Chapter 4: Clustering
I
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
What is Cluster Analysis?
Finding groups of objects such that the objects in a group will be similar (or related) to one another and different from (or unrelated to) the objects in other groups
Inter-cluster distances are maximized
Intra-cluster distances are minimized
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Applications of Cluster Analysis
Understanding
Group related documents for browsing, group genes and proteins that have similar functionality, or group stocks with similar price fluctuations
Summarization
Reduce the size of large data sets
Clustering precipitation in Australia
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
What is not Cluster Analysis?
Simple segmentation
Dividing students into different registration groups alphabetically, by last name
Results of a query
Groupings are a result of an external specification
Clustering is a grouping of objects based on the data
Supervised classification
Have class label information
Association Analysis
Local vs. global connections
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Notion of a Cluster can be Ambiguous
How many clusters?
Four Clusters
Two Clusters
Six Clusters
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Types of Clusterings
A clustering is a set of clusters
Important distinction between hierarchical and partitional sets of clusters
Partitional Clustering
A division of data objects into non-overlapping subsets (clusters) such that each data object is in exactly one subset
Hierarchical clustering
A set of nested clusters organized as a hierarchical tree
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Partitional Clustering
Original Points
A Partitional Clustering
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Hierarchical Clustering
Traditional Hierarchical Clustering
Non-traditional Hierarchical Clustering
Non-traditional Dendrogram
Traditional Dendrogram
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Types of Clusters
Well-separated clusters
Center-based clusters
Contiguous clusters
Density-based clusters
Property or Conceptual
Described by an Objective Function
02/14/2018 Introduction to Data Mining, 2nd Edition ‹#›
Types of Clusters: Well-Separated
Well-Separated Clusters:
A cluster is a set of points such that any point in a cluster is closer (or more similar) to every other point in the cluster than to any point not ...
Cluster analysis is an unsupervised machine learning technique used to group unlabeled data points into clusters based on similarities. It is useful for applications such as understanding relationships between data points, data summarization, and identifying hidden patterns. The document discusses different types of cluster analysis including partitional clustering which divides data into disjoint clusters and hierarchical clustering which produces nested clusters organized as a tree structure. It also covers various cluster characteristics, algorithms like K-means, and factors that impact cluster analysis results.
The document discusses cluster analysis and various clustering algorithms. It begins with an introduction to cluster analysis, describing what clusters are and some applications of clustering techniques. It then categorizes major clustering methods into partitioning methods, hierarchical methods, density-based methods, and others. Finally, it describes two partitioning algorithms - k-means and k-medoids - in more detail. The k-means algorithm is explained as selecting initial cluster centers, assigning all objects to the closest center, and updating centers repeatedly until convergence.
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.
This document provides an overview of cluster analysis and clustering algorithms. It defines cluster analysis as grouping objects such that objects within a group are similar to each other and different from objects in other groups. The document discusses different types of clusters, including well-separated, prototype-based, contiguity-based, and density-based clusters. It also covers hierarchical and partitional clustering. Finally, it describes the widely used k-means clustering algorithm and its objective function.
- Hierarchical clustering produces nested clusters organized as a hierarchical tree called a dendrogram. It can be either agglomerative, where each point starts in its own cluster and clusters are merged, or divisive, where all points start in one cluster which is recursively split.
- Common hierarchical clustering algorithms include single linkage (minimum distance), complete linkage (maximum distance), group average, and Ward's method. They differ in how they calculate distance between clusters during merging.
- K-means is a partitional clustering algorithm that divides data into k non-overlapping clusters based on minimizing distance between points and cluster centroids. It is fast but sensitive to initialization and assumes spherical clusters of similar size and density.
Clustering is the process of grouping similar data points together. There are different types of clustering algorithms like k-means clustering, DBSCAN clustering, and self-organizing maps. K-means clustering divides data into k groups by finding cluster centroids. It assigns data points to the nearest centroid and recalculates centroids until clusters stabilize. DBSCAN clustering finds core data points that have many nearby points and groups them into clusters based on density. Self-organizing maps use neural networks to place similar data objects close to each other on a 2D grid based on their similarity to centroid values that are updated during training.
The document provides an overview of different clustering methods including partitioning methods like k-means and k-medoids, hierarchical methods like agglomerative and divisive, and density-based methods like DBSCAN and OPTICS. It discusses the basic concepts of clustering, requirements for effective clustering like scalability and ability to handle different data types and shapes. It also summarizes clustering algorithms like BIRCH that aim to improve scalability for large datasets.
Cluster analysis is an unsupervised machine learning technique used to group unlabeled data points into clusters based on similarities. It is useful for applications such as understanding relationships between data points, data summarization, and identifying hidden patterns. The document discusses different types of cluster analysis including partitional clustering which divides data into disjoint clusters and hierarchical clustering which produces nested clusters organized as a tree structure. It also covers various cluster characteristics, algorithms like K-means, and factors that impact cluster analysis results.
The document discusses cluster analysis and various clustering algorithms. It begins with an introduction to cluster analysis, describing what clusters are and some applications of clustering techniques. It then categorizes major clustering methods into partitioning methods, hierarchical methods, density-based methods, and others. Finally, it describes two partitioning algorithms - k-means and k-medoids - in more detail. The k-means algorithm is explained as selecting initial cluster centers, assigning all objects to the closest center, and updating centers repeatedly until convergence.
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.
This document provides an overview of cluster analysis and clustering algorithms. It defines cluster analysis as grouping objects such that objects within a group are similar to each other and different from objects in other groups. The document discusses different types of clusters, including well-separated, prototype-based, contiguity-based, and density-based clusters. It also covers hierarchical and partitional clustering. Finally, it describes the widely used k-means clustering algorithm and its objective function.
- Hierarchical clustering produces nested clusters organized as a hierarchical tree called a dendrogram. It can be either agglomerative, where each point starts in its own cluster and clusters are merged, or divisive, where all points start in one cluster which is recursively split.
- Common hierarchical clustering algorithms include single linkage (minimum distance), complete linkage (maximum distance), group average, and Ward's method. They differ in how they calculate distance between clusters during merging.
- K-means is a partitional clustering algorithm that divides data into k non-overlapping clusters based on minimizing distance between points and cluster centroids. It is fast but sensitive to initialization and assumes spherical clusters of similar size and density.
Clustering is the process of grouping similar data points together. There are different types of clustering algorithms like k-means clustering, DBSCAN clustering, and self-organizing maps. K-means clustering divides data into k groups by finding cluster centroids. It assigns data points to the nearest centroid and recalculates centroids until clusters stabilize. DBSCAN clustering finds core data points that have many nearby points and groups them into clusters based on density. Self-organizing maps use neural networks to place similar data objects close to each other on a 2D grid based on their similarity to centroid values that are updated during training.
The document provides an overview of different clustering methods including partitioning methods like k-means and k-medoids, hierarchical methods like agglomerative and divisive, and density-based methods like DBSCAN and OPTICS. It discusses the basic concepts of clustering, requirements for effective clustering like scalability and ability to handle different data types and shapes. It also summarizes clustering algorithms like BIRCH that aim to improve scalability for large datasets.
This document provides an introduction to document clustering and clustering algorithms. It discusses how clustering can be used in information retrieval applications like organizing search results and improving search recall. It also covers different types of clustering algorithms like partitioning algorithms (such as K-means) and hierarchical algorithms. Key steps of the K-means and hierarchical agglomerative clustering algorithms are described.
Cluster analysis is used to group similar objects together and separate dissimilar objects. It has applications in understanding data patterns and reducing large datasets. The main types are partitional which divides data into non-overlapping subsets, and hierarchical which arranges clusters in a tree structure. Popular clustering algorithms include k-means, hierarchical clustering, and graph-based clustering. K-means clustering assigns data points to centroids to minimize distance within clusters. Hierarchical clustering creates nested clusters and can be visualized using dendrograms. Both have benefits and limitations depending on the dataset.
Cluster analysis is used to group similar objects together and separate dissimilar objects. It has applications in understanding data patterns and reducing large datasets. The main types are partitional which divides data into non-overlapping subsets, and hierarchical which arranges clusters in a tree structure. Popular clustering algorithms include k-means, hierarchical clustering, and graph-based clustering. K-means partitions data into k clusters by minimizing distances between points and cluster centroids, but requires specifying k and is sensitive to initial centroid positions. Hierarchical clustering creates nested clusters without needing to specify the number of clusters, but has higher computational costs.
Cluster analysis is used to group similar objects together and separate dissimilar objects. It has applications in understanding data patterns and reducing large datasets. The main types are partitional which divides data into non-overlapping subsets, and hierarchical which arranges clusters in a tree structure. Popular clustering algorithms include k-means, hierarchical clustering, and graph-based clustering. K-means partitions data into k clusters by minimizing distances between points and cluster centroids, but requires specifying k and is sensitive to initial centroid positions. Hierarchical clustering creates nested clusters without needing to specify the number of clusters, but has higher computational costs.
This document introduces data exploration and reduction techniques in XLMiner, including principal component analysis and cluster analysis. Principal component analysis transforms correlated variables into uncorrelated principal components to reduce data size while maintaining variability. Cluster analysis assigns objects to clusters to group similar objects together and different objects apart. The document demonstrates how to perform k-means clustering and hierarchical clustering in XLMiner.
This document introduces data exploration and reduction techniques in XLMiner, including principal component analysis and cluster analysis. Principal component analysis transforms correlated variables into uncorrelated principal components to reduce data size while maintaining variability. Cluster analysis assigns objects to clusters to group similar objects together and different objects apart. The document demonstrates how to perform k-means clustering and hierarchical clustering in XLMiner.
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.
This document provides an overview of supervised and unsupervised learning, with a focus on clustering as an unsupervised learning technique. It describes the basic concepts of clustering, including how clustering groups similar data points together without labeled categories. It then covers two main clustering algorithms - k-means, a partitional clustering method, and hierarchical clustering. It discusses aspects like cluster representation, distance functions, strengths and weaknesses of different approaches. The document aims to introduce clustering and compare it with supervised learning.
This document provides an overview of hierarchical clustering methods. It discusses agglomerative hierarchical clustering methods like AGNES that start by treating each object as a separate cluster and merge them into larger clusters. It also discusses divisive hierarchical clustering methods like DIANA that start with all objects in one cluster and split them into smaller clusters. A dendrogram is used to visualize the nested clustering formed at different levels in the hierarchical clustering tree. The document also discusses different measures for calculating the distance between clusters during the merging or splitting process.
Data Mining Primitives, Languages & SystemsNiloy Sikder
The document summarizes the key primitives and concepts in data mining. It discusses the main primitives as task-relevant data, knowledge types to be mined, and background knowledge. It describes various data mining techniques including data selection, grouping, characterization, discrimination, associations and correlations. It also discusses classification methods and prediction. Finally, it covers data mining system architectures and languages such as DMQL and OLE DB.
Clustering is an unsupervised machine learning technique used to group unlabeled data points. There are two main approaches: hierarchical clustering and partitioning clustering. Partitioning clustering algorithms like k-means and k-medoids attempt to partition data into k clusters by optimizing a criterion function. Hierarchical clustering creates nested clusters by merging or splitting clusters. Examples of hierarchical algorithms include agglomerative clustering, which builds clusters from bottom-up, and divisive clustering, which separates clusters from top-down. Clustering can group both numerical and categorical data.
This document provides an introduction and overview of document clustering techniques in information retrieval. It discusses motivations for clustering documents, such as improving search recall and organizing search results. It covers common clustering algorithms like K-means and hierarchical clustering, how they work, and considerations like choosing the number of clusters. The document uses examples and diagrams to illustrate clustering concepts and algorithms.
This document provides an introduction and overview of document clustering techniques in information retrieval. It discusses motivations for clustering documents, such as improving search recall and organizing search results. It covers common clustering algorithms like K-means and hierarchical clustering, how they work, and considerations like choosing the number of clusters. The document uses examples and diagrams to illustrate clustering concepts and algorithms.
