ܦܘܫܩܐ ܕܥܢܩܘ̈ܕܐ

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ܟܡܕܒܩܚ ܥܡܘ̈ܕܐ ܬܪܝܨܐܝܬ ܡܢ ܓܠܝܘܢܐ (worksheet)، ܥܡ ܣܕܪܐ ܕܪ̈ܫܐ (header row) – ܟܪܫܡܚ ܓܘ ܐܩܣܠ، ܟܢܣܚܚ، ܘܟܡܕܒܩܚ ܠܐܟܐ:

ܡܫܬܚܠܦ̈ܢܐ ܘܬܘܩ̈ܢܐ (Variables and settings)

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ܐܡܬܝ ܕܫܘܐܠܐ ܥܠ ܐܢܫ̈ܐ ܝܠܗ ܘܠܐ ܥܠ ܡܫܬܚܠܦ̈ܢܐ (variables): ܐܢ ܐܝܬ ܒܝܢܝ ܡܦܢܝ̈ܢܐ (respondents) ܙܢ̈ܐ ܦܪ̈ܝܫܐ – ܡܦܠܚ̈ܢܐ ܕܠܐ ܥܒܕܐ ܩܘܒܠ ܒܪ̈ܘܝܐ ܕܡܐ ܕܐܝܬ، ܝܠܘܦ̈ܐ ܕܣܢܝܩ̈ܐ ܝܢܐ ܠܥܘܕܪܢܐ ܩܘܒܠ ܐܢܝ ܕܠܐ – ܕܠܐ ܕܐܢܝ ܓܘ̈ܕܐ ܣܝܡ̈ܐ ܗܘܝ ܡܢ ܩܕܡ. Wendler ܘ-Gröttrup (2021، ܦܐܬܐ 623) ܟܡܦܪܫܝ ܦܘܫܩܐ ܕܥܢܩܘ̈ܕܐ (cluster analysis) ܐܝܟ ܟܢܫܐ ܕܐܘܪ̈ܚܬܐ ܕܡܫܬܚܠܦ̈ܢܐ ܣܓܝ̈ܐܐ ܕ"ܡܬܦܠܚܝ ܩܐ ܡܫܟܚܬܐ ܕܓܘ̈ܕܐ ܕܡܕܡ̈ܐ 'ܕܡܝ̈ܐ'"، ܘܟܡܢܝ ܒܝܢܝ ܡܬ̈ܠܐ ܦܘܠܓܐ ܕܙܒܘ̈ܢܐ ܓܘ ܙܒܢܬܐ ܘܡܫܟܚܬܐ ܕ"ܓܘ̈ܕܐ ܙܥܘܪ̈ܐ ܕܝܠܘܦ̈ܐ ܥܡ ܣܢܝܩܘ̈ܬܐ ܡܝܚܕ̈ܝܬܐ" ܠܦܘܬ ܒܝܬܐ ܘܥܒܕܐ (ܦܐܬܐ 624). ܦܘܪܫܢܐ ܪܫܝܐ ܝܬܝܪ، ܓܘ ܨܘܪܬܐ 7.1 ܕܝܗܘܢ: ܦܘܫܩܐ ܕܥܢܩܘ̈ܕܐ ܟܟܢܫ ܣܕܪ̈ܐ (ܡܫܬܘܬܦ̈ܢܐ)، ܘܦܘܫܩܐ ܕܥܠ̈ܠܬܐ (factor analysis) ܟܟܢܫ ܥܡܘ̈ܕܐ (ܦܣܘ̈ܩܐ). ܐܢ ܫܘܐܠܐ ܝܠܗ ܐܝܢܝ ܦܣܘ̈ܩܐ (items) ܟܡܫܚܝ ܚܕ ܡܕܡ – ܦܘܫܩܐ ܕܥܠ̈ܠܬܐ ܡܫܪܪܢܐ ܘܦܐܬܐ ܝܘܠܦܢܐ ܕܠܐ ܡܕܒܪܢܐ (PCA) ܘܡܚܫܒܢܐ ܦܘܫܩܐ ܕܥܠ̈ܠܬܐ ܡܒܨܝܢܐ؛ ܐܢ ܓܘ̈ܕܐ ܝܕܝܥ̈ܐ ܝܢܐ ܡܢ ܩܕܡ ܘܟܒܥܝܚ ܐܢ ܦܪ̈ܝܫܐ ܝܢܐ – ܦܘܫܩܐ ܕܫܘܚܠܦܐ. ܠܐܟܐ ܠܝܬ ܡܫܬܚܠܦܢܐ ܡܢܒܝܐ ܘܠܝܬ "ܦܘܢܝܐ ܬܪܝܨܐ": ܐܗܐ ܝܘܠܦܢܐ ܕܠܐ ܡܕܒܪܢܐ (unsupervised learning) ܝܠܗ، ܘ"ܡܛܠ ܕܐܝܬܘܬܐ ܓܘ ܓܘܕܐ ܠܐ ܝܕܝܥܬܐ ܝܠܗ ܡܢ ܩܕܡ، ܐܠܓܘܪܝܬܡ ܠܐ ܡܨܐ ܕܝܠܦ ܡܢܝ̈ܘܬܐ ܕܓܘ̈ܕܐ" (Wendler & Gröttrup, 2021, p. 627).

ܦܐܬܐ ܝܘܠܦܢܐ ܕܠܐ ܡܕܒܪܢܐ ܟܚܐܪ ܒܟܠ ܐܘܪ̈ܚܬܐ ܕܥܢܩܘ̈ܕܐ (k-ܡܨܥܝ̈ܬܐ، ܕܕܪ̈ܓܐ، Gaussian mixtures، DBSCAN، TwoStep) ܘܒܫܘܐܠܐ ܐܢ ܐܝܬ ܥܢܩܘ̈ܕܐ ܒܟܠܗ. ܐܗܐ ܦܐܬܐ ܟܪܗܛ ܐܘܪܚܐ ܕܡܬܚܙܝܐ ܣܓܝ ܝܬܝܪ ܓܘ ܦܢܩܝ̈ܬܐ ܕܫܘܐ̈ܠܐ (questionnaires): Ward ܘܒܬܪܗ k-ܡܨܥܝ̈ܬܐ (k-means)، ܥܠ ܡܫܬܚܠܦ̈ܢܐ ܟܡܝ̈ܝܐ. ܡܫܬܚܠܦ̈ܢܐ ܕܓܢ̈ܣܐ (categorical؛ ܡܬܠܐ ܓܢܣܐ، ܕܘܟܬܐ ܕܝܘܠܦܢܐ) ܠܐ ܟܥܐܠܝ ܓܘ ܚܘܫܒܢܐ ܕܪܘܚܩܐ ܠܐܟܐ؛ ܐܢܝ ܡܬܦܠܚ̈ܐ ܝܢܐ ܒܬܪ ܗܕܐ ܩܐ ܫܘܪܪܐ ܘܬܚܘܝܬܐ ܕܥܢܩܘ̈ܕܐ.

ܣܘܟܠܐ: ܬܪܝܢ ܕܪ̈ܓܐ

ܕܪܓܐ ܐ – ܥܢܩܘ̈ܕܐ ܕܕܪ̈ܓܐ ܟܢܫܢ̈ܝܐ (agglomerative hierarchical clustering). ܟܫܪܝܚ ܡܢ ܟܠ ܡܫܬܘܬܦܢܐ ܐܝܟ ܥܢܩܘܕܐ ܦܪܝܫܐ، ܘܓܘ ܟܠ ܦܣܥܬܐ ܟܚܝܕܚ ܬܪܝܢ ܥܢܩܘ̈ܕܐ ܩܪܝܒ̈ܐ ܝܬܝܪ، ܗܕܟܐ ܕܟܠܗܘܢ ܓܘ ܚܕ ܥܢܩܘܕܐ (Hastie et al., 2009, p. 520; Everitt et al., 2011, p. 73). ܟܠ ܕܪܓܐ ܦܘܠܓܐ ܝܠܗ، ܘܐܝܠܢܐ ܟܠܗ dendrogram ܝܠܗ: "ܪܘܡܐ ܕܟܠ ܩܛܪܐ ܦܘܚܡܝܐ ܝܠܗ ܠܠܐ ܕܡܝܘܬܐ ܒܝܢܝ ܬܪܝܢ ܒܢ̈ܘܗܝ"، ܘܦܣܩܐ ܦܫܝܛܐ ܕܐܝܠܢܐ ܟܝܗܒ ܦܘܠܓܐ ܠܥܢܩܘ̈ܕܐ (Hastie et al., 2009, p. 521). ܡܘܕܝ ܝܠܗ "ܩܪܝܒ̈ܐ ܝܬܝܪ" ܐܡܬܝ ܕܡܠܬܐ ܥܠ ܓܘ̈ܕܐ ܝܠܗ؟ ܐܗܐ ܝܠܗ ܐܣܘܪܐ (linkage؛ Everitt et al., 2011, pp. 76-77, Figure 4.3): ܝܚܝܕܝܐ (single؛ ܙܘܓܐ ܩܪܝܒܐ)، ܓܡܝܪܐ (complete؛ ܙܘܓܐ ܪܚܝܩܐ)، ܡܨܥܝܬܐ (average؛ ܟܠ ܙܘ̈ܓܐ) – ܘWard، ܕܓܘܗ ܟܚܝܕܚ ܬܪܝܢ ܥܢܩܘ̈ܕܐ ܕܓܘ ܚܘܝܕܐ ܕܝܗܘܢ "ܬܘܣܦܬܐ ܓܘ ܓܘܡܪܐ ܕܡܪ̈ܒܥܐ ܓܘ ܥܢܩܘ̈ܕܐ… ܥܠ ܟܠ ܡܫܬܚܠܦ̈ܢܐ" ܙܥܘܪܬܐ ܝܬܝܪ ܝܠܗ (Ward, 1963، ܐܝܟ ܕܡܬܡܠܠ ܓܘ Everitt et al., 2011، ܦܐܬܐ 78، ܠܘܚܐ 4.1؛ ܘܗܕܟܐ ܐܦ Wendler & Gröttrup, 2021، ܦܐܬܐ 648، ܠܘܚܐ 7.24). Ward ܝܠܗ ܐܣܘܪܐ ܕܦܐܫ ܠܢܝܫܐ ܕ-k-ܡܨܥܝ̈ܬܐ: ܬܪܘܝܗܝ ܟܡܙܥܪܝ ܚܕ ܡܕܡ – ܓܘܡܪܐ ܕܪܘ̈ܚܩܐ ܡܪ̈ܒܥܐ (sum of squared distances) ܡܢ ܡܨܥ̈ܝܐ ܓܘ ܥܢܩܘ̈ܕܐ. ܐܣܘܪܐ ܝܚܝܕܝܐ ܟܡܨܠܐ ܠ"ܫܫܠܬܐ" ܕܡܫܬܘܬܦ̈ܢܐ ܠܥܢܩܘ̈ܕܐ ܐܪ̈ܝܟܐ، ܘܓܡܝܪܐ – ܠܥܢܩܘ̈ܕܐ ܟܢܝܫ̈ܐ ܕܚܒܪ̈ܘܗܝ ܒܥܕܢ̈ܐ ܩܪܝܒ̈ܐ ܝܢܐ ܝܬܝܪ ܠܥܢܩܘܕܐ ܐܚܪܢܐ (Hastie et al., 2009, pp. 524-525).

ܕܪܓܐ ܒ – k-ܡܨܥܝ̈ܬܐ (k-means). ܠܥܢܩܘ̈ܕܐ ܕܕܪ̈ܓܐ ܐܝܬ ܚܣܝܪܘܬܐ ܒܓܘ ܒܢܝܢܐ: "ܦܘܠܓܐ ܕܡܕܡ ܠܥܢܩܘܕܐ ܡܝܚܕܐ ܡܢ ܡܬܘܡ ܠܐ ܡܨܐ ܕܡܫܚܠܦ" (Wendler & Gröttrup, 2021, p. 628, Table 7.1) – ܡܫܬܘܬܦܢܐ ܕܚܝܕܐ ܝܠܗ ܩܕܡ ܥܡ ܫܒܒܐ ܟܦܐܫ ܬܡܢ ܐܦ ܐܢ ܒܚܪܬܐ ܩܪܝܒܐ ܝܠܗ ܝܬܝܪ ܠܡܨܥܝܐ ܕܥܢܩܘܕܐ ܐܚܪܢܐ. k-ܡܨܥܝ̈ܬܐ ܟܬܪܨ ܐܗܐ: ܐܢ ܐܝܬ ܡܨܥ̈ܝܐ، ܟܦܠܓܚ ܟܠ ܡܫܬܘܬܦܢܐ ܠܡܨܥܝܐ ܩܪܝܒܐ؛ ܐܢ ܐܝܬ ܦܘܠܓܐ، ܟܚܫܒܚ ܟܠ ܡܨܥܝܐ ܐܝܟ ܡܨܥܝܬܐ (mean)؛ ܘܟܡܬܢܚ ܗܕܟܐ ܕܚܕ ܦܘܠܓܐ ܠܐ ܟܡܫܚܠܦ (Hastie et al., 2009، ܦܐܬܐ 510، ܐܠܓܘܪܝܬܡ 14.1). ܟܠ ܦܣܥܬܐ ܟܡܙܥܪܐ ܓܘܡܪܐ ܕܪܘ̈ܚܩܐ ܡܪ̈ܒܥܐ، ܘܡܛܠ ܗܕܐ ܟܢܘܫܝܐ ܫܪܝܪܐ ܝܠܗ – "ܐܝܢܐ ܦܐܪܐ ܡܨܐ ܕܗܘܐ ܛܝܡܐ ܬܚܬܝܐ ܕܘܟܝܐ ܠܐ ܛܒܐ ܝܬܝܪ"، ܘܬܠܝܐ ܝܠܗ ܒܢܘܩܙܐ ܕܫܘܪܝܐ (ܬܡܢ). ܠܐܟܐ ܬܪܝܢ ܕܪ̈ܓܐ ܟܦܓܥܝ: Géron (2022, Chapter 9, "Centroid initialization methods") – "ܐܢ ܒܓܕܫܐ ܝܕܥܝܬܘܢ ܒܩܘܪܒܐ ܐܝܟܐ ܣܢܝܩ̈ܐ ܝܢܐ ܡܨܥ̈ܝܐ (ܡܬܠܐ، ܐܢ ܪܗܛܠܘܟܘܢ ܩܕܡ ܐܠܓܘܪܝܬܡ ܐܚܪܢܐ ܕܥܢܩܘ̈ܕܐ)، ܡܨܝܬܘܢ ܕܣܝܡܝܬܘܢ ܠܗܘܢ" ܐܝܟ ܢܘܩܙܐ ܕܫܘܪܝܐ. ܘܗܕܟܐ ܐܦ Everitt ܘܚܒܪ̈ܘܗܝ (2011، ܦܐܬܐ 122): ܦܘܠܓܐ ܕܫܘܪܝܐ "ܡܨܐ ܕܗܘܐ ܦܐܪܐ ܕܡܦܠܚܢܘܬܐ ܩܕܡܝܬܐ ܕܐܘܪܚܐ ܐܚܪܬܐ ܕܥܢܩܘ̈ܕܐ، ܒܠܟܝ ܚܕܐ ܡܢ ܐܘܪ̈ܚܬܐ ܕܕܪ̈ܓܐ". ܠ-Kassambara (2017، ܦܐܬܐ 163) ܐܝܬ ܫܡܐ ܩܐ ܐܗܐ ܐܘܪܚܐ – "hierarchical k-means"، ܐܘܪܚܐ ܡܚܠܛܬܐ "ܩܐ ܡܛܝܒܘܬܐ ܕܦܐܪ̈ܐ ܕ-k-ܡܨܥܝ̈ܬܐ"، ܕܝܠܝܕܬܐ ܝܠܗ ܡܢ ܗܕܐ ܕ"ܦܘܫܪܐ ܐܚܪܝܐ ܕ-k-ܡܨܥܝ̈ܬܐ ܪܓܝܫܐ ܝܠܗ ܣܓܝ ܠܓܘܒܝܐ ܓܕܫܢܝܐ ܕܫܘܪܝܐ ܕܡܨܥ̈ܝܐ ܕܥܢܩܘ̈ܕܐ"، ܘܬܠܬܐ ܦܣܥ̈ܬܐ ܕܝܠܗ ܚܬܝܬܐܝܬ ܐܢܝ ܝܢܐ: ܥܢܩܘ̈ܕܐ ܕܕܪ̈ܓܐ ܘܦܣܩܐ ܕܐܝܠܢܐ ܠ-k ܥܢܩܘ̈ܕܐ، ܡܨܥܝܐ ܕܝܘܩܪܐ ܕܟܠ ܚܕ، ܘ-k-ܡܨܥܝ̈ܬܐ ܡܢ ܐܢܝ ܡܨܥ̈ܝܐ – ܐܡܬܝ ܕ"k-ܡܨܥܝ̈ܬܐ ܒܕ ܡܛܝܒ ܦܘܠܓܐ ܕܫܘܪܝܐ"، ܘܡܛܠ ܗܕܐ ܐܝܢܐ "ܡܨܐ ܕܗܘܐ ܦܪܝܫܐ ܩܠܝܠ" ܡܢ ܐܚܪܝܐ. ܐܗܐ ܦܐܬܐ ܚܬܝܬܐܝܬ ܐܗܐ ܟܥܒܕ: ܟܦܣܩ ܐܝܠܢܐ ܕ-Ward ܠ-k ܥܢܩܘ̈ܕܐ، ܟܚܫܒ ܡܨܥܝܐ ܕܝܘܩܪܐ ܕܟܠ ܚܕ، ܘܟܪܗܛ ܡܢܝܗܘܢ k-ܡܨܥܝ̈ܬܐ. ܦܐܪܐ ܦܣܝܩܐ ܝܠܗ (deterministic) – ܗܢ ܝܗܝ̈ܒܬܐ، ܗܢ ܦܐܪܐ، ܓܘ ܟܠ ܪܗܛܐ – ܘܠܝܬ ܓܘܗ ܫܘܪܝܐ ܓܕܫܢܝܐ ܕ-Wendler ܘ-Gröttrup (2021، ܦܐܬܐ 677) ܟܡܙܗܪܝ ܡܢܗ ܓܘ K-Means ܕ-Modeler، ܕܟܫܪܐ ܡܢ ܣܕܪܐ ܩܕܡܝܐ ܘܡܛܠ ܗܕܐ "ܦܐܪܐ ܬܠܝܐ ܝܠܗ ܒܟܡܝܘܬܐ ܡܝܚܕܬܐ ܒܛܟܣܐ ܕܡܕܡ̈ܐ". ܡܢ ܓܢܒܐ ܐܚܪܢܐ، ܚܕ ܢܘܩܙܐ ܕܫܘܪܝܐ ܐܦ ܬܚܘܡܐ ܝܠܗ: "ܦܐܪ̈ܐ ܕܐܘܪܚܐ ܕܡܛܝܒܘܬܐ (optimization) ܡܨܝ ܕܗܘܝ ܡܥܒܕ̈ܐ ܣܓܝ ܡܢ ܓܘܒܝܐ ܕܦܘܠܓܐ ܕܫܘܪܝܐ"، ܘ"ܛܒ ܝܠܗ ܕܪܗܛܚ ܐܠܓܘܪܝܬܡ ܕܡܛܝܒܘܬܐ ܟܡܐ ܓܗ̈ܐ ܥܡ ܦܘܠ̈ܓܐ ܦܪ̈ܝܫܐ ܕܫܘܪܝܐ"؛ ܘܐܢ ܦܘܠ̈ܓܐ ܦܪ̈ܝܫܐ ܕܫܘܪܝܐ ܟܝܗܒܝ ܟܢܘ̈ܫܝܐ ܦܪ̈ܝܫܐ ܣܓܝ، ܐܗܐ ܒܥܕܢܐ ܣܓܝ ܪܘܫܡܐ ܝܠܗ ܕܡܢܝܢܐ ܕܥܢܩܘ̈ܕܐ ܛܥܝܐ ܝܠܗ، ܝܢ ܕܠܝܬ ܣܗܕܘܬܐ ܓܠܝܬܐ ܩܐ ܥܢܩܘ̈ܕܐ (Marriott, 1982, as cited in Everitt et al., 2011, p. 122). ܐܗܐ ܦܐܬܐ ܟܬܢܐ ܬܪܝܢ ܒܘܚܪ̈ܢܐ ܙܥܘܪ̈ܐ ܓܘ ܐܗܐ ܐܘܪܚܐ – ܦܘܚܡܐ ܒܝܢܝ ܬܪܝܢ ܕܪ̈ܓܐ ܘܒܘܚܪܢܐ ܕܫܘܢܝܐ ܝܚܝܕܝܐ (ܠܬܚܬ)؛ ܪܗܛܐ ܡܢ ܫܘܪ̈ܝܐ ܓܕܫܢ̈ܝܐ ܣܓܝ̈ܐܐ – ܓܘ R، ܥܡ nstart.

Wendler ܘ-Gröttrup (2021، ܦܐܬܐ 628، ܠܘܚܐ 7.1) ܟܟܢܫܝ ܛܝܒ̈ܘܬܐ ܘܚܣܝܪ̈ܘܬܐ ܕܬܪܬܝܢ ܫܪ̈ܒܬܐ، ܘܐܘܪܚܐ ܕܬܪܝܢ ܕܪ̈ܓܐ ܟܫܩܠܐ ܡܢ ܟܠ ܚܕܐ ܡܐ ܕܛܒܬܐ ܝܠܗ ܒܗ: ܡܢ ܕܕܪ̈ܓܐ – ܐܝܠܢܐ ܕܟܡܚܘܐ ܒܢܝܢܐ ܓܘ ܟܠ ܕܪ̈ܓܐ ܘܦܐܪܐ "ܦܣܝܩܐ (ܠܐ ܓܕܫܢܝܐ)"، ܘܡܢ k-ܡܨܥܝ̈ܬܐ – ܪܓܝܓܘܬܐ ܕ"ܦܘܠܓܐ ܡܢ ܪܝܫ ܕܡܕܡ̈ܐ ܠܥܢܩܘ̈ܕܐ ܐܚܪ̈ܢܐ". ܦܘܚܡܐ ܒܝܢܝ ܬܪܝܢ ܕܪ̈ܓܐ ܐܦ ܒܘܚܪܢܐ ܙܥܘܪܐ ܝܠܗ: ܐܗܐ ܦܐܬܐ ܟܬܢܐ ܟܡܐ ܡܫܬܘܬܦ̈ܢܐ ܫܚܠܦܠܗܘܢ ܥܢܩܘܕܐ ܓܘ ܕܪܓܐ ܒ ܘܡܚܘܝܢܐ ܕ-Rand ܡܬܘܪܨܐ (ARI) ܒܝܢܝ ܬܪܝܢ ܦܘܠ̈ܓܐ – 1 ܫܠܡܘܬܐ ܓܡܝܪܬܐ ܝܠܗ. ܐܗܐ ܒܘܚܪܢܐ ܕܐܠܓܘܪܝܬܡ ܝܠܗ ܥܠ ܗܢ ܝܗܝ̈ܒܬܐ، ܠܐ ܕܩܝܡܘܬܐ ܕܦܘܫܪܐ ܓܘ ܕܘܡܝܐ ܐܚܪܢܐ – ܘܥܠ ܐܗܐ ܓܘ ܦܣܩܐ ܫܪ̈ܛܐ ܘܙܘܗܪ̈ܐ.

ܡܬܩܢܘܬܐ ܘܪܘܚܩܐ

ܪܘܚܩܐ. ܬܪܬܝܢ ܐܘܪ̈ܚܬܐ ܟܦܠܚ̈ܐ ܥܠ ܪܘܚܩܐ ܕ-Euclid ܒܝܢܝ ܡܫܬܘܬܦ̈ܢܐ (participants) – ܫܪܫܐ ܕܓܘܡܪܐ ܕܡܪ̈ܒܥܐ ܕܦܘܪ̈ܫܢܐ ܓܘ ܟܠ ܡܫܬܚܠܦ̈ܢܐ، "ܡܚܘܝܢܐ ܕܪܘܚܩܐ ܕܡܬܚܙܐ ܣܓܝ ܝܬܝܪ" (Everitt et al., 2011, p. 49, Equation 3.4) – ܘܥܠ ܡܪܒܥܐ ܕܝܠܗ، ܕ"ܪܓܝܫܐ ܝܠܗ ܝܬܝܪ ܠܛܝ̈ܡܐ ܢܘܟܪ̈ܝܐ (outliers)" (Wendler & Gröttrup, 2021, pp. 630, 637). ܡܢ ܐܟܐ ܟܐܬܐ ܫܘܐܠܐ ܪܫܝܐ ܓܘ ܐܗܐ ܦܐܬܐ: ܡܘܕܝ ܟܥܐܠ ܓܘ ܪܘܚܩܐ، ܘܒܐܝܢܐ ܡܬܩܠܐ. Hastie ܘܚܒܪ̈ܘܗܝ (2009، ܦܐܬܐ 506) ܟܦܣܩܝ ܕ"ܓܘܒܝܐ ܕܡܚܘܝܢܐ ܬܪܝܨܐ ܕܠܐ ܕܡܝܘܬܐ ܪܫܝܐ ܝܠܗ ܣܓܝ ܝܬܝܪ ܩܐ ܨܘܠܚܢܐ ܓܘ ܥܢܩܘ̈ܕܐ ܡܢ ܓܘܒܝܐ ܕܐܠܓܘܪܝܬܡ ܕܥܢܩܘ̈ܕܐ".

ܡܬܩܢܘܬܐ (standardization) – ܐܝܢ، ܒܥܕܢܐ ܣܓܝ. ܓܘ ܪܘܚܩܐ ܕ-Euclid، ܡܫܬܚܠܦܢܐ ܥܡ ܦܬܝܘܬܐ (range) ܪܒܬܐ ܟܡܫܠܛ: ܫܥ̈ܐ ܒܝܘܡܐ (0-12) ܒܕ "ܛܦܝ" ܠܡܢܝܢܐ ܕܣܒܠܬܐ (1-5)، ܘܥܢܩܘ̈ܕܐ ܒܕ ܗܘܝ ܒܥܒܕܐ ܥܢܩܘ̈ܕܐ ܕܫܥ̈ܐ. ܡܛܠ ܗܕܐ، ܐܝܟ ܕܟܟܬܒܝ Everitt ܘܚܒܪ̈ܘܗܝ (2011، ܦܐܬܐ 67)، "ܠܐ ܒܕ ܗܘܐ ܡܣܬܟܠܢܐ ܕܚܫܒܚ، ܡܬܠܐ، ܡܬܩܠܐ ܒܟܝܠܘܓܪ̈ܡܐ، ܪܘܡܐ ܒܡܝܛܪ̈ܐ ܘܙܘܥܬܐ ܒܣܒܠܬܐ ܕܐܪܒܥܐ ܢܘܩ̈ܙܐ ܐܝܟ ܫܘ̈ܝܐ"، ܘ"ܦܘܫܪܐ ܕܡܬܦܝܣ ܣܓܝ… ܦܫܝܛܐܝܬ ܡܬܩܢܘܬܐ ܕܟܠ ܡܫܬܚܠܦܢܐ ܠܫܘܚܠܦܐ (variance) 1 ܩܕܡ ܟܠ ܦܘܫܩܐ"؛ Wendler ܘ-Gröttrup (2021، ܦܐ̈ܬܐ 656-657) – ܡܬܩܢܘܬܐ ܕܡܫܬܚܠܦ̈ܢܐ ܡܢܝܢ̈ܝܐ "ܪܫܝܬܐ ܝܠܗ ܣܓܝ ܩܐ ܐܘܪܚܐ ܕܥܢܩܘ̈ܕܐ… ܩܐ ܕܛܝ̈ܡܐ ܗܘܝ ܡܬܦܚܡ̈ܢܐ، ܣܢܝܩ ܝܠܗ ܕܡܬܩܢܚ ܠܗܘܢ"؛ ܘ-Géron (2022, Chapter 9, "Limits of k-means") – "ܪܫܝܐ ܝܠܗ ܕܡܬܩܢܚ ܡܢܝ̈ܘܬܐ ܕܝܗܝ̈ܒܬܐ ܩܕܡ ܕܪܗܛܚ k-ܡܨܥܝ̈ܬܐ، ܐܢ ܠܐ ܥܢܩܘ̈ܕܐ ܡܨܝ ܕܗܘܝ ܡܬܝܚ̈ܐ ܣܓܝ". ܓܘܒܝܐ ܡܩܕܡܐ (default) ܓܘ ܐܗܐ ܦܐܬܐ ܡܢܝ̈ܢܐ z ܝܢܐ (ܡܨܥܝܬܐ 0 ܘܣܛܝܘܬܐ ܬܩܝܢܬܐ 1 ܩܐ ܟܠ ܡܫܬܚܠܦܢܐ)، ܘܗܝܓܝ ܠܟܠ ܡܫܬܚܠܦܢܐ ܐܝܬ ܡܬܩܠܐ ܫܘܝܐ ܓܘ ܪܘܚܩܐ.

ܡܬܩܢܘܬܐ – ܠܐ ܟܠ ܙܒܢܐ. Hastie ܘܚܒܪ̈ܘܗܝ (2009، ܦܐܬܐ 506) ܟܡܙܗܪܝ ܡܢ ܐܗܐ ܢܡܘܣܐ: ܝܗܒܐ ܕܡܥܒܕܢܘܬܐ ܫܘܝܬܐ ܠܟܠ ܡܫܬܚܠܦ̈ܢܐ "ܥܡ ܕܡܨܐ ܕܡܬܚܙܐ ܡܣܬܟܠܢܐ، ܘܒܥܕܢ̈ܐ ܣܓܝ̈ܐܐ ܡܬܦܝܣܐ ܝܠܗ، ܡܨܐ ܕܡܚܠܫ ܣܓܝ". ܐܢ ܚܕ ܡܫܬܚܠܦܢܐ ܟܦܪܫ ܛܒ ܒܝܢܝ ܓܘ̈ܕܐ ܘܐܚܪܢܐ ܩܠܐ ܝܠܗ (noise)، ܡܬܩܢܘܬܐ ܟܡܪܒܝܐ ܡܬܩܠܐ ܕܩܠܐ، ܘ"ܒܕ ܡܨܠܐ ܕܡܛܫܝܐ ܓܘ̈ܕܐ ܗܕܟܐ ܕܐܠܓܘܪܝܬܡ ܕܥܢܩܘ̈ܕܐ ܠܐ ܡܨܐ ܕܓܠܐ ܠܗܘܢ" – ܓܘ ܨܘܪܬܐ 14.5 ܕܝܗܘܢ ܬܪܬܝܢ ܓܘ̈ܕܐ ܦܪ̈ܝܫܐ ܛܒ ܟܐܙܠܝ ܒܬܪ ܡܬܩܢܘܬܐ. Everitt ܘܚܒܪ̈ܘܗܝ (2011، ܦܐ̈ܬܐ 66-67) ܟܐܡܪܝ ܗܢ ܡܕܡ: ܡܬܩܢܘܬܐ ܓܒܐ ܡܝܚܕܐ ܝܠܗ ܕܝܗܒܐ ܕܡܬ̈ܩܠܐ، ܘ"ܐܡܬܝ ܕܡܬܩܢܚ ܡܫܬܚܠܦ̈ܢܐ ܩܕܡ ܦܘܫܩܐ، ܒܨܘܝܬܐ ܟܣܒܪܐ ܕܪܫܝܘܬܐ ܕܡܫܬܚܠܦܢܐ ܟܢܚܬܐ ܐܝܟ ܕܫܘܚܠܦܐ ܕܝܠܗ ܟܣܠܩ"؛ ܡܬ̈ܩܠܐ ܠܦܘܬ ܣܛܝ̈ܘܬܐ ܬܩܝ̈ܢܬܐ "ܟܡܬܚܙܝ ܕܠܐ ܦܠܚ̈ܐ ܝܢܐ"؛ ܦܘܠܓܐ ܥܠ ܦܬܝܘܬܐ ܕܡܫܬܚܠܦܢܐ ܛܒ ܝܬܝܪ ܝܠܗ ܡܢ ܡܢܝ̈ܢܐ z ܓܘ ܡܦܠܚ̈ܢܘܬܐ ܣܓܝ̈ܐܐ (Milligan & Cooper, 1988، ܐܝܟ ܕܡܬܡܠܠܝ ܬܡܢ)؛ ܘܛܒ ܝܬܝܪ ܝܠܗ ܕܡܬܩܢܚ ܠܦܘܬ ܫܘܚܠܦܐ ܓܘ ܓܘ̈ܕܐ ܘܠܐ ܠܦܘܬ ܫܘܚܠܦܐ ܟܠܢܝܐ – ܐܝܢܐ ܓܘ̈ܕܐ ܚܬܝܬܐܝܬ ܐܢܝ ܝܢܐ ܕܟܒܨܝܚ ܥܠܝܗܝ. ܘܐܡܬܝ ܕܟܠ ܡܫܬܚܠܦ̈ܢܐ ܡܢ ܩܕܡ ܓܘ ܚܕܐ ܣܒܠܬܐ (scale) ܝܢܐ (ܡܬܠܐ ܐܪܒܥܐ ܡܢܝ̈ܢܐ ܕܣܒ̈ܠܬܐ 1-5)، ܡܬܩܢܘܬܐ ܟܡܫܚܠܦܐ ܡܬ̈ܩܠܐ ܠܦܘܬ ܣܛܝ̈ܘܬܐ ܬܩܝ̈ܢܬܐ ܓܘ ܕܘܡܝܐ – ܠܐ ܟܠ ܙܒܢܐ ܡܐ ܕܒܥܝܚ. ܐܗܐ ܦܐܬܐ ܟܡܩܪܒ ܡܢܝ̈ܢܐ z ܝܢ ܡܫܘ̈ܚܬܐ ܫܪ̈ܫܝܬܐ؛ ܡܬܩܢܘܬܐ ܠܦܘܬ ܦܬܝܘܬܐ ܠܐ ܝܠܗ ܓܘ ܐܗܐ ܦܐܬܐ. ܦܣܩܐ ܬܠܝܐ ܝܠܗ ܒܡܫܬܚܠܦ̈ܢܐ ܘܒܫܘܐܠܐ ("ܠܝܬ ܚܠܦܐ ܩܐ ܚܘܫܒܐ ܙܗܝܪܐ ܓܘ ܟܠ ܥܘܩܬܐ"، ܬܡܢ)؛ ܓܘ ܐܗܐ ܦܐܬܐ ܟܪܫܡܚ ܝܢ ܟܫܩܠܚ ܪܘܫܡܐ ܡܢ "ܡܬܩܢܘܬܐ ܕܡܫܬܚܠܦ̈ܢܐ"، ܘܡܦܩܢܐ ܟܡܚܘܐ ܡܘܕܝ ܓܒܝܐ ܝܠܗ. ܐܡܬܝ ܕܡܬܩܢܘܬܐ ܛܦܝܬܐ ܝܠܗ ܘܦܬܝ̈ܘܬܐ ܕܡܫܬܚܠܦ̈ܢܐ ܦܪ̈ܝܫܐ ܝܢܐ ܣܓܝ – ܐܗܐ ܦܐܬܐ ܟܡܙܗܪ.

