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K-means Clustering

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Anna Fensel
Anna Fensel

Lecture slides presenting K-means clustering algorithm.

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K-means Clustering Ass.-Prof. Dr.rer.nat Anna Fensel
Outline » Introduction, learning goals » Motivation and example » Clustering » K-means clustering algorithm definition, functions, iteration process, pseudocode » Computational complexity » Extensions » Tools » Application examples » Conclusions » References Page 2
Outline » Introduction, learning goals » Motivation and example » Clustering » K-means clustering algorithm definition, functions, iteration process, pseudocode » Computational complexity » Extensions » Tools » Application examples » Conclusions » References Page 3
What you should be able to do after this lecture? » Understand the concept of clustering, and particularly k-means clustering » Explain the k-means clustering algorithm » Provide diverse usage examples of the k- means clustering algorithm » Understand different challenges in the use of the k-means clustering algorithm and its extensions/variations Page 4
Motivation: Why clustering? What is clustering? Page 5
Motivation: Why clustering? What is clustering? » Finding “natural” groupings between objects » We want to find similar objects (f.e. documents) to treat them in the same way We aim at: » High intra-cluster similarity » Low inter-cluster similarity Page 6
Motivating example: Web document search » A web search engine often returns thousands of pages in response to a broad query, making it difficult for users to browse or to identify relevant information. » Clustering methods can be used to automatically group the retrieved documents into a list of meaningful categories. Page 7
Is clustering typically …? A. Supervised B. Unsupervised Page 8
Is clustering typically …? A. Supervised B. Unsupervised Page 9
Outline » Introduction, learning goals » Motivation and example » Clustering » K-means clustering algorithm definition, functions, iteration process, pseudocode » Computational complexity » Extensions » Tools » Application examples » Conclusions » References Page 10
What is K-means clustering? » k-means clustering aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean » Works for n-dimensional spaces as well Page 11
How do we measure similarity? Give examples of similarity measures. » … » … Page 12
How do we measure similarity? Give examples of similarity measures. » Similarity is subjective » Its measure therefore depends on the data, the use case, the users » In practice it is not always straightforward which metrics work well - then “trial and error” can be followed » Examples of similarity measures: Euclidean, Manhattan, cosine distance Page 13
How does K-means do the clustering? K-means is a very important/basic flat clustering algorithm. Its objective is to minimize the average squared Euclidean distance of values from their cluster centers where a cluster center is defined as the mean or centroid of the values in a cluster :
Selection of centroids » The first step of K-means is to select as initial cluster centers K randomly selected documents, the seeds. » The algorithm then moves the cluster centers around in space in order to minimize RSS (the function that defines how central the centroids are). Page 15
Page 16Ort I Name I Datum K-means illustration step-by-step (1 dimensional) From: StatQuest: K-means clustering: https://www.youtube.com/watch?v=4b5d3muPQmA
Calculating the “centrality” of the centroids [Manning & Schütze, 2008] Page 17 A measure of how well the centroids represent the members of their clusters is the residual sum of squares or RSS, the squared distance of each vector from its centroid summed over all vectors: Our goal is to minimize RSS (i.e. the average squared distance) till it is possible.
K-means algorithm summary [Manning & Schütze, 2008] Page 18
K-means – iteration process [Manning & Schütze, 2008] » reassigning documents to the cluster with the closest centroid, » recomputing each centroid based on the current members of its cluster.
How to set where to terminate the algorithm? [Manning & Schütze, 2008] Page 20 • A fixed number of iterations has been completed. • This condition limits the runtime of the clustering algorithm, but in some cases the quality of the clustering will be poor because of an insufficient number of iterations. • Assignment of documents to clusters (the partitioning function ) does not change between iterations. • Except for cases with a bad local minimum, this produces a good clustering, but runtimes may be unacceptably long. • Centroids do not change between iterations. • This is equivalent to not changing. • Terminate when RSS falls below a threshold. • This criterion ensures that the clustering is of a desired quality after termination. In practice, we need to combine it with a bound on the number of iterations to guarantee termination. • Terminate when the decrease in RSS falls below a threshold . • For small , this indicates that we are close to convergence. Again, we need to combine it with a bound on the number of iterations to prevent very long runtimes.
A variation: How do you know how many clusters you should make? » It is possible to try different cluster numbers. » And check where the variation stabilizes to decide on the number of clusters. Page 21 From: StatQuest: K-means clustering: https://www.youtube.com/watch?v=4b5d3muPQmA
Outline » Introduction, learning goals » Motivation and example » Clustering » K-means clustering algorithm definition, functions, iteration process, pseudocode » Computational complexity » Extensions » Tools » Application examples » Conclusions » References Page 22
K-means computational aspects » K-means converges, but there is unfortunately no guarantee that a global minimum in the objective function will be reached. This is a particular problem if a document set contains many outliers , documents that are far from any other documents and therefore do not fit well into any cluster. We may end up with a singleton cluster (a cluster with only one document) even though there is probably a clustering with lower RSS. Page 23
What is the time complexity K-means? [Manning & Schütze, 2008] » Most of the time is spent on computing vector distances. One such operation costs . » The reassignment step computes KN distances, so its overall complexity is . » In the recomputation step, each vector gets added to a centroid once, so the complexity of this step is . » For a fixed number of iterations I, the overall complexity is therefore . Page 24
Extensions There are numerous extensions to the K-means clustering f.e. » K-means clustering can be generalized e.g. into a Gaussian mixture model. » Efficiency problem can be addressed e.g. by K-medoids , a variant of K-means that computes medoids instead of centroids as cluster centers. The medoid of a cluster as the value that is closest to the centroid. Distance computations are faster in this case. Page 25
Tool support A number of tools implementing k-means clustering are available: » Open source e.g. Apache Spark Torch, R, and » Proprietary e.g. MATLAB, Mathematica, SAP HANA Page 26
Outline » Introduction, learning goals » Motivation and example » Clustering » K-means clustering algorithm definition, functions, iteration process, pseudocode » Computational complexity » Extensions » Tools » Application examples » Conclusions » References Page 27
Application example: Image compression » Aim: compress an image in size » Question: with how many dimensional space we are working here? Page 28 Image credit: Shinichi Tamura @ Slideshare
Application example: Retail – recommendation and yield management » User profiles/personas: similar purchase behavior… Page 29 » Product profiles: similar selling patterns » Deciding when to discount product groups Question: with how many dimensional space we are working here?
Outline » Introduction, learning goals » Motivation and example » Clustering » K-means clustering algorithm definition, functions, iteration process, pseudocode » Computational complexity » Extensions » Tools » Application examples » Conclusions » References Page 30
Summary » Provide 5 most important points you have learnt from today’s lecture. » … » … » … » … » … » (and let’s compare the points) Page 31
References » Hartigan, J. A., & Wong, M. A. (1979). Algorithm AS 136: A k-means clustering algorithm. Journal of the Royal Statistical Society. Series C (Applied Statistics), 28(1), 100-108. » Manning С, R. P., & Schütze, H. (2008). Introduction to information retrieval. Cambridge University Press. » K-Means Clustering at Wikipedia: https://en.wikipedia.org/wiki/K- means_clustering » StatQuest: K-means clustering: https://www.youtube.com/watch?v=4b5d3muPQmA » Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 16.1. Gaussian Mixture Models and k-Means Clustering". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8. Page 32
www.sti-innsbruck.at Thank you for attention. Questions?