This document provides an introduction and overview of document clustering techniques in information retrieval. It discusses motivations for clustering documents, such as improving search recall and organizing search results. It covers common clustering algorithms like K-means and hierarchical clustering, how they work, and considerations like choosing the number of clusters. The document uses examples and diagrams to illustrate clustering concepts and algorithms.
This document provides an introduction and overview of document clustering techniques in information retrieval. It discusses motivations for clustering documents, different document representations, evaluation criteria, and clustering algorithms including partitional algorithms like K-means and hierarchical algorithms. It provides examples and discusses issues like determining the optimal number of clusters to generate. The overall summary is that document clustering groups similar documents together to help with tasks like document navigation, improving search recall, and organizing search results.
This document summarizes a presentation on cluster analysis given by Tekendra Nath Yogi. It defines cluster analysis and describes several clustering methods and algorithms, including k-means clustering. It also discusses applications of cluster analysis in fields like business intelligence, image recognition, web search, and biology. Requirements for effective clustering algorithms are outlined.
This document compares hierarchical and non-hierarchical clustering algorithms. It summarizes four clustering algorithms: K-Means, K-Medoids, Farthest First Clustering (hierarchical algorithms), and DBSCAN (non-hierarchical algorithm). It describes the methodology of each algorithm and provides pseudocode. It also describes the datasets used to evaluate the performance of the algorithms and the evaluation metrics. The goal is to compare the performance of the clustering methods on different datasets.
Cluster analysis is an unsupervised machine learning technique used to group unlabeled data points into clusters based on similarities. It involves finding groups of objects such that objects within a cluster are more similar to each other than objects in different clusters. The key goals of cluster analysis are to maximize intra-cluster similarity while minimizing inter-cluster similarity. Common applications of cluster analysis include market segmentation, document classification, and identifying homogeneous groups in biological data.
Childhood Abuse and Delinquency 150 Words Research regarding.docxTawnaDelatorrejs
Childhood Abuse and Delinquency 150 Words
Research regarding spanking children has had mixed results, do you think spanking contributes to delinquency or helps to prevent it? Justify your response.
Please remember to use netiquette when responding to your classmates
.
Childrens StoryKnowing how to address a variety of situations in .docxTawnaDelatorrejs
Children's Story
Knowing how to address a variety of situations in the early childhood setting and effectively partnering with parents to do so are important skills for all teachers and caregivers. For this assignment, you will choose one of the following scenarios:
Shane has a difficult time separating from his mother each morning. At drop off, he clings to her and screams uncontrollably. After she leaves, Shane continues to scream and cry until you are able to soothe him.
Lisa often gets frustrated when trying to play with other children. She takes toys from their hands and even hits children with the toys.
Next, address each of the following points according to the teaching approach/setting that best reflects your style in your desired classroom setting (e.g. Montessori, Reggio Emilia, Waldorf, traditional preschool, etc.):
Outline a specific plan for addressing the discipline or guidance scenario.
Explain how your plan would support the teaching approach/setting.
Describe how you will create an effective partnership with parents to address the discipline or guidance scenario.
Describe one or two possible obstacles you might encounter when implementing your plan.
Discuss how you will address these obstacles.
The paper should be three to four pages in addition to the title page and the reference page. Use at least two scholarly sources in addition to your text. Your paper should also be formatted according to APA style as outlined in the Ashford Writing Center.
Description
:
Total Possible Score
: 6.00
Outlines a Specific Plan for Addressing the Discipline or Guidance Scenario
Total: 1.25
Distinguished - Outlines in detail a specific plan for addressing the discipline or guidance scenario. The plan is well supported by scholarly sources.
Proficient - Outlines a specific plan for addressing the discipline or guidance scenario. The plan is supported by scholarly sources but is missing minor details.
Basic - Vaguely outlines a plan for addressing the discipline or guidance scenario; however, the plan may not be sufficiently supported by scholarly sources and is missing relevant details.
Below Expectations - Attempts to outline a plan for addressing the scenario; however, the plan is not sufficiently supported by scholarly sources and is missing significant details.
Non-Performance - The outline of a specific plan is either nonexistent or lacks the components described in the assignment instructions.
Explains How the Plan Supports the Teaching Approach/Setting
Total: 0.50
Distinguished - Clearly and comprehensively explains how the plan supports the chosen teaching approach/setting. The explanation is well supported by scholarly sources.
Proficient - Explains how the plan supports the chosen teaching approach/setting. The explanation is supported by scholarly sources but is slightly underdeveloped.
Basic - Briefly explains how the plan supports the chosen teaching approach/setting. The explanation may not be sufficiently supported by s.
This document provides an introduction to document clustering and clustering algorithms. It discusses how clustering can be used in information retrieval applications like organizing search results and improving search recall. It also covers different types of clustering algorithms like partitioning algorithms (such as K-means) and hierarchical algorithms. Key steps of the K-means and hierarchical agglomerative clustering algorithms are described.
Cluster analysis is used to group similar objects together and separate dissimilar objects. It has applications in understanding data patterns and reducing large datasets. The main types are partitional which divides data into non-overlapping subsets, and hierarchical which arranges clusters in a tree structure. Popular clustering algorithms include k-means, hierarchical clustering, and graph-based clustering. K-means clustering assigns data points to centroids to minimize distance within clusters. Hierarchical clustering creates nested clusters and can be visualized using dendrograms. Both have benefits and limitations depending on the dataset.
Cluster analysis is used to group similar objects together and separate dissimilar objects. It has applications in understanding data patterns and reducing large datasets. The main types are partitional which divides data into non-overlapping subsets, and hierarchical which arranges clusters in a tree structure. Popular clustering algorithms include k-means, hierarchical clustering, and graph-based clustering. K-means partitions data into k clusters by minimizing distances between points and cluster centroids, but requires specifying k and is sensitive to initial centroid positions. Hierarchical clustering creates nested clusters without needing to specify the number of clusters, but has higher computational costs.
Cluster analysis is used to group similar objects together and separate dissimilar objects. It has applications in understanding data patterns and reducing large datasets. The main types are partitional which divides data into non-overlapping subsets, and hierarchical which arranges clusters in a tree structure. Popular clustering algorithms include k-means, hierarchical clustering, and graph-based clustering. K-means partitions data into k clusters by minimizing distances between points and cluster centroids, but requires specifying k and is sensitive to initial centroid positions. Hierarchical clustering creates nested clusters without needing to specify the number of clusters, but has higher computational costs.
This document introduces data exploration and reduction techniques in XLMiner, including principal component analysis and cluster analysis. Principal component analysis transforms correlated variables into uncorrelated principal components to reduce data size while maintaining variability. Cluster analysis assigns objects to clusters to group similar objects together and different objects apart. The document demonstrates how to perform k-means clustering and hierarchical clustering in XLMiner.
This document introduces data exploration and reduction techniques in XLMiner, including principal component analysis and cluster analysis. Principal component analysis transforms correlated variables into uncorrelated principal components to reduce data size while maintaining variability. Cluster analysis assigns objects to clusters to group similar objects together and different objects apart. The document demonstrates how to perform k-means clustering and hierarchical clustering in XLMiner.
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.
This document provides an overview of supervised and unsupervised learning, with a focus on clustering as an unsupervised learning technique. It describes the basic concepts of clustering, including how clustering groups similar data points together without labeled categories. It then covers two main clustering algorithms - k-means, a partitional clustering method, and hierarchical clustering. It discusses aspects like cluster representation, distance functions, strengths and weaknesses of different approaches. The document aims to introduce clustering and compare it with supervised learning.
This document provides an overview of hierarchical clustering methods. It discusses agglomerative hierarchical clustering methods like AGNES that start by treating each object as a separate cluster and merge them into larger clusters. It also discusses divisive hierarchical clustering methods like DIANA that start with all objects in one cluster and split them into smaller clusters. A dendrogram is used to visualize the nested clustering formed at different levels in the hierarchical clustering tree. The document also discusses different measures for calculating the distance between clusters during the merging or splitting process.
Data Mining Primitives, Languages & SystemsNiloy Sikder
The document summarizes the key primitives and concepts in data mining. It discusses the main primitives as task-relevant data, knowledge types to be mined, and background knowledge. It describes various data mining techniques including data selection, grouping, characterization, discrimination, associations and correlations. It also discusses classification methods and prediction. Finally, it covers data mining system architectures and languages such as DMQL and OLE DB.
Clustering is an unsupervised machine learning technique used to group unlabeled data points. There are two main approaches: hierarchical clustering and partitioning clustering. Partitioning clustering algorithms like k-means and k-medoids attempt to partition data into k clusters by optimizing a criterion function. Hierarchical clustering creates nested clusters by merging or splitting clusters. Examples of hierarchical algorithms include agglomerative clustering, which builds clusters from bottom-up, and divisive clustering, which separates clusters from top-down. Clustering can group both numerical and categorical data.
This document provides an introduction and overview of document clustering techniques in information retrieval. It discusses motivations for clustering documents, such as improving search recall and organizing search results. It covers common clustering algorithms like K-means and hierarchical clustering, how they work, and considerations like choosing the number of clusters. The document uses examples and diagrams to illustrate clustering concepts and algorithms.
This document provides an introduction and overview of document clustering techniques in information retrieval. It discusses motivations for clustering documents, such as improving search recall and organizing search results. It covers common clustering algorithms like K-means and hierarchical clustering, how they work, and considerations like choosing the number of clusters. The document uses examples and diagrams to illustrate clustering concepts and algorithms.
This document provides an introduction and overview of document clustering techniques in information retrieval. It discusses motivations for clustering documents, such as improving search recall and organizing search results. It covers common clustering algorithms like K-means and hierarchical clustering, how they work, and considerations like choosing the number of clusters. The document uses examples and diagrams to illustrate clustering concepts and algorithms.
This document provides an introduction and overview of document clustering techniques in information retrieval. It discusses motivations for clustering documents, different document representations, evaluation criteria, and clustering algorithms including partitional algorithms like K-means and hierarchical algorithms. It provides examples and discusses issues like determining the optimal number of clusters to generate. The overall summary is that document clustering groups similar documents together to help with tasks like document navigation, improving search recall, and organizing search results.
This document summarizes a presentation on cluster analysis given by Tekendra Nath Yogi. It defines cluster analysis and describes several clustering methods and algorithms, including k-means clustering. It also discusses applications of cluster analysis in fields like business intelligence, image recognition, web search, and biology. Requirements for effective clustering algorithms are outlined.
This document compares hierarchical and non-hierarchical clustering algorithms. It summarizes four clustering algorithms: K-Means, K-Medoids, Farthest First Clustering (hierarchical algorithms), and DBSCAN (non-hierarchical algorithm). It describes the methodology of each algorithm and provides pseudocode. It also describes the datasets used to evaluate the performance of the algorithms and the evaluation metrics. The goal is to compare the performance of the clustering methods on different datasets.
Cluster analysis is an unsupervised machine learning technique used to group unlabeled data points into clusters based on similarities. It involves finding groups of objects such that objects within a cluster are more similar to each other than objects in different clusters. The key goals of cluster analysis are to maximize intra-cluster similarity while minimizing inter-cluster similarity. Common applications of cluster analysis include market segmentation, document classification, and identifying homogeneous groups in biological data.
Childhood Abuse and Delinquency 150 Words Research regarding.docxTawnaDelatorrejs
Childhood Abuse and Delinquency 150 Words
Research regarding spanking children has had mixed results, do you think spanking contributes to delinquency or helps to prevent it? Justify your response.
Please remember to use netiquette when responding to your classmates
.
Childrens StoryKnowing how to address a variety of situations in .docxTawnaDelatorrejs
Children's Story
Knowing how to address a variety of situations in the early childhood setting and effectively partnering with parents to do so are important skills for all teachers and caregivers. For this assignment, you will choose one of the following scenarios:
Shane has a difficult time separating from his mother each morning. At drop off, he clings to her and screams uncontrollably. After she leaves, Shane continues to scream and cry until you are able to soothe him.
Lisa often gets frustrated when trying to play with other children. She takes toys from their hands and even hits children with the toys.
