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: Penggerombolan Provinsi di Indonesia Berdasarkan Kasus Kriminalitas Golongan Kejahatan Konvensional Menggunakan Metode K-Medoid
Abstrak: Tindakan kejahatan atau kriminalitas adalah suatu tindakan seseorang atau sekelompok orang yang melanggar hukum dan merugikan orang lain yang menjadi korban, baik dari segi materil (ekonomis) maupun psikologisnya. Kejahatan konvensional merupakan salah satu golongan kejahatan yang paling mendominasi di Indonesia. Banyaknya kasus kejahatan konvensional meningkat setiap tahunnya, maka perlu dilakukan penggerombolan provinsi di Indonesia berdasarkan kasus kriminalitas golongan kejahatan konvensional. Penggerombolan ini dilakukan agar jenis kejahatan konvensional yang terjadi di suatu provinsi dapat diidentifikasi. Hasil penelitian ini diharapkan dapat membantu pemerintah dan kepolisian dalam rangka menekan tingkat kejahatan konvensional (conventional crime rate). Secara eksplorasi diketahui bahwa ada beberapa provinsi yang menjadi pencilan pada beberapa jenis kejahatan, sehingga penggerombolannya menggunakan metode k-medoid dengan jarak Euclidean. Penentuan ukuran gerombol optimal dan evaluasi hasil penggerombolan dilakukan berdasarkan nilai koefisien silhouette, indeks Dunn dan indeks connectivity. Penelitian ini menghasilkan tiga gerombol dengan nilai koefisien silhouette 0,27; indeks Dunn sebesar 0,22 dan indeks connectivity sebesar 17,86., Crime is an act of a person or group of people who violate the law and harm other people who are victims, both in terms of material (economic) and psychological. Conventional crime is one of the most dominating crime groups in Indonesia. As the number of conventional crime cases increases every year, it is necessary to cluster provinces in Indonesia based on conventional criminal cases. This clustering is carried out so that the types of conventional crimes that occur in a province can be identified. The results of this study are expected to help the government and the police to reduce the conventional crime rate. Explorationally, it is known that several provinces are outliers in several types of crime. Hence, the clustering uses the k-medoid method with Euclidean distance. Determination of the optimal cluster size and evaluation of the clustering results are carried out based on the silhouette coefficient, Dunn index, and connectivity index. This study resulted in three clusters with a silhouette coefficient value of 0,27; Dunn index of 0,22 and connectivity index of 17,86.
Keyword: cluster analysis, conventional crime, k-medoid

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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