ܝܘܠܦܢܐ ܕܠܐ ܡܕܒܪܢܐ: ܥܢܩܘ̈ܕܐ، ܡܢܘ̈ܬܐ ܘܥܠ̈ܠܬܐ

ܡܕܒܪܢܐ (guide)

ܡܐ ܐܝܬ ܓܘ ܐܗܐ ܦܐܬܐ (On this page)

ܬܪܝܢ ܦܐܬ̈ܐ ܐܚ̈ܐ – ܝܘܠܦܢܐ ܕܡܟܝܢܐ ܩܐ ܒܨܘܝܐ ܕܟܢܫܐ ܘܡܚܘ̈ܝܢܐ ܕܥܒܕܐ ܘܫܘܪܪܐ – ܥܣܝ̈ܩܐ ܝܢܐ ܒܝܘܠܦܢܐ ܡܕܒܪܢܐ: ܐܝܬ ܡܫܬܚܠܦܢܐ ܬܠܝܐ (dependent variable)، ܡܘܕܝܠ (model) ܟܝܠܦ ܠܡܢܒܝܐ ܠܗ، ܘܟܫܪܪܚ ܠܗ ܥܠ ܝܗܝ̈ܒܬܐ ܕܠܐ ܚܙܐ. ܐܗܐ ܦܐܬܐ ܥܣܝܩܐ ܝܠܗ ܒܡܐ ܕܟܗܘܐ ܐܡܬܝ ܕܠܝܬ ܡܫܬܚܠܦܢܐ ܬܠܝܐ – ܘܒܨܘܝܐ ܕܟܢܫܐ ܟܥܒܕ ܐܗܐ ܡܢ ܫܢ̈ܝܐ ܕܠܐ ܩܪܝܬܐ ܠܗ ܗܟܢܐ: ܦܘܫܩܐ ܕܥܠ̈ܠܬܐ (factor analysis) ܥܠ ܦܣܘ̈ܩܐ (items) ܕܦܢܩܝܬܐ ܕܫܘܐ̈ܠܐ، ܦܘܫܩܐ ܕܥܢܩܘ̈ܕܐ (cluster analysis) ܕܟܦܠܓ ܡܫܬܘܬܦ̈ܢܐ (participants) ܠܛܘ̈ܦܣܐ، ܦܘܫܩܐ ܕܡܢܘ̈ܬܐ ܪ̈ܫܝܬܐ (principal components) ܕܟܒܨܪ ܥܣܪܝܢ ܡܫܬܚܠܦ̈ܢܐ ܠܬܠܬܐ. ܓܘ ܟܬܒ̈ܐ ܕܝܘܠܦܢܐ ܕܡܟܝܢܐ ܟܠ ܐܢܝ̈ ܟܡܬܩܪܝܢ ܝܘܠܦܢܐ ܕܠܐ ܡܕܒܪܢܐ (unsupervised learning؛ ܓܘ ܥܒܪܝܐ ܐܦ "ܝܘܠܦܢܐ ܕܠܐ ܒܚܝܪܐ")، ܘܐܝܬ ܠܦܐܬܐ ܬܪܝܢ ܢܝܫ̈ܐ: ܠܡܬܪܓܡ – ܠܡܚܘܝܐ ܕܥܢܩܘܕܐ ܕ-Modeler ܝܠܗ ܗܘ CLUSTER ܕ-SPSS ܝܠܗ ܗܘ kmeans() ܕ-R – ܘܠܡܙܗܪ، ܡܛܠ ܕܟܐ ܟܓܐܒܐ ܗܝ ܡܕܥܡ ܝܚܝܕܝܐ ܕܟܢܛܪ ܠܒܨܘܝܐ ܓܘ ܡܘܕܝܠ ܕܡܢܒܝܢܘܬܐ: ܡܚܘܝܢܐ (measure). ܠܝܬ ܓܘܕܐ ܕܒܘܚܪܢܐ (test set) ܕܐܡܪܐ ܐܢ ܬܠܬܐ ܥܠ̈ܠܬܐ "ܬܪ̈ܝܨܐ" ܝܢܐ ܝܢ ܐܢ ܐܪܒܥܐ ܥܢܩܘ̈ܕܐ ܐܝܬܝܗܘܢ. ܛܟܣܐ: ܡܐ ܟܥܒܕ ܝܘܠܦܢܐ "ܕܠܐ ܡܕܒܪܢܐ"، ܘܩܐ ܡܘܢܐ ܐܗܐ ܚܠܬܐ ܠܐ ܢܝܚܬܐ ܝܠܗ (1)؛ ܒܨܪܐ ܕܡܫܘܚ̈ܬܐ – ܦܘܫܩܐ ܕܡܢܘ̈ܬܐ ܪ̈ܫܝܬܐ (2)؛ ܦܘܪܫܐ ܕ-software ܟܡܛܫܐ – ܡܢܘ̈ܬܐ ܠܐ ܝܢܐ ܥܠ̈ܠܬܐ (3)؛ ܥܢܩܘ̈ܕܐ – ܐܘܪ̈ܚܬܐ ܘܡܐ ܕܟܠ ܚܕܐ ܟܣܒܪܐ (4)؛ ܟܡܐ ܥܢܩܘ̈ܕܐ، ܘܐܢ ܒܟܠܗ (5)؛ ܡܐ ܟܥܒܕܚ ܥܡ ܥܢܩܘܕܐ ܒܬܪ ܕܡܨܚܚ ܠܗ (6)؛ ܐܝܟܢܐ ܟܠ ܡܕܡ ܟܡܬܩܪܐ ܓܘ ܪ̈ܫܝܡܬܐ ܕ-SPSS ܘܕ-Modeler (7)؛ ܘܐܝܟܢܐ ܟܥܒܕܚ ܐܗܐ ܓܘ R (8). ܡܒܘ̈ܥܐ ܝܢܐ ܚܡܫܐ ܟܬܒ̈ܐ ܕܝܘܠܦܢܐ ܕܬܪܝܢ ܦܐܬ̈ܐ ܐܚ̈ܐ – Géron (2022، ܨܚܚ̈ܐ 8-9)، Hastie، Tibshirani ܘ-Friedman (2009، ܨܚܚܐ 14)، Bishop (2006، ܨܚܚ̈ܐ 9 ܘ-12)، Truong (2026، ܨܚܚ̈ܐ 12-13) ܘ-Gutierrez (2015، ܨܚܚܐ 8) – ܘܥܡܗܘܢ، ܡܢ ܓܢܒܐ ܦܣܝܟܘܡܝܛܪܝܐ، ܒܨܝܪܐ ܕܥܝܕܐ ܕܦܘܫܩܐ ܕܥܠ̈ܠܬܐ ܓܘ ܦܣܝܟܘܠܘܓܝܐ (Costello & Osborne, 2005)، ܘܡܐܡܪ̈ܐ ܫܪ̈ܫܝܐ ܕܬܠܬܐ ܢܡܘ̈ܣܐ ܕܦܐܬܐ ܟܡܦܩܕܐ ܥܠܝܗܝ: ܦܘܫܩܐ ܒܓܢܒ ܚܕ̈ܕܐ (parallel analysis) (Horn, 1965)، silhouette (Rousseeuw, 1987) ܘ-gap statistic (Tibshirani, Walther & Hastie, 2001)؛ ܘܡܢ ܓܢܒܐ ܕ-SPSS Statistics – ܨܚܚ̈ܐ ܕܦܘܫܩܐ ܕܥܠ̈ܠܬܐ ܓܘ ܬܪܝܢ ܟܬܒ̈ܐ ܕܣܐܝܛ ܟܡܝܬܐ ܓܘ ܦܐܬ̈ܐ ܕܡܚܫܒܢ̈ܐ: Field (2024، ܨܚܚܐ 18) ܘ-Pallant (2020، ܨܚܚܐ 15)؛ ܘܡܢ ܓܢܒܐ ܕ-SPSS Modeler – ܬܚܘܝܬܐ ܕ-IBM ܘܟܬܒܐ ܕܝܘܠܦܢܐ ܐܟܕܡܝܐ ܕ-Wendler ܘ-Gröttrup (2021، ܨܚܚ̈ܐ 6-7). ܡܥܝܪܢܘܬܐ ܥܠ ܡܝܬܝܬܐ: ܐܝܟ ܓܘ ܦܐܬ̈ܐ ܐܚ̈ܐ – Géron، Gutierrez، Field ܘ-Pallant ܠܦܘܬ ܨܚܚܐ ܘܦܣܩܐ (ܢܘܣ̈ܚܐ ܕܠܐ ܦܐܬ̈ܐ)، ܫܪܟܐ ܠܦܘܬ ܦܐܬܐ.