ܣܒܠܬܐ ܕ-Likert. ܪܘܚܩܐ ܟܣܒܪ ܕܦܘܪܫܢܐ ܒܝܢܝ 1 ܘ-2 ܫܘܝܐ ܝܠܗ ܠܦܘܪܫܢܐ ܒܝܢܝ 4 ܘ-5. Wendler ܘ-Gröttrup (2021، ܦܐܬܐ 635، ܠܘܚܐ 7.8) ܟܡܥܗܕܝ ܕܓܘ ܣܒܠܬܐ ܕ-Likert "ܣܢܝܩ ܝܠܗ ܕܦܫܩܚ ܪܘ̈ܚܩܐ ܒܙܗܝܪܘܬܐ، ܝܢ ܠܐ ܒܟܠܗ"، ܘܕܫܘܐܠܐ ܦܠܝܓܐ ܝܠܗ ܓܘ ܟܬܒ̈ܐ. ܠܦܘܬ ܥܝܕܐ ܕܣܐܝܛ (site convention؛ ܦܐܬܐ ܡܫܬܚܠܦ̈ܢܐ ܘܕܪ̈ܓܐ ܕܡܫܘܚܬܐ)، ܣܒܠܬܐ ܕܦܣܘ̈ܩܐ ܣܓܝ̈ܐܐ (multi-item scale) – ܡܨܥܝܬܐ ܕܟܡܐ ܦܣܘ̈ܩܐ – ܡܬܚܫܒܐ ܝܠܗ ܐܝܟ ܡܫܬܚܠܦܢܐ ܕܪܘ̈ܚܩܐ ܫܘ̈ܝܐ (interval variable)، ܘܦܣܘܩܐ ܕ-Likert ܝܚܝܕܝܐ – ܠܐ. ܡܢ ܐܟܐ ܦܘܝܣܐ: ܟܢܘܫܝܐ ܠܦܘܬ ܡܢܝ̈ܢܐ ܕܣܒ̈ܠܬܐ (ܦܐܬܐ ܐܠܦܐ ܕ-Cronbach ܘܡܢܝܢܐ ܕܣܒܠܬܐ)، ܠܐ ܠܦܘܬ ܦܣܘ̈ܩܐ ܝܚܝܕ̈ܝܐ.

ܟܡܐ ܥܢܩܘ̈ܕܐ؟

ܐܗܐ ܝܠܗ ܦܣܩܐ ܥܣܩܐ ܓܘ ܦܘܫܩܐ، ܘܠܝܬ ܠܗ ܒܘܚܪܢܐ ܕܪܫܝܡܘܬܐ (significance). Hastie ܘܚܒܪ̈ܘܗܝ (2009، ܦܐܬܐ 518) ܟܦܪܫܝ ܒܝܢܝ ܬܪܝܢ ܡܢܝ̈ܘܬܐ: ܓܘ ܦܘܠܓܐ (segmentation)، "k ܒܥܕܢܐ ܣܓܝ ܣܝܡܐ ܝܠܗ ܐܝܟ ܡܢܬܐ ܕܥܘܩܬܐ" (ܟܘܡܦܢܝܐ ܥܡ k ܙܒܘܢ̈ܐ)؛ ܘܐܡܬܝ ܕܥܢܩܘ̈ܕܐ ܬܚܘܝܬܐ ܝܢܐ ܕ"ܓܘ̈ܕܐ ܟܝܢܝ̈ܐ"، k ܠܐ ܝܕܝܥܐ ܝܠܗ ܘܣܢܝܩ ܝܠܗ ܕܫܥܪܚ ܠܗ ܡܢ ܝܗܝ̈ܒܬܐ. ܐܗܐ ܦܐܬܐ ܟܪܗܛ ܦܘܫܪܐ ܕܬܪܝܢ ܕܪ̈ܓܐ ܩܐ ܟܠ k ܡܢ 1 ܗܕܟܐ ܕܡܢܝܢܐ ܥܠܝܐ ܕܓܒܠܘܟܘܢ، ܘܟܛܒܥ ܬܠܬܐ ܡܫܘ̈ܚܬܐ ܚܕ ܓܢܒ ܐܚܪܢܐ:

Elbow. ܓܘܡܪܐ ܕܪܘ̈ܚܩܐ ܡܪ̈ܒܥܐ ܡܢ ܡܨܥ̈ܝܐ ܓܘ ܥܢܩܘ̈ܕܐ، W_k، ܟܠ ܙܒܢܐ ܟܢܚܬ ܐܝܟ ܕ-k ܟܪܒܐ – ܓܘ ܣܘܦܐ، ܟܠ ܡܫܬܘܬܦܢܐ ܥܢܩܘܕܐ ܩܐ ܓܢܗ ܘ-W = 0 (Hastie et al., 2009, p. 519). ܐܝܢܐ ܐܢ ܒܫܪܪܐ ܐܝܬ k* ܓܘ̈ܕܐ، ܡܚܬܬܐ ܒܕ ܗܘܝܐ ܚܪܝܦܬܐ ܗܕܟܐ ܕ-k* ܘܢܝܚܬܐ ܒܬܪܗ، ܡܛܠ ܕܒܬܪܗ ܟܠ ܥܢܩܘܕܐ ܡܘܣܦܐ ܒܣ ܟܦܠܓ ܓܘܕܐ ܟܝܢܝܬܐ ܠܬܪܬܝܢ؛ ܟܒܨܝܚ ܥܠ "ܟܦܦܐ" (kink) ܓܘ ܣܪܛܐ ܟܦܝܦܐ – "ܘܐܝܟ ܓܒ̈ܐ ܐܚܪ̈ܢܐ ܕܐܘܪ̈ܚܬܐ ܕܥܢܩܘ̈ܕܐ، ܐܗܐ ܐܘܪܚܐ ܩܠܝܠ heuristic ܝܠܗ" (Hastie et al., 2009, p. 519)؛ Everitt ܘܚܒܪ̈ܘܗܝ (2011، ܦܐܬܐ 126) ܟܡܘܣܦܝ ܕܐܝܢܐ "ܡܨܐ ܕܗܘܐ ܣܓܝ ܕܝܠܢܝܐ (subjective)"، ܘ"ܪܒܐ" ܒܥܕܢ̈ܐ ܣܓܝ̈ܐܐ "ܬܠܝܐ ܝܠܗ ܒܣܟܘܝ̈ܐ ܩܕܡܝ̈ܐ ܕܡܦܠܚܢܐ". ܓܘ ܠܘܚܐ ܡܚܘܝܬܐ ܝܠܗ ܐܦ ܐܝܟ ܫܘܚܠܦܐ ܡܦܫܩܐ (explained variance)، 1 − W_k/W₁ – ܡܢܬܐ ܕܫܘܚܠܦܐ ܟܠܢܝܐ ܕܦܘܠܓܐ ܟܡܦܫܩ (ܐܝܟ R²؛ Wendler & Gröttrup, 2021، ܦܐܬܐ 649، ܠܘܚܐ 7.25).

Silhouette. ܩܐ ܟܠ ܡܫܬܘܬܦܢܐ: a – ܪܘܚܩܐ ܡܨܥܝܐ ܠܚܒܪ̈ܐ ܕܥܢܩܘܕܐ ܕܝܠܗ، b – ܪܘܚܩܐ ܡܨܥܝܐ ܠܚܒܪ̈ܐ ܕܥܢܩܘܕܐ ܩܪܝܒܐ ܕܒܬܪܗ، ܘ-silhouette s = (b − a)/max(a, b)، ܒܝܢܝ 1− ܘ-1+. Rousseeuw (1987، ܦܐܬܐ 56)، ܕܦܪܫܠܗ ܠܡܚܘܝܢܐ: s ܩܪܝܒܐ ܠ-1 – ܡܫܬܘܬܦܢܐ "ܟܢܝܫܐ ܝܠܗ ܛܒ"؛ ܚܕܪ 0 – "ܓܒܐ ܡܨܥܝܐ"، ܕ"ܠܐ ܓܠܝܐ ܝܠܗ ܒܟܠܗ" ܠܐܝܢܐ ܡܢ ܬܪܝܢ ܥܢܩܘ̈ܕܐ ܕܝܠܗ ܝܠܗ؛ ܩܪܝܒܐ ܠ-1− – ܩܪܝܒܐܝܬ ܡܨܝܚ ܕܐܡܪܚ ܕܐܝܢܐ "ܦܘܪܫܐ ܠܓܢ̈ܣܐ ܕܝܠܗ ܠܐ ܬܪܝܨܐ ܝܠܗ". ܩܐ ܡܫܬܘܬܦܢܐ ܕܒܓܢܗ ܝܠܗ ܓܘ ܥܢܩܘܕܐ ܛܝܡܐ ܕ-silhouette 0 ܝܠܗ – ܓܘܒܝܐ "ܕܨܒܝܢܐ ܝܠܗ ܒܫܪܪܐ، ܐܝܢܐ ܛܝܡܐ ܕܨܦܪܐ ܟܡܬܚܙܐ ܡܨܥܝܐ ܝܬܝܪ" (ܬܡܢ). ܡܚܘܝܢܐ ܦܐܫ ܐܡܬܝ ܕܪܘ̈ܚܩܐ ܓܘ ܣܒܠܬܐ ܕܦܘܚܡܐ (ratio scale) ܝܢܐ (ܐܝܟ ܪܘܚܩܐ ܕ-Euclid) ܘܐܡܬܝ ܕ"ܟܒܨܝܚ ܥܠ ܥܢܩܘ̈ܕܐ ܟܢܝܫ̈ܐ ܘܦܪ̈ܝܫܐ ܓܠܝܐܝܬ" (ܦܐܬܐ 55)، ܘܡܨܥܝܬܐ ܕ-silhouette "ܡܨܝܐ ܕܡܬܦܠܚܐ ܩܐ ܓܘܒܝܐ ܕܡܢܝܢܐ ‘ܦܐܝܐ’ ܕܥܢܩܘ̈ܕܐ" (ܦܐܬܐ 53). ܡܨܥܝܬܐ ܕ-silhouette ܥܠ ܟܠ ܡܫܬܘܬܦ̈ܢܐ ܡܚܘܝܢܐ ܝܠܗ ܕܛܝܒܘܬܐ ܕܦܘܠܓܐ، ܘ-k ܥܡ ܡܨܥܝܬܐ ܪܡܬܐ ܝܬܝܪ ܡܬܩܪܒܢܐ ܝܠܗ؛ ܩܐ ܬܚܘ̈ܡܐ ܐܝܬ ܬܪܝܢ ܡܒܘ̈ܥܐ ܕܒܪܫܐ ܫܠܡ̈ܐ ܝܢܐ: Kaufman ܘ-Rousseeuw ܟܚܙܝ "ܦܘܪܫܐ ܡܣܬܟܠܢܐ" ܓܘ ܡܨܥܝܬܐ ܕ-silhouette ܠܥܠ ܡܢ 0.5 (Kaufman & Rousseeuw, 1990، ܐܝܟ ܕܡܬܡܠܠܝ ܓܘ Everitt et al., 2011، ܦܐܬܐ 129)، ܘܟܬܒܐ ܕ-SPSS Modeler، ܐܝܟ ܕܡܝܬܝ Wendler ܘ-Gröttrup (2021، ܦܐܬܐ 649)، ܟܣܕܪ ܗܕܟܐ ܕ-.25 ܟܪܝܗܐ، ܒܝܢܝ .25 ܘ-.50 ܡܨܥܝܐ، ܠܥܠ ܡܢ .50 ܛܒܐ (IBM, 2015، ܐܝܟ ܕܡܬܡܠܠ ܓܘܝܗܘܢ). Silhouette ܠܐ ܦܪܝܫܐ ܝܠܗ ܩܐ k = 1، ܘܡܛܠ ܗܕܐ ܠܐ ܡܨܐ ܕܐܡܪ "ܠܝܬ ܠܐܟܐ ܥܢܩܘ̈ܕܐ".

Gap statistic (Tibshirani et al., 2001، ܐܝܟ ܕܡܬܡܠܠܝ ܓܘ Hastie et al., 2009، ܦܐܬܐ 519) "ܟܦܚܡ ܣܪܛܐ ܟܦܝܦܐ log W_k ܥܡ ܣܪܛܐ ܟܦܝܦܐ ܕܟܐܬܐ ܡܢ ܝܗܝ̈ܒܬܐ ܕܒܕܝܪ̈ܐ ܝܢܐ ܫܘܝܐܝܬ ܥܠ ܡܪܒܥܐ ܡܓܪܓܐ ܕܟܚܒܫ ܝܗܝ̈ܒܬܐ"، ܝܥܢܝ ܡܢ ܝܗܝ̈ܒܬܐ ܕܠܝܬ ܓܘܝܗܝ ܥܢܩܘ̈ܕܐ. Gap ܝܠܗ ܦܘܪܫܢܐ ܒܝܢܝ ܬܪܘܝܗܝ؛ "ܒܥܒܕܐ ܐܗܐ ܐܘܪܚܐ ܡܢ ܓܢܗ ܝܠܗ ܩܐ ܡܫܟܚܬܐ ܕ'ܟܦܦܐ'"، ܘܐܝܢܐ "ܟܦܠܚ ܛܒ ܐܦ ܐܡܬܝ ܕܝܗܝ̈ܒܬܐ ܟܢܦܠܝ ܓܘ ܚܕ ܥܢܩܘܕܐ، ܘܓܘ ܐܗܐ ܓܒܐ ܒܕ ܡܨܠܐ ܕܫܥܪ ܡܢܝܢܐ ܛܒܐ ܝܬܝܪ ܕܥܢܩܘ̈ܕܐ ܐܝܟ ܚܕ – ܓܒܐ ܕܓܘܗ ܐܘܪ̈ܚܬܐ ܐܚܪ̈ܢܝܬܐ ܣܓܝ ܟܡܓܕܠܝ" (ܬܡܢ). ܢܡܘܣܐ (ܫܘܝܘܬܐ 14.39 ܬܡܢ): k ܙܥܘܪܐ ܝܬܝܪ ܕܓܘܗ gap ܠܐ ܢܚܝܬܐ ܝܠܗ ܡܢ gap ܓܘ k + 1 ܒܨܝܪ ܛܥܝܘܬܐ ܬܩܝܢܬܐ (standard error) ܕܝܠܗ. ܐܗܐ ܦܐܬܐ ܟܒܢܐ ܝܗܝ̈ܒܬܐ ܫܘ̈ܝܬܐ ܒܡܘܠܕܢܐ ܕ-R (ܙܪܥܐ 2026 ܐܝܟ ܓܘܒܝܐ ܡܩܕܡܐ، 100 ܕܘܡ̈ܝܐ)، ܗܕܟܐ ܕܦܐܪܐ ܕܡܐ ܝܠܗ ܠܦܐܪܐ ܕ-clusGap ܓܘ ܚܙܡܐ cluster ܥܡ ܗܢ ܙܪܥܐ (ܡܕܒܪܢܐ ܕ-R ܠܬܚܬ). ܐܗܐ ܝܠܗ ܡܫܘܚܬܐ ܝܚܝܕܝܬܐ ܡܢ ܬܠܬܐ ܕܡܨܝܐ ܕܡܦܢܝܐ "ܚܕ": Everitt ܘܚܒܪ̈ܘܗܝ (2011، ܦܐܬܐ 129) ܟܡܚܝܠܝ ܕܐܝܢܐ "ܟܡܨܐ ܠܫܘܥܪܐ ܕܛܝܒܘܬܐ ܕܦܘܫܪܐ ܕܚܕ ܥܢܩܘܕܐ"، ܝܥܢܝ ܠܡܒܩܪܐ ܐܢ ܐܝܬ ܒܟܠܗ ܣܗܕܘܬܐ ܩܐ ܥܢܩܘ̈ܕܐ ܦܪ̈ܝܫܐ – ܘܡܛܠ ܗܕܐ ܐܝܢܐ ܝܠܗ ܓܘܒܝܐ ܡܩܕܡܐ ܩܐ ܓܘܒܝܐ ܕ-k ܓܘ ܡܦܩܢܐ.