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K-means Clustering

  • 2. Outline » Introduction, learning goals » Motivation and example » Clustering » K-means clustering algorithm definition, functions, iteration process, pseudocode » Computational complexity » Extensions » Tools » Application examples » Conclusions » References Page 2
  • 3. Outline » Introduction, learning goals » Motivation and example » Clustering » K-means clustering algorithm definition, functions, iteration process, pseudocode » Computational complexity » Extensions » Tools » Application examples » Conclusions » References Page 3
  • 4. What you should be able to do after this lecture? » Understand the concept of clustering, and particularly k-means clustering » Explain the k-means clustering algorithm » Provide diverse usage examples of the k- means clustering algorithm » Understand different challenges in the use of the k-means clustering algorithm and its extensions/variations Page 4
  • 5. Motivation: Why clustering? What is clustering? Page 5
  • 6. Motivation: Why clustering? What is clustering? » Finding “natural” groupings between objects » We want to find similar objects (f.e. documents) to treat them in the same way We aim at: » High intra-cluster similarity » Low inter-cluster similarity Page 6
  • 7. Motivating example: Web document search » A web search engine often returns thousands of pages in response to a broad query, making it difficult for users to browse or to identify relevant information. » Clustering methods can be used to automatically group the retrieved documents into a list of meaningful categories. Page 7
  • 8. Is clustering typically …? A. Supervised B. Unsupervised Page 8
  • 9. Is clustering typically …? A. Supervised B. Unsupervised Page 9
  • 10. Outline » Introduction, learning goals » Motivation and example » Clustering » K-means clustering algorithm definition, functions, iteration process, pseudocode » Computational complexity » Extensions » Tools » Application examples » Conclusions » References Page 10
  • 11. What is K-means clustering? » k-means clustering aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean » Works for n-dimensional spaces as well Page 11
  • 12. How do we measure similarity? Give examples of similarity measures. » … » … Page 12
  • 13. How do we measure similarity? Give examples of similarity measures. » Similarity is subjective » Its measure therefore depends on the data, the use case, the users » In practice it is not always straightforward which metrics work well - then “trial and error” can be followed » Examples of similarity measures: Euclidean, Manhattan, cosine distance Page 13
  • 14. How does K-means do the clustering? K-means is a very important/basic flat clustering algorithm. Its objective is to minimize the average squared Euclidean distance of values from their cluster centers where a cluster center is defined as the mean or centroid of the values in a cluster :
  • 15. Selection of centroids » The first step of K-means is to select as initial cluster centers K randomly selected documents, the seeds. » The algorithm then moves the cluster centers around in space in order to minimize RSS (the function that defines how central the centroids are). Page 15
  • 16. Page 16Ort I Name I Datum K-means illustration step-by-step (1 dimensional) From: StatQuest: K-means clustering: https://www.youtube.com/watch?v=4b5d3muPQmA
  • 17. Calculating the “centrality” of the centroids [Manning & Schütze, 2008] Page 17 A measure of how well the centroids represent the members of their clusters is the residual sum of squares or RSS, the squared distance of each vector from its centroid summed over all vectors: Our goal is to minimize RSS (i.e. the average squared distance) till it is possible.
  • 18. K-means algorithm summary [Manning & Schütze, 2008] Page 18
  • 19. K-means – iteration process [Manning & Schütze, 2008] » reassigning documents to the cluster with the closest centroid, » recomputing each centroid based on the current members of its cluster.
