مقایسهٔ روش‌ها

	آماره شکاف	شاخص کالینسکی-هاراباس	شاخص دیویس-بولدین	روش آرنج	Inertia (Within-Cluster Sum of Squares)
حوزه	ارزیابی مدل	ارزیابی مدل	ارزیابی مدل	ارزیابی مدل	ارزیابی مدل
خانواده	MCDM	MCDM	MCDM	MCDM	MCDM
سال پیدایش	2001	1974	1979	1953	1967
پدیدآور	Robert Tibshirani, Guenther Walther, Trevor Hastie	Tadeusz Calinski, Jerzy Harabasz	David L. Davies, Donald W. Bouldin	Robert Thorndike	Stuart Lloyd, James MacQueen
نوع	Statistical criterion	Cluster quality metric	Cluster quality metric	Heuristic optimization criterion	Clustering quality metric
منبع بنیادین	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.	Calinski, T., & Harabasz, J. (1974). A dendrite method for cluster analysis. Communications in Statistics, 3(1), 1-27.	Davies, D. L., & Bouldin, D. W. (1979). A cluster separation measure. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1(2), 224-227.	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.
نام‌های دیگر	gap index, Tibshirani gap statistic	variance ratio criterion, pseudo F-statistic, CH index	DBI, Davies Bouldin index	elbow analysis, knee detection	WCSS, within-cluster sum of squares, cluster cohesion
مرتبط	5	5	5	5	5
خلاصه	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 Calinski-Harabasz Index, also called the Variance Ratio Criterion, was introduced by Calinski and Harabasz in 1974. It is a metric that measures the ratio of between-cluster variance to within-cluster variance, adjusted for the number of clusters and data points. Higher values indicate better-separated, more compact clusters.	The Davies-Bouldin Index, introduced by Davies and Bouldin in 1979, is a metric for evaluating clustering quality based on the average similarity between each cluster and its most similar neighboring cluster. Lower values indicate better clustering, with a minimum of 0 representing perfectly separated, non-overlapping clusters.	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.