Next, address each of the following points according to the teaching approach/setting that best reflects your style in your desired classroom setting (e.g. Montessori, Reggio Emilia, Waldorf, traditional preschool, etc.):
Outline a specific plan for addressing the discipline or guidance scenario.
Explain how your plan would support the teaching approach/setting.
Describe how you will create an effective partnership with parents to address the discipline or guidance scenario.
Describe one or two possible obstacles you might encounter when implementing your plan.
Discuss how you will address these obstacles.
The paper should be three to four pages in addition to the title page and the reference page. Use at least two scholarly sources in addition to your text. Your paper should also be formatted according to APA style as outlined in the Ashford Writing Center.
Description
:
Total Possible Score
: 6.00
Outlines a Specific Plan for Addressing the Discipline or Guidance Scenario
Total: 1.25
Distinguished - Outlines in detail a specific plan for addressing the discipline or guidance scenario. The plan is well supported by scholarly sources.
Proficient - Outlines a specific plan for addressing the discipline or guidance scenario. The plan is supported by scholarly sources but is missing minor details.
Basic - Vaguely outlines a plan for addressing the discipline or guidance scenario; however, the plan may not be sufficiently supported by scholarly sources and is missing relevant details.
Below Expectations - Attempts to outline a plan for addressing the scenario; however, the plan is not sufficiently supported by scholarly sources and is missing significant details.
Non-Performance - The outline of a specific plan is either nonexistent or lacks the components described in the assignment instructions.
Explains How the Plan Supports the Teaching Approach/Setting
Total: 0.50
Distinguished - Clearly and comprehensively explains how the plan supports the chosen teaching approach/setting. The explanation is well supported by scholarly sources.
Proficient - Explains how the plan supports the chosen teaching approach/setting. The explanation is supported by scholarly sources but is slightly underdeveloped.
Basic - Briefly explains how the plan supports the chosen teaching approach/setting. The explanation may not be sufficiently supported by s.
Children build their identities based on what they are exposed to, a.docxTawnaDelatorrejs
Children build their identities based on what they are exposed to, as well as how adults and peers interact with them. After having read this Module's materials, let's discuss this further.
What do you think are the most influential factors in the building of multicultural identities in children?
How do you raise children to be sensitive, multicultural adults
.
Child poverty and homelessness are two of the most complex problems .docxTawnaDelatorrejs
Child poverty and homelessness have increased significantly in recent decades in the US. The number of children living in poverty grew from 11.6 million to 15 million between 2000 and 2015, with over 20% of children now living below the federal poverty level. Additionally, between 1-2% of children experience homelessness at some point, a number that rose due to the recent recession. Growing up in poverty puts children at greater risk of physical, cognitive, emotional and social problems. However, politicians and policymakers often disagree on the causes of and solutions to child poverty, leading to vigorous public debate.
Child abuse and neglect are critical issues inherent in the field of.docxTawnaDelatorrejs
Child abuse and neglect are critical issues inherent in the field of human services. You will likely encounter clients who are abused and neglected. Review the characteristics of neglected children in Chapter 4, and answer the following questions:
How does the presence of child abuse or neglect affect a child’s normal development?
How might you respond to a child who indicates that he or she is being abused or neglected?
What agencies would you contact and why?
.
Check.DescriptionI need help with this one-page essay Please!Co.docxTawnaDelatorrejs
Check.
Description:
I need help with this one-page essay Please!Compare and contrast the postcolonial elements that define the works of a range of world authors, including Derek Walcott, Chinua Achebe, Deepika Bahri, W.B. Yeats, Seamus Heaney, E. M. Forster, Salman Rushdie, and Arundhati Roy.
.
Check the paper you write and add your perspective I forgot to say s.docxTawnaDelatorrejs
Check the paper you write and add your perspective I forgot to say some instructions. put some opinion about torah
Write a 3 page paper on what you have learned about Judaism that new for you and which is somehow significant to your understanding about this religion and how it affected your thinking.
Could you add some perspectives to paper you wrote...
i dont want you write new paper just add some opinion to paper
.
Check out attachments and read instructions before you make Hand Sh.docxTawnaDelatorrejs
"Check out attachments and read instructions before you make Hand Shake. Otherwise, I can't sign the agreement"
The most
IMPORTANT
things for me:
1)
Use very simple language, I'm an international student
.
2) Follow ALL instructions carefully 100%.
3) Finish it
on time
.
4) Last but not least,
Originality
.
====
I will run the paper through Copyscape that homework market provides, and the result MUST be = ZERO.
Thanks in advance,
.
check out the attachment, it has prompt, use the 4 website to quote .docxTawnaDelatorrejs
check out the attachment, it has prompt, use the 4 website to quote AND paraphrase (both are required) that i pasted on there. 800 words. APA style
download the attachment and follow the requiremen
1. A Swiveling Proxy That Will Even Wear a Tutu
By ROBBIE BROWNJUNE 7, 2013
http://www.nytimes.com/2013/06/08/education/for-homebound-students-a-robot-proxy-in-the-classroom.html?_r=0
2. How One Boy With Autism Became BFF With Apple’s Siri
By JUDITH NEWMANOCT. 17, 2014
http://www.nytimes.com/2014/10/19/fashion/how-apples-siri-became-one-autistic-boys-bff.html
3. The Ethical Frontiers of Robotics
Noel Sharkey*
http://webpages.uncc.edu/~jmconrad/ECGR4161-2011-05/notes/Science_Article_Robotics_Ethics2.pdf
4. THE ROBOTIC MOMENT
sherry turkle
In late November 2005, I took my daughter Rebecca, then fourteen, to the Darwin exhibition
at the American Museum of Natural History in New York. From the moment you step into
the museum and come face-to-face with a full-size dinosaur, you become part of a celebration
of life on Earth, what Darwin called “endless forms most beautiful.” Millions upon millions of
now lifeless specimens represent nature’s invention in every corner of the globe. There could
be no better venue for documenting Darwin’s life and thought and his theory of evolution by
natural selection, the central truth that underpins contemporary biology. The exhibition aimed
to please and, a bit defensively in these days of attacks on the theory of evolution, wanted to
convince.
At the exhibit’s entrance were two giant tortoises from the Galápagos Islands, the bestknown
inhabitants of the archipelago where Darwin did his most famous investigations. The
museum had been advertising these tortoises as wonders, curiosities, and marvels. Here,
among the plastic models at the museum, was the life that Darwin saw more than a century
and a half ago. One tortoise was hidden from view; the other rested in its cage, utterly still.
Rebecca inspected the visible tortoise thoughtfully for a while and then said matter-of-factly,
“They could have used a robot.” I was taken aback and asked what she meant. She said she
thought it was a shame to bring the turtle all this way from its island home in the Pacific, when
it was just going to sit there in the museum, motionless, doing nothing. Rebecca was both
concerned for the imprisoned turtle and unmoved by its authenticity.
It was Thanksgiving weekend. The line was long, the crowd frozen in place. I began to talk
with some of the other parents and children. My question—“Do you care that the turtle is
alive?”—was a welcome diversion from the boredom of the wait. A ten-year-old girl told me
that she would prefer a robot turtle because aliveness comes with aesthetic inconvenience:
“Its water looks dirty. Gross.” More usually, votes for the robots echoed my daughter’s sentiment
that in this setting, aliveness didn’t seem worth the trouble. A twelve-year-old girl was
adam.
Charles Mann is not only interested in how American societies arrive.docxTawnaDelatorrejs
Charles Mann is not only interested in how American societies arrived, developed, and
evolved, but also how they adapted to the multiple environments of the Americas. How
did indigenous Americans find ways to overcome environmental obstacles? What
techniques, attitudes, or actions did indigenous Americans share? What techniques were
unique to certain areas? Why did some communities and societies thrive in the years
before 1492 while others fell apart and disbanded into new groups or the landscape? How did scholars of the eighteenth, nineteenth, and twentieth centuries differ on their ideas of American Indian development?
.
Check out attachments and read instructions before you make Hand Sha.docxTawnaDelatorrejs
Check out attachments and read instructions before you make Hand Shake.
Otherwise
, I can't sign the agreement"
The most
IMPORTANT
things for me:
1)
Use very simple language, I'm an international student
.
2) Follow ALL instructions carefully 100%.
3) Finish it
on time
.
4) Last but not least, Originality.
====
I will run the paper through Copyscape that homework market provides, and the result MUST be = ZERO.
.
Chapters 5-8. One very significant period in Graphic Design History .docxTawnaDelatorrejs
Chapters 5-8. One very significant period in Graphic Design History was the Renaissance. Maybe a person or object of art made you start thinking about how it was done. here's the link for the chaper that u need to look at
https://www.youtube.com/watch?v=3vCNvvQwCos&list=PLxPtyllY6Cx_Xar71rcNFqX2bDB7Wzfll
.
childrens right in Pakistan.6 pagesat least 7 referencesAPA s.docxTawnaDelatorrejs
children's right in Pakistan.
6 pages
at least 7 references
APA style
References, citation needed
outline:
1.
Country in context
2.
Demographics
3.
History
4.
Culture and socio-economic context: official language, religion,
5.
Legislation/policies addressing rights
6.
Health status of child
7.
Education
8.
Well-being and quality of life: human develop index
9.
Status of children with special needs
10.
summary
.
CHAPTER ONEIntroductionLearning Objectives• Be able to concept.docxTawnaDelatorrejs
CHAPTER ONEIntroduction
Learning Objectives
• Be able to conceptualize the “information explosion” and how it relates to the brain sciences.
• Be able to describe pharmacodynamics and pharmacokinetics.
• Be able to articulate the benefits of an integrative approach to psychopharmacology.
ENCOURAGEMENT TO THE READER
Some of you may begin this book with some anxiety because this is a new area for you. You may imagine that psychopharmacology is exclusively a “hard science,” and perhaps you don't think of yourself as a “hard science” kind of person. You may even feel uncertain about your ability to master basic psychopharmacological concepts. First, let us assure you one more time that our goal is to make this topic accessible to readers who are practicing as or studying to be mental health professionals, many of whom may not have a background in the physical or organic sciences. Second, we recommend to those teaching a course in psychopharmacology that, because of the rapid nature of change in the field, teaching styles that rely on memorization are of limited use in this area. We recommend helping students master basic concepts and then applying these concepts to cases. To facilitate that process, we supply cases and objectives/review questions for main sections of the book. Finally, we invite you students to join us in an incredible journey centering on the most complex organ known to humanity—the human mind and brain. We hope you can revel in the complexity of the brain and the sheer magnitude of its power. We hope you can resist the temptation to want simple and concrete answers to many of the questions this journey will raise. We also hope you learn to appreciate the ambiguous nature of “mind” and its relationship to the brain. As authors and researchers who have traveled this path before us will attest, there are no simple or even known answers to many of the questions that arise (Grilly & Salmone, 2011; Schatzberg & Nemeroff, 1998). We encourage a mixture of trying to comprehend the information while dwelling in the mystery that is the context for the information. Before moving on, we offer a mantra to help you implement this recommendation.
A MANTRA
Even though psychopharmacology is in its embryonic stage, it is a vast and complex topic. Several years ago I (Ingersoll) engaged in some multicultural counseling training with Paul Pederson. In that training, Dr. Pederson commented, “Culture is complex, and complexity is our friend.” We offer a paraphrase as a mantra for psychopharmacology students: “Reality is complex, and complexity is our friend.” We remind the reader of this mantra throughout the book. You might try saying it aloud right now: “Reality is complex, and complexity is our friend.” If you reach a passage in this book that is challenging for you or that arouses anxiety, stop, take a deep breath, and practice the mantra.
The primary audience for this book is mental health clinicians who may not have had much training in biology.
