustering is more effective than Hard c-Means because the objective
function is less likely to get caught in a local minimum [7]. Furthermore, it runs in ( )cNO time
and ( )cO space. However, it is susceptible to outliers [9]. The robust algorithm presented in the
next section addresses this problem.
Robust Fuzzy c-Means Clustering
Real world data sets often contain outliers. These extraneous data elements are usually
very far away from the larger cluster centers. Consider a data set with two large well-defined
clusters and one small outlying cluster that is very far away from the other two. Due to the 2
ikd
term in ( )vUJ , (3), the distance of a data point from its exemplar will have a quadratic effect on
the value of objective function. Since FCM attempts to minimize ( )vUJ , , it will attempt to
reduce the impact of the outliers’ large 2
ikd values by placing an exemplar over the outlying
cluster. This minimizes the objective function, but does not correctly identify the larger cluster
centers.
8. 6
Kersten’s implementation of Choi and Krishnapuram’s Robust Fuzzy c-Means Clustering
Algorithm (RFCM) takes steps to solve this problem [9]. Huber’s m-estimator is used to reduce
the influence of outliers. Huber’s function ρ is defined as:
( )
>−
≤
=
1,
1,
2
1
2
2
1
xifx
xifx
xρ . (4)
The 2
ikd term is replaced with ( )γρ ikd where γ is a scaling constant. As a result, the influence
of the distance between cluster centers and data elements is quadratic when the data element is
close to the exemplar and linear when the data element is far away from the exemplar. The
objective function to be minimized becomes:
( ) ( )∑∑= =
=
N
k
c
i
ik
m
ik duvUJ c
1 1
, γρ . (5)
The membership values of each element are given by:
( )
( )
1
1
1
1 −
=
−
= ∑
c
j
m
jk
ik
ik
c
d
d
u
γρ
γρ
. (6)
Using this function, the membership of a data element kx in cluster iv is assigned in proportion
to the distance between kx and iv . In other words, the data element will have a larger
membership in clusters that are closer to it.
The center of a cluster is computed by determining the average value of all the points in
that cluster. Since a point’s membership in a cluster is fuzzy, the mean must be adjusted by the
membership values iku . Therefore, the locations of the exemplars are computed by using the
weighted mean given by:
9. 7
( )
( )∑
∑
=
=
= N
k
ik
m
ik
N
k
ikik
m
ik
i
dwu
xdwu
v
c
c
1
1
γ
γ
(7)
where Huber’s weighting function ( ) ( ) xxxw ρ′= . In this case
( )
>
≤
=
1,1
1,1
xx
x
xw . (8)
Huber’s w function has the effect of reducing the influence of data points that are far away from
the cluster centers thereby making the algorithm robust to outliers.
In order for the ρ and w functions to work properly, all distances must be adjusted by a
scaling constant γ [9]. The experiments in this paper use the median absolute deviation about
the median (MAD) [11] to compute γ . The MAD is a robust estimator similar to the standard
deviation. All distances are divided by three times the MAD before Huber’s functions are
applied, i.e. MAD3⋅=γ . As a result, when ρ is applied, data points have a quadratic influence
when they are MAD3⋅ or less from the exemplar and linear influence when they are greater than
MAD3⋅ away. One should note that computing the MAD takes ( )NNO lg time (on average)
and ( )NO space. The normalization of the data using an estimator like the MAD is crucial to
making the algorithm run properly.
Except for the calculation of the scaling constant and the application of Huber’s
functions, RFCM is identical to FCM. However, RFCM is not as susceptible to the influence of
outliers [9].
Determining the Number of Clusters
Robust Fuzzy n-Means Clustering
One problem with RFCM is that the user must input the desired number of clusters.
10. 8
Quite often the optimal number of clusters is not known prior to execution. The Robust Fuzzy n-
Means Algorithm (RFNM) presented in this paper retains the robustness of RFCM, yet does not
require a priori knowledge of the proper number of clusters.
RFNM requires the user to provide a maximum number mc of clusters. The algorithm
begins by executing the RFCM algorithm with mc clusters. During every iteration cluster
centers that are close together are considered for merging. Several methods for merging have
been explored, including Validity-Guided Clustering described in [1] and Competitive Clustering
described in [5]. However, the merging criteria should be robust and efficient.
