Clustering high-dimensional data

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Clustering high-dimensional data is the cluster analysis of data with anywhere from a few dozen to many thousands of dimensions. Such high-dimensional spaces of data are often encountered in areas such as medicine, where DNA microarray technology can produce many measurements at once, and the clustering of text documents, where, if a word-frequency vector is used, the number of dimensions equals the size of the vocabulary.

Contents

Problems

Four problems need to be overcome for clustering in high-dimensional data: [1]

Recent research indicates that the discrimination problems only occur when there is a high number of irrelevant dimensions, and that shared-nearest-neighbor approaches can improve results. [2]

Approaches

Approaches towards clustering in axis-parallel or arbitrarily oriented affine subspaces differ in how they interpret the overall goal, which is finding clusters in data with high dimensionality. [1] An overall different approach is to find clusters based on pattern in the data matrix, often referred to as biclustering, which is a technique frequently utilized in bioinformatics.

Subspace clustering

Example 2D space with subspace clusters SubspaceClustering.png
Example 2D space with subspace clusters

Subspace clustering aims to look for clusters in different combinations of dimensions (i.e., subspaces) and unlike many other clustering approaches does not assume that all of the clusters in a dataset are found in the same set of dimensions. [3] Subspace clustering can take bottom-up or top-down approaches. Bottom-up methods (such as CLIQUE) heuristically identify relevant dimensions by dividing the data space into a grid structure, selecting dense units, and then iteratively linking them if they are adjacent and dense. [3]

The adjacent image shows a mere two-dimensional space where a number of clusters can be identified. In the one-dimensional subspaces, the clusters (in subspace ) and , , (in subspace ) can be found. cannot be considered a cluster in a two-dimensional (sub-)space, since it is too sparsely distributed in the axis. In two dimensions, the two clusters and can be identified.

The problem of subspace clustering is given by the fact that there are different subspaces of a space with dimensions. If the subspaces are not axis-parallel, an infinite number of subspaces is possible. Hence, subspace clustering algorithms utilize some kind of heuristic to remain computationally feasible, at the risk of producing inferior results. For example, the downward-closure property (cf. association rules) can be used to build higher-dimensional subspaces only by combining lower-dimensional ones, as any subspace T containing a cluster, will result in a full space S also to contain that cluster (i.e. S ⊆ T), an approach taken by most of the traditional algorithms such as CLIQUE, [4] SUBCLU. [5] It is also possible to define a subspace using different degrees of relevance for each dimension, an approach taken by iMWK-Means, [6] EBK-Modes [7] and CBK-Modes. [8]

Projected clustering

Projected clustering seeks to assign each point to a unique cluster, but clusters may exist in different subspaces. The general approach is to use a special distance function together with a regular clustering algorithm.

For example, the PreDeCon algorithm checks which attributes seem to support a clustering for each point, and adjusts the distance function such that dimensions with low variance are amplified in the distance function. [9] In the figure above, the cluster might be found using DBSCAN with a distance function that places less emphasis on the -axis and thus exaggerates the low difference in the -axis sufficiently enough to group the points into a cluster.

PROCLUS uses a similar approach with a k-medoid clustering. [10] Initial medoids are guessed, and for each medoid the subspace spanned by attributes with low variance is determined. Points are assigned to the medoid closest, considering only the subspace of that medoid in determining the distance. The algorithm then proceeds as the regular PAM algorithm.

If the distance function weights attributes differently, but never with 0 (and hence never drops irrelevant attributes), the algorithm is called a "soft"-projected clustering algorithm.

Projection-based clustering

Projection-based clustering is based on a nonlinear projection of high-dimensional data into a two-dimensional space. [11] Typical projection-methods like t-distributed stochastic neighbor embedding (t-SNE), [12] or neighbor retrieval visualizer (NerV) [13] are used to project data explicitly into two dimensions disregarding the subspaces of higher dimension than two and preserving only relevant neighborhoods in high-dimensional data. In the next step, the Delaunay graph [14] between the projected points is calculated, and each vertex between two projected points is weighted with the high-dimensional distance between the corresponding high-dimensional data points. Thereafter the shortest path between every pair of points is computed using the Dijkstra algorithm. [15] The shortest paths are then used in the clustering process, which involves two choices depending on the structure type in the high-dimensional data. [11] This Boolean choice can be decided by looking at the topographic map of high-dimensional structures. [16] In a benchmarking of 34 comparable clustering methods, projection-based clustering was the only algorithm that always was able to find the high-dimensional distance or density-based structure of the dataset. [11] Projection-based clustering is accessible in the open-source R package "ProjectionBasedClustering" on CRAN. [17]