Chapter TenThe Federal JudiciaryBrian M. MurphyLearnin.docxTawnaDelatorrejs
Chapter Ten
The Federal Judiciary
Brian M. Murphy
Learning Objectives
After covering the topic of the federal judiciary, students should
understand:
1. The relationship of state courts to the federal judiciary.
2. The jurisdiction of federal courts.
3. The structure of the federal judicial system.
4. The procedures of the U.S. Supreme Court.
5. The powers of the federal judiciary.
Abstract
The udicial y e i he i ed a e i a ed he d c ri e
federalism. Two court systems exist side-by-side, national and state, and
each has a distinct set of powers. State courts, for the most part, are
responsible for handling the legal issues that arise under their own laws. It
is primarily when a federal uestion is presented that the federal udicial
system can become in ol ed in a state court. therwise, state udiciaries
are generally autonomous even from one another. The Constitution
precisely outlines the types of cases that can be heard by federal courts,
yet it is almost impossible to force a federal court to hear a case that falls
under its urisdiction if the udge s wants to avoid it. The authority of
the U.S. Supreme Court has slowly grown over time, largely through the
power of udicial review. onetheless, federalism has managed to remain
a signi cant barrier against federal courts becoming too powerful. The
udicial system designed by the framers continues to survive and function
after 200 years.
Introduction
The federal judicial system is the least commonly known and least
understood branch of American government. In 2007, 78% could not
name the current Chief Justice of the U.S. Supreme Court but 66% were
able to identify at least one of the judges on the T show American
Idol (Jamieson, 2007). Much of judicial work is conducted out of the
limelight and courts are not considered an important in uence in the daily
lives of people. It is clear the framers believed that the federal judicial
system would be the weakest of the three branches because, as Alexander
amilton wrote, it has no in uence over either the sword or the purse
(Hamilton, 1961, 465). In other words, courts cannot command an army
(or even police) to ensure that decisions are enforced or allocate money to
implement one of their rulings. Judges must depend on the other branches
in order to get anything done. According to an oft-repeated story, President
Andrew Jackson supposedly mocked a decision by Chief Justice John
Marshall with the words, John Marshall has made his decision, now let
him enforce it’’ (Schwartz, 1993, 94).
But times and the role of the federal judiciary have changed. One
scholar even concluded that the United States is now operating under a
government by judiciary’’ because the U.S. Supreme Court can revise
the Constitution by how it interprets the wording (Berger, 1997). As Chief
Justice Charles vans Hughes once uipped, e are under a Constitution,
but the Constitution is what the judges say it is’’ (Hughes, 1916, 185). .
Chapter 9 provides a discussion of the challenges of identifying ELL.docxTawnaDelatorrejs
Chapter 9 provides a discussion of the challenges of identifying ELLs’ as having a learning disability or being gifted with their lower than grade-level proficiency in English. After reading Chapter 9, write a post that addresses the following questions:
What kinds of disabilities might an ELL have?
What are the challenges of determining whether an ELL has a learning ability or is gifted?
What kinds of interventions are used once an ELL has been identified as having a learning disability?
What kinds of interventions are used once an ELL is determined to be gifted?
If you were teaching a class with some ELLs in it, what signals would you look for in the behavior or they ELLs to determine whether they might need to be tested for learning disabilities or being gifted?
How might you adapt your curriculum for an ELL student with a learning disability or who is gifted?
.
Chapter 8 -- Crimes
1. Conduct that may be a misdemeanor in one state may be a felony in another state.
2. A required element for a crime is that the criminal party voluntarily commits the prohibited act (think “gun to head”).
3. A person cannot commit a crime if the person does not know that his or her conduct is criminal (think “Honduran bony fish or short lobster).
4. The Fourth Amendment prohibits ALL government searches of businesses.
5. Traditionally, extortion involves wrongful demands made by public officials.
6. A company cannot be found guilty of a crime that is committed by its agent.
7. If an employee wrongfully keeps money that was entrusted to the employee by his or employer, the employee has committed the crime of embezzlement.
8. Government officers do not need a search warrant in order to inspect property that is in "plain view".
9. The Constitution guarantees individuals the right to a speedy trial in criminal cases.
10. The Digital Millennium Copyright Act allows a person to thwart encryption devices that copy right holders place on copyrighted material if the person has purchased the copyrighted item in question.
Chapter 9 -- Torts
11. One wrongful act may be both a crime and a tort.
12. A person is not entitled to recover for EVERY injury or loss that is caused by another person.
13. In general, tort liability will not be imposed for an involuntary act even if the act harms another.
14. Under tort law, one owes a duty to society to conform his or her conduct to a required standard (think: does society sue the tortfeasor does the “somebody done me wrong” individual plaintiff sue the tortfeasor?).
15. The U.S. government cannot be sued for harm caused by the negligence of federal employees.
16. In some states, a plaintiff may recover for emotional distress that is negligently caused by another.
17. Companies can now make commercial use of the name or likeness of celebrities without first obtaining the celebrities permission to do so because most states do not recognize the tort of invasion of the right to publicity.
Chapter 10
18.
Patents are granted by state governments, not by the federal government.
19.
Trademarks may be protected for up to three years prior to the time that they are actually used.
20. A “term” acquires a secondary meaning when, through prolonged use, the public has come to associate that term with a particular product.
21. In general, mere ideas and concepts cannot be copyrighted or patented.
22.
A trade secret may be disclosed without losing its legal .
chapter 5 Making recommendations for I studied up to this .docxTawnaDelatorrejs
chapter 5
Making recommendations for I studied up to this point, what should now be study after I have written about what I found. All chapter 5 about chapter 4 what all things I discovered, what senses do they make to you what would you have study more if you have more time, what I think about , what I found
.
Chapter 4. Terris, Daniel. (2005) Ethics at Work Creating Virtue at.docxTawnaDelatorrejs
Chapter 4. Terris, Daniel. (2005) Ethics at Work: Creating Virtue at an American Corporation. Brandeis University Press. Apply critical thinking skills
in evaluating Lockheed Martin's efforts.
1. What do you think about the notion presented by Terris that Lockheed's ethics program does little to prevent ethical breaches at the highest level of the organization?
2. Are the efforts put forth—such as making sure higher level executives participate in training—enough to help executives navigate what Terris calls the 'ethical minefield' faced by leadership in such an organization?
3. What are some things that could be done to address the issue related to ethics at higher executive levels of the organization?
4. Terris points out that the company's program is overly focused on individuals and that it doesn't really address group dynamics that can impact ethical situations. For instance, there can be a tendency for groups to ‘go with the flow’ of the group decision making process and overlook ethical issues in the process. What would you recommend that Lockheed Martin do to address this situation?
(Hint: reviewing p. 128 and the following pages – before section headed “Personal Responsibility, Collective Innocence” - of the text might be helpful).
Assignment Expectations: Write a 4- to 5-page paper, not including title page or references page addressing the issue.
Your paper should be double-spaced and in 12-point type size.
Your paper should have a separate cover page and a separate reference page. Make sure you cite your sources.
.
Chapter 41. Read in the text about Alexanders attempt to fuse Gre.docxTawnaDelatorrejs
Chapter 4
1. Read in the text about Alexander's attempt to fuse Greek and Eastern cultures (116-120 -see box Alexander meets an Indian King, 115). Then go to:
Alexander the Great
- a from a BBC documentary. The video will have to be opened in a new window.
Write a brief review after watching the documentary (You don't have to watch the entire hour). What does Wood have to say about the scope of Alexander the Great's accomplishments? Does watching a video set in the actual landscape of Macedonia and Turkey help understand the history of an ancient civilization? How?
2. Go to:
Building of the Parthenon
and
Optical 'tricks' at the Parthenon
to see the accomplishments of Greek architects and politicians. What is the connection between Athenian politics and the building of the Parthenon? What illusions were utlitzed by the architects and engineers to emphasize the grandeur of the Parthenon?
Chapter 5
Select TWO of the following questions and complete the links assignments: Remember to mention source material in your response.
(Select 3 for extra credit
1. Go to:
Roman Writers view their world
and choose 2 authors to write an essay on entertainments and past times of Roman citizens and how eyewitnesses wrote about their world. Who are they? What position did they hold in Roman society? Is this important to their view point?
2. Go to
Christian symbolism
and
Colors in religious art
and write about how a largely illiterate (slave and lower class Romans and client state residents) society could learn about this new "Christian" religion through art, symbolism and color. How would this help the conversion process?
3. Go to
Sights along the Silk Road
. Click on the interactive maps and visit several of the stops along the Silk Road. What did you find? Learn? Then go to :
Silk Road Project
. Click on "Music and Artists." Then "Listen to Music."
Click on a title for ex: "Arabian" to listen to sample of the music and instrument. Write on your findings.
You may have to update your "Flash" player to hear music
.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Thinking of getting a dog? Be aware that breeds like Pit Bulls, Rottweilers, and German Shepherds can be loyal and dangerous. Proper training and socialization are crucial to preventing aggressive behaviors. Ensure safety by understanding their needs and always supervising interactions. Stay safe, and enjoy your furry friends!
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
1. Chapter 4: Clustering
I
02/14/2018 Introduction to Data Mining, 2nd Edition
‹#›
What is Cluster Analysis?
Finding groups of objects such that the objects in a group will
be similar (or related) to one another and different from (or
unrelated to) the objects in other groups
2. Inter-cluster distances are maximized
Intra-cluster distances are minimized
02/14/2018 Introduction to Data Mining, 2nd Edition
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Applications of Cluster Analysis
Understanding
Group related documents for browsing, group genes and
proteins that have similar functionality, or group stocks with
similar price fluctuations
Summarization
Reduce the size of large data sets
3. Clustering precipitation in Australia
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What is not Cluster Analysis?
Simple segmentation
Dividing students into different registration groups
alphabetically, by last name
Results of a query
Groupings are a result of an external specification
Clustering is a grouping of objects based on the data
Supervised classification
Have class label information
Association Analysis
Local vs. global connections
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Notion of a Cluster can be Ambiguous
6. Six Clusters
02/14/2018 Introduction to Data Mining, 2nd Edition
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Types of Clusterings
A clustering is a set of clusters
Important distinction between hierarchical and partitional sets
of clusters
Partitional Clustering
A division of data objects into non-overlapping subsets
(clusters) such that each data object is in exactly one subset
Hierarchical clustering
A set of nested clusters organized as a hierarchical tree
02/14/2018 Introduction to Data Mining, 2nd Edition
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Partitional Clustering
7. Original Points
A Partitional Clustering
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Hierarchical Clustering
Traditional Hierarchical Clustering
Non-traditional Hierarchical Clustering
Non-traditional Dendrogram
Traditional Dendrogram
02/14/2018 Introduction to Data Mining, 2nd Edition
8. ‹#›
Types of Clusters
Well-separated clusters
Center-based clusters
Contiguous clusters
Density-based clusters
Property or Conceptual
Described by an Objective Function
02/14/2018 Introduction to Data Mining, 2nd Edition
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Types of Clusters: Well-Separated
Well-Separated Clusters:
A cluster is a set of points such that any point in a cluster is
closer (or more similar) to every other point in the cluster than
to any point not in the cluster.
3 well-separated clusters
02/14/2018 Introduction to Data Mining, 2nd Edition
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9. Types of Clusters: Center-Based
Center-based
A cluster is a set of objects such that an object in a cluster is
closer to the “center” of a cluster, than to the center of any
other cluster
The center of a cluster is a centroid.
The average of all the points in the cluster is a medoid
4 center-based clusters
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Types of Clusters: Contiguity-Based
Contiguous Cluster (Nearest neighbor or Transitive)
A point in a cluster is closer (or more similar) to one or more
other points in the cluster than to any point not in the cluster.
8 contiguous clusters
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Types of Clusters: Density-Based
Density-based
A cluster is a dense region of points, which is separated by low -
density regions, from other regions of high density.
Used when the clusters are irregular or intertwined, and when
noise and outliers are present.
6 density-based clusters
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Types of Clusters: Conceptual Clusters
Shared Property or Conceptual Clusters
Finds clusters that share some common property or represent a
particular concept.
.
2 Overlapping Circles
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Types of Clusters: Objective Function
Clusters Defined by an Objective Function
Finds clusters that minimize or maximize an objective function.
Enumerate all possible ways of dividing the points into clusters
and evaluate the `goodness' of each potential set of clusters by
using the given objective function. (NP Hard)
Can have global or local objectives.
