b1c29854-d625-4125-bf30-986cc35c0c3020210301053308720wseamdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON COMPUTERS2224-28721109-275010.37394/23205http://wseas.org/wseas/cms.action?id=40262720202720201910.37394/23205.2020.19http://wseas.org/wseas/cms.action?id=23186Data Analysis Using Representation Theory and Clustering AlgorithmsSubohAlkhushayniComputer Information Science Department, Minnesota State University, Mankato, Mankato, MN, 56001, USATaeyoungChoiComputer Information Science Department, Minnesota State University, Mankato, Mankato, MN, 56001, USADu’aAlzaleqComputer Information Science Department, Minnesota State University, Mankato, Mankato, MN, 56001, USAThis work aims to expand the knowledge of the area of data analysis through both persistence homology, as well as representations of directed graphs. To be specific, we looked for how we can analyze homology cluster groups using agglomerative Hierarchical Clustering algorithms and methods. Additionally, the Wine data, which is offered in R studio, was analyzed using various cluster algorithms such as Hierarchical Clustering, K-Means Clustering, and PAM Clustering. The goal of the analysis was to find out which cluster's method is proper for a given numerical data set. By testing the data, we tried to find the agglomerative hierarchical clustering method that will be the optimal clustering algorithm among these three; K-Means, PAM, and Random Forest methods. By comparing each model's accuracy value with cultivar coefficients, we came with a conclusion that K-Means methods are the most helpful when working with numerical variables. On the other hand, PAM clustering and Gower with random forest are the most beneficial approaches when working with categorical variables. All these tests can determine the optimal number of clustering groups, given the data set, and by doing the proper analysis. Using those the project, we can apply our method to several industrial areas such that clinical, business, and others. For example, people can make different groups based on each patient who has a common disease, required therapy, and other things in the clinical society. Additionally, for the business area, people can expect to get several clustered groups based on the marginal profit, marginal cost, or other economic indicators.312021312021310320https://www.wseas.org/multimedia/journals/computers/2020/a765105-1407.pdf10.37394/23205.2020.19.38https://www.wseas.org/multimedia/journals/computers/2020/a765105-1407.pdfG. Carlsson, "Topology and data". In: Bulletin of the American Mathematical Society 46.2 (2009) pp. 255-308.10.1007/978-3-319-42545-0F. Chaze, V. de Silva, M. Glisse, and S.Y. Oudot. The Structure an stability of persistence modules. Research Report arXiv:1207.3674 [math.AT] To appear as volume of SpringerBriefs in Mathematics. 2012.K. Meeham, D. Meyer. 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Retrieved from https://www.maths.ed.ac.uk/~v1ranick/papers/edelhare.pdfHerbert Edelsbrunner∗ and Dmitriy Morozov†. Persistent Homology: Theory and Practice. Retrieved from https://pdfs.semanticscholar.org/cf6d/43b39d66a6c3f061afeb73327312ca9cc4cb.pdfPeter Bubenik University of Florida Department of Mathematics. Topology for Data Science 1: An Introduction to Topological Data Analysis. https://people.clas.ufl.edu/peterbubenik/files/abacus_1.pdfPeter Bubenik, Department of Mathematics Cleveland State University. Statistical Topological Data Analysis using Persistence Landscapes. http://www.jmlr.org/papers/volume16/bubenik15a/bubenik15a.pdfAnon, (2019). [online] Available at: https://www.quora.com/What-are-the-most-relevant-findings-and-limitations-of-Topological-Data-Analysis [Accessed 25 Oct. 2019].k-Means Advantages and Disadvantages | Clustering in Machine Learning. (n.d.). Retrieved from https://developers.google.com/machine-learning/clustering/algorithm/advantages-disadvantages.Marina Santini, Department of Linguistics and Philology Uppsals University, Advantages & Disadvantages of K-Means and Hierarchical clustering (2016), retrieved from http://santini.se/teaching/ml/2016/Lect_10/10c_UnsupervisedMethods.pdfWhat are the Strengths and Weaknesses of Hierarchical Clustering? (n.d.). Retrieved from https://www.displayr.com/strengths-weaknesses-hierarchical-clustering/Hierarchical clustering algorithm - Data Clustering Algorithms. (n.d.). Retrieved from https://sites.google.com/site/dataclusteringalgorithms/hierarchical-clustering-algorithm.K-Means Advantages and Disadvantages | Clustering in Machine Learning. (n.d.). Retrieved from https://developers.google.com/machine-learning/clustering/algorithm/advantages-disadvantages.Marina Santini, Department of Linguistics and Philology Uppsals University, Advantages & Disadvantages of K-Means and Hierarchical clustering (2016), retrieved from http://santini.se/teaching/ml/2016/Lect_10/10c_UnsupervisedMethods.pdfUnknown. (1970, January 1). K-Means Clustering Advantages and Disadvantages. Retrieved from http://playwidtech.blogspot.com/2013/02/k-means-clustering-advantages-and.html.Keppel, J., & Schmalz, S. (2017, November 27). Anomaly Detection: (Dis-)advantages of k-means clustering - inovex-Blog. Retrieved from https://www.inovex.de/blog/disadvantages-of-k-means-clustering/B. Rieck1,2 and H. Leitte1, exploring and comparing clustering's of multivariate data sets using persistent homology, file:///E:/2019%20Proejct/reasearch3.pdf