Bootstrap-based clustering

Bootstrap aggregation (bagging) can be used to create multiple clusters and aggregate the findings. This is done by taking random subsamples of the data, performing a cluster analysis on each of them and then aggregating the results of the clusterings to generate a dissimilarity measure which can then be used to explore and cluster the original data. [18] [19] Since high-dimensional data are likely to have many non-informative features, weights can be used during the bagging process to increase the impact of the more informative aspects. This produces "ABC dissimilarities" which can then be used to explore and cluster the original data and also to assess which features appear to be more impactful in defining the clusters. [20] [21] [22]

Hybrid approaches

Not all algorithms try to either find a unique cluster assignment for each point or all clusters in all subspaces; many settle for a result in between, where a number of possibly overlapping, but not necessarily exhaustive set of clusters are found. An example is FIRES, which is from its basic approach a subspace clustering algorithm, but uses a heuristic too aggressive to credibly produce all subspace clusters. [23] Another hybrid approach is to include a human-into-the-algorithmic-loop: Human domain expertise can help to reduce an exponential search space through heuristic selection of samples. This can be beneficial in the health domain where, e.g., medical doctors are confronted with high-dimensional descriptions of patient conditions and measurements on the success of certain therapies. An important question in such data is to compare and correlate patient conditions and therapy results along with combinations of dimensions. The number of dimensions is often very large, consequently one needs to map them to a smaller number of relevant dimensions to be more amenable for expert analysis. This is because irrelevant, redundant, and conflicting dimensions can negatively affect effectiveness and efficiency of the whole analytic process. [24]

Correlation clustering

Another type of subspaces is considered in Correlation clustering (Data Mining).

Software

Related Research Articles

Vector quantization (VQ) is a classical quantization technique from signal processing that allows the modeling of probability density functions by the distribution of prototype vectors. Developed in the early 1980s by Robert M. Gray, it was originally used for data compression. It works by dividing a large set of points (vectors) into groups having approximately the same number of points closest to them. Each group is represented by its centroid point, as in k-means and some other clustering algorithms. In simpler terms, vector quantization chooses a set of points to represent a larger set of points.

<span class="mw-page-title-main">Nonlinear dimensionality reduction</span> Summary of algorithms for nonlinear dimensionality reduction

Nonlinear dimensionality reduction, also known as manifold learning, refers to various related techniques that aim to project high-dimensional data onto lower-dimensional latent manifolds, with the goal of either visualizing the data in the low-dimensional space, or learning the mapping itself. The techniques described below can be understood as generalizations of linear decomposition methods used for dimensionality reduction, such as singular value decomposition and principal component analysis.

In data mining and statistics, hierarchical clustering is a method of cluster analysis that seeks to build a hierarchy of clusters. Strategies for hierarchical clustering generally fall into two categories:

Dimensionality reduction, or dimension reduction, is the transformation of data from a high-dimensional space into a low-dimensional space so that the low-dimensional representation retains some meaningful properties of the original data, ideally close to its intrinsic dimension. Working in high-dimensional spaces can be undesirable for many reasons; raw data are often sparse as a consequence of the curse of dimensionality, and analyzing the data is usually computationally intractable. Dimensionality reduction is common in fields that deal with large numbers of observations and/or large numbers of variables, such as signal processing, speech recognition, neuroinformatics, and bioinformatics.

<span class="mw-page-title-main">Cluster analysis</span> Grouping a set of objects by similarity

Cluster analysis or clustering is the task of grouping a set of objects in such a way that objects in the same group are more similar to each other than to those in other groups (clusters). It is a main task of exploratory data analysis, and a common technique for statistical data analysis, used in many fields, including pattern recognition, image analysis, information retrieval, bioinformatics, data compression, computer graphics and machine learning.