Hierarchical clustering algorithms typically have local
objectives
Partitional algorithms typically have global objectives
A variation of the global objective function approach is to fit
the data to a parameterized model.
Parameters for the model are determined from the data.
Mixture models assume that the data is a ‘mixture' of a number
of statistical distributions.
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Characteristics of the Input Data Are Important
Type of proximity or density measure
Central to clustering
Depends on data and application
Data characteristics that affect proximity and/or density are
Dimensionality
Sparseness
Attribute type
Special relationships in the data
12. For example, autocorrelation
Distribution of the data
Noise and Outliers
Often interfere with the operation of the clustering algorithm
02/14/2018 Introduction to Data Mining, 2nd Edition
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Clustering Algorithms
K-means and its variants
Hierarchical clustering
Density-based clustering
02/14/2018 Introduction to Data Mining, 2nd Edition
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K-means Clustering
Partitional clustering approach
Number of clusters, K, must be specified
Each cluster is associated with a centroid (center point)
Each point is assigned to the cluster with the closest centroid
The basic algorithm is very simple
13. 02/14/2018 Introduction to Data Mining, 2nd Edition
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Example of K-means Clustering
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Example of K-means Clustering
02/14/2018 Introduction to Data Mining, 2nd Edition
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K-means Clustering – Details
Initial centroids are often chosen randomly.
14. Clusters produced vary from one run to another.
The centroid is (typically) the mean of the points in the cluster.
‘Closeness’ is measured by Euclidean distance, cosine
similarity, correlation, etc.
K-means will converge for common similarity measures
mentioned above.
Most of the convergence happens in the first few iterations.
Often the stopping condition is changed to ‘Until relatively few
points change clusters’
Complexity is O( n * K * I * d )
n = number of points, K = number of clusters,
I = number of iterations, d = number of attributes
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Evaluating K-means Clusters
Most common measure is Sum of Squared Error (SSE)
For each point, the error is the distance to the nearest cluster
To get SSE, we square these errors and sum them.
x is a data point in cluster Ci and mi is the representative point
for cluster Ci
can show that mi corresponds to the center (mean) of the
cluster
Given two sets of clusters, we prefer the one with the smallest
error
One easy way to reduce SSE is to increase K, the number of
clusters
A good clustering with smaller K can have a lower SSE than a
poor clustering with higher K
15. 02/14/2018 Introduction to Data Mining, 2nd Edition
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Two different K-means Clusterings
Sub-optimal Clustering
Optimal Clustering
Original Points
02/14/2018 Introduction to Data Mining, 2nd Edition
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Limitations of K-means
K-means has problems when clusters are of differing
Sizes
Densities
Non-globular shapes
K-means has problems when the data contains outliers.
02/14/2018 Introduction to Data Mining, 2nd Edition
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Limitations of K-means: Differing Sizes
16. Original Points
K-means (3 Clusters)
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Limitations of K-means: Differing Density
Original Points
K-means (3 Clusters)
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Limitations of K-means: Non-globular Shapes
Original Points
K-means (2 Clusters)
17. 02/14/2018 Introduction to Data Mining, 2nd Edition
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Overcoming K-means Limitations
Original PointsK-means Clusters
One solution is to use many clusters.
Find parts of clusters, but need to put together.
02/14/2018 Introduction to Data Mining, 2nd Edition
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Overcoming K-means Limitations
Original PointsK-means Clusters
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Overcoming K-means Limitations
18. Original PointsK-means Clusters
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Importance of Choosing Initial Centroids
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Importance of Choosing Initial Centroids
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19. ‹#›
Importance of Choosing Initial Centroids …
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Importance of Choosing Initial Centroids …
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Problems with Selecting Initial Points
If there are K ‘real’ clusters then the chance of selecting one
centroid from each cluster is small.
Chance is relatively small when K is large
If clusters are the same size, n, then
For example, if K = 10, then probability = 10!/1010 = 0.00036
Sometimes the initial centroids will readjust themselves in
20. ‘right’ way, and sometimes they don’t
Consider an example of five pairs of clusters
02/14/2018 Introduction to Data Mining, 2nd Edition
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10 Clusters Example
Starting with two initial centroids in one cluster of each pair of
clusters
02/14/2018 Introduction to Data Mining, 2nd Edition
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10 Clusters Example
Starting with two initial centroids in one cluster of each pair of
clusters
02/14/2018 Introduction to Data Mining, 2nd Edition
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21. Solution
s to Initial Centroids Problem
Multiple runs
Helps, but probability is not on your side
Sample and use hierarchical clustering to determine initial
centroids
Select more than k initial centroids and then select among these
initial centroids
Select most widely separated
Generate a larger number of clusters and then perform a
hierarchical clustering
02/14/2018 Introduction to Data Mining, 2nd Edition
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Hierarchical Clustering
Produces a set of nested clusters organized as a hierarchical tree
Can be visualized as a dendrogram
A tree like diagram that records the sequences of merges or
splits
22. 02/14/2018 Introduction to Data Mining, 2nd Edition
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Strengths of Hierarchical Clustering
Do not have to assume any particular number of clusters
Any desired number of clusters can be obtained by ‘cutting’ the
dendrogram at the proper level
They may correspond to meaningful taxonomies
Example in biological sciences (e.g., animal kingdom)
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Hierarchical Clustering
Two main types of hierarchical clustering
Agglomerative:
23. Start with the points as individual clusters
At each step, merge the closest pair of clusters until only one
cluster (or k clusters) left
Divisive:
Start with one, all-inclusive cluster
At each step, split a cluster until each cluster contains an
individual point (or there are k clusters)
Traditional hierarchical algorithms use a similarity or distance
matrix
Merge or split one cluster at a time
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Agglomerative Clustering Algorithm
Most popular hierarchical clustering technique
Basic algorithm is straightforward
Compute the proximity matrix
Let each data point be a cluster
24. Repeat
Merge the two closest clusters
Update the proximity matrix
Until only a single cluster remains
Key operation is the computation of the proximity of two
clusters
Different approaches to defining the distance between clusters
distinguish the different algorithms
02/14/2018 Introduction to Data Mining, 2nd Edition
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Proximity Matrix
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‹#›
Starting Situation
28. C4
C5
Proximity Matrix
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Intermediate Situation
We want to merge the two closest clusters (C2 and C5) and
update the proximity matrix.
36. Proximity Matrix
MIN
MAX
Group Average
Distance Between Centroids
Other methods driven by an objective function
Ward’s Method uses squared error
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How to Define Inter-Cluster Similarity
38. Proximity Matrix
MIN
MAX
Group Average
Distance Between Centroids
Other methods driven by an objective function
Ward’s Method uses squared error
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How to Define Inter-Cluster Similarity
41. Proximity Matrix
MIN
MAX
Group Average
Distance Between Centroids
Other methods driven by an objective function
Ward’s Method uses squared error
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How to Define Inter-Cluster Similarity
43. Proximity Matrix
MIN
MAX
Group Average
Distance Between Centroids
Other methods driven by an objective function
Ward’s Method uses squared error
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MIN or Single Link
Proximity of two clusters is based on the two closest points in
the different clusters
Determined by one pair of points, i.e., by one link in the
proximity graph
Example:
45. 2
3
4
5
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Strength of MIN
Original Points
Six Clusters
Can handle non-elliptical shapes
02/14/2018 Introduction to Data Mining, 2nd Edition
46. ‹#›
Limitations of MIN
Original Points
Two Clusters
Sensitive to noise and outliers
Three Clusters
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MAX or Complete Linkage
Proximity of two clusters is based on the two most distant
points in the different clusters
Determined by all pairs of points in the two clusters
Distance Matrix:
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Hierarchical Clustering: MAX
Nested Clusters
Dendrogram
1
2
3
4
5
6
1
2
48. 5
3
4
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Strength of MAX
Original Points
Two Clusters
Less susceptible to noise and outliers
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Limitations of MAX
49. Original Points
Two Clusters
Tends to break large clusters
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Group Average
Proximity of two clusters is the average of pairwise proximity
between points in the two clusters.
Need to use average connectivity for scalability since total
proximity favors large clusters
Distance Matrix:
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Hierarchical Clustering: Group Average
Nested Clusters
Dendrogram
1
2
3
4
5
6
1
51. 2
5
3
4
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Hierarchical Clustering: Group Average
Compromise between Single and Complete Link
Strengths
Less susceptible to noise and outliers
Limitations
Biased towards globular clusters
02/14/2018 Introduction to Data Mining, 2nd Edition
52. ‹#›
Cluster Similarity: Ward’s Method
Similarity of two clusters is based on the increase in squared
error when two clusters are merged
Similar to group average if distance between points is distance
squared
Less susceptible to noise and outliers
Biased towards globular clusters
Hierarchical analogue of K-means
Can be used to initialize K-means
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Hierarchical Clustering: Comparison
Group Average
Ward’s Method
56. 4
5
6
1
2
3
4
5
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Hierarchical Clustering: Time and Space requirements
O(N2) space since it uses the proximity matrix.
N is the number of points.
O(N3) time in many cases
There are N steps and at each step the size, N2, proximity
57. matrix must be updated and searched
Complexity can be reduced to O(N2 log(N) ) time with some
cleverness
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Hierarchical Clustering: Problems and Limitations
Once a decision is made to combine two clusters, it cannot be
undone
No global objective function is directly minimized
Different schemes have problems with one or more of the
following:
Sensitivity to noise and outliers
Difficulty handling clusters of different sizes and non-globular
shapes
Breaking large clusters
58. 02/14/2018 Introduction to Data Mining, 2nd Edition
‹#›
Cluster Validity
For supervised classification we have a variety of measures to
evaluate how good our model is
Accuracy, precision, recall
For cluster analysis, the analogous question is how to evaluate
the “goodness” of the resulting clusters?
But “clusters are in the eye of the beholder”!
Then why do we want to evaluate them?
To avoid finding patterns in noise
To compare clustering algorithms
To compare two sets of clusters
To compare two clusters
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‹#›
59. Clusters found in Random Data
Random Points
K-means
Complete Link
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‹#›
Determining the clustering tendency of a set of data, i.e.,
distinguishing whether non-random structure actually exists in
the data.
Comparing the results of a cluster analysis to externally known
results, e.g., to externally given class labels.
Evaluating how well the results of a cluster analysis fit the data
without reference to external information.
- Use only the data
Comparing the results of two different sets of cluster analyses
to determine which is better.
Determining the ‘correct’ number of clusters.
60. For 2, 3, and 4, we can further distinguish whether we want to
evaluate the entire clustering or just individual clusters.
Different Aspects of Cluster Validation
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‹#›
Numerical measures that are applied to judge various aspects of
cluster validity, are classified into the following three types.
External Index: Used to measure the extent to which cluster
labels match externally supplied class labels.
Entropy
Internal Index: Used to measure the goodness of a clustering
structure without respect to external information.
Sum of Squared Error (SSE)
Relative Index: Used to compare two different clusterings or
clusters.
Often an external or internal index is used for this functi on,
e.g., SSE or entropy
Sometimes these are referred to as criteria instead of indices
However, sometimes criterion is the general strategy and index
61. is the numerical measure that implements the criterion.
Measures of Cluster Validity
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‹#›
Two matrices
Proximity Matrix
Ideal Similarity Matrix
One row and one column for each data point
An entry is 1 if the associated pair of points belong to the same
cluster
An entry is 0 if the associated pair of points belongs to different
clusters
Compute the correlation between the two matrices
Since the matrices are symmetric, only the correlation between
n(n-1) / 2 entries needs to be calculated.
High correlation indicates that points that belong to the same
cluster are close to each other.
Not a good measure for some density or contiguity based
clusters.
Measuring Cluster Validity Via Correlation
62. 02/14/2018 Introduction to Data Mining, 2nd Edition
‹#›
Measuring Cluster Validity Via Correlation
Correlation of ideal similarity and proximity matrices for the K-
means clusterings of the following two data sets.
Corr = -0.9235
Corr = -0.5810
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‹#›
Order the similarity matrix with respect to cluster labels and
inspect visually.