Merging Criterion
If two clusters are “close” together they should be merged. Two clusters are close if the
distance between their centers is small compared to their compactness. The notion of
compactness [12] is the weighted mean square deviation of the cluster. It can be thought of as
the average “radius” squared. The compactness of a cluster is defined in terms of its variation
and cardinality.
The variation of a cluster is a measure of the cluster’s dispersion. One can think of it as
the fuzzy variance. Formally, the variation is defined by [12]:
∑=
=
N
k
ik
m
iki du c
1
2
σ (9)
The fuzzy cardinality of a cluster is a measure of the cluster’s size. The more data
elements that belong to the cluster the larger the cluster’s cardinality will be. Often, the fuzzy
cardinality is used as a divisor when calculating the fuzzy mean. Formally the fuzzy cardinality
is defined by [12]:
∑=
=
N
k
iki un
1
. (10)
11. 9
The compactness of a cluster is the ratio of its variation and cardinality [12]:
i
i
i
n
σ
π = . (11)
To make the compactness formula robust to outliers Huber’s ρ function (4) is inserted into the
equation. Finally, the cardinality of the cluster must take the weighting exponent cm into
account. Therefore, the robust compactness of a cluster iv is defined as
( )
∑
∑
=
=
= n
k
m
ik
N
k
ik
m
ik
i
c
c
u
du
1
1
γρ
π . (12)
RFNM uses a modified version of separation [12] to measure how far clusters are apart.
Formally, the separation between two clusters qv and rv is defined as the Euclidean distance
between the clusters’ centers:
2rqqr vvs −= . (13)
The merging criterion uses a merge ratio, which is similar to the validity index defined in
[12]. The merge ratio will be small when exemplars are close together relative to their
compactness. Formally, it is the ratio of the separation squared over the compactness:
q
qr
qr
s
π
ω
2
= . (14)
Once again, to make the formula robust Huber’s function is substituted:
( )
q
qr
qr
s
π
γρ
ω = . (15)
During every iteration of RFCM, the merge ratio qrω is calculated for every cluster
vvq ∈ and vvr ∈ . If αω ≤qr , where α is some constant, then the clusters centered at qv and
12. 10
rv are merged. Choosing a value of 1<α means that in order for two clusters to be merged, the
distance between the clusters’ centers must be less than the compactness (radius) of the clusters.
Experimentally, values of [ ]3.0,1.0∈α work well.
Merging Mechanics
Once the decision is made to join two clusters, they must be combined in a meaningful
way. The new exemplar should exist on a line segment that runs between the two old exemplars.
The new center will be placed closer to the cluster with the larger fuzzy cardinality.
The placement of the new exemplar is accomplished by using a parameter p:
rq
q
nn
n
p
+
= (16)
where qv and rv are the centers of the two clusters to be merged. The location of the new
exemplar is calculated using a combination formula:
( ) rqn vppvv −+= 1 (17)
where nv is the center of the new cluster. The old exemplars are removed from v and replaced
with the new center nv . The next iteration of the algorithm will compute the membership values
of X in the new cluster.
The RFNM Algorithm
The Robust Fuzzy n-Means algorithm is based on the FCM algorithm described in [10].
It uses an alternating optimization (AO) technique for minimizing the objective function (5).
FCM has been modified to be robust and unsupervised. The mc exemplars begin at locations
determined by the user. During execution, these exemplars gravitate toward the data set’s “true”
cluster centers. Some exemplars may merge along the way. Ideally, the algorithm will terminate
with exactly one exemplar positioned near the center of each cluster. The user provides the
13. 11
following input:
∞ℵ∈mc initial (maximum) number of clusters
[ )∞∈ ,1cm weighting exponent
( )1,0∈α merging criterion constant
( )∞∈ ,0ε stopping constant (small positive number)
( )∞∈ ,0γ scaling constant (example: three times the MAD)
{ }mcvvvv ,,, 21 K= initial placement of the exemplars (cluster centers)
Algorithm: ( )vmcRFNM cm ,,,,, γεα
Step 1. Let mcc =
Step 2. Let csss ,,, 21 K equal cvvv ,,, 21 K respectively.
Step 3. Calculate the new membership matrix U by the following procedure: for each
Xxk ∈ , if 0
2
>− ik vx for all ci ℵ∈ , then compute iku using equation (6). If
0
2
=− ik vx for some cIi ℵ⊆∈ , then define iku for Ii ∈ by any nonnegative
real numbers satisfying ∑∈
=
Ii
iku 1 and define 0=iku for Ii c −ℵ∈ .