The curse of dimensionality refers to various phenomena that arise when analyzing and organizing data in high-dimensional spaces that do not occur in low-dimensional settings such as the three-dimensional physical space of everyday experience. The expression was coined by Richard E. Bellman when considering problems in dynamic programming. The curse generally refers to issues that arise when the number of datapoints is small relative to the intrinsic dimension of the data.

Biclustering, block clustering, Co-clustering or two-mode clustering is a data mining technique which allows simultaneous clustering of the rows and columns of a matrix. The term was first introduced by Boris Mirkin to name a technique introduced many years earlier, in 1972, by John A. Hartigan.

k-means clustering is a method of vector quantization, originally from signal processing, that aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean, serving as a prototype of the cluster. This results in a partitioning of the data space into Voronoi cells. k-means clustering minimizes within-cluster variances, but not regular Euclidean distances, which would be the more difficult Weber problem: the mean optimizes squared errors, whereas only the geometric median minimizes Euclidean distances. For instance, better Euclidean solutions can be found using k-medians and k-medoids.

Medoids are representative objects of a data set or a cluster within a data set whose sum of dissimilarities to all the objects in the cluster is minimal. Medoids are similar in concept to means or centroids, but medoids are always restricted to be members of the data set. Medoids are most commonly used on data when a mean or centroid cannot be defined, such as graphs. They are also used in contexts where the centroid is not representative of the dataset like in images, 3-D trajectories and gene expression. These are also of interest while wanting to find a representative using some distance other than squared euclidean distance.

The k-medoids problem is a clustering problem similar to k-means. The name was coined by Leonard Kaufman and Peter J. Rousseeuw with their PAM algorithm. Both the k-means and k-medoids algorithms are partitional and attempt to minimize the distance between points labeled to be in a cluster and a point designated as the center of that cluster. In contrast to the k-means algorithm, k-medoids chooses actual data points as centers, and thereby allows for greater interpretability of the cluster centers than in k-means, where the center of a cluster is not necessarily one of the input data points. Furthermore, k-medoids can be used with arbitrary dissimilarity measures, whereas k-means generally requires Euclidean distance for efficient solutions. Because k-medoids minimizes a sum of pairwise dissimilarities instead of a sum of squared Euclidean distances, it is more robust to noise and outliers than k-means.

Nearest neighbor search (NNS), as a form of proximity search, is the optimization problem of finding the point in a given set that is closest to a given point. Closeness is typically expressed in terms of a dissimilarity function: the less similar the objects, the larger the function values.

In data analysis, anomaly detection is generally understood to be the identification of rare items, events or observations which deviate significantly from the majority of the data and do not conform to a well defined notion of normal behavior. Such examples may arouse suspicions of being generated by a different mechanism, or appear inconsistent with the remainder of that set of data.

Density-based spatial clustering of applications with noise (DBSCAN) is a data clustering algorithm proposed by Martin Ester, Hans-Peter Kriegel, Jörg Sander and Xiaowei Xu in 1996. It is a density-based clustering non-parametric algorithm: given a set of points in some space, it groups together points that are closely packed together, marking as outliers points that lie alone in low-density regions . DBSCAN is one of the most common, and most commonly cited, clustering algorithms.

Consensus clustering is a method of aggregating results from multiple clustering algorithms. Also called cluster ensembles or aggregation of clustering, it refers to the situation in which a number of different (input) clusterings have been obtained for a particular dataset and it is desired to find a single (consensus) clustering which is a better fit in some sense than the existing clusterings. Consensus clustering is thus the problem of reconciling clustering information about the same data set coming from different sources or from different runs of the same algorithm. When cast as an optimization problem, consensus clustering is known as median partition, and has been shown to be NP-complete, even when the number of input clusterings is three. Consensus clustering for unsupervised learning is analogous to ensemble learning in supervised learning.

Silhouette refers to a method of interpretation and validation of consistency within clusters of data. The technique provides a succinct graphical representation of how well each object has been classified. It was proposed by Belgian statistician Peter Rousseeuw in 1987.

Ordering points to identify the clustering structure (OPTICS) is an algorithm for finding density-based clusters in spatial data. It was presented by Mihael Ankerst, Markus M. Breunig, Hans-Peter Kriegel and Jörg Sander. Its basic idea is similar to DBSCAN, but it addresses one of DBSCAN's major weaknesses: the problem of detecting meaningful clusters in data of varying density. To do so, the points of the database are (linearly) ordered such that spatially closest points become neighbors in the ordering. Additionally, a special distance is stored for each point that represents the density that must be accepted for a cluster so that both points belong to the same cluster. This is represented as a dendrogram.

