Judul: Comparison of K-Medoid and Robust Sparse K-Means for Clustering Provinces in Indonesia Based on Crime Indicators
Abstrak: Kriminalitas adalah tindakan melanggar hukum yang dilakukan dengan sadar dan sengaja untuk menguntungkan diri sendiri dan merugikan orang lain. Pada tahun 2021, Indonesia menempati peringkat 72 dari 142 negara di dunia dengan indeks kriminalitas sebesar 46,23. Semakin tinggi tingkat kriminalitas pada suatu daerah, semakin banyak kejahatan yang terjadi sehingga menyebabkan masyarakat kehilangan rasa aman. Banyak kejahatan provinsi di Indonesia beragam yang menunjukkan adanya kesenjangan akan keamanan. Kesenjangan banyak kejahatan antarprovinsi menimbulkan pencilan pada beberapa daerah. Tujuan penelitian adalah menggerombolkan provinsi berdasarkan indikator kriminalitas dengan memerhatikan pencilan menggunakan K-Medoid dan Robust Sparse K-Means. Data yang digunakan pada penelitian ini berasal dari 34 provinsi dengan 16 peubah numerik indikator kriminalitas tahun 2021. K-Medoid menghasilkan empat gerombol optimal dengan banyak anggota dari masing-masing gerombolnya adalah 22, 4, 1, dan 7. Gerombol 1, 2, dan 3 dikategorikan sebagai daerah dengan angka kejahatan rendah. Gerombol 3 merupakan daerah dengan kepadatan penduduk dan PDRB tinggi ekstrem dibandingkan gerombol lain. Gerombol 4 dikategorikan sebagai daerah dengan angka kejahatan sangat tinggi. Hasil penggerombolan optimum ditentukan berdasarkan rasio sw⁄sb minimum. K Medoid lebih baik dibandingkan RSK-Means karena memiliki rasio sw⁄sb lebih kecil., Crime is an unlawful act carried out consciously and deliberately to benefit oneself and harm others. In 2021, Indonesia was ranked 72 out of 142 countries in the world with a crime index of 46,23. The higher the crime rate in an area, the more crimes that occur, causing people to feel insecure. The number of crimes per province in Indonesia varies, showing a gap in security. This range of variation leads to an extreme crime rate index or outliers in some areas. The aim of this study is to cluster provinces based on crime indicators by paying attention to outliers using K Medoid and Robust Sparse K-Means. The data used in this study were from 34 provinces with 16 numerical variables of crime indicators in 2021. K-Medoid produces four optimal clusters with the number of members in each cluster being 22, 4, 1, and 7. Clusters 1, 2, and 3 are categorized as areas with a low crime rate. Cluster 3 is an area with an extremely high population density and GRDP compared to other clusters. Cluster 4 is categorized as an area with a very high crime rate. The best clustering results are determined based on the minimum sw⁄sb ratio. K Medoid has better performance than RSK Means with a smaller sw⁄sb ratio.
Keyword: gerombol, kriminalitas, k-medoid, pencilan, robust sparse k-means

Judul: Optimization of Provincial Clustering Based on Sustainable Development Goals with Hierarchical Clustering, K-Medoids, and DBSCAN
Abstrak: Analisis gerombol merupakan teknik statistika untuk mengidentifikasi gerombol bermakna dari objek data multivariat berdasarkan ukuran kemiripan. Selain memilih ukuran kemiripan, ketepatan dalam menerapkan metode pembentukan gerombol juga menjadi hal fundamental. Karakteristik data dapat memengaruhi performa analisis gerombol. Data multidimensional dengan jumlah dimensi yang besar memiliki tantangan adanya curse of dimensionality. Salah satu cara mengatasi masalah tersebut adalah dengan melakukan reduksi dimensi. Penelitian ini mengevaluasi performa metode reduksi dimensi analisis komponen utama dan t-SNE, serta membandingkan tiga metode penggerombolan yaitu metode hierarki, metode partisi dengan algoritma K-Medoids, dan metode berbasis kepekatan dengan algoritma Density-Based Spatial Clustering of Applications with Noise (DBSCAN) pada data multidimensi, studi kasus penggerombolan provinsi berdasarkan tujuan pembangunan berkelanjutan (TPB). Evaluasi kinerja hasil penggerombolan didasarkan pada tiga indeks validitas. Hasil penelitian menunjukkan bahwa kombinasi t-SNE dengan algoritma hierarki pautan tunggal atau dengan DBSCAN menghasilkan gerombol paling optimal dengan nilai Silhouette tertinggi (0,56), indeks Davies-Bouldin terendah (0,47), dan indeks Calinski-Harabasz tertinggi (75,04) dibandingkan hasil penggerombolan algoritma lainnya. Penggerombolan optimal tersebut menghasilkan tujuh gerombol provinsi dengan karakteristik pencapaian TPB yang berbeda. Terdapat dua gerombol yang menunjukkan pencapaian TPB rendah, sehingga memerlukan perhatian ekstra sesuai dengan karakteristiknya masing-masing. Gerombol-gerombol tersebut terdiri atas provinsi di wilayah timur Indonesia., Cluster analysis is a statistical technique used to identify meaningful clusters from multivariate data objects based on similarity measures. In addition to choosing the right similarity measures, the accuracy in applying clustering methods is also fundamental. The characteristics of the data can influence the performance of cluster analysis. Multidimensional data with many dimensions face the challenge known as the curse of dimensionality. Dimensionality reduction helps address this issue. This study evaluates the performance of two dimensionality reduction methods, principal component analysis and t-SNE, and compares three clustering methods: hierarchical clustering, K-Medoids, and Density-Based Spatial Clustering of Applications with Noise (DBSCAN). The study focuses on clustering provinces based on sustainable development goals (SDGs). Clustering performance is evaluated using three validity indices. Results indicate that combining t-SNE with the single-linkage hierarchical algorithm or DBSCAN produces the most optimal clusters. These methods achieve the highest Silhouette score (0.56), the lowest Davies-Bouldin index (0.47), and the highest Calinski-Harabasz index (75.04) compared to other algorithms. The optimal clustering results in seven distinct clusters of provinces with different SDGs achievement characteristics. Two clusters, comprising provinces in eastern Indonesia, show low SDGs achievement and require extra attention based on their specific characteristics.
Keyword: cluster analysis, DBSCAN, principal component analysis, sustainable development goals, t-SNE

Judul: Hamiltonian formulation for linear waves motion in two layers fluid
Abstrak: The surface wave could be considered as a wave that separates two fluids, namely water and air. Based on this assumption, it is introduced the interfacial wave, a wave between two layers of fluid with different density. The formulation of interfacial waves motion begins with deriving the base equation of irrotational ideal fluid. Furthermore, according to irrotational fluid assumption, the base equation can be stated in velocity potential. In this derivation, the fluids domain is assumed to be restricted by rigid lid boundary conditions, both at the upper and lower limit. Therefore, the interfacial waves motion can be explained in a hamiltonian formulation. In the hamiltonian formulation, total energy is defined as the sum of kinetic and potential energy. The hamiltonian system is obtained from reduction of kinetic energy by using the Dirichlet Neumann Operator. The resulted kinetic energy equation is nonlinear. Therefore, this form is linearized by first part of the Taylor expansion. This linearization gives a dispersion relation of linear wave. Based on this dispersion relation, the phase speed of the linear wave depends on the density ratio of the two layers fluid.
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