Step 4. Merge clusters that are close together. For every vvq ∈ and vvr ∈ and rq ≠
do the following: calculate qrω using equation (15); if αω ≤qr then compute
nv using equations (16) and (17); let nq vv = ; remove rv from v and decrement
c by 1. NOTE: Any cluster can only be merged once per iteration.
Step 5. Calculate the c cluster centers cvvv ,,, 21 K using equation (7) and the given value
of cm .
14. 12
Step 6. If a merge took place in Step 4 then return to Step 2. Otherwise if
ε≤−
ℵ∈
ii
i
sv
c
max then stop. Otherwise, return to Step 2.
On average, this algorithm has linear time complexity. Steps 2, 5 and 6 have a total
maximum running time of ( )cbacm ++ where a, b and c are constants. The maximum running
time of Step 3 is Nck m ⋅⋅ and Step 4 will run in 2
mcl ⋅ time (worst case), where k and l are
constants. Thus, the total running time of this algorithm has an upper bound of
( )[ ]cbacNckclt mmm +++⋅⋅+⋅ 2
where t is the number of iterations of the algorithm. In most
cases the size of the data set N will be significantly larger than mc and t. Therefore, the N term
will overwhelm the 2
mc and t terms yielding a running time complexity of ( )NcO m .
The memory overhead of this algorithm is also linear. Storing the exemplar vectors v and
s requires ( )mcO space in the worst case. The membership matrix U requires ( )NcO m ⋅ space.
The memory required for the data set X is not considered because it is not overhead. The total
memory overhead is ( )NcO m . Often times, the data set to be clustered is very large. Therefore,
storing something of size N will cost much memory. However, clever coding and some slight
modifications will allow the algorithm to run in ( )mcO space. The following modifications
compute the membership matrix U, exemplars v and the merge ratio ω on the fly without storing
U in memory:
Algorithm: ( )vmcRFNMFast cm ,,,,, γεα
Step 1. Let mcc = .
Step 2. Let csss ,,, 21 K equal cvvv ,,, 21 K respectively.
Step 3. Calculate, but do not store, the new membership matrix U by the following
15. 13
procedure: for each Xxk ∈ , if 0
2
>− ik vx for all ci ℵ∈ , then compute iku
using equation (6). If 0
2
=− ik vx for some cIi ℵ⊆∈ , then define iku for
Ii ∈ by any nonnegative real numbers satisfying ∑∈
=
Ii
iku 1 and define 0=iku
for Ii c −ℵ∈ . Simultaneously, calculate and keep the running sums used in
equations (7), (15) and (16).
Step 4. For every vvq ∈ and vvr ∈ and rq ≠ do the following: calculate qrω using
equation (15) and the running sums from Step 3; if αω ≤qr then compute nv
using equations (16) and (17) and the running sum from Step 3; let nq vv = ;
remove rv from v and decrement c by 1. If any two clusters were merged, then
return to Step 3.
Step 5. Calculate the c cluster centers cvvv ,,, 21 K using equation (7), the given value of
cm and the running sums from Step 3.
Step 6. If a merge took place in Step 4 then return to Step 2. Otherwise if
ε≤−
ℵ∈
ii
i
sv
c
max then stop. Otherwise, return to Step 2.
This “fast” version of RFNM actually has the same time complexity as the normal
algorithm. However, it has a much lower memory complexity. Since it does not store the
membership matrix U, the N term can be dropped from the space complexity. Consequently, the
memory overhead is ( )mcO , which represents only storing the exemplar vectors. On average,
this “fast” algorithm will execute quicker because its lower memory overhead reduces the risk of
page faults. RFNM provides robust unsupervised learning with a linear running time and low
memory overhead. This makes it ideally suited for real-time data processing applications.
16. 14
Testing
Exemplar Placement (Gaussian Tests 1 and 2)
The RFNM algorithm described in the previous section was tested using a five-
dimensional Gaussian scatter of random data. The test data has two cluster centers equidistant
from the origin. The first two tests start with six exemplars ( )6=mc , 75.1=cm and 3.0=α .
The positioning of the initial exemplars is critical. Figure 1 shows a 2D plot of the movement of
the exemplars.
Figure 1. Gaussian Test 1, 6=mc .