ܦܘܝܣܐ ܟܠܢܝܐ ܝܠܗ "ܕܠܐ ܡܣܡܟܚ ܥܠ ܚܕ ܢܡܘܣܐ ܩܐ ܓܘܒܝܐ ܕܡܢܝܢܐ ܕܓܘ̈ܕܐ، ܐܠܐ ܕܡܪܟܒܚ ܦܐܪ̈ܐ ܕܟܡܐ ܐܘܪ̈ܚܬܐ" (Everitt et al., 2011, p. 130) – ܘܡܛܠ ܗܕܐ ܠܘܚܐ ܟܡܚܘܐ ܠܬܠܬܝܗܘܢ. ܐܡܬܝ ܕܡܫܘ̈ܚܬܐ ܦܠܝܓ̈ܐ ܝܢܐ – ܘܐܗܐ ܟܗܘܐ – ܦܣܩܐ ܟܥܒܪ ܠܒܨܘܝܬܐ: Géron (2022، ܨܚܚܐ 9) ܟܡܚܘܐ ܡܬܠܐ ܕܓܘܗ silhouette ܩܠܝܠ ܟܓܒܐ k=4، ܘܗܘ ܟܓܒܐ 5 ܡܛܠ ܕܥܢܩܘ̈ܕܐ ܒܪ̈ܒܘܬܐ ܕܡ̈ܝܐ ܝܢܐ؛ ܘ-Wendler ܘ-Gröttrup (2021، ܦܐܬܐ 650) ܟܡܥܗܕܝ ܕ"ܐܘܪ̈ܚܬܐ ܕܥܢܩܘ̈ܕܐ ܠܐ ܟܝܗܒܝ ܚܕ ܥܘܕܪܢܐ ܓܘ ܡܫܟܚܬܐ ܕܬܚܘܝܬܐ ܦܐܝܬܐ ܩܐ ܟܠ ܥܢܩܘܕܐ" – ܦܘܫܪܐ ܕܠܐ ܡܨܝܚ ܕܡܚܘܚ ܠܗ ܒܡ̈ܠܐ ܠܐ ܝܠܗ ܦܘܫܪܐ ܛܒܐ، ܐܦ ܐܢ ܡܚܘ̈ܝܢܐ ܕܝܠܗ ܪܡ̈ܐ ܝܢܐ. ܢܡܘܣܐ ܕܨܒܥܐ √(n/2) (Mardia et al., 1979، ܐܝܟ ܕܡܬܡܠܠܝ ܓܘ Wendler & Gröttrup, 2021، ܦܐܬܐ 649) ܓܣܐ ܝܠܗ ܣܓܝ ܩܐ ܓܘܒܝܐ؛ ܐܝܢܐ ܟܡܬܚܙܐ ܓܘ ܩܘܛܝܐ ܕܩܪܝܢܐ ܒܣ ܐܝܟ ܢܘܩܙܐ ܕܦܘܚܡܐ.

ܩܪܝܢܐ ܕܡܦܩܢܐ (How to read the output)

ܠܘܚܐ 1 – ܓܘܒܝܐ ܕܡܢܝܢܐ ܕܥܢܩܘ̈ܕܐ. ܩܐ ܟܠ k: W_k، ܫܘܚܠܦܐ ܡܦܫܩܐ (explained variance)، ܡܨܥܝܬܐ ܕ-silhouette، gap ܘܛܥܝܘܬܐ ܬܩܝܢܬܐ ܕܝܠܗ. ܣܕܪܐ ܕ-gap statistic ܟܓܒܐ ܪܫܝܡܐ ܝܠܗ، ܘܗܕܟܐ ܐܦ k ܥܡ silhouette ܪܡܐ ܝܬܝܪ. ܨܘܪܬܐ 1 ܟܡܚܘܝܐ ܬܠܬܐ ܣܪ̈ܛܐ ܟܦܝܦ̈ܐ ܚܕ ܓܢܒ ܐܚܪܢܐ.

ܡܢܝܢܐ ܕܥܢܩܘ̈ܕܐ ܩܐ ܦܘܫܪܐ. ܠܬܚܬ ܡܢ ܠܘܚܐ 1 ܟܓܒܚ k. ܓܘܒܝܐ ܡܩܕܡܐ (default) ܓܘܒܝܐ ܕ-gap statistic ܝܠܗ، ܘܐܢ ܓܒܐ 1 – k ܥܡ silhouette ܪܡܐ ܝܬܝܪ، ܥܡ ܙܘܗܪܐ. ܫܘܚܠܦܐ ܕܓܘܒܝܐ ܟܚܫܒ ܡܢ ܪܝܫ ܟܠ ܡܐ ܕܟܐܬܐ ܒܬܪܗ، ܕܠܐ ܡܬܢܝܬܐ ܕ-gap statistic.

ܨܘܪܬܐ 2 – dendrogram ܕܕܪܓܐ ܐ، ܥܡ ܣܪܛܐ ܕܦܣܩܐ ܩܐ k ܓܒܝܐ. ܣܪܛܐ ܙܩܝܦܐ (vertical axis) ܪܘܡܐ ܕܚܘܝܕܐ ܝܠܗ (ܒܪܘ̈ܚܩܐ ܕ-Ward): ܚܘܝ̈ܕܐ ܪܡ̈ܐ، ܠܥܠ ܡܢ ܣܕܪܐ ܕܚܘܝ̈ܕܐ ܢܚ̈ܝܬܐ، ܡܬܩܪ̈ܒܢܐ ܝܢܐ ܩܐ ܥܢܩܘ̈ܕܐ ܟܝܢ̈ܝܐ (Hastie et al., 2009, p. 521). SPSS ܟܡܚܘܐ ܗܢ ܐܝܠܢܐ ܥܠ ܣܒܠܬܐ ܡܬܩܢܬܐ ܕ-0-25 ("Rescaled Distance Cluster Combine"; Wendler & Gröttrup, 2021, p. 645) – ܐܣܟܝܡܐ ܕܡܐ ܝܠܗ. ܓܘ ܕܘܡܝܐ ܕܝܬܝܪ ܡܢ 120 ܡܫܬܘܬܦ̈ܢܐ ܛܪ̈ܦܐ ܠܐ ܟܡܬܨܝܪܝ ܚܕ-ܚܕ: ܐܝܠܢܐ ܡܚܘܝܐ ܝܠܗ ܠܥܠ ܡܢ 40 ܓܘ̈ܕܐ ܕܬܚܬܗ، ܥܡ ܡܢܝܢܐ ܕܡܫܬܘܬܦ̈ܢܐ ܓܘ ܟܠ ܓܘܕܐ.

ܠܘܚܐ 2 – ܪܒܘܬܐ ܘܛܝܒܘܬܐ ܕܥܢܩܘ̈ܕܐ: ܡܢܝܢܐ ܕܡܫܬܘܬܦ̈ܢܐ ܘܡܢܬܐ ܒܡܐܐ ܕܝܗܘܢ، ܡܨܥܝܬܐ ܕ-silhouette ܓܘ ܟܠ ܥܢܩܘܕܐ ܘܡܢܝܢܐ ܕܡܫܬܘܬܦ̈ܢܐ ܥܡ silhouette ܬܚܝܬܝܐ (ܕܩܪܝܒ̈ܐ ܝܢܐ ܝܬܝܪ ܠܥܢܩܘܕܐ ܐܚܪܢܐ)، ܘܪܒܘܬܐ ܓܘ ܥܢܩܘܕܐ ܕ-Ward ܩܕܡ ܕܪܓܐ ܒ. ܓܘ ܡܥܝܪܢܘܬܐ: ܟܡܐ ܡܫܬܘܬܦ̈ܢܐ ܫܚܠܦܠܗܘܢ ܥܢܩܘܕܐ ܓܘ ܕܪܓܐ ܒ، ܘܡܚܘܝܢܐ ܕ-Rand ܡܬܘܪܨܐ ܒܝܢܝ ܬܪܝܢ ܕܪ̈ܓܐ.

ܠܘܚܐ 3 – ܬܚܘܝܬܐ ܕܥܢܩܘ̈ܕܐ (cluster profile): ܩܐ ܟܠ ܡܫܬܚܠܦܢܐ – ܡܨܥܝܬܐ ܘܣܛܝܘܬܐ ܬܩܝܢܬܐ ܓܘ ܟܠ ܥܢܩܘܕܐ ܘܓܘ ܕܘܡܝܐ ܟܠܗ، ܒܡܫܘ̈ܚܬܐ ܫܪ̈ܫܝܬܐ، ܘ-η² – ܡܢܬܐ ܕܫܘܚܠܦܐ ܕܡܫܬܚܠܦܢܐ ܕܒܝܢܝ ܥܢܩܘ̈ܕܐ. ܨܘܪܬܐ 3 ܟܡܚܘܝܐ ܗܢ ܡܨܥܝ̈ܬܐ ܒܡܢܝ̈ܢܐ z، ܗܕܟܐ ܕܒܚܕ ܚܙܝܐ ܟܚܙܝܚ ܒܡܘܕܝ ܟܠ ܥܢܩܘܕܐ ܪܡܐ ܝܢ ܢܚܝܬܐ ܝܠܗ ܡܢ ܡܨܥܝܬܐ. ܡܢ ܐܟܐ ܟܐܬܐ ܫܡܐ ܕܒܕ ܝܗܒܝܬܘܢ ܠܟܠ ܥܢܩܘܕܐ، ܘܐܗܐ ܦܘܫܩܐ ܕܝܘܟܘܢ ܝܠܗ، ܠܐ ܦܐܪܐ ܕܚܘܫܒܢܐ (Wendler & Gröttrup, 2021, p. 650).

ܨܘܪܬܐ 4 – silhouette: ܥܡܘܕܐ ܩܐ ܟܠ ܡܫܬܘܬܦܢܐ، ܡܛܟܣܬܐ ܓܘ ܟܠ ܥܢܩܘܕܐ – "ܐܣܟܝܡܐ ܕܣܟܝܢܐ" ܩܐ ܟܠ ܥܢܩܘܕܐ. ܪܘܡܐ ܕܐܣܟܝܡܐ ܡܢܝܢܐ ܕܡܫܬܘܬܦ̈ܢܐ ܓܘ ܥܢܩܘܕܐ ܝܠܗ ܘܦܬܝܘܬܐ ܕܝܠܗ ܛܝ̈ܡܐ ܕ-silhouette ܕܝܗܘܢ؛ ܥܢܩܘܕܐ ܕܣܓܝ ܡܢ ܚܒܪ̈ܘܗܝ ܠܬܚܬ ܡܢ ܡܨܥܝܬܐ ܟܠܢܝܬܐ ܝܢܐ (ܣܪܛܐ ܡܦܣܩܐ) ܥܢܩܘܕܐ ܟܪܝܗܐ ܝܠܗ (Géron, 2022, Chapter 9). ܦܣܩܐ ܕܡܫܟܚ̈ܢܘܬܐ – ܒܣܘܪܝܬ ܘܒܐܢܓܠܝܙܝܐ؛ ܩܘܛܝܐ ܕ"ܩܪܝܢܐ ܕܦܐܪܐ" – ܡܫܘ̈ܚܬܐ ܫܠܡ̈ܐ ܝܢܐ؟، ܛܝܒܘܬܐ ܕܥܢܩܘ̈ܕܐ، ܡܘܕܝ ܙܝܥܠܗ ܒܝܢܝ ܕܪ̈ܓܐ، ܘܡܘܕܝ ܟܥܒܕܚ ܗܫܐ. ܡܚܬܬܐ ܕܝܗܝ̈ܒܬܐ ܥܡ ܥܡܘܕܐ ܕܥܢܩܘܕܐ – ܗܢ ܦܐܝܠ ܥܡ ܥܡܘܕܐ ܡܘܣܦܬܐ، ܩܐ ܬܚܘܝܬܐ ܘܩܐ ܫܘܪܪܐ ܓܘ ܦܐ̈ܬܐ ܐܚܪ̈ܢܐ.

ܩܝܡܘܬܐ (stability) ܓܘ ܓܘܒܝܐ ܕܕܘܡܝܐ ܡܢ ܪܝܫ – ܒܚܕ ܕܘܫܐ. ܠܬܚܬ ܡܢ ܨܘܪܬܐ ܕ-silhouette ܐܝܬ button "ܚܘܫܒܢܐ ܕܩܝܡܘܬܐ" (100 ܙܘ̈ܓܐ ܘܕܘ̈ܡܝܐ ܐܝܟ ܓܘܒܝܐ ܡܩܕܡܐ، ܒܗܘ ܙܪܥܐ ܐܝܟ gap statistic). ܠܘܚܐ 4 ܟܝܗܒ ܩܐ ܟܠ k ܦܘܠܓܐ ܕܡܚܘܝܢܐ ܕ-Rand ܡܬܘܪܨܐ ܒܝܢܝ ܦܘܫܪ̈ܐ ܕܬܪܝܢ ܕܘܡ̈ܝܐ ܕ-bootstrap – ܛܝܡܐ ܬܚܬܝܐ، ܪ̈ܘܒܥܐ، ܛܝܡܐ ܡܨܥܝܐ (median) ܘܛܝܡܐ ܥܠܝܐ – ܘܨܘܪܬܐ 5 ܟܡܚܘܝܐ ܠܗ ܓܘ ܨܘܪܬܐ ܕܩܘܛܝܐ (box plot)، ܥܡ ܢܘܩܙܐ ܩܐ ܟܠ ܙܘܓܐ. ܠܘܚܐ 5 ܟܥܒܪ ܡܢ ܦܘܠܓܐ ܟܠܗ ܠܦܘܫܪܐ ܓܒܝܐ: ܩܐ ܟܠ ܥܢܩܘܕܐ – ܡܨܥܝܬܐ ܕܕܡܝܘܬܐ ܕ-Jaccard ܥܡ ܥܢܩܘܕܐ ܕܕܡܐ ܠܗ ܓܘ ܕܘܡ̈ܝܐ ܕ-bootstrap، ܟܡܐ ܓܗ̈ܐ "ܦܪܩܠܗ" ܘܟܡܐ ܓܗ̈ܐ "ܐܬܐ ܠܐܬܪܗ"، ܘܦܘܫܩܐ ܠܦܘܬ Hennig (2023). ܫܘܚܠܦܐ ܕ-k ܟܚܫܒ ܡܢ ܪܝܫ ܠܘܚܐ 5 ܒܣ. ܐܡܬܝ ܕܩܝܡܘܬܐ ܚܫܝܒܬܐ ܝܠܗ، ܟܥܐܠܐ ܐܦ ܓܘ ܦܣܩܐ ܕܡܫܟܚ̈ܢܘܬܐ ܘܓܘ ܩܘܛܝܐ ܕܩܪܝܢܐ. ܐܘܪ̈ܚܬܐ – ܓܘ ܦܣܩܐ ܕܒܬܪܗ.