SUBCLU is an algorithm for clustering high-dimensional data by Karin Kailing, Hans-Peter Kriegel and Peer Kröger. It is a subspace clustering algorithm that builds on the density-based clustering algorithm DBSCAN. SUBCLU can find clusters in axis-parallel subspaces, and uses a bottom-up, greedy strategy to remain efficient.

<span class="mw-page-title-main">ELKI</span> Data mining framework

ELKI is a data mining software framework developed for use in research and teaching. It was originally at the database systems research unit of Professor Hans-Peter Kriegel at the Ludwig Maximilian University of Munich, Germany, and now continued at the Technical University of Dortmund, Germany. It aims at allowing the development and evaluation of advanced data mining algorithms and their interaction with database index structures.

In anomaly detection, the local outlier factor (LOF) is an algorithm proposed by Markus M. Breunig, Hans-Peter Kriegel, Raymond T. Ng and Jörg Sander in 2000 for finding anomalous data points by measuring the local deviation of a given data point with respect to its neighbours.

Arthur Zimek is a professor in data mining, data science and machine learning at the University of Southern Denmark in Odense, Denmark.

References

  1. 1 2 Kriegel, H. P.; Kröger, P.; Zimek, A. (2009). "Clustering high-dimensional data". ACM Transactions on Knowledge Discovery from Data. 3: 1–58. doi:10.1145/1497577.1497578. S2CID   17363900.
  2. ↑ Houle, M. E.; Kriegel, H. P.; Kröger, P.; Schubert, E.; Zimek, A. (2010). Can Shared-Neighbor Distances Defeat the Curse of Dimensionality? (PDF). Scientific and Statistical Database Management. Lecture Notes in Computer Science. Vol. 6187. p. 482. doi:10.1007/978-3-642-13818-8_34. ISBN   978-3-642-13817-1.
  3. 1 2 Parsons, Lance; Haque, Ehtesham; Liu, Huan (2004-06-01). "Subspace clustering for high dimensional data: a review". ACM SIGKDD Explorations Newsletter. 6 (1): 90–105. doi:10.1145/1007730.1007731. ISSN   1931-0145.
  4. ↑ Agrawal, R.; Gehrke, J.; Gunopulos, D.; Raghavan, P. (2005). "Automatic Subspace Clustering of High Dimensional Data". Data Mining and Knowledge Discovery. 11: 5–33. CiteSeerX   10.1.1.131.5152 . doi:10.1007/s10618-005-1396-1. S2CID   9289572.
  5. ↑ Kailing, K.; Kriegel, H. P.; Kröger, P. (2004). Density-Connected Subspace Clustering for High-Dimensional Data . Proceedings of the 2004 SIAM International Conference on Data Mining. pp.  246. doi: 10.1137/1.9781611972740.23 . ISBN   978-0-89871-568-2.
  6. ↑ De Amorim, R.C.; Mirkin, B. (2012). "Minkowski metric, feature weighting and anomalous cluster initializing in K-Means clustering". Pattern Recognition. 45 (3): 1061. Bibcode:2012PatRe..45.1061C. doi:10.1016/j.patcog.2011.08.012.
  7. ↑ Carbonera, Joel Luis; Abel, Mara (November 2014). "An Entropy-Based Subspace Clustering Algorithm for Categorical Data". 2014 IEEE 26th International Conference on Tools with Artificial Intelligence. IEEE. pp. 272–277. doi:10.1109/ictai.2014.48. ISBN   9781479965724. S2CID   7208538.
  8. ↑ Carbonera, Joel Luis; Abel, Mara (2015). "CBK-Modes: A Correlation-based Algorithm for Categorical Data Clustering". Proceedings of the 17th International Conference on Enterprise Information Systems. SCITEPRESS - Science and Technology Publications. pp. 603–608. doi:10.5220/0005367106030608. ISBN   9789897580963.