  • 20. How to set where to terminate the algorithm? [Manning & Schütze, 2008] Page 20 • A fixed number of iterations has been completed. • This condition limits the runtime of the clustering algorithm, but in some cases the quality of the clustering will be poor because of an insufficient number of iterations. • Assignment of documents to clusters (the partitioning function ) does not change between iterations. • Except for cases with a bad local minimum, this produces a good clustering, but runtimes may be unacceptably long. • Centroids do not change between iterations. • This is equivalent to not changing. • Terminate when RSS falls below a threshold. • This criterion ensures that the clustering is of a desired quality after termination. In practice, we need to combine it with a bound on the number of iterations to guarantee termination. • Terminate when the decrease in RSS falls below a threshold . • For small , this indicates that we are close to convergence. Again, we need to combine it with a bound on the number of iterations to prevent very long runtimes.
  • 21. A variation: How do you know how many clusters you should make? » It is possible to try different cluster numbers. » And check where the variation stabilizes to decide on the number of clusters. Page 21 From: StatQuest: K-means clustering: https://www.youtube.com/watch?v=4b5d3muPQmA
  • 22. Outline » Introduction, learning goals » Motivation and example » Clustering » K-means clustering algorithm definition, functions, iteration process, pseudocode » Computational complexity » Extensions » Tools » Application examples » Conclusions » References Page 22
  • 23. K-means computational aspects » K-means converges, but there is unfortunately no guarantee that a global minimum in the objective function will be reached. This is a particular problem if a document set contains many outliers , documents that are far from any other documents and therefore do not fit well into any cluster. We may end up with a singleton cluster (a cluster with only one document) even though there is probably a clustering with lower RSS. Page 23
  • 24. What is the time complexity K-means? [Manning & Schütze, 2008] » Most of the time is spent on computing vector distances. One such operation costs . » The reassignment step computes KN distances, so its overall complexity is . » In the recomputation step, each vector gets added to a centroid once, so the complexity of this step is . » For a fixed number of iterations I, the overall complexity is therefore . Page 24
  • 25. Extensions There are numerous extensions to the K-means clustering f.e. » K-means clustering can be generalized e.g. into a Gaussian mixture model. » Efficiency problem can be addressed e.g. by K-medoids , a variant of K-means that computes medoids instead of centroids as cluster centers. The medoid of a cluster as the value that is closest to the centroid. Distance computations are faster in this case. Page 25
  • 26. Tool support A number of tools implementing k-means clustering are available: » Open source e.g. Apache Spark Torch, R, and » Proprietary e.g. MATLAB, Mathematica, SAP HANA Page 26
  • 27. Outline » Introduction, learning goals » Motivation and example » Clustering » K-means clustering algorithm definition, functions, iteration process, pseudocode » Computational complexity » Extensions » Tools » Application examples » Conclusions » References Page 27
  • 28. Application example: Image compression » Aim: compress an image in size » Question: with how many dimensional space we are working here? Page 28 Image credit: Shinichi Tamura @ Slideshare
  • 29. Application example: Retail – recommendation and yield management » User profiles/personas: similar purchase behavior… Page 29 » Product profiles: similar selling patterns » Deciding when to discount product groups Question: with how many dimensional space we are working here?
  • 30. Outline » Introduction, learning goals » Motivation and example » Clustering » K-means clustering algorithm definition, functions, iteration process, pseudocode » Computational complexity » Extensions » Tools » Application examples » Conclusions » References Page 30
  • 31. Summary » Provide 5 most important points you have learnt from today’s lecture. » … » … » … » … » … » (and let’s compare the points) Page 31
  • 32. References » Hartigan, J. A., & Wong, M. A. (1979). Algorithm AS 136: A k-means clustering algorithm. Journal of the Royal Statistical Society. Series C (Applied Statistics), 28(1), 100-108. » Manning С, R. P., & Schütze, H. (2008). Introduction to information retrieval. Cambridge University Press. » K-Means Clustering at Wikipedia: https://en.wikipedia.org/wiki/K- means_clustering » StatQuest: K-means clustering: https://www.youtube.com/watch?v=4b5d3muPQmA » Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 16.1. Gaussian Mixture Models and k-Means Clustering". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8. Page 32
  • 33. www.sti-innsbruck.at Thank you for attention. Questions?