ܫܪ̈ܛܐ ܘܙܘܗܪ̈ܐ (Assumptions and warnings)

ܐܠܓܘܪܝܬܡ ܟܠ ܙܒܢܐ ܟܡܕܥܪ ܥܢܩܘ̈ܕܐ. "ܐܘܪ̈ܚܬܐ ܕܥܢܩܘ̈ܕܐ ܐܝܟ ܐܢܝ ܟܠ ܙܒܢܐ ܟܡܛܝܝ ܠ-k ܓܘ̈ܕܐ، ܡܐ ܕܗܘܝ ܝܗܝ̈ܒܬܐ" (Rousseeuw, 1987, p. 55): k-ܡܨܥܝ̈ܬܐ ܒܕ ܡܕܥܪ ܬܠܬܐ ܥܢܩܘ̈ܕܐ ܐܦ ܡܢ ܝܗܝ̈ܒܬܐ ܫܘ̈ܝܬܐ ܓܡܝܪܐܝܬ، ܕܠܐ ܚܕ ܒܢܝܢܐ؛ ܘ"ܐܘܪ̈ܚܬܐ ܕܕܪ̈ܓܐ ܟܐܠܨܝ ܒܢܝܢܐ ܕܕܪ̈ܓܐ ܐܢ ܐܝܬ ܐܢ ܠܝܬ ܒܢܝܢܐ ܐܝܟ ܐܗܐ ܒܥܒܕܐ ܓܘ ܝܗܝ̈ܒܬܐ" (Hastie et al., 2009, p. 522). ܡܛܠ ܗܕܐ gap statistic ܓܘܒܝܐ ܡܩܕܡܐ ܝܠܗ ܩܐ ܓܘܒܝܐ ܕ-k، ܘܡܛܠ ܗܕܐ ܩܘܛܝܐ ܕܩܪܝܢܐ ܟܡܙܗܪ ܐܡܬܝ ܕ-silhouette ܟܪܝܗܐ ܝܠܗ.

ܩܝܡܘܬܐ: ܐܢ ܦܘܫܪܐ ܒܕ ܕܥܪ ܓܘ ܕܘܡܝܐ ܐܚܪܢܐ؟ Leisch (2016، ܦܐܬܐ 637)، ܓܘ ܨܚܚܐ ܥܠ ܒܘܚܪܢܐ ܕܩܝܡܘܬܐ ܓܘ ܟܬܒܐ ܕܐܝܕܐ ܕ-Hennig ܘܚܒܪ̈ܘܗܝ، ܟܦܪܫ ܒܝܢܝ ܬܪܝܢ ܡܒܘ̈ܥܐ ܕܓܕܫܢܘܬܐ ܓܘ ܟܠ ܦܘܠܓܐ: ܕܘܡܝܐ – ܓܘܕܐ ܙܥܘܪܬܐ ܓܕܫܢܝܬܐ ܡܢ ܟܢܫܐ ܟܠܢܝܐ (population) – ܘܐܠܓܘܪܝܬܡ، ܕ"ܝܕܝܥܐ ܝܠܗ ܕܟܐܠܨ ܒܢܝܢܐ ܥܠ ܝܗܝ̈ܒܬܐ ܐܦ ܐܡܬܝ ܕܐܢܝ ܠܐ ܒܢܝ̈ܐ ܝܢܐ ܛܒ" (Dolnicar & Leisch, 2010، ܐܝܟ ܕܡܬܡܠܠܝ ܬܡܢ)؛ ܘܡܛܠ ܗܕܐ "ܡܦܠܚܢܘܬܐ ܕܦܘܠܓܐ ܝܚܝܕܝܐ، ܕܟܐܬܐ ܡܢ ܚܘܫܒܢܐ ܝܚܝܕܝܐ ܕܐܠܓܘܪܝܬܡ ܡܝܚܕܐ، ܟܡܩܝܡܐ ܠܒܨܘܝ̈ܐ ܓܘ ܣܘܟܢܐ ܕܡܦܠܚܢܘܬܐ ܕܦܘܫܪܐ ܓܕܫܢܝܐ، ܘܠܐ ܕܦܘܫܪܐ ܫܪܝܪܐ". ܐܘܪܚܐ ܕܬܪܝܢ ܕܪ̈ܓܐ ܟܡܒܛܠܐ ܓܕܫܢܘܬܐ ܕܐܠܓܘܪܝܬܡ (ܠܝܬ ܫܘܪܝܐ ܓܕܫܢܝܐ)، ܐܝܢܐ ܠܐ ܕܕܘܡܝܐ. ܩܝܡܘܬܐ ܪܫܝܬܐ ܝܠܗ ܡܝܚܕܐܝܬ ܐܡܬܝ ܕܟܦܣܩܚ ܥܠ ܐܣܐܣܐ ܕܦܘܠܓܐ – ܓܘ ܦܘܠܓܐ ܕܫܘ̈ܩܐ ܐܝܬܝܗ ܚܕܐ ܡܢ ܫܬܐ ܡܫܘ̈ܚܬܐ ܩܐ ܫܘܥܪܐ ܕܡܢܘ̈ܬܐ (Wedel & Kamakura, 1998، ܐܝܟ ܕܡܬܡܠܠܝ ܬܡܢ، ܦܐܬܐ 639) – ܘܠܐ ܝܠܗ ܕܡܐ ܠܐܝܬܘܬܐ ܕܥܢܩܘ̈ܕܐ ܦܪ̈ܝܫܐ: ܓܘ ܨܘܪܬܐ 28.1 ܐܪܒܥܐ ܥܢܩܘ̈ܕܐ ܕ-k-ܡܨܥܝ̈ܬܐ ܥܠ ܝܗܝ̈ܒܬܐ ܫܘ̈ܝܬܐ ܓܘ ܡܪܒܥܐ ܩܝܡ̈ܐ ܝܢܐ ܣܓܝ، ܘܥܠ ܥܓܘܠܐ – ܠܐ ܩܝܡ̈ܐ ܒܟܠܗ (ܦܐܬܐ 638). ܐܘܪܚܐ ܕܒܘܚܪܢܐ ܓܘܒܝܐ ܕܕܘܡܝܐ ܡܢ ܪܝܫ ܝܠܗ: ܟܓܒܚ ܓܕܫܢܐܝܬ ܡܢ ܝܗܝ̈ܒܬܐ ܕܘܡ̈ܝܐ ܕ-bootstrap، ܟܟܢܫܚ ܟܠ ܚܕ ܠܥܢܩܘ̈ܕܐ، ܘܟܦܚܡܚ ܦܘܠ̈ܓܐ (ܦܐ̈ܬܐ 639-640)؛ ܨܚܚܐ ܟܡܚܘܐ ܬܠܬܐ ܐܘܪ̈ܚܬܐ – ܚܝܠܐ ܕܡܢܒܝܢܘܬܐ (prediction strength؛ Tibshirani & Walther, 2005)، ܩܝܡܘܬܐ ܩܐ ܟܠ ܥܢܩܘܕܐ ܒܡܚܘܝܢܐ ܕ-Jaccard (Hennig, 2007)، ܘ-ARI ܒܝܢܝ ܙܘ̈ܓܐ ܕܕܘܡ̈ܝܐ ܕ-bootstrap (Dolnicar & Leisch, 2010؛ ܟܠܗܘܢ ܐܝܟ ܕܡܬܡܠܠܝ ܬܡܢ، ܦܐ̈ܬܐ 641-643) – ܘܟܟܢܫ ܕ"ܓܘܒܝܐ ܕܕܘܡܝܐ ܡܢ ܪܝܫ ܟܥܒܕ ܦܘܪܫܢܐ ܪܒܐ، ܐܝܢܐ ܐܘܪܚܐ ܚܬܝܬܬܐ ܠܐ ܝܠܗ ܗܕܟܐ ܪܫܝܬܐ" (ܦܐܬܐ 650). ܡܨܝܚ ܐܦ ܕܓܒܚ k ܠܦܘܬ ܩܝܡܘܬܐ، ܐܝܢܐ ܒܙܗܝܪܘܬܐ: "ܒܥܕܢ̈ܐ ܣܓܝ̈ܐܐ ܠܐ ܡܨܝܚ ܕܡܒܛܠܚ ܕܐܦ ܠܦܘܠܓܐ ܐܚܪܢܐ ܡܢ ܫܪܝܪܐ ܐܝܬ ܩܝܡܘܬܐ ܪܡܬܐ" (ܦܐܬܐ 639). ܐܗܐ ܦܐܬܐ ܟܪܗܛ، ܒܚܕ ܕܘܫܐ، ܐܘܪܚܐ ܕܬܠܝܬܝܬܐ ܘܕܬܪܝܢܝܬܐ.

ܩܝܡܘܬܐ ܕܦܘܠܓܐ ܟܠܗ (ܠܘܚܐ 4 ܘܨܘܪܬܐ 5). ܓܘ ܐܘܪܚܐ ܕ-Dolnicar ܘ-Leisch (2010، ܐܝܟ ܕܡܬܡܠܠܝ ܓܘ Leisch, 2016، ܦܣܩܐ 28.2.3) ܟܓܒܚ ܓܕܫܢܐܝܬ ܬܪܝܢ ܕܘܡ̈ܝܐ ܕ-bootstrap ܒܪܒܘܬܐ ܕܕܘܡܝܐ، ܟܟܢܫܚ ܟܠ ܚܕ ܠܥܢܩܘ̈ܕܐ، ܟܦܠܓܚ ܟܠ ܡܫܬܘܬܦ̈ܢܐ ܫܪ̈ܫܝܐ ܠܡܨܥܝܐ ܩܪܝܒܐ ܓܘ ܟܠ ܚܕ ܡܢ ܬܪܘܝܗܝ، ܘܟܦܚܡܚ ܬܪܝܢ ܦܘܠ̈ܓܐ ܒܡܚܘܝܢܐ ܕ-Rand ܡܬܘܪܨܐ؛ ܟܡܬܢܚ ܐܗܐ ܣܓܝ ܓܗ̈ܐ، ܘܟܠ ܙܘܓܐ ܟܝܗܒ ܛܝܡܐ ܚܐܪܐ. ܐܗܐ ܐܘܪܚܐ "ܒܦܠܓܐ" ܝܠܗ ܒܝܢܝ ܬܪܬܝܢ ܐܚܪ̈ܢܝܬܐ: ܟܡܫܚܐ ܦܘܠܓܐ ܟܠܗ، ܐܝܢܐ ܟܡܨܐ ܠܥܢܩܘ̈ܕܐ ܡܝܚ̈ܕܐ، ܒܠܟܝ ܙܥܘܪ̈ܐ، ܕܗܘܝ ܠܐ ܩܝܡ̈ܐ (ܦܐܬܐ 642). ܟܚܙܝܚ ܦܘܠܓܐ، ܓܘ ܨܘܪܬܐ ܕܩܘܛܝܐ، ܘ"ܛܝ̈ܡܐ ܩܪܝܒ̈ܐ ܠܚܕ ܟܡܚܘܝ ܩܝܡܘܬܐ ܪܡܬܐ" (ܦܐܬܐ 643). ܓܘ ܡܬܠܐ ܥܒܝܕܐ ܕ-Leisch، ܓܘ ܡܢܝܢܐ ܬܪܝܨܐ ܕܥܢܩܘ̈ܕܐ ܟܠ ܡܚܘ̈ܝܢܐ ܫܘ̈ܝܐ ܝܢܐ ܠ-1، ܘܓܘ ܡܢܝܢܐ ܛܥܝܐ ܦܘܠܓܐ ܟܙܝܥ ܠܣܡܠܐ، ܘܒܥܕܢ̈ܐ ܐܝܬ ܠܗ ܬܪܝܢ ܪ̈ܘܡܐ (ܦܐܬܐ 645). ܓܘ ܐܗܐ ܦܐܬܐ، ܗܢ ܙܘ̈ܓܐ ܡܬܦܠܚ̈ܐ ܝܢܐ ܩܐ ܟܠ ܛܝ̈ܡܐ ܕ-k، ܗܕܟܐ ܕܦܘܚܡܐ ܒܝܢܝ ܛܝ̈ܡܐ ܕ-k ܥܠ ܗܢ ܕܘܡ̈ܝܐ ܝܠܗ.

ܩܝܡܘܬܐ ܕܟܠ ܥܢܩܘܕܐ (ܠܘܚܐ 5). ܓܘ ܐܘܪܚܐ ܕ-Hennig (2007, as cited in Leisch, 2016, Section 28.2.2) ܟܒܚܪܚ ܦܘܠܓܐ ܓܒܝܐ، ܠܐ k ܒܐܘܪܚܐ ܟܠܢܝܬܐ (ܦܐܬܐ 646): ܓܘ ܟܠ ܕܘܡܝܐ ܕ-bootstrap ܟܫܩܠܚ ܡܫܬܘܬܦ̈ܢܐ ܕܓܒܝ̈ܐ ܝܢܐ، ܟܠ ܚܕ ܚܕ ܓܗܐ، ܟܟܢܫܚ ܠܗܘܢ ܠܥܢܩܘ̈ܕܐ، ܘܩܐ ܟܠ ܥܢܩܘܕܐ ܫܪܫܝܐ ܟܒܨܝܚ ܥܠ ܥܢܩܘܕܐ ܕܕܡܐ ܠܗ ܝܬܝܪ ܓܘ ܕܘܡܝܐ ܠܦܘܬ ܕܡܝܘܬܐ ܕ-Jaccard – ܡܢܝܢܐ ܕܡܫܬܘܬܦ̈ܢܐ ܡܫܘܬܦ̈ܐ ܡܦܠܓܐ ܥܠ ܡܢܝܢܐ ܕܡܫܬܘܬܦ̈ܢܐ ܓܘ ܬܪܘܝܗܝ ܐܟܚܕܐ. ܡܨܥܝܬܐ ܕܕܡܝܘܬܐ ܥܠ ܕܘܡ̈ܝܐ ܡܚܘܝܢܐ ܕܩܝܡܘܬܐ ܕܥܢܩܘܕܐ ܝܠܗ. Hennig (2023)، ܓܘ ܟܬܒܐ ܕܦܘܢܩܣܝܐ clusterboot ܕܟܬܒܠܗ ܓܘ ܚܙܡܐ fpc، ܟܝܗܒ ܢܡܘ̈ܣܐ ܕܨܒܥܐ: ܥܢܩܘܕܐ ܬܪܝܨܐ ܘܩܝܡܐ ܣܢܝܩ ܝܠܗ ܠܡܨܥܝܬܐ ܕ-.75 ܘܠܥܠ؛ ܒܝܢܝ .60 ܘ-.75 ܥܢܩܘܕܐ ܟܡܚܘܐ ܛܘܦܣܐ ܓܘ ܝܗܝ̈ܒܬܐ، ܐܝܢܐ ܐܝܢܝ ܡܫܬܘܬܦ̈ܢܐ ܕܝܠܗ ܝܢܐ – ܡܫܟܟܐ ܝܠܗ ܣܓܝ؛ ܠܬܚܬ ܡܢ .60 – ܠܐ ܡܣܡܟܚ ܥܠܗ؛ ܘ"ܩܝܡܐ ܣܓܝ" – .85 ܘܠܥܠ. ܛܝܡܐ ܕ-.50 ܘܠܬܚܬ ܓܘ ܕܘܡܝܐ ܝܚܝܕܝܐ ܡܬܚܫܒܐ ܝܠܗ "ܦܪܩܐ" ܕܥܢܩܘܕܐ، ܘܩܐ ܐܗܐ ܐܝܬ ܬܪܝܨܘܬܐ ܬܐܘܪܝܬܐ (Hennig, 2008, as cited in García-Escudero et al., 2016, p. 668). ܘܓܘ ܗܢ ܟܬܒܐ: ܩܝܡܘܬܐ ܠܐ ܝܠܗ ܡܫܘܚܬܐ ܝܚܝܕܝܬܐ ܩܐ ܬܪܝܨܘܬܐ (validity)، ܘܐܦ ܐܘܪܚܐ ܩܫܝܬܐ ܣܓܝ ܡܨܝܐ ܕܝܗܒܐ ܥܢܩܘ̈ܕܐ ܩܝܡ̈ܐ ܕܠܐ ܬܪܝܨ̈ܐ ܝܢܐ.

