Linganisha mbinu

	Takwimu ya Pengo	Njia ya Kiwiko	Inertia
Nyanja	Tathmini ya Modeli	Tathmini ya Modeli	Tathmini ya Modeli
Familia	MCDM	MCDM	MCDM
Mwaka wa asili	2001	1953	1967
Mwanzilishi	Robert Tibshirani, Guenther Walther, Trevor Hastie	Robert Thorndike	Stuart Lloyd, James MacQueen
Aina	Statistical criterion	Heuristic optimization criterion	Clustering quality metric
Chanzo asilia	Tibshirani, R., Walther, G., & Hastie, T. (2001). Estimating the number of clusters in a data set via the gap statistic. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 63(2), 411-423.	Hastie, T., Tibshirani, R., & Friedman, J. (2009). The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer Series in Statistics.	Lloyd, S. P. (1982). Least squares quantization in PCM. IEEE Transactions on Information Theory, 28(2), 129-137.
Majina mbadala	gap index, Tibshirani gap statistic	elbow analysis, knee detection	WCSS, within-cluster sum of squares, cluster cohesion
Zinazohusiana	5	5	5
Muhtasari	The Gap Statistic, developed by Tibshirani, Walther, and Hastie in 2001, is a principled statistical method for determining the optimal number of clusters in a dataset. It compares the observed within-cluster sum of squares to the expected value under a null hypothesis of no clustering structure, providing a theoretically grounded approach to cluster number selection.	The Elbow Method is a heuristic for selecting the optimal number of clusters in partitional clustering. Introduced by Robert Thorndike in 1953, it involves fitting clustering models for increasing numbers of clusters and plotting the within-cluster sum of squares (WCSS) against the number of clusters. The 'elbow' occurs where the rate of WCSS decrease sharply changes, suggesting an optimal cluster count.	Inertia, also called Within-Cluster Sum of Squares (WCSS), is a measure of cluster cohesion that quantifies how tightly points are grouped around their cluster centroids. Lower values indicate more compact, cohesive clusters. Inertia is the primary objective function for k-means clustering and has been a fundamental metric since the method's introduction.