Using Similarity Matrix for Cluster Validation
63. 02/14/2018 Introduction to Data Mining, 2nd Edition
‹#›
Using Similarity Matrix for Cluster Validation
Clusters in random data are not so crisp
K-means
02/14/2018 Introduction to Data Mining, 2nd Edition
‹#›
Using Similarity Matrix for Cluster Validation
Clusters in random data are not so crisp
Complete Link
64. 02/14/2018 Introduction to Data Mining, 2nd Edition
‹#›
Clusters in more complicated figures aren’t well separated
Internal Index: Used to measure the goodness of a clustering
structure without respect to external information
SSE
SSE is good for comparing two clusterings or two clusters
(average SSE).
Can also be used to estimate the number of clusters
Internal Measures: SSE
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‹#›
Internal Measures: SSE
SSE curve for a more complicated data set
65. SSE of clusters found using K-means
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‹#›
Correlation of ideal similarity and proximity matrices for the K-
means clusterings of the following two data sets.
Statistical Framework for Correlation
Corr = -0.9235
Corr = -0.5810
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‹#›
Cluster Cohesion: Measures how closely related are objects in a
66. cluster
Example: SSE
Cluster Separation: Measure how distinct or well-separated a
cluster is from other clusters
Example: Squared Error
Cohesion is measured by the within cluster sum of squares
(SSE)
Separation is measured by the between cluster sum of squares
Where |Ci| is the size of cluster i
Internal Measures: Cohesion and Separation
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‹#›
68. K=2 clusters:
K=1 cluster:
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‹#›
A proximity graph based approach can also be used for cohesion
and separation.
Cluster cohesion is the sum of the weight of all links within a
cluster.
Cluster separation is the sum of the weights between nodes in
the cluster and nodes outside the cluster.
Internal Measures: Cohesion and Separation
69.
70. cohesion
separation
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‹#›
“The validation of clustering structures is the most difficult
and frustrating part of cluster analysis.
Without a strong effort in this direction, cluster analysis will
remain a black art accessible only to those true believers who
have experience and great courage.”
Algorithms for Clustering Data, Jain and Dubes
Final Comment on Cluster Validity
71. 02/14/2018 Introduction to Data Mining, 2nd Edition
‹#›
Discovered Clusters Industry Group
1
Applied-Matl-DOWN,Bay-Network-Down,3-COM-DOWN,
Cabletron-Sys-DOWN,CISCO-DOWN,HP-DOWN,
DSC-Comm-DOWN,INTEL-DOWN,LSI-Logic-DOWN,
Micron-Tech-DOWN,Texas-Inst-Down,Tellabs-Inc-Down,
Natl-Semiconduct-DOWN,Oracl-DOWN,SGI-DOWN,
Sun-DOWN
Technology1-DOWN
2
Apple-Comp-DOWN,Autodesk-DOWN,DEC-DOWN,
ADV-Micro-Device-DOWN,Andrew-Corp-DOWN,
Computer-Assoc-DOWN,Circuit-City-DOWN,
Compaq-DOWN, EMC-Corp-DOWN, Gen-Inst-DOWN,
Motorola-DOWN,Microsoft-DOWN,Scientific-Atl-DOWN
Technology2-DOWN
3
Fannie-Mae-DOWN,Fed-Home-Loan-DOWN,
73. JOURNAL OF APPLIED BEHAVIOR ANALYSIS
WHAT MAKES EXTINCTION WORK: AN ANALYSIS OF
PROCEDURAL FORM AND FUNCTION
BRIAN A. IWATA
THE UNIVERSITY OF FLORIDA
GARY M. PACE
THE KENNEDY INSTITUTE, BALTIMORE, MARYLAND
GLYNNIS EDWARDS COWDERY
WE CARE, INC., BERKELEY, CALIFORNIA
AND
RAYMOND G. MILTENBERGER
NORTH DAKOTA STATE UNIVERSITY
We examined methods for determining how extinction should be
applied to different functions of
self-injurious behavior (SIB). Assessment data indicated that
74. the head banging of 3 children with
developmental disabilities was maintained by different
reinforcement contingencies: One subject's
SIB was positively reinforced by attention from adults, the 2nd
subject's SIB was negatively reinforced
by escape from educational tasks, and the 3rd subject's SIB
appeared to be automatically reinforced
or "self-stimulatory" in nature. Three functional variations of
extinction-EXT (attention), EXT
(escape), and EXT (sensory)-were evaluated, and each subject
was exposed to at least two of these
variations in reversal or multiple baseline designs. Reductions
in SIB were observed only when
implementation of "extinction" involved the discontinuation of
reinforcement previously shown to
be responsible for maintaining the behavior. These results
highlight important differences among
treatment techniques based on the same behavioral principle
(extinction) when applied to topo-
graphically similar but functionally dissimilar responses, and
further illustrate the practical impli-
cations of a functional analysis of behavior disorders for
designing, selecting, and classifying ther-
apeutic interventions.
DESCRIPTORS: extinction, functional analysis, self-injurious
75. behavior
Research on the functional analysis of severe
behavior disorders has produced a variety of inter-
ventions based on the modification of antecedent
events that occasion behavior problems as well as
the consequences that maintain them (see Iwata,
Vollmer, & Zarcone, 1990, and Mace, Lalli, &
Lalli, 1991, for recent reviews). Much of the em-
phasis in treatment has been on strengthening new
stimulus-response-consequence relationships. By
contrast, relatively little attention has been paid to
eliminating reinforcement that maintained the be-
This research was supported in part by a grant from the
Developmental Disabilities Planning Council and the Florida
Department of Health and Rehabilitative Services.
Reprints may be obtained from Brian Iwata, Psychology
Department, University of Florida, Gainesville, Florida 3261 1.
havior problem in the first place, even though many
of the interventions described in the literature im-
plicitly contained such a provision. That is, pun-
ishment of Response A is almost always combined
76. with the cessation of reinforcement; likewise, treat-
ment involving reinforcement of Response B as a
replacement for Response A usually coincides with
the termination of reinforcement for Response A.
Moreover, data from several recent studies suggest
that the effects of reinforcement-based interventions
may be limited unless extinction (withholding the
behavior's maintaining reinforcers) is included as
part of the treatment program. It has been shown,
for example, that interventions based on the de-
velopment of communicative responses (Carr &
Durand, 1985) and on behavioral momentum
(Mace & Belfiore, 1990) produced either mixed
131
1994127,131-144 NUMBER I (SPRING 1994)
BRIAN A. IWATA et al.
results (Wacker et al., 1990) or no treatment effect
(Zarcone, Iwata, Hughes, & Vollmer, 1993) when
implemented without extinction, and that the crit-
77. ical feature of procedures such as differential rein-
forcement of other behavior (DRO) contingencies
appears to be the extinction component rather than
the reinforcement component (Mazaleski, Iwata,
Vollmer, Zarcone, & Smith, 1993).
The assessment and treatment of self-injurious
behavior (SIB) provide an important context for
the examination of extinction, because it has been
demonstrated that SIB can be maintained through
a variety of operant mechanisms. Nevertheless, be-
cause the "discontinuation of reinforcement" for
behavior problems in general often involves ces-
sation of ongoing events, this procedure has become
so common that some textbooks define extinction
solely through reference to "ignoring," time-out,
and other examples of stimulus termination (e.g.,
LaVigna & Donnellan, 1986). However, research
has shown that procedural time-out can serve dif-
ferent functions, not all of which are behavior re-
ducing (Plummer, Baer, & LeBlanc, 1977; Solnick,
Rincover, & Peterson, 1977), and the same is true
of procedural approaches to defining extinction.
Ignoring misbehavior may represent extinction in
some cases but not in others; conversely, the correct
78. application of extinction may require termination
of events in some cases but continuation in others.
The procedures that define extinction in a given
situation are determined by the specific nature of
the reinforcement to be "discontinued."
In cases in which SIB was maintained by social-
positive reinforcement in the form of adult atten-
tion, extinction consisted of withholding attention
or terminating it contingent on the occurrence of
SIB (e.g., Day, Rea, Schussler, Larsen, & Johnson,
1988; Lovaas & Simmons, 1969). By contrast, SIB
maintained through social-negative reinforcement
in the form of escape from task demands has been
extinguished by preventing escape; in other words,
continuing and not terminating the ongoing sit-
uation (e.g., Iwata, Pace, Kalsher, Cowdery, &
Cataldo, 1990; Repp, Felce, & Barton, 1988).
Finally, SIB apparently maintained by nonsocial,
automatic reinforcement (e.g., sensory stimulation)
has shown extinction-like decreases when the in-
dividuals wore equipment that allowed the behav-
ior to occur, but attenuated its consequences (e.g.,
Rincover & Devany, 1982). Thus, at least three
79. functional variations of extinction have been used
as treatment for SIB, each designed to terminate a
different source of reinforcement, and each ame-
nable to a number of procedural modifications and
descriptive labels.
Although extinction procedures are clearly unique
from the standpoint of both form (specific therapist
actions) and function (the maintaining contingen-
cies to which they apply), these variations are not
well differentiated through existing terminology.
Planned ignoring, the label perhaps most often
applied to extinction of attention-maintained be-
havior (Nelson & Rutherford, 1983), provides an
adequate description of the therapist's response, but
it does not describe the underlying behavioral pro-
cess (extinction), nor does it identify the source of
reinforcement being withheld (attention). Escape
extinction, a term used to describe extinction of
escape-maintained behavior (Iwata, Pace, Kalsher,
Cowdery, & Cataldo, 1990), specifies the relevant
process as well as the reinforcer, but not the pro-
cedure. Finally, sensory extinction, which refers to
a variety of techniques designed to attenuate stim-
ulation directly produced by a response (Rincover,
80. 1978; Rincover, Cook, Peoples, & Packard, 1979),
has the same descriptive characteristics as escape
extinction (it specifies process and reinforcer, but
not procedure); in addition, the wording is awk-
ward because nothing "sensory" is extinguished.
In an attempt to promote brevity as well as con-
sistency of terminology, we have adopted through-
out this paper the convention of referring to dif-
ferentfunctional variations of extinction by using
the abbreviation "EXT" followed in parentheses
by the source of reinforcement withheld, as in EXT
(attention), EXT (escape), and EXT (sensory). We
chose "EXT" (attention) over "attention EXT"
because the former places emphasis on behavioral
process, with the reinforcer incidental, whereas the
latter connotes that attention is extinguished.
This terminology is still incomplete because it
does not differentiate procedural variations within
132
VARIATIONS OF EXTINCTION
81. a given functional class of extinction techniques.
For example, EXT (attention) may take different
forms depending on the situational context in which
it is applied. If the target behavior occurs in the
absence of ongoing social interaction, EXT (atten-
tion) requires no response from the therapist and
simply consists of withholding attention that pre-
viously followed the behavior. A more complicated
situation exists if the target behavior occurs while
the therapist is interacting with the client. In this
case, withholding reinforcement [EXT (attention)]
would consist of terminating the interaction.' Be-
cause similar variations are also characteristic of
both EXT (escape) and EXT (sensory), we did not
attempt to specify any particular procedure by way
of descriptive label; this practice is consistent with
terms such as DRO, time-out, punishment, and so
forth, all of which require further elaboration.
A reasonable conclusion based on descriptions
of behavior, procedure, and outcome reported in
the above studies is that extinction designed for
SIB serving one function may have little or no
therapeutic effect on SIB serving a different func-
82. tion. Given the variations of extinction described
previously and the maintaining contingencies for
which they might be used, it is possible to construct
a contingency matrix such as that found in Figure
1, which allows predictions about behavioral out-
come. For example, elimination of social reinforce-
ment for SIB using EXT (attention) or EXT (es-
cape) will not interfere with the delivery of sensory
(automatic) reinforcement, and EXT (sensory)
would have no effect on social sources of reinforce-
ment. Most seriously, EXT (attention) applied to
escape-maintained SIB (i.e., stimulus removal con-
tingent on escape behavior) and, conversely, EXT
(escape) applied to attention-maintained SIB (i.e.,
stimulus continuation contingent on attention-get-
ting behavior) may be countertherapeutic and se-
riously exacerbate the behavior problem. At the
In this example, it is unclear if the subsequent change
in behavior reflects withholding of contingent reinforcement
that maintained SIB (extinction) or removal of ongoing re-
inforcement (time-out). In either case, and from a practical
standpoint, the only means of not reinforcing the behavior
consists of terminating interaction.