ܓܘ ܦܐܝܠ ܕܡܬܠܐ (100 ܙܘ̈ܓܐ، ܙܪܥܐ 2026) ܠܘܚܐ 4 ܟܡܚܘܐ ܚܬܝܬܐܝܬ ܡܐ ܕܨܚܚܐ ܟܡܚܘܐ. ܩܐ k = 3 ܛܝܡܐ ܡܨܥܝܐ ܕ-ARI .95 ܘܪܘܒܥܐ ܬܚܬܝܐ .94 – ܩܝܡܐ. ܩܐ k = 2 ܛܝܡܐ ܡܨܥܝܐ .87، ܐܝܢܐ ܪܘܒܥܐ ܬܚܬܝܐ .33 – ܦܘܠܓܐ ܐܝܬ ܠܗ ܬܪܝܢ ܪ̈ܘܡܐ، ܝܥܢܝ ܬܪܝܢ ܦܘܠ̈ܓܐ ܦܪ̈ܝܫܐ ܠܬܪܝܢ ܥܢܩܘ̈ܕܐ ܟܡܬܚܪܝ ܚܕ ܥܡ ܐܚܪܢܐ. Leisch ܟܡܚܘܐ ܦܘܠܓܐ ܐܝܟ ܐܗܐ ܐܝܟ ܪܘܫܡܐ ܕ"ܟܡܐ ܛܝ̈ܡܐ ܥܠ̈ܝܐ ܕܘܟ̈ܝܐ": ܛܝܡܐ ܛܒܐ ܕܘܟܝܐ ܓܘ ܝܗܝ̈ܒܬܐ ܫܪ̈ܫܝܬܐ ܡܨܐ ܕܗܘܐ ܛܝܡܐ ܛܒܐ ܟܠܢܝܐ ܓܘ ܕܘܡܝܐ ܕ-bootstrap (ܦܐܬܐ 643). ܡܢ k = 4 ܛܝܡܐ ܡܨܥܝܐ ܟܢܚܬ: .76، .62، ܘܗܕܟܐ ܕ-.50 ܓܘ k = 8. ܓܘ ܠܘܚܐ 5، ܠܬܠܬܐ ܥܢܩܘ̈ܕܐ ܡܨܥܝܬܐ ܕ-Jaccard .98، .98 ܘ-.96 – ܩܝܡ̈ܐ ܣܓܝ، ܘܚܕ ܡܢܝܗܘܢ ܠܐ ܦܪܩܠܗ ܓܘ ܚܕ ܕܘܡܝܐ. ܘܙܘܗܪܐ ܐܚܪܢܐ ܡܢ ܨܚܚܐ: ܡܨܝܚ ܕܓܒܚ k ܠܦܘܬ ܩܝܡܘܬܐ، ܐܝܢܐ ܒܙܗܝܪܘܬܐ – "ܒܥܕܢ̈ܐ ܣܓܝ̈ܐܐ ܠܐ ܡܨܝܚ ܕܡܒܛܠܚ ܕܐܦ ܠܦܘܠܓܐ ܐܚܪܢܐ ܡܢ ܫܪܝܪܐ ܐܝܬ ܩܝܡܘܬܐ ܪܡܬܐ" (ܦܐܬܐ 639). ܓܘ ܒܘܨܝܐ ܕܫܘܐ̈ܠܐ (survey) ܕܬܝܪܘܬܐ ܓܘ ܨܚܚܐ، ܦܘܠܓܐ ܠ-5 ܩܝܡܐ ܝܬܝܪ ܝܠܗ ܗܘܐ، ܐܝܢܐ ܠܡܢܘ̈ܬܐ ܕܝܠܗ ܠܝܬ ܗܘܐ ܬܚܘܝܬܐ ܓܠܝܬܐ، ܘܡܛܠ ܗܕܐ ܠܐ ܓܒܝܐ ܝܠܗ (ܦܐܬܐ 648)؛ ܘܓܘ ܦܘܠܓܐ ܠ-9 ܡܢܬܐ ܡܢ ܥܢܩܘ̈ܕܐ ܩܝܡ̈ܐ ܝܢܐ ܗܘܘ ܝܬܝܪ ܡܢ ܐܚܪ̈ܢܐ، ܘܩܝܡ̈ܐ ܕܓܘܝܗܘܢ ܐܢܝ ܝܢܐ ܗܘܘ ܕܡܢܚܝܢ̈ܐ (ܦܐܬܐ 649).

ܐܣܟܝܡܐ ܕܥܢܩܘ̈ܕܐ. k-ܡܨܥܝ̈ܬܐ ܘ-Ward ܟܓܒܝ ܥܢܩܘ̈ܕܐ "ܐܣܦܝܪ̈ܝܐ" ܒܪܒܘܬܐ ܕܡܝܬܐ: "k-ܡܨܥܝ̈ܬܐ ܠܐ ܟܦܠܚ ܛܒ ܐܡܬܝ ܕܐܝܬ ܠܥܢܩܘ̈ܕܐ ܪ̈ܒܘܬܐ ܦܪ̈ܝܫܐ، ܥܒܝܘ̈ܬܐ ܦܪ̈ܝܫܐ ܝܢ ܐܣܟܝ̈ܡܐ ܠܐ ܐܣܦܝܪ̈ܝܐ" (Géron, 2022, Chapter 9, "Limits of k-means")؛ Ward "ܟܡܨܠܐ ܕܡܫܟܚ ܥܢܩܘ̈ܕܐ ܐܣܦܝܪ̈ܝܐ ܒܪܒܘܬܐ ܕܡܝܬܐ" ܘ"ܪܓܝܫܐ ܝܠܗ ܠܛܝ̈ܡܐ ܢܘܟܪ̈ܝܐ" (Everitt et al., 2011, p. 78, Table 4.1)، "ܒܥܕܢ̈ܐ ܣܓܝ̈ܐܐ ܟܡܬܚܙܐ ܕܟܦܠܚ ܛܒ، ܐܝܢܐ ܡܨܐ ܕܐܠܨ ܒܢܝܢܐ ܐܣܦܝܪܝܐ ܐܝܟܐ ܕܠܝܬ" (ܦܐܬܐ 84)، ܘܓܘ ܒܘܨ̈ܝܐ ܕܡܬܠܐ ܥܒܝܕܐ (simulation) ܨܠܚܠܗ ܐܡܬܝ ܕܥܢܩܘ̈ܕܐ ܒܪܒܘܬܐ ܕܡܝܬܐ ܝܢܐ ܗܘܘ ܘܓܕܠܠܗ ܐܡܬܝ ܕܐܢܝ ܒܪ̈ܒܘܬܐ ܦܪ̈ܝܫܐ ܝܢܐ ܗܘܘ (ܦܐܬܐ 83)؛ ܓܘ Ward "ܪ̈ܒܘܬܐ ܕܥܢܩܘ̈ܕܐ ܕܡ̈ܝܐ ܝܢܐ ܒܩܘܪܒܐ" (Wendler & Gröttrup, 2021, p. 648)، ܘ-k-ܡܨܥܝ̈ܬܐ "ܟܡܨܠܐ ܕܥܒܕ ܥܢܩܘ̈ܕܐ ܒܪܒܘܬܐ ܕܡܝܬܐ" (ܦܐܬܐ 677). ܓܘܕܐ ܙܥܘܪܬܐ ܘܦܪܝܫܬܐ ܒܫܪܪܐ ܡܨܝܐ ܕܡܬܒܠܥܐ؛ ܩܐ ܥܢܩܘ̈ܕܐ ܡܓܪ̈ܓܐ (elliptical) – Gaussian mixture (ܦܐܬܐ ܝܘܠܦܢܐ ܕܠܐ ܡܕܒܪܢܐ، ܦܣܩܐ 4).

ܛܝ̈ܡܐ ܢܘܟܪ̈ܝܐ (outliers). ܡܫܬܘܬܦܢܐ ܚܕ ܪܚܝܩܐ ܟܓܪܫ ܡܨܥܝܐ ܠܓܒܗ (ܡܨܥܝܬܐ) ܘܟܡܪܒܐ ܓܘܡܪܐ ܕܡܪ̈ܒܥܐ؛ ܓܘ ܐܣܘܪܐ ܝܚܝܕܝܐ ܛܝ̈ܡܐ ܢܘܟܪ̈ܝܐ ܟܦܝܫܝ ܡܫܬܘܬܦ̈ܢܐ ܝܚܝܕ̈ܝܐ، ܘܓܘ ܓܡܝܪܐ ܐܢܝ ܟܡܬܒܠܥܝ ܓܘ ܥܢܩܘ̈ܕܐ (Wendler & Gröttrup, 2021, p. 648, Table 7.24). ܥܢܩܘܕܐ ܕܚܕ ܝܢ ܬܪܝܢ ܡܫܬܘܬܦ̈ܢܐ ܓܘ ܦܘܫܪܐ ܩܪܝܒܐܝܬ ܟܠ ܙܒܢܐ ܛܝܡܐ ܢܘܟܪܝܐ ܝܠܗ، ܠܐ ܙܢܐ – ܟܒܚܪܚ ܓܘ ܦܐܬܐ ܛܝ̈ܡܐ ܢܘܟܪ̈ܝܐ ܘܫܪ̈ܛܐ.

ܬܚܘ̈ܡܐ ܕܥܒܕܐ. ܛܝ̈ܡܐ ܚܣܝ̈ܪܐ (missing values): ܫܩܠܬܐ ܕܣܕܪܐ ܟܠܗ (listwise deletion) ܥܠ ܡܫܬܚܠܦ̈ܢܐ ܕܥܢܩܘܕܐ (ܐܗܐ ܦܐܬܐ ܟܡܘܕܥ ܟܡܐ ܫܩܝܠ̈ܐ ܝܢܐ؛ ܩܐ ܚܠܦ̈ܐ – Hastie et al., 2009، ܦܐܬܐ 507، ܘܦܐܬܐ ܛܝ̈ܡܐ ܚܣܝ̈ܪܐ). ܪܒܘܬܐ: ܗܕܟܐ ܕ-3,000 ܡܫܬܘܬܦ̈ܢܐ، ܡܛܠ ܕܥܢܩܘ̈ܕܐ ܕܕܪ̈ܓܐ ܟܢܛܪܝ ܟܠ ܪܘ̈ܚܩܐ ܒܝܢܝ ܙܘ̈ܓܐ ܕܡܫܬܘܬܦ̈ܢܐ؛ gap statistic ܟܪܗܛ ܦܘܫܩܐ ܟܠܗ ܥܠ ܟܠ ܛܘܦܣܐ ܕܦܘܚܡܐ، ܘܡܛܠ ܗܕܐ ܓܘ 3,000 ܡܫܬܘܬܦ̈ܢܐ ܘ-100 ܕܘܡ̈ܝܐ ܐܝܢܐ ܟܫܩܠ ܐܝܟ ܚܕܐ ܩܛܝܢܬܐ (ܡܨܝܬܘܢ ܕܡܙܥܪܝܬܘܢ ܡܢܝܢܐ ܕܕܘܡ̈ܝܐ). ܦܐܪܐ ܦܣܝܩܐ ܝܠܗ (deterministic)، ܐܝܢܐ ܐܡܬܝ ܕܐܝܬ ܡܫܬܘܬܦ̈ܢܐ ܕܡ̈ܝܐ ܝܢ ܪܘ̈ܚܩܐ ܫܘ̈ܝܐ ܚܬܝܬܐܝܬ (ܡܬܚܙܐ ܣܓܝ ܓܘ ܦܣܘ̈ܩܐ ܕ-Likert)، ܛܟܣܐ ܕܚܘܝ̈ܕܐ ܕܪܘܡܐ ܫܘܝܐ ܦܣܝܩܐ ܝܠܗ ܠܦܘܬ ܛܟܣܐ ܕܣܕܪ̈ܐ، ܐܝܟ ܓܘ R. ܡܫܬܚܠܦ̈ܢܐ ܣܢܝܩ̈ܐ ܝܢܐ ܕܗܘܝ ܟܡܝ̈ܝܐ: ܡܫܬܚܠܦܢܐ ܕܓܢ̈ܣܐ ܒܪܡܙܢܘܬܐ ܡܢܝܢܝܬܐ (1 = ܓܒܪܐ، 2 = ܐܢܬܬܐ) ܟܥܐܠ ܐܝܟ ܡܢܝܢܐ ܘܟܥܒܕ ܠܪܘܚܩܐ ܕܠܐ ܣܘܟܠܐ – ܐܗܐ ܦܐܬܐ ܟܡܙܗܪ.

ܡܘܕܝ ܠܐ ܟܚܫܒܐ ܐܗܐ ܦܐܬܐ. ܐܣܘܪ̈ܐ (linkages) ܐܚܪ̈ܢܐ (ܝܚܝܕܝܐ، ܓܡܝܪܐ، ܡܨܥܝܬܐ)، ܪܘ̈ܚܩܐ ܐܚܪ̈ܢܐ (Manhattan، cosine)، ܡܫܬܚܠܦ̈ܢܐ ܕܓܢ̈ܣܐ ܘܡܚܠܛ̈ܐ (TwoStep ܥܡ log-likelihood)، Gaussian mixtures ܘ-BIC، DBSCAN، ܒܘܚܪ̈ܢܐ ܕܪܫܝܡܘܬܐ ܒܝܢܝ ܥܢܩܘ̈ܕܐ (ܚܙܝܡܘܢ ܓܘ ܛܥܝ̈ܘܬܐ ܕܡܬܚܙܝܢ ܣܓܝ)، ܩܝܡܘܬܐ ܒܐܘܪ̈ܚܬܐ ܐܚܪ̈ܢܝܬܐ ܕܓܘܒܝܐ ܕܕܘܡܝܐ (ܚܝܠܐ ܕܡܢܒܝܢܘܬܐ، ܕܘܡ̈ܝܐ ܙܥܘܪ̈ܐ، ܡܘܣܦܐ ܕܩܠܐ)، ܘܫܘܪܪܐ ܒܪܝܐ ܡܢ ܓܢܗ – ܥܡܘܕܐ ܕܥܢܩܘܕܐ ܟܡܚܬܚ ܘܟܒܚܪܚ ܓܘ ܦܐ̈ܬܐ ܟܝ ܡܪܒܥܐ ܘ-ܦܘܫܩܐ ܕܫܘܚܠܦܐ.

ܘܕܟܝ ܟܥܒܕܚ ܐܗܐ ܒ-SPSS؟ (And how do you do it in SPSS?)

ܓܘ SPSS Statistics ܬܪܝܢ ܕܪ̈ܓܐ ܬܪܬܝܢ ܦܪܘܣܝܕܘܪ̈ܐ (procedures) ܦܪ̈ܝܫܐ ܝܢܐ ܬܚܘܬ Analyze › Classify. ܫܒ̈ܝܠܐ ܘܓܘܒܝ̈ܐ ܕܒܬܪ ܗܕܐ ܡܘܕܥܢܘܬܐ ܟܠܢܝܬܐ ܝܢܐ ܥܠ ܬܘܟܢܬܐ (software)، ܐܝܟ ܓܘ ܦܐܬܐ ܝܘܠܦܢܐ ܕܠܐ ܡܕܒܪܢܐ، ܦܣܩܐ 7؛ ܡܒܘ̈ܥܐ ܕܐܗܐ ܦܐܬܐ ܥܠ SPSS Modeler ܝܢܐ ܘܠܐ ܥܠ Statistics. (1) ܡܬܩܢܘܬܐ: Analyze › Descriptive Statistics › Descriptives ܥܡ Save standardized values as variables – ܟܗܘܝ ܡܫܬܚܠܦ̈ܢܐ ܒܫܡܐ Z…. (2) Ward: Hierarchical Cluster ܥܠ ܡܫܬܚܠܦ̈ܢܐ ܕ-Z؛ ܓܘ button Method – Ward's method ܥܡ Squared Euclidean distance؛ ܓܘ button Plots – Dendrogram؛ ܓܘ button Save – Single solution ܥܡ k. ܥܡܘܕܐ ܕ-Coefficients ܓܘ ܠܘܚܐ Agglomeration Schedule ܕ-SPSS ܠܐ ܝܠܗ ܓܘ ܣܒܠܬܐ ܕܪܘܡܐ ܕܐܝܠܢܐ ܓܘ R ܘܓܘ ܐܗܐ ܦܐܬܐ، ܐܝܢܐ ܛܟܣܐ ܕܚܘܝ̈ܕܐ ܣܢܝܩ ܝܠܗ ܕܗܘܐ ܗܢ ܛܟܣܐ، ܠܒܪ ܡܢ ܚܘܝ̈ܕܐ ܕܪܘܡܐ ܫܘܝܐ. (3) k-ܡܨܥܝ̈ܬܐ: K-Means Cluster ܥܠ ܗܢ ܡܫܬܚܠܦ̈ܢܐ، ܥܡ ܡܢܝܢܐ ܕܥܢܩܘ̈ܕܐ، ܘܡܨܥ̈ܝܐ ܕܫܘܪܝܐ ܡܢ ܦܐܝܠ (Read initial)؛ ܡܨܥ̈ܝܐ ܟܚܫܒܚ ܐܝܟ ܡܨܥܝ̈ܬܐ ܕܡܫܬܚܠܦ̈ܢܐ ܕ-Z ܠܦܘܬ ܥܢܩܘܕܐ ܕ-Ward (Data › Aggregate). ܕܠܐ ܡܨܥ̈ܝܐ ܕܫܘܪܝܐ، SPSS ܟܓܒܐ ܠܗܘܢ ܒܓܢܗ ܡܢ ܝܗܝ̈ܒܬܐ، ܘܦܐܪܐ ܡܨܐ ܕܗܘܐ ܦܪܝܫܐ ܡܢ ܦܐܪܐ ܕܐܗܐ ܦܐܬܐ.