The “true” cluster centers, which are computed using the sample mean, exist at
( )0,0,0,0,0.1− and ( )0,0,0,0,0.1 . In Figure 1, two exemplars (labeled A and B) merge together at
17. 15
point C and then converge to approximately ( )0,0,0,1.0,3.1 −− . Two more exemplars (D and E)
merge together at F and converge to ( )0,0,0,1.0,4.1≈ . The last two exemplars (G and H) were
initialized on the y-axis. They merge and converge very close to the origin (point I). The test
data is almost symmetric. Since the middle exemplars (G and H) started equidistant from two
nearly symmetric clusters, they were never drawn to one cluster or the other. In this case the
algorithm does not converge properly.
Figure 2. Gaussian Test 2, 6=mc .
The second test uses the same data set and starting parameters except two of the initial
exemplars (Figure 2: D and G) are offset 5.0+ along the x-axis. Figure 2 plots the movement of
the exemplars. Notice the algorithm converges to the desired two cluster centers ( )0,0,0,0,0.1± .
18. 16
Exemplars A and B merge together into exemplar C, which then converges to the cluster center
at ( )0,0,0,0,1− .
The exemplar trace on the right side of the y-axis is more interesting. Exemplars D and E
merge together and become F. Exemplar G then merges with F to become H. Finally, H and I
merge into exemplar J, which then converges to the cluster center at ( )0,0,0,0,1 . Since the
middle exemplars (D and G) were initialized slightly closer the right hand cluster, they were
drawn toward that cluster’s center. By comparing the results of test 1 and test 2, one can see that
the initial placement of the exemplars can change the results dramatically.
Robustness Testing (Cauchy Test 1)
A second set of two-dimensional test data was randomly generated using a Cauchy
distribution. This data set has two well-defined main clusters, but also has several outliers that
are very far away from the main clusters. The presence of the outliers obviously increases the
compactness of the clusters (makes them less compact). Consequently, the exemplars tend to
merge very quickly. To compensate for this, lower values of 25.1=cm and 2.0=α were
chosen. Additionally, the initial exemplars were started a little further away from the origin. To
reduce the influence of outliers, the sample median is used to determine the “true” cluster
centers: approximately ( )0,3.1± . Figure 3 shows a trace of the six exemplars.
The merging sequence in Figure 3 is very similar to the previous test. Exemplars A and
B merge into C and converge to ( )0,8.1− . On the right side of the y-axis, exemplars D and E
merge into F. Next, G and F merge into H. Finally, H and I merge into J and converge to
( )0,5.1 . Notice, the algorithm converges near the two desired cluster centers of ( )0,3.1± .
However, it does not converge exactly because the outliers still have some influence on the
exemplars. The total error is 0.7.
19. 17
Figure 3. Cauchy Test 1 (RFNM), 6=mc .
Additionally, the proper choice of α is very important. Choosing a merge ratio that is
too low ( )1.0<α will cause the exemplars to not merge. Conversely, setting the merge ratio too
high ( )4.0>α will cause all of the exemplars to merge together into one cluster center. In both
cases, the true cluster centers are never found. Thus, choosing a good merge ratio is crucial.
Robust n-Means vs. c-Means Clustering (Cauchy Test 1)
For comparison purposes, a standard RFCM algorithm ( )0=α was run on the same set of
data with the same parameters. Figure 4 shows the trace. Exemplar A converges to ( )0,5.1 , and
exemplar B converges to ( )0,8.1− . In other words, the two exemplars in this example converge
to the same cluster centers as the exemplars in the previous test. Clearly, the RFNM algorithm
20. 18
performs as well as RFCM.
Figure 4. Cauchy Test 1 (RFCM), 2== ccm .
The values of the objective functions (5) for both methods were plotted against time (see
Figure 5). Both methods converge to the same value ( )830≈ within the same number of
iterations. Notice the RFNM algorithm (left), with an initial 6=mc and final 2=c , yields an
increasing value of ( )vUJ , . This is because reducing the number of clusters actually causes an
increase in the objective function. However, RFNM still reaches the optimal solution without
requiring the user to input the desired number of clusters.
Catching Outliers (Cauchy Test 2)
The previous examples start with the exemplars near the cluster centers. In the next
21. 19
example, eight exemplars ( )8=mc begin far away from the origin. One again, the Cauchy data
set is used. Figure 6 shows a close-up of the exemplar trace. Exemplars A and B merge into C.