83. FUNCTIONAL VARIATIONS OF EXTINCTION
0
LL
0
z
LU
0
z
1
z
0
0
z
Ir.2 f ifis_---_..Treatment
Figure 1. Expected outcomes when functional variations
of extinction are applied to different maintaining contingen-
cies for SIB.
present time, evidence directly supporting these pre-
dictions can be found in only a few studies. Research
on the treatment of self-injurious escape behavior,
84. for example, sometimes has included a baseline
condition in which time-out from educational tasks
was made contingent on SIB (Iwata, Pace, Kalsher,
Cowdery, & Cataldo, 1990; Repp et al., 1988;
Steege et al., 1990). In effect, this condition pro-
cedurally amounted to EXT (attention) but was
associated with increases in escape-maintained SIB.
In the present study, we provide an empirical
evaluation of the key predictions found in Figure
1 by conducting a comparative analysis of extinc-
tion techniques. The relative effects of three func-
tional variations of extinction were examined when
applied to the same topography of SIB maintained
by three different contingencies of reinforcement.
METHOD
Subjects and Setting
Subjects were selected for participation based on
(a) their having similar topographies of SIB, and
(b) the outcomes of functional analysis baselines
indicating that their SIB was maintained by dif-
ferent sources of reinforcement. Three children with
85. developmental disabilities who met these criteria
were included as participants. All of them exhibited
head banging that produced contusions or lacera-
tions on the head and face. Donnie, a 7-year-old
boy with severe mental retardation, was ambulatory
133
BRIAN A. IWATA et al.
and could feed himself. He did not follow vocal
instructions, although he had a fairly good imitative
repertoire, and he communicated his needs by
pointing. Donnie's parents could not identify any
situations that were predictive of either high or low
levels of SIB. His previous treatments consisted of
response interruption, redirection to another activ-
ity, time-out, and differential reinforcement. Jack,
a 12-year-old boy with severe mental retardation,
was nonambulatory but could move his wheelchair
over short distances, feed himself, and manipulate
small objects. He followed a few instructions and
communicated primarily through idiosyncratic ges-
86. ture (i.e., reaching and making a high-pitched vocal
noise). Jack's parents and teachers reported that his
SIB seemed to occur as part of a tantrum, although
they could not reliably describe the types of situ-
ations in which tantrums occurred. His previous
treatments consisted of time-out and differential
reinforcement. Millie, an 8-year-old girl with mod-
erate mental retardation, was ambulatory, could
feed herself, and occasionally followed simple in-
structions. Although she did not exhibit any ex-
pressive language, she appeared to be socially re-
sponsive to adults (e.g., she approached adults when
they were nearby, smiled at them, etc.). Millie's
parents indicated that her SIB occurred "most of
the time," but became worse when she did not
"get her way." Her previous treatments consisted
of verbal reprimands, response interruption, man-
ual restraint, redirection, and differential reinforce-
ment. She was also placed in a hard seizure helmet
as a means of protection when her SIB was deemed
to be "uncontrollable."
Sessions were usually conducted individually in
therapy rooms containing one-way observation win-
dows, but occasionally (due to scheduling neces-
87. sities) were run in a large group area where two or
three other clients and one or two other staff mem-
bers were present but located in a different part of
the room (i.e., at least 6 m away). This periodic
change in location seemed to have no effect on the
subjects' behavior. Sessions were 15 min in duration
and usually were conducted 5 days per week, with
four to eight sessions daily and at least 15-min
breaks between sessions.
Response Measurement and Reliability
SIB was defined as any audible contact between
the head or face and either a fist or an object (e.g.,
furniture or walls). Data were recorded using one
of two methods: (a) on paper during continuous
10-s intervals, which were cued by cassette tape;
or (b) on a hand-held computer (Panasonic Model
RL-H1800). All data were converted into per-
centage of 10-s intervals during which one or more
instances of SIB occurred. Data were also collected
on a variety of experimenter behaviors (delivery of
instructions or attention, withdrawal of instructions
or attention, placement and removal of apparatus)
as a means of monitoring procedural consistency;
88. these data indicated that experimenter compliance
exceeded 90% across all categories.
Interobserver agreement was assessed by having
a second observer simultaneously but independently
record data during 23% of all sessions. Agreement
percentages were calculated based on interval-by-
interval comparison of observers' records by divid-
ing the number of agreements by the number of
agreements plus disagreements and multiplying by
100%. Mean agreement was 92% across subjects,
with all values exceeding 86%.
Experimental Designs and Sequence
During the initial baseline, subjects were exposed
to a series of assessment conditions in a multiele-
ment design (Sidman, 1960; Ulman & Sulzer-Aza-
roff, 1975) to identify the maintaining variables
for their SIB. Subsequently, each subject was ex-
posed to two or more functional variations of ex-
tinction by way of reversal or multiple baseline
designs. Because the specific conditions and their
order of presentation were determined by subjects'
baseline performances, more complete details are
89. provided in the Results section for each subject.
Functional Analysis Baseline
Subjects were exposed to four conditions based
on Iwata, Dorsey, Slifer, Bauman, and Richman
(1982). A brief description of each condition is
provided here.
Attention. This condition was designed to test
134
VARIATIONS OF EXTINCTION
for positive reinforcement (attention) as a main-
taining variable for SIB. An experimenter initially
placed the subject in contact with toys that were
in the room, then proceeded to ignore the subject
while seeming to do paperwork. Contingent on the
occurrence of SIB, the experimenter expressed con-
cern (e.g., "Stop. Don't do that; you'll hurt your-
self.") and briefly interrupted the response.
90. Demand. This condition was a test for negative
reinforcement (escape from demands) as a main-
taining variable. An experimenter presented aca-
demic tasks (e.g., object identification, sorting,
drawing, and other tasks requiring simple motor
responses) comparable to those found in the sub-
ject's individual educational plan. Tasks were pre-
sented in a discrete-trial format approximately once
every 30 s. The experimenter used modeling and
physical guidance as supplementary prompts to
produce compliance, delivered praise contingent on
correct responses, and implemented a 30-s time-
out contingent on the occurrence of SIB. The time-
out consisted of removing the materials from the
table and turning away from the subject.
Alone. By restricting access to social and material
sources of stimulation, this condition provided a
test for SIB maintained by nonsocial (automatic)
reinforcement. The subject was observed while alone
in the room (which was empty except for a chair
or couch).
Play. This condition served as a control. The
91. experimenter provided continuous access to toys,
delivered praise approximately every 30 s contin-
gent on the absence of SIB, and ignored occurrences
of SIB.
Extinction Conditions
Three variations of extinction were evaluated,
each designed to discontinue a different source of
reinforcement for SIB. An attempt was made to
implement each type of extinction with each sub-
ject, but this was not always possible if (a) little or
no SIB occurred in a given baseline condition, or
(b) the baseline condition itself was not amenable
to implementation of the procedure as described.
For example, it was not possible to prevent escape
from tasks (i.e., to withhold negative reinforce-
ment) during the alone baseline, because no tasks
were presented initially.
EXT (attention). This variation of extinction
was designed to discontinue positive reinforcement
in the form of attention delivered by the experi-
menter. EXT (attention) already was a component
92. of three of the baseline conditions: during the de-
mand condition, attention was withdrawn contin-
gent on SIB; during the alone condition, no adult
was present to deliver any attention; and during
the play condition, attention was withheld contin-
gent on SIB. Thus, the attention condition was the
only baseline on which EXT (attention) could be
introduced as a new manipulation. This extinction
contingency consisted of having the experimenter
ignore (not attend to) SIB, and was combined with
experimenter attention for the absence of SIB, which
was delivered on a 30-s DRO schedule. Occur-
rences of SIB reset the DRO interval.
EXT (escape). This variation of extinction dis-
continued negative reinforcement in the form of
escape from tasks. It was implemented during the
demand condition, which was the only condition
containing a task requirement. During EXT (es-
cape), learning trials continued as during baseline
but were not terminated contingent on SIB. Instead,
the experimenter physically guided the subject to
comply with the instruction and continued the ses-
sion according to schedule. It has been noted pre-
viously (Iwata, Pace, Kalsher, Cowdery, & Cataldo,
93. 1990) that the physical guidance component of
this intervention may represent a punishment con-
tingency. However, (a) guidance was the inevitable
consequence of noncompliance even during base-
line, and (b) behavior change observed following
implementation of this procedure in the above study
as well as in others (e.g., Goh & Iwata, 1994;
Repp et al., 1988) showed either temporary burst-
ing or gradual decreases consistent with extinction
processes.
EXT (sensory). This variation of extinction was
designed to attenuate the sensory consequences pro-
duced by subjects' head banging, and could be
implemented across any of the baseline conditions.
Each subject was fitted with an oversized helmet
(either a seizure or a lacrosse helmet), which was
135
BRIAN A. IWATA et al.
reinforced with additional padding. During EXT
94. (sensory), the helmet was placed on the subject's
head at the beginning of a session and remained
in place throughout (subjects never attempted to
remove the helmets). An extension of this "non-
contingent" helmet condition was implemented for
Donnie and is described below.
RESULTS
Donnie
Figure 2 shows the results obtained for Donnie.
He exhibited a considerable amount of SIB across
all of the baseline conditions, thus permitting a
series of manipulations on each baseline. Moderate
levels of SIB were observed during the demand,
attention, and play conditions (lower three panels),
whereas SIB occurred almost continuously during
the alone condition (upper panel) containing no
access to social stimulation, play materials, or tasks.
These data suggested that Donnie's SIB was main-
tained by factors independent of the social and
physical environment and were consistent with be-
havior reinforced by directly produced, automatic
consequences.
95. On the alone baseline (top panel), EXT (sensory)
was implemented in a reversal (ABAB) design, and
results showed decreases in SIB associated with the
helmet intervention. The final condition on the
alone baseline involved two modifications of the
EXT (sensory) procedure. First, play materials that
Donnie was observed to manipulate during the
original play baseline were introduced. Second, the
protective helmet was applied only contingent on
the occurrence of SIB. As long as SIB did not occur,
Donnie was free to play with the toys and move
about without wearing the helmet. When SIB oc-
curred, an experimenter, who was in the room but
did not interact with Donnie during the session,
placed the helmet on Donnie for a 2-min time-
out, during which the toys were removed from
sight. These procedural modifications were based
on results reported by Dorsey, Iwata, Reid, and
Davis (1982), who found that it was possible to
transfer control of SIB from a noncontingent to a
contingent helmet condition. SIB increased initially
when this change was made on the alone baseline,
but eventually decreased to a level similar to that
96. observed during the EXT (sensory) conditions.
On the demand baseline (second panel from the
top), the introduction of EXT (sensory) was asso-
ciated with a rapid and large decrease in Donnie's
SIB. This reduction occurred in spite of the fact
that the original baseline contingency for SIB-
EXT (attention) or, alternatively, escape from the
task contingent on SIB-remained in effect. Next,
EXT (sensory) was removed and EXT (escape) was
implemented; this resulted in an increase in SIB to
its baseline level. Finally, the contingent helmet
procedure was combined with EXT (escape). The
experimenter presented learning trials as before and,
contingent on SIB, guided Donnie through the task
and then placed the helmet on him. The helmet
remained on for approximately 2 min (four learning
trials). This condition was associated with a decrease
in SIB similar to that seen during the previous EXT
(sensory) condition.
On the attention baseline (third panel from the
top), the introduction of EXT (attention) had no
effect on Donnie's SIB. During the next condition,
EXT (sensory), SIB decreased even though it pro-
97. duced attention from the experimenter. The final
procedure implemented on this baseline consisted
of EXT (attention) combined with a 2-min time-
out in the helmet contingent on occurrences of SIB,
and was associated with a reduction in SIB similar
to that seen in the EXT (sensory) condition.