ܣܝܢܬܐܟܣ (syntax) ܕܐܗܐ ܦܐܬܐ ܟܒܢܐ ܟܥܒܕ ܬܠܬܐ ܦܣܥ̈ܬܐ: DESCRIPTIVES /SAVE، CLUSTER ܥܡ /METHOD WARD /MEASURE SEUCLID ܘ-/SAVE CLUSTER(k)، ܘ-QUICK CLUSTER ܥܡ /INITIAL – ܡܨܥ̈ܝܐ ܕ-Ward ܕܐܗܐ ܦܐܬܐ ܚܫܒܠܗ، ܣܝܡ̈ܐ ܐܝܟ ܡܢܝ̈ܢܐ – ܘܥܡ /METHOD KMEANS(NOUPDATE)، ܝܥܢܝ ܚܘܕܬܐ ܕܡܨܥ̈ܝܐ ܒܣ ܓܘ ܣܘܦܐ ܕܟܠ ܥܒܪܬܐ ܥܠ ܝܗܝ̈ܒܬܐ، ܐܝܟ ܓܘ ܐܠܓܘܪܝܬܡ ܕܐܗܐ ܦܐܬܐ. ܣܝܢܬܐܟܣ ܠܐ ܪܗܛܐ ܝܠܗ ܓܘ SPSS. ܓܘ PSPP، ܬܘܟܢܬܐ ܚܐܪܬܐ ܕܟܩܪܝܐ ܣܝܢܬܐܟܣ ܕ-SPSS، ܒܣ ܡܢܬܐ ܩܕܡܝܬܐ (ܡܬܩܢܘܬܐ) ܒܚܝܪܬܐ ܝܠܗ، ܡܛܠ ܕܠܝܬ ܓܘܗ ܦܘܩܕܢܐ CLUSTER ܘܠܝܬ ܓܘܗ /INITIAL. ܠ-silhouette، ܠ-gap statistic ܘܠܒܘܚܪ̈ܢܐ ܕܩܝܡܘܬܐ ܓܘ ܓܘܒܝܐ ܕܕܘܡܝܐ ܡܢ ܪܝܫ SPSS Statistics ܠܐ ܟܚܫܒ – ܩܐ ܐܢܝ R ܝܢ ܐܗܐ ܦܐܬܐ.

ܘܕܟܝ ܟܥܒܕܚ ܐܗܐ ܒ-R؟ (And how do you do it in R?)

ܓܘ R ܐܣܝܐ: scale ܩܐ ܡܬܩܢܘܬܐ، hclust(dist(z), method = "ward.D2") ܩܐ Ward (ward.D2 ܝܠܗ Ward ܥܠ ܪܘ̈ܚܩܐ ܕ-Euclid؛ ward.D ܥܬܝܩܐ ܟܣܒܪ ܕܡܢ ܩܕܡ ܡܪܒܥܠܘܟܘܢ ܠܗܘܢ)، cutree ܩܐ ܦܣܩܐ، ܘ-kmeans ܥܡ centers = ܡܨܥ̈ܝܐ ܕ-Ward ܘ-algorithm = "Lloyd" – ܐܠܓܘܪܝܬܡ ܕܐܗܐ ܦܐܬܐ (ܓܘܒܝܐ ܡܩܕܡܐ ܕ-R ܝܠܗ Hartigan-Wong، ܕ"ܟܡܫܪܪ ܕܠܝܬ ܫܘܢܝܐ ܝܚܝܕܝܐ ܕܚܙܝܬܐ ܡܢ ܓܘܕܐ ܠܓܘܕܐ ܐܚܪܬܐ ܕܒܕ ܡܙܥܪ ܠܢܝܫܐ"؛ Hastie et al., 2009، ܦܐܬܐ 510. ܓܘ ܦܐܝܠ ܕܡܬܠܐ، ܩܐ k = 2 ܘ-3 ܬܪܝܢ ܐܠܓܘܪܝܬ̈ܡܐ ܟܝܗܒܝ ܗܢ ܦܘܫܪܐ؛ ܩܐ k ܪܒܐ ܝܬܝܪ Hartigan-Wong ܒܥܕܢ̈ܐ ܟܡܫܟܚ ܦܘܫܪܐ ܥܡ W_k ܙܥܘܪܐ ܩܠܝܠ – ܘܐܗܐ ܦܐܬܐ ܟܬܢܐ ܓܘ ܩܘܛܝܐ ܕܩܪܝܢܐ ܟܡܐ ܡܫܬܘܬܦ̈ܢܐ ܕܫܘܢܝܐ ܝܚܝܕܝܐ ܕܝܗܘܢ ܒܕ ܡܙܥܪ ܗܘܐ W_k). ܗܢ ܐܘܪܚܐ ܐܝܟ ܦܘܢܩܣܝܐ ܡܛܝܒܬܐ: hkmeans ܓܘ ܚܙܡܐ factoextra (Kassambara, 2017, pp. 163-164) – Ward (ward.D2) ܘܒܬܪܗ kmeans، ܐܝܢܐ ܒܓܘܒܝ̈ܐ ܡܩܕ̈ܡܐ ܕ-R (Hartigan-Wong، ܘ-iter.max = 10)؛ ܓܘ ܦܐܝܠ ܕܡܬܠܐ ܐܝܢܐ ܕܡܝܐ ܝܠܗ ܠܐܗܐ ܦܐܬܐ ܩܐ k = 2 ܘ-3، ܘܩܐ k ܪܒܐ ܝܬܝܪ ܟܝܗܒܐ W_k ܙܥܘܪܐ ܩܠܝܠ. ܚܙܡܐ cluster ܟܝܗܒ silhouette ܘ-clusGap؛ ܥܡ d.power = 2، spaceH0 = "original" ܘ-set.seed ܫܘܝܐ ܠܙܪܥܐ ܕܐܗܐ ܦܐܬܐ – ܗܢ gap ܘܗܢ ܛܥܝܘܬܐ ܬܩܝܢܬܐ ܐܝܟ ܓܘ ܠܘܚܐ 1. ܓܘ ܣܘܦܐ ܕܣܩܪܝܦܬ (script) – ܬܪܝܢ ܒܘܚܪ̈ܢܐ ܕܩܝܡܘܬܐ ܕܠܘ̈ܚܐ 4-5: 100 ܙܘ̈ܓܐ ܕܕܘܡ̈ܝܐ ܕ-bootstrap ܥܡ ARI ܩܐ ܟܠ k، ܘ-clusterboot ܡܢ ܚܙܡܐ fpc ܥܡ ܦܘܫܪܐ ܕܬܪܝܢ ܕܪ̈ܓܐ ܐܝܟ ܐܘܪܚܐ ܕܥܢܩܘ̈ܕܐ؛ ܒܗܘ ܙܪܥܐ ܐܢܝ ܟܝܗܒܝ ܚܬܝܬܐܝܬ ܛܝ̈ܡܐ ܕܐܗܐ ܦܐܬܐ. ܣܩܪܝܦܬ ܪܗܛܐ ܝܠܗ ܐܝܟ ܕܐܝܬܘܗܝ ܥܠ ܦܐܝܠ ܕܡܬܠܐ (ܛܝ̈ܡܐ ܓܘ ܡܥܝܪ̈ܢܘܬܐ). ܢܡܘܣܐ ܕܣܐܝܛ (site rule) ܩܐ ܣܘܪܝܬ ܒ-R: ܦܐܝܠ UTF-8، ܟܠ ܣܕܪܐ ܕܡܥܝܪܢܘܬܐ ܟܫܪܐ ܒܡܠܬܐ ܕܣܘܪܝܬ، ܘܫܡ̈ܗܐ ܕܡܫܬܚܠܦ̈ܢܐ ܒܐܢܓܠܝܙܝܐ (ܦܐܝܠ ܐܢܓܠܝܙܝܐ).

# ܦܘܫܩܐ ܕܥܢܩܘ̈ܕܐ – ܦܪܘܦܣܘܪ ܣܛܐܛ. ܐܗܐ ܦܐܝܠ ܢܛܝܪܐ ܝܠܗ ܒ-UTF-8؛ ܟܠ ܣܕܪܐ ܕܡܥܝܪܢܘܬܐ ܟܫܪܐ ܒܡܠܬܐ ܕܣܘܪܝܬ
library(readxl)    # ܩܪܝܢܐ ܕܐܩܣܠ
library(cluster)   # ܡܚܘ̈ܝܢܐ silhouette ܘ-gap statistic
library(mclust)    # ܡܚܘܝܢܐ ܕ-Rand ܡܬܘܪܨܐ (adjustedRandIndex)
library(fpc)       # ܩܝܡܘܬܐ ܕܟܠ ܥܢܩܘܕܐ (clusterboot)

# ܛܥܢܐ: ܛܘܦܣܐ ܬܩܝܢܐ – ܣܕܪܐ = ܡܫܬܘܬܦܢܐ، ܥܡܘܕܐ = ܡܫܬܚܠܦܢܐ؛ ܓܠܝܘܢܐ "Data"
d <- read_excel("C:/Rdata/DATA_cluster_example_eng.xlsx", sheet = "Data")
v <- c("passive_use", "social_comparison", "content_creation", "social_connection", "daily_hours")
x <- as.data.frame(d[complete.cases(d[, v]), v])           # ܫܩܠܬܐ ܕܣܕܪܐ ܟܠܗ (148 ܡܢ 150)
z <- scale(x)                                             # ܡܬܩܢܘܬܐ: ܡܢܝ̈ܢܐ z (ܡܦܠܓ n − 1)

# ܕܪܓܐ ܐ: Ward ܥܠ ܪܘ̈ܚܩܐ ܕ-Euclid – ܐܝܠܢܐ، ܘܪܘܡܐ ܕܚܘܝ̈ܕܐ ܐܚܪ̈ܝܐ
hc <- hclust(dist(z), method = "ward.D2")
rev(tail(hc$height, 5))                                   # ܪ̈ܘܡܐ: 22.80 17.44 7.15 6.99 5.71 – ܫܘܪܬܐ ܒܬܪ ܬܠܝܬܝܐ

# ܕܪܓܐ ܒ: ܩܐ ܟܠ k – ܦܣܩܐ ܕܐܝܠܢܐ، ܡܨܥ̈ܝܐ ܕܝܘܩܪܐ ܐܝܟ ܡܨܥ̈ܝܐ ܕܫܘܪܝܐ، ܘ-k-ܡܨܥܝ̈ܬܐ (Lloyd)
two_stage <- function(z, k, hc = hclust(dist(z), "ward.D2")) {
  w <- cutree(hc, k)
  centers <- apply(z, 2, function(col) tapply(col, w, mean))
  kmeans(z, centers = centers, iter.max = 100, algorithm = "Lloyd")
}
res <- lapply(2:8, function(k) two_stage(z, k, hc))
W   <- c(sum(z^2), sapply(res, function(m) m$tot.withinss))           # ܛܝ̈ܡܐ W_k، ܥܡ k = 1
sil <- sapply(res, function(m) mean(silhouette(m$cluster, dist(z))[, "sil_width"]))
round(cbind(k = 1:8, W = W, R2 = 1 - W / W[1], silhouette = c(NA, sil)), 3)   # ܠܘܚܐ: R² .365/.575/.605؛ silhouette .337/.371/.274 (k = 2/3/4)

# ܡܚܘܝܢܐ gap statistic: ܗܝ ܦܘܢܩܣܝܐ، 100 ܕܘܡ̈ܝܐ ܕܦܘܚܡܐ ܫܘ̈ܝܐ ܓܘ ܦܬܝܘܬܐ ܕܟܠ ܡܫܬܚܠܦܢܐ؛ set.seed = ܙܪܥܐ ܕܐܗܐ ܦܐܬܐ
set.seed(2026)
gap <- clusGap(z, FUNcluster = function(x, k) two_stage(x, k), K.max = 8, B = 100,
               d.power = 2, spaceH0 = "original")
print(gap, method = "Tibs2001SEmax")                      # ܦܐܪܐ ܕ-gap: k = 3؛ gap 0.404/0.607/0.863/0.814, SE ≈ 0.035

# ܦܘܫܪܐ ܓܒܝܐ: ܪ̈ܒܘܬܐ، ܘܬܚܘܝܬܐ ܒܡܫܘ̈ܚܬܐ ܫܪ̈ܫܝܬܐ
km <- res[[2]]                                            # k = 3
km$size                                                   # ܪ̈ܒܘܬܐ: 59 51 38
round(aggregate(x, by = list(cluster = km$cluster), FUN = mean), 2)
summary(silhouette(km$cluster, dist(z)))$clus.avg.widths  # ܡܨܥܝܬܐ ܕ-silhouette ܩܐ ܥܢܩܘܕܐ: .370 .369 .373
d$cluster <- NA; d$cluster[complete.cases(d[, v])] <- km$cluster   # ܥܡܘܕܐ ܕܥܢܩܘܕܐ ܩܐ ܦܐܝܠ

# ܗܢ ܦܘܫܪܐ ܒܦܘܢܩܣܝܐ ܡܛܝܒܬܐ: hkmeans ܓܘ ܚܙܡܐ factoextra (Kassambara, 2017، ܨܚܚܐ 16) – Ward ܘܒܬܪܗ k-ܡܨܥܝ̈ܬܐ، ܐܝܢܐ ܒ-Hartigan-Wong
# ܘܕܠܐ iter.max = 100؛ ܓܘ ܦܐܝܠ ܕܡܬܠܐ ܕܡܐ ܝܠܗ ܩܐ k = 2-3، ܘܩܐ k = 4 ܘܠܥܠ W ܙܥܘܪܐ ܩܠܝܠ
# factoextra::hkmeans(z, 3)$size                          # 59 51 38

# ܩܝܡܘܬܐ ܓܘ ܓܘܒܝܐ ܕܕܘܡܝܐ ܡܢ ܪܝܫ (Leisch, 2016، ܨܚܚܐ 28) – ܐܝܟ ܠܘ̈ܚܐ 4-5 ܓܘ ܐܗܐ ܦܐܬܐ، ܒܗܘ ܙܪܥܐ
# (ܐ) ܦܘܫܪܐ ܟܠܗ (Dolnicar & Leisch, 2010، ܐܝܟ ܕܡܬܡܠܠܝ ܬܡܢ): ܬܪܝܢ ܕܘܡ̈ܝܐ ܕ-bootstrap، ܥܢܩܘ̈ܕܐ ܓܘ ܟܠ ܚܕ ܩܐ ܟܠ k، ܦܘܠܓܐ ܕܟܠ ܡܫܬܘܬܦ̈ܢܐ
# ܫܪ̈ܫܝܐ ܠܡܨܥܝܐ ܩܪܝܒܐ ܓܘ ܟܠ ܚܕ ܡܢ ܬܪܘܝܗܝ، ܘ-ARI ܒܝܢܝ ܬܪܝܢ ܦܘܠ̈ܓܐ؛ 100 ܙܘ̈ܓܐ – ܗܢ ܙܘ̈ܓܐ ܩܐ ܟܠ k
nearest <- function(z, centers) apply(z, 1, function(r) which.min(colSums((t(centers) - r)^2)))
set.seed(2026)
stab <- t(replicate(100, {
  s1 <- z[sample.int(nrow(z), replace = TRUE), ]
  s2 <- z[sample.int(nrow(z), replace = TRUE), ]
  sapply(2:8, function(k) adjustedRandIndex(nearest(z, two_stage(s1, k)$centers), nearest(z, two_stage(s2, k)$centers)))
}))
colnames(stab) <- paste0("k=", 2:8)
round(apply(stab, 2, quantile, c(.25, .5, .75)), 3)       # ܠܘܚܐ 4: ܛܝܡܐ ܡܨܥܝܐ k=2 .868 (ܪܘܒܥܐ ܬܚܬܝܐ .333)، k=3 .945 (.939)، k=4 .760، k=5 .621 ܘܗܕܟܐ ܕ-.503 ܓܘ k=8
boxplot(stab, xlab = "k", ylab = "ARI")                   # ܨܘܪܬܐ 5 (ܕܠܐ ܢܘܩ̈ܙܐ)