At the same time exemplars D and E merge into F. Finally, C and F merge into G and converge
to ( )0,5.1− . Exemplar H converges to ( )0,5.1 without merging. The total error is 0.4, and the
final number of clusters is 5=c .
Figure 5. RFNM vs. RFCM (Cauchy Test 1).
Figure 7 shows a trace of the same test on a larger scale. One can see exemplars A, B, D,
E and H move toward the main clusters, merge and converge to the cluster centers (see Figure 6).
Furthermore, exemplars I, J and K converge near clusters of outliers: approximately located at
( )1,10− , ( )25,0 and ( )17,1 −− respectively. These outlying clusters have very low fuzzy
cardinalities (less than 10% of the main clusters). In the final analysis, one could classify
exemplars with low cardinalities as clusters of outliers. Depending on the application, it may be
useful to discover outlying clusters. Otherwise, they can be ignored and removed from the final
partition.
The use of Huber’s functions in RFNM reduces the influence of outliers, but it does not
22. 20
eliminate their influence. However, notice that the exemplars in Cauchy test 2 (Figure 6) are
closer to the desired centers of ( )0,3.1± than the exemplars in Cauchy test 1 (Figure 3). In fact,
test 2 yields improvement of 0.3 in total error over test 1. This is because the second test placed
exemplars near the outliers (see Figure 7), which has the effect of reducing the influence of those
outliers on the two main clusters. As a result, the true centers of the main cluster are identified
with greater accuracy. Furthermore, the final value of the objective function is lower:
approximately 611 as opposed to 830 from test 1. Of course, the larger number of exemplars
( )5=c in test 2 accounts for much of this decrease. Figure 8 shows a plot of the objective
function. A good initial placement of the exemplars will improve the robustness of the algorithm
Figure 5. Cauchy Test 2 (Zoomed-In), 8=mc .
23. 21
and yield better results.
Figure 6. Cauchy Test 2 (Zoomed Out), 8=mc .
Conclusion
The goal of this paper is to provide a robust algorithm that will find an optimal partition
without knowing the proper number of clusters. Ideally, the user should be able to partition a
data set without any a priori knowledge of the data’s structure. Robust Fuzzy n-Means
Clustering provides a good start toward this goal.
Experiments with Gaussian data have demonstrated that RFNM can accurately find the
desired number of clusters and their centers. Furthermore, the first Cauchy test has shown that
RFNM provides results which are identical to the results reached by RFCM. Thus, RFNM is as
24. 22
accurate and robust as RFCM, yet it does not increase the time complexity. Finally, clever
initialization of the exemplars allows RFNM to identify outlying clusters (Cauchy test 2). This
in turn improves the accuracy of the final results. Clearly, RFNM provides robust accurate
results without requiring prior knowledge of the data’s structure.
Figure 7. Cauchy Test 2 (Objective Function).
Although RFNM is an improvement over other algorithms, it does have some
shortcomings. First, it is not completely unsupervised because the user’s choices of cm and α
will have significant effects on the results. Data sets with outliers, for example, require lower
values cm and α than sets with compact well-separated clusters. Future research should
examine ways of preprocessing the target data in order to determine the ideal clustering
parameters so that the entire process can be fully automated.
Second, the initial positioning of the exemplars is crucial to getting optimal results.
Placing the exemplars exactly between two cluster centers, for example, may cause those
exemplars to not converge. One possible solution is to place the initial exemplars very far away
25. 23
from the cluster centers. This will allow the exemplars to compete equally for cardinality. In
other words, one exemplar will not have an advantage simply because it was initially placed
close to a cluster of points. However, if the exemplars are initialized too far away from the
cluster centers, then the main clusters and the outliers will have equal influence. As a result, the
exemplars may skip over the outliers all together. Determining an automated, yet reliable way of
initializing the exemplars would be very beneficial. Future research in this area should be
considered.
Third, the preprocessing requirements of RFNM can be costly. The experiments in this
paper use the MAD to compute the scaling constant γ . This operation takes ( )NNO lg time and
uses ( )NO space. Research into more efficient preprocessing techniques may be useful.
Robust Fuzzy n-Means Clustering has a wide range of applications in image and data
processing. It requires less user supervision than many other algorithms, but it is not completely
unsupervised. However, in several situations the RFNM algorithm provides a good solution in
linear time.
26. 24
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