On the play baseline (bottom panel), which con-
tained an EXT (attention) component, no decrease
in SIB was observed until EXT (sensory) was in-
troduced. Subsequently, SIB increased temporarily
but decreased again when the procedure was changed
to a 2-min time-out in the helmet contingent on
SIB.
Jack
The initial baseline of Figure 3 shows the results
of Jack's assessment. He exhibited little or no SIB
except during the demand condition, which con-
tained two distinctive features: the presentation of
task demands and a brief time-out from the task
contingent on SIB. The time-out component pro-
136
98. DONNIE
100- Baseline
800
n:n -a
40 -
20 -
VARIATIONS OF EXTINCTION
r EXT(sensory) 1
BL I Toys + Conting(
-a- Ati
-a-- -D
-.*-- Al4
-a- Pli
100. 20
0 20 40 60 80 100
137
co)
-J
cn
w
H
z
uL
0
Lu
w
z
101. w
0L
20 40 60 80 1 00
SESSIONS
Figure 2. Percentage of intervals of SIB exhibited by Donnie
across experimental conditions.
ou -1
A
II
BRIAN A. IWATA et al.
EXT(sensory)
100
co
103. 20
0
-H0.
SESSIONS
Figure 3. Percentage of intervals of SIB exhibited by Jack
across experimental conditions.
cedurally resembled EXT (attention) but appeared
to function as (negative) reinforcement for escape
behavior. Because little or no SIB was observed
during other conditions, all subsequent manipu-
lations were made on the demand baseline. EXT
(sensory) and EXT (escape) were implemented con-
currently and resulted in a large decrease in Jack's
SIB. A partial reversal was then conducted in which
EXT (escape) was removed while EXT (sensory)
remained in effect; this change resulted in an in-
crease in Jack's SIB to its original baseline level.
SIB decreased again when EXT (escape) was re-
introduced, and it remained low following the re-
moval of EXT (sensory).
104. Millie
Figure 4 shows the results obtained for Millie.
During the initial baseline assessment, her SIB oc-
curred almost exclusively during the attention con-
dition, suggesting that the behavior was maintained
by social-positive reinforcement. Very little SIB was
observed during the play and demand conditions,
both of which contained a brief time-out from
attention contingent on SIB, and SIB decreased
throughout the alone condition, which contained
no attention whatsoever. Thus, Millie's assessment
data revealed not only a reinforcement effect during
the attention condition but also an apparent ex-
tinction effect during others.
In light of these results, subsequent interventions
were implemented only on the attention baseline.
EXT (attention) with attention delivered a 30-s
DRO schedule was introduced concurrent with EXT
(sensory) and resulted in a large decrease in Millie's
SIB. Next, while EXT (sensory) remained in effect,
a reversal was conducted in which EXT (attention)
105. + DRO was removed; this partial return to base-
line was associated with an immediate and large
increase in SIB. When EXT (attention) + DRO
was reinstated, SIB decreased to near-zero levels.
EXT (sensory) was then removed while EXT (at-
tention) + DRO remained in effect, and no increase
was observed in SIB. During this final condition,
the length of the DRO interval was gradually in-
creased to 1, 2, and finally 5 min, and SIB remained
low across all schedule changes.
138
VARIATIONS OF EXTINCTION
EXT(sensory)
_,4,0MILLIE
EXT(attn) + DRO
Baseline BL
-a Attention
*- Demand
106. -0- Alone
0 Play
20 40 60 80
SESSIONS
Figure 4. Percentage of intervals of SIB exhibited by Millie
across experimental conditions.
Summary of Key Comparisons
Table 1 contains a summary of the results from
selected baseline and extinction conditions for Don-
nie, Jack, and Millie. The data reflect reductions
or increases in SIB expressed as mean percentages
of baseline responding (EXT mean/baseline mean)
for relevant and irrelevant extinction conditions
compared to their appropriate baselines. Because
SIB during relevant extinction conditions usually
showed decreasing trends, comparisons based on
overall condition means would not accurately depict
end-of-treatment responding. Therefore, all per-
centages are based on the last five sessions of a
condition. For example, under "Relevant EXT,"
107. Donnie's mean percentage of SIB during the last
five sessions of EXT (sensory) occurred at 0.5% of
the mean SIB during the last five sessions of his
alone baseline, whereas under "Irrelevant EXT,"
his five-session mean percentage of SIB during EXT
(escape) occurred at 81.9% of his demand baseline
mean. These comparisons show that SIB for all
subjects was reduced to below 15% of its baseline
Table 1
Mean Change (Reduction or Increase) in SIB Expressed as a
Percentage of Baseline Responding for Relevant and
Irrelevant Extinction Conditions Compared to Their Relevant
Baselines. Percentages Are Based on the Last Five Sessions
of a Condition
Relevant EXT / baseline Irrelevant EXT / baseline
Subject Conditions compared % of BL Conditions compared %
of BL
Donnie EXT (sensory) / alone BL 0.5 EXT (escape) / demand
BL 81.9
EXT (sensory) / demand BL 6.9 EXT (attn) / attn …
108. Chapter 4: Clustering
I
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What is Cluster Analysis?
Finding groups of objects such that the objects in a group will
be similar (or related) to one another and different from (or
unrelated to) the objects in other groups
110. Intra-cluster distances are minimized
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Applications of Cluster Analysis
Understanding
Group related documents for browsing, group genes and
proteins that have similar functionality, or group stocks with
similar price fluctuations
Summarization
Reduce the size of large data sets
Clustering precipitation in Australia
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What is not Cluster Analysis?
Simple segmentation
Dividing students into different registration groups
alphabetically, by last name
Results of a query
Groupings are a result of an external specification
Clustering is a grouping of objects based on the data
Supervised classification
Have class label information
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Notion of a Cluster can be Ambiguous
115. Six Clusters
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Types of Clusterings
A clustering is a set of clusters
Important distinction between hierarchical and partitional sets
of clusters
Partitional Clustering
A division of data objects into non-overlapping subsets
(clusters) such that each data object is in exactly one subset
116. Hierarchical clustering
A set of nested clusters organized as a hierarchical tree
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Partitional Clustering
117. Original Points
A Partitional Clustering
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Hierarchical Clustering
Traditional Hierarchical Clustering
Non-traditional Hierarchical Clustering
Non-traditional Dendrogram
Traditional Dendrogram
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Other Distinctions Between Sets of Clusters
Exclusive versus non-exclusive
In non-exclusive clusterings, points may belong to multiple
clusters.
Can represent multiple classes or ‘border’ points
Fuzzy versus non-fuzzy
In fuzzy clustering, a point belongs to every cluster with some
weight between 0 and 1
Weights must sum to 1
Probabilistic clustering has similar characteristics
Partial versus complete
In some cases, we only want to cluster some of the data
Heterogeneous versus homogeneous
Clusters of widely different sizes, shapes, and densities
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119. ‹#›
Types of Clusters
Well-separated clusters
Center-based clusters
Contiguous clusters
Density-based clusters
Property or Conceptual
Described by an Objective Function
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Types of Clusters: Well-Separated
Well-Separated Clusters:
A cluster is a set of points such that any point in a cluster is
closer (or more similar) to every other point in the cluster than
to any point not in the cluster.
120. 3 well-separated clusters
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Types of Clusters: Center-Based
Center-based
A cluster is a set of objects such that an object in a cluster is
closer to the “center” of a cluster, than to the center of any
other cluster
The center of a cluster is a centroid.
The average of all the points in the cluster is a medoid
4 center-based clusters
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Types of Clusters: Contiguity-Based
Contiguous Cluster (Nearest neighbor or Transitive)
A point in a cluster is closer (or more similar) to one or more
other points in the cluster than to any point not in the cluster.
8 contiguous clusters
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Types of Clusters: Density-Based
Density-based
A cluster is a dense region of points, which is separated by low -
density regions, from other regions of high density.
Used when the clusters are irregular or intertwined, and when
noise and outliers are present.
6 density-based clusters
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Types of Clusters: Conceptual Clusters
123. Shared Property or Conceptual Clusters
Finds clusters that share some common property or represent a
particular concept.
.
2 Overlapping Circles
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Types of Clusters: Objective Function
Clusters Defined by an Objective Function
Finds clusters that minimize or maximize an objective function.
Enumerate all possible ways of dividing the points into clusters
and evaluate the `goodness' of each potential set of clusters by
using the given objective function. (NP Hard)
Can have global or local objectives.
Hierarchical clustering algorithms typically have local
objectives
Partitional algorithms typically have global objectives
A variation of the global objective function approach is to fit
the data to a parameterized model.
124. Parameters for the model are determined from the data.
Mixture models assume that the data is a ‘mixture' of a number
of statistical distributions.
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Characteristics of the Input Data Are Important
Type of proximity or density measure
Central to clustering
Depends on data and application
Data characteristics that affect proximity and/or density are
Dimensionality
Sparseness
Attribute type
Special relationships in the data
For example, autocorrelation
Distribution of the data
Noise and Outliers
Often interfere with the operation of the clustering algorithm
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Clustering Algorithms
K-means and its variants
Hierarchical clustering
Density-based clustering
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K-means Clustering
Partitional clustering approach
126. Number of clusters, K, must be specified
Each cluster is associated with a centroid (center point)
Each point is assigned to the cluster with the closest centroid
The basic algorithm is very simple
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Example of K-means Clustering
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127. Example of K-means Clustering
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K-means Clustering – Details
Initial centroids are often chosen randomly.
Clusters produced vary from one run to another.
The centroid is (typically) the mean of the points in the cluster.
‘Closeness’ is measured by Euclidean distance, cosine
similarity, correlation, etc.
K-means will converge for common similarity measures
mentioned above.
Most of the convergence happens in the first few iterations.
Often the stopping condition is changed to ‘Until relatively few
points change clusters’
128. Complexity is O( n * K * I * d )
n = number of points, K = number of clusters,
I = number of iterations, d = number of attributes
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Evaluating K-means Clusters
Most common measure is Sum of Squared Error (SSE)
For each point, the error is the distance to the nearest cluster
To get SSE, we square these errors and sum them.
x is a data point in cluster Ci and mi is the representative point
for cluster Ci
can show that mi corresponds to the center (mean) of the
cluster
Given two sets of clusters, we prefer the one with the smallest
error
One easy way to reduce SSE is to increase K, the number of
clusters
A good clustering with smaller K can have a lower SSE than a
129. poor clustering with higher K
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Two different K-means Clusterings
Sub-optimal Clustering
Optimal Clustering
Original Points
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Limitations of K-means
K-means has problems when clusters are of differing
130. Sizes
Densities
Non-globular Shapes
K-means has problems when the data contains outliers/noise.
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Limitations of K-means: Differing Sizes
Original Points
K-means (3 Clusters)
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131. Limitations of K-means: Differing Density
Original Points
K-means (3 Clusters)
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Limitations of K-means: Non-globular Shapes
Original Points
K-means (2 Clusters)
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132. ‹#›
Overcoming K-means Limitations
Original PointsK-means Clusters
One solution is to use many clusters.
Find parts of clusters, but need to put together.
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Overcoming K-means Limitations
Original PointsK-means Clusters
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Overcoming K-means Limitations
Original PointsK-means Clusters
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Importance of Choosing Initial Centroids
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Importance of Choosing Initial Centroids
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Importance of Choosing Initial Centroids …
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Importance of Choosing Initial Centroids …
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Problems with Selecting Initial Points
If there are K ‘real’ clusters then the chance of selecting one
136. centroid from each cluster is small.
Chance is relatively small when K is large
If clusters are the same size, n, then
For example, if K = 10, then probability = 10!/1010 = 0.00036
Sometimes the initial centroids will readjust themselves in
‘right’ way, and sometimes they don’t
Consider an example of five pairs of clusters
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10 Clusters Example
Starting with two initial centroids in one cluster of each pair of
clusters
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10 Clusters Example
Starting with two initial centroids in one cluster of each pair of
clusters
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