# (ܒ) ܟܠ ܥܢܩܘܕܐ (Hennig, 2007، ܐܝܟ ܕܡܬܡܠܠ ܬܡܢ): clusterboot ܡܢ ܚܙܡܐ fpc، ܥܡ ܦܘܫܪܐ ܕܬܪܝܢ ܕܪ̈ܓܐ ܐܝܟ ܦܘܢܩܣܝܐ ܕܥܢܩܘ̈ܕܐ
hkm <- function(data, k) {
  cl <- two_stage(as.matrix(data), k)$cluster
  list(result = cl, nc = k, clusterlist = lapply(1:k, function(i) cl == i), partition = cl, clustermethod = "Ward + k-means")
}
cb <- clusterboot(z, B = 100, bootmethod = "boot", clustermethod = hkm, k = 3, seed = 2026, count = FALSE)
round(cb$bootmean, 3)                                     # ܠܘܚܐ 5، ܡܨܥܝܬܐ ܕ-Jaccard: .981 .975 .963
cb$bootbrd; cb$bootrecover                                # ܦܪܩܠܗ (≤ .50) / ܐܬܐ ܠܐܬܪܗ (> .75)، ܡܢ 100: 0 0 0 / 100 100 100

ܬܘܢܝܐ ܕܡܫܟܚ̈ܢܘܬܐ (reporting the results)

ܓܘ ܦܘܫܩܐ ܕܥܢܩܘ̈ܕܐ (cluster analysis) ܐܝܬ ܣܓܝ ܦܣ̈ܩܐ ܕܒܨܘܝܬܐ، ܘܡܛܠ ܗܕܐ ܬܘܢܝܐ ܣܢܝܩ ܝܠܗ ܕܡܨܐ ܠܡܬܢܝܐ ܕܝܗܘܢ ܡܢ ܪܝܫ: ܐܝܢܝ ܡܫܬܚܠܦ̈ܢܐ (variables) ܥܠܠܗܘܢ ܘܩܐ ܡܘܕܝ؛ ܐܢ ܡܬܩ̈ܢܐ ܝܢܐ؛ ܪܘܚܩܐ ܘܐܘܪܚܐ (Ward، ܘܒܬܪܗ k-ܡܨܥܝ̈ܬܐ (k-means) ܡܢ ܡܨܥ̈ܝܐ ܕ-Ward)؛ ܕܟܝ ܓܒܝܐ ܝܠܗ ܡܢܝܢܐ ܕܥܢܩܘ̈ܕܐ – ܡܫܘ̈ܚܬܐ ܘܛܝ̈ܡܐ ܕܝܗܘܢ، ܐܦ ܐܡܬܝ ܕܦܠܝܓ̈ܐ ܝܢܐ؛ ܪܒܘܬܐ ܕܟܠ ܥܢܩܘܕܐ؛ ܠܘܚܐ ܕܬܚܘܝܬܐ ܥܡ ܡܨܥܝ̈ܬܐ ܘܣܛܝ̈ܘܬܐ ܬܩܝ̈ܢܬܐ؛ ܡܨܥܝܬܐ ܕ-silhouette؛ ܐܢ ܒܚܝܪܬܐ ܝܠܗ ܩܝܡܘܬܐ (stability) – ܐܘܪܚܐ، ܡܢܝܢܐ ܕܕܘܡ̈ܝܐ ܘܦܐܪܐ؛ ܘܐܢ ܥܒܝܕܐ ܝܠܗ ܫܘܪܪܐ – ܒܡܘܕܝ ܥܢܩܘ̈ܕܐ ܦܪ̈ܝܫܐ ܝܢܐ ܓܘ ܡܫܬܚܠܦ̈ܢܐ ܕܠܐ ܫܬܘܬܦܠܗܘܢ ܓܘ ܒܢܝܢܐ ܕܝܗܘܢ. APA (2020، ܦܣܩܐ 6.43) ܟܒܥܐ ܕܡܬܢܚ "ܡܘܕܥܢܘܬܐ ܒܣܦܩ ܩܐ ܕܩܪܘܝܐ ܡܨܐ ܠܡܫܪܪܐ ܕܦܘܫ̈ܩܐ". ܫܡ̈ܗܐ ܕܥܢܩܘ̈ܕܐ ("ܡܦܠܚ̈ܢܐ ܕܠܐ ܥܒܕܐ") ܦܘܫܩܐ ܝܢܐ، ܘܟܡܬܢܚ ܠܗܘܢ ܥܡ ܡܨܥܝ̈ܬܐ ܕܐܢܝ ܩܝܡ̈ܐ ܝܢܐ ܥܠܝܗܝ. ܢܡܘ̈ܣܐ ܕܐܣܟܝܡܐ (APA, 2020, Section 6.36): ܡܨܥܝ̈ܬܐ ܘܣܛܝ̈ܘܬܐ ܬܩܝ̈ܢܬܐ ܥܡ ܨܦܪܐ ܩܕܡ ܢܘܩܙܐ، ܡܨܥܝܬܐ ܕ-silhouette ܘ-η² ܕܠܐ. ܦܣܩܐ ܕܡܫܟܚ̈ܢܘܬܐ ܕܓܘ ܡܦܩܢܐ ܗܕܟܐ ܒܢܝܐ ܝܠܗ، ܒܣܘܪܝܬ ܘܒܐܢܓܠܝܙܝܐ.

ܛܥܝ̈ܘܬܐ ܕܡܬܚܙܝܢ ܣܓܝ (Common mistakes)

ܛܥܝܘܬܐ ܕܠܐ ܡܬܩܢܘܬܐ ܕܡܫܬܚܠܦ̈ܢܐ ܒܡܫܘ̈ܚܬܐ ܦܪ̈ܝܫܬܐ – ܘܩܒܠܬܐ ܕܥܢܩܘ̈ܕܐ ܕܡܫܬܚܠܦܢܐ ܕܐܝܬ ܠܗ ܦܬܝܘܬܐ (range) ܪܒܬܐ. ܘܡܬܩܢܘܬܐ ܕܠܐ ܚܘܫܒܐ – ܐܡܬܝ ܕܡܫܬܚܠܦܢܐ ܕܩܠܐ ܟܩܒܠ ܡܬܩܠܐ ܕܡܫܬܚܠܦܢܐ ܕܟܦܪܫ (Hastie et al., 2009, p. 506).

ܟܢܘܫܝܐ ܠܦܘܬ ܥܣܪ̈ܐ ܕܦܣܘ̈ܩܐ ܕ-Likert ܝܚܝܕ̈ܝܐ ܒܕܘܟܬܐ ܕܡܢܝ̈ܢܐ ܕܣܒ̈ܠܬܐ – ܪܘܚܩܐ ܥܠ ܦܣܘ̈ܩܐ (items) ܝܚܝܕ̈ܝܐ ܦܠܝܓܐ ܝܠܗ (Wendler & Gröttrup, 2021, p. 635)، ܘܦܣܘ̈ܩܐ ܣܓܝ̈ܐܐ ܕܚܕܐ ܣܒܠܬܐ ܟܝܗܒܝ ܠܗ ܡܬܩܠܐ ܥܘܝܦܐ.

ܡܥܠܬܐ ܕ"ܟܠ ܡܐ ܕܐܝܬ"، ܥܡ ܡܫܬܚܠܦ̈ܢܐ ܕܬܚܘܝܬܐ (background variables) (ܓܢܣܐ، ܥܘܡܪܐ) ܘܡܫܬܚܠܦ̈ܢܐ ܕܦܐܪܐ – ܘܒܬܪ ܗܕܐ "ܡܫܟܚܬܐ" ܕܥܢܩܘ̈ܕܐ ܦܪ̈ܝܫܐ ܝܢܐ ܓܘܝܗܘܢ. ܡܫܬܚܠܦܢܐ ܕܫܬܘܬܦܠܗ ܓܘ ܒܢܝܢܐ ܕܥܢܩܘ̈ܕܐ ܠܐ ܡܨܐ ܕܫܪܪ ܠܗܘܢ.

ܓܘܒܝܐ ܕ-k ܠܦܘܬ ܡܐ ܕܟܡܬܚܙܐ ܛܒܐ ܕܠܐ ܬܘܢܝܐ ܕܡܫܘ̈ܚܬܐ، ܝܢ ܓܘܒܝܐ ܕ-k ܥܡ silhouette ܪܡܐ ܝܬܝܪ ܐܡܬܝ ܕܐܝܢܐ .20 ܝܠܗ – ܝܥܢܝ ܐܡܬܝ ܕܠܝܬ ܒܢܝܢܐ ܓܠܝܐ.

ܒܘܚܪ̈ܢܐ F ܝܢ t ܒܝܢܝ ܥܢܩܘ̈ܕܐ ܥܠ ܡܫܬܚܠܦ̈ܢܐ ܕܥܢܩܘܕܐ، ܘܬܘܢܝܐ ܕ"ܥܢܩܘ̈ܕܐ ܦܪ̈ܝܫܐ ܝܢܐ ܪܫܝܡܐܝܬ". ܐܠܓܘܪܝܬܡ ܒܢܐ ܠܥܢܩܘ̈ܕܐ ܗܕܟܐ ܕܦܪ̈ܝܫܐ ܗܘܝ ܚܬܝܬܐܝܬ ܓܘ ܐܢܝ ܡܫܬܚܠܦ̈ܢܐ، ܘܡܛܠ ܗܕܐ ܪܫܝܡܘܬܐ (significance) ܫܪܝܪܬܐ ܝܠܗ ܡܢ ܩܕܡ ܘܠܐ ܟܡܘܣܦܐ ܡܘܕܥܢܘܬܐ. ܐܗܐ ܦܐܬܐ ܟܛܒܥ η² ܐܝܟ ܡܚܘܝܢܐ ܕܬܚܘܝܬܐ، ܕܠܐ p.

ܦܘܫܩܐ ܕ-dendrogram ܐܝܟ "ܒܢܝܢܐ ܕܝܗܝ̈ܒܬܐ". "ܐܘܪ̈ܚܬܐ ܕܕܪ̈ܓܐ ܦܪ̈ܝܫܬܐ، ܘܐܦ ܫܘ̈ܚܠܦܐ ܙܥܘܪ̈ܐ ܓܘ ܝܗܝ̈ܒܬܐ، ܡܨܝ ܕܡܛܝܝ ܠ-dendrogram ܦܪܝܫܐ ܣܓܝ" (Hastie et al., 2009, p. 521).

ܣܦܩܘܬܐ ܒܚܕ ܪܗܛܐ. ܦܘܠܓܐ ܚܕ ܡܢ ܕܘܡܝܐ ܚܕ ܡܨܐ ܕܗܘܐ "ܦܘܫܪܐ ܓܕܫܢܝܐ، ܘܠܐ ܦܘܫܪܐ ܫܪܝܪܐ" (Leisch, 2016, p. 637)؛ ܩܕܡ ܕܡܬܢܚ ܙܢ̈ܐ – ܟܒܚܪܚ ܩܝܡܘܬܐ ܓܘ ܓܘܒܝܐ ܕܕܘܡܝܐ ܡܢ ܪܝܫ (resampling؛ ܠܘ̈ܚܐ 4-5).

ܚܙܝܐ ܕܥܢܩܘܕܐ ܐܝܟ ܓܘܕܐ ܫܪܝܪܬܐ – ܙܢܐ ܕܐܝܬ ܓܘ ܟܢܫܐ ܟܠܢܝܐ (population) – ܕܠܐ ܫܘܪܪܐ: ܥܢܩܘܕܐ ܬܚܘܝܬܐ ܝܠܗ ܕܐܗܐ ܕܘܡܝܐ، ܥܡ ܐܢܝ ܡܫܬܚܠܦ̈ܢܐ ܘܐܗܐ ܪܘܚܩܐ (ܦܐܬܐ ܝܘܠܦܢܐ ܕܠܐ ܡܕܒܪܢܐ، ܦܣܩܐ 6).

ܡܒܘ̈ܥܐ

American Psychological Association. (2020). Publication manual of the American Psychological Association (7th ed.). https://doi.org/10.1037/0000165-000

Everitt, B. S., Landau, S., Leese, M., & Stahl, D. (2011). Cluster analysis (5th ed.). Wiley. https://doi.org/10.1002/9780470977811

García-Escudero, L. A., Gordaliza, A., Matrán, C., Mayo-Iscar, A., & Hennig, C. (2016). Robustness and outliers. In C. Hennig, M. Meilă, F. Murtagh, & R. Rocci (Eds.), Handbook of cluster analysis (pp. 653-677). Chapman & Hall/CRC.

Géron, A. (2022). Hands-on machine learning with Scikit-Learn, Keras, and TensorFlow: Concepts, tools, and techniques to build intelligent systems (3rd ed.). O'Reilly Media.

Hastie, T., Tibshirani, R., & Friedman, J. (2009). The elements of statistical learning: Data mining, inference, and prediction (2nd ed.). Springer. https://doi.org/10.1007/978-0-387-84858-7

Hennig, C. (2023). fpc: Flexible procedures for clustering (R package version 2.2.11) [Computer software]. https://CRAN.R-project.org/package=fpc

Kassambara, A. (2017). Practical guide to cluster analysis in R: Unsupervised machine learning (Ed. 1). STHDA.

Leisch, F. (2016). Resampling methods for exploring cluster stability. In C. Hennig, M. Meilă, F. Murtagh, & R. Rocci (Eds.), Handbook of cluster analysis (pp. 637-652). Chapman & Hall/CRC.

Rousseeuw, P. J. (1987). Silhouettes: A graphical aid to the interpretation and validation of cluster analysis. Journal of Computational and Applied Mathematics, 20, 53-65. https://doi.org/10.1016/0377-0427(87)90125-7

Wendler, T., & Gröttrup, S. (2021). Data mining with SPSS Modeler: Theory, exercises and solutions (2nd ed.). Springer. https://doi.org/10.1007/978-3-030-54338-9

Ward (1963)، Kaufman ܘ-Rousseeuw (1990)، Marriott (1982)، Milligan ܘ-Cooper (1988) – ܐܝܟ ܕܡܬܡܠܠܝ ܓܘ Everitt ܘܐܚܪ̈ܢܐ (2011)؛ IBM (2015) ܘ-Mardia ܘܐܚܪ̈ܢܐ (1979) – ܐܝܟ ܕܡܬܡܠܠܝ ܓܘ Wendler ܘ-Gröttrup (2021)؛ Tibshirani ܘܐܚܪ̈ܢܐ (2001) – ܐܝܟ ܕܡܬܡܠܠܝ ܓܘ Hastie ܘܐܚܪ̈ܢܐ (2009) ܘܓܘ Everitt ܘܐܚܪ̈ܢܐ (2011)؛ Dolnicar ܘ-Leisch (2010)، Wedel ܘ-Kamakura (1998)، Tibshirani ܘ-Walther (2005) ܘ-Hennig (2007) – ܐܝܟ ܕܡܬܡܠܠܝ ܓܘ Leisch (2016)؛ Hennig (2008) – ܐܝܟ ܕܡܬܡܠܠ ܓܘ García-Escudero ܘܐܚܪ̈ܢܐ (2016). Hennig (2023) – ܟܬܒܐ ܕ-clusterboot (ܦܐܬܐ ܕܥܘܕܪܢܐ ܓܘ ܚܙܡܐ)، ܕܩܪܝܐ ܝܠܗ ܓܘ ܢܘܣܚܐ ܫܪܫܝܐ. ܠܦܘܬ ܢܡܘܣܐ ܕܣܐܝܛ (site rule)، ܓܘ ܪܫܝܡܬܐ ܕܡܒܘ̈ܥܐ ܟܡܬܚܙܝ ܒܣ ܡܒܘ̈ܥܐ ܕܩܪܝ̈ܐ ܝܢܐ ܓܘ ܢܘܣܚܐ ܫܪܫܝܐ. Géron (2022) ܓܘ ܢܘܣܚܐ ܕܝܓܝܛܠܝܐ ܕܠܐ ܡܢܝ̈ܢܐ ܕܦܐ̈ܬܐ – ܡܬܠܠܬܐ ܠܦܘܬ ܨܚܚܐ ܘܦܣܩܐ؛ Everitt ܘܐܚܪ̈ܢܐ (2011)، Hastie ܘܐܚܪ̈ܢܐ (2009)، Wendler ܘ-Gröttrup (2021)، Rousseeuw (1987)، Leisch (2016)، García-Escudero ܘܐܚܪ̈ܢܐ (2016) ܘ-Kassambara (2017) – ܠܦܘܬ ܦܐܬܐ ܛܒܝܥܬܐ. ܚܘܫܒܢܐ ܓܘ ܐܗܐ ܦܐܬܐ ܒܚܝܪܐ ܝܠܗ ܩܘܒܠ R 4.3.3 (hclust، cutree، kmeans) ܘܚܙܡ̈ܐ cluster (silhouette، clusGap) ܘ-mclust (adjustedRandIndex)، ܘܩܝܡܘܬܐ – ܩܘܒܠ sample.int ܕ-R ܘ-fpc (clusterboot).