  9. ↑ Böhm, C.; Kailing, K.; Kriegel, H. -P.; Kröger, P. (2004). Density Connected Clustering with Local Subspace Preferences (PDF). Fourth IEEE International Conference on Data Mining (ICDM'04). p. 27. doi:10.1109/ICDM.2004.10087. ISBN   0-7695-2142-8.
  10. ↑ Aggarwal, C. C.; Wolf, J. L.; Yu, P. S.; Procopiuc, C.; Park, J. S. (1999). "Fast algorithms for projected clustering". ACM SIGMOD Record. 28 (2): 61. CiteSeerX   10.1.1.681.7363 . doi:10.1145/304181.304188.
  11. 1 2 3 Thrun, M. C., & Ultsch, A. : Using Projection based Clustering to Find Distance and Density based Clusters in High-Dimensional Data, J. Classif., pp. 1-33, doi: 10.1007/s00357-020-09373-2.
  12. ↑ Van der Maaten, L., & Hinton, G.: Visualizing Data using t-SNE, Journal of Machine Learning Research, Vol. 9(11), pp. 2579-2605. 2008.
  13. ↑ Venna, J., Peltonen, J., Nybo, K., Aidos, H., & Kaski, S.: Information retrieval perspective to nonlinear dimensionality reduction for data visualization, The Journal of Machine Learning Research, Vol. 11, pp. 451-490. 2010.
  14. ↑ Delaunay, B.: Sur la sphere vide, Izv. Akad. Nauk SSSR, Otdelenie Matematicheskii i Estestvennyka Nauk, Vol. 7(793-800), pp. 1-2. 1934.
  15. ↑ Dijkstra, E. W.: A note on two problems in connexion with graphs, Numerische mathematik, Vol. 1(1), pp. 269-271. 1959.
  16. ↑ Thrun, M. C., & Ultsch, A.: Uncovering High-Dimensional Structures of Projections from Dimensionality Reduction Methods, MethodsX, Vol. 7, pp. 101093, doi: 10.1016/j.mex.20200.101093,2020.
  17. ↑ "CRAN - Package ProjectionBasedClustering". Archived from the original on 2018-03-17.
  18. ↑ Dudoit, S. and Fridlyand, J. (2003). Bagging to improve the accuracy of a clustering procedure. Bioinformatics, 19/9, 1090–1099. doi:10.1093/bioinformatics/btg038.
  19. ↑ Strehl, A. & Ghosh, J. (2002). Cluster ensembles - a knowledge reuse framework for combining multiple partitions. Journal of Machine Learning Research. 3. 583-617. 10.1162/153244303321897735.
  20. ↑ Amaratunga, D., Cabrera, J. & Kovtun, V.. (2008). Microarray learning with ABC. Biostatistics. 9. 128-36. 10.1093/biostatistics/kxm017.
  21. ↑ Amaratunga, D. & Cabrera, J. & Lee, Y.S. (2014). Resampling-based similarity measures for high-dimensional data. Journal of Computational Biology. 22. 10.1089/cmb.2014.0195.
  22. ↑ Cherkas, Y., Amaratunga, D., Raghavan, N., Sasaki, J. and McMillian, M. (2016). ABC gene-ranking for prediction of drug-induced cholestasis in rats, Toxicology Reports, 3: 252–261.
  23. ↑ Kriegel, H.; Kröger, P.; Renz, M.; Wurst, S. (2005). A Generic Framework for Efficient Subspace Clustering of High-Dimensional Data (PDF). Fifth IEEE International Conference on Data Mining (ICDM'05). p. 250. doi:10.1109/ICDM.2005.5. ISBN   0-7695-2278-5.
  24. ↑ Hund, M.; Böhm, D.; Sturm, W.; Sedlmair, M.; Schreck, T.; Keim, D.A.; Majnaric, L.; Holzinger, A. (2016). "Visual analytics for concept exploration in subspaces of patient groups: Making sense of complex datasets with the Doctor-in-the-loop". Brain Informatics. 3 (4): 233–247. doi:10.1007/s40708-016-0043-5. PMC   5106406 . PMID   27747817.
  25. ↑ Thrun, M. C., & Stier, Q.: Fundamental Clustering Algorithms Suite, SoftwareX, Vol. 13(C), pp. 100642, doi: 10.1016/j.softx.2020.100642, 2021.
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