1. ܕܠܐ ܡܠܦܢܐ: ܡܐ ܝܠܗ، ܘܩܐ ܡܘܢܐ ܚܠܬܐ ܠܐ ܢܝܚܬܐ ܝܠܗ

Hastie ܘܚܒܪ̈ܘܗܝ ܟܡܫܪܝ ܨܚܚܐ 14 ܒܦܘܪܫܐ: ܝܘܠܦܢܐ ܡܕܒܪܢܐ (supervised learning) ܝܠܗ "ܝܘܠܦܢܐ ܥܡ ܡܠܦܢܐ" – ܠ"ܬܠܡܝܕܐ" ܐܝܬ ܦܘܢܝܐ ܩܐ ܟܠ ܚܙܝܬܐ (observation) ܘܡܠܦܢܐ ܟܡܬܪܨ؛ ܝܘܠܦܢܐ ܕܠܐ ܡܕܒܪܢܐ (unsupervised learning) ܝܠܗ "ܝܘܠܦܢܐ ܕܠܐ ܡܠܦܢܐ" – ܐܝܬ N ܚܙ̈ܝܬܐ ܕܘܩܛܘܪܐ X، ܘ"ܢܝܫܐ ܝܠܗ ܠܡܦܠܛ ܬܪܝܨܐܝܬ ܬܚܘܝ̈ܬܐ ܕܐܗܐ ܦܘܠܓܐ، ܕܠܐ ܥܘܕܪܢܐ ܕܡܕܒܪܢܐ ܝܢ ܡܠܦܢܐ ܕܝܗܒ ܦܘܢܝ̈ܐ ܬܪ̈ܝܨܐ ܝܢ ܡܫܘܚܬܐ ܕܛܥܝܘܬܐ ܩܐ ܟܠ ܚܙܝܬܐ" (Hastie et al., 2009, p. 486). ܘܒܬܪ ܗܕܐ ܦܬܓܡܐ ܕܡܢܗ ܟܦܐܫ ܟܠ ܫܪܟܐ ܕܦܐܬܐ: "ܓܘ ܝܘܠܦܢܐ ܡܕܒܪܢܐ ܐܝܬ ܡܫܘܚܬܐ ܓܠܝܬܐ ܕܡܨܠܚܢܘܬܐ، ܝܢ ܕܚܣܝܪܘܬܐ ܕܝܠܗ̇… ܓܘ ܣܝܩܐ ܕܝܘܠܦܢܐ ܕܠܐ ܡܕܒܪܢܐ ܠܝܬ ܡܫܘܚܬܐ ܬܪܝܨܬܐ ܕܡܨܠܚܢܘܬܐ ܐܝܟ ܗܕܐ. ܥܣܩܐ ܝܠܗ ܠܡܫܪܪܐ ܬܪܝܨܘܬܐ (validity) ܕܦܠܛ̈ܐ ܕܟܦܝܫܝ ܡܢ ܡܦܩܢܐ (output) ܕܣܓܝܐܐ ܡܢ ܐܠܓܘܪܝܬܡ̈ܐ. ܐܠܨܐ ܝܠܗ ܠܡܣܢܕܐ ܥܠ ܡܠ̈ܐ ܕܦܘܫܩܐ ܕܡܨܝܕܢܘܬܐ (heuristic) ܠܐ ܒܣ ܩܐ ܡܙܕܩܐ ܐܠܓܘܪܝܬܡ̈ܐ… ܐܠܐ ܐܦ ܩܐ ܕܝܢܐ ܕܛܝܒܘܬܐ ܕܦܐܪ̈ܐ. ܐܗܐ ܚܠܬܐ ܠܐ ܢܝܚܬܐ ܡܝܬܝܠܗ̇ ܠܫܘܦܥܐ ܪܒܐ ܕܐܘܪ̈ܚܬܐ ܡܩܪ̈ܒܬܐ، ܡܛܠ ܕܡܥܕܪܢܘܬܐ ܡܠܬܐ ܕܪܥܝܢܐ ܝܠܗ̇ ܘܠܐ ܡܨܝܚ ܠܡܫܪܪܐ ܠܗ̇ ܬܪܝܨܐܝܬ" (ܦܐܬܐ 487). Gutierrez ܟܐܡܪ ܐܗܐ ܒܦܫܝܛܘܬܐ: "ܥܡ ܝܘܠܦܢܐ ܕܠܐ ܡܕܒܪܢܐ ܠܝܬ ܐܘܪܚܐ ܠܡܫܪܪܐ ܥܒܕܐ ܕܝܘܟܘܢ، ܡܛܠ ܕܠܝܬ ܦܘܢܝܐ ܬܪܝܨܐ. ܐܗܐ ܝܠܗ ܥܠܬܐ ܕܟܡܬܩܪܐ ܝܘܠܦܢܐ ܕܠܐ ܡܕܒܪܢܐ" (Gutierrez, 2015، ܨܚܚܐ 8، ܦܬܚܐ). Géron ܟܡܝܬܐ ܠ-Yann LeCun: ܐܢ ܗܘܢܐ ܝܠܗ ܟܥܟܐ، ܝܘܠܦܢܐ ܕܠܐ ܡܕܒܪܢܐ ܝܠܗ ܟܥܟܐ، ܡܕܒܪܢܐ – ܚܦܝܐ، ܘܝܘܠܦܢܐ ܕܚܝܠܢܘܬܐ (reinforcement) – ܓܦܬܐ؛ "ܣܓܝܐܘܬܐ ܪܒܬܐ ܕܝܗܝ̈ܒܬܐ ܕܐܝܬ ܠܐ ܝܢܐ ܪ̈ܫܝܡܐ ܒܪܘܫܡܐ" (Géron, 2022، ܨܚܚܐ 9، ܦܬܚܐ).

ܡܐ ܟܒܨܚ ܐܡܬܝ ܕܠܝܬ y. Hastie ܘܚܒܪ̈ܘܗܝ ܟܦܪܫܝ ܠܦܘܬ ܡܐ ܕܐܘܪܚܐ ܟܒܨܝܐ ܓܘ ܦܘܠܓܐ ܕ-X: ܡܢܘ̈ܬܐ ܪ̈ܫܝܬܐ، ܡܫܘܚܬܐ ܕܣܓܝ ܡܫܘܚ̈ܬܐ (multidimensional scaling) ܘܕܡܝ̈ܐ ܕܝܠܗܝ "ܟܡܢܣܝ ܠܡܦܪܫܐ ܐܦ̈ܐ ܕܡܫܘܚ̈ܬܐ ܒܨܝܪ̈ܐ ܓܘ ܦܬܝܘܬܐ ܕ-X ܕܟܡܚܘܝܐ ܐܬܪ̈ܘܬܐ ܕܝܗܝ̈ܒܬܐ ܥܒܝ̈ܐ"، ܘܡܛܠ ܗܕܐ ܟܐܡܪܝ "ܐܢ ܡܫܬܚܠܦ̈ܢܐ ܡܨܝܐ ܝܢܐ ܠܡܬܚܫܒܐ ܦܘܢܩܣܝ̈ܘܢܐ (functions) ܕܓܘܕܐ ܙܥܘܪܬܐ ܒܢܘܬ ܕܡܫܬܚܠܦ̈ܢܐ 'ܟܣ̈ܝܐ'"؛ ܦܘܫܩܐ ܕܥܢܩܘ̈ܕܐ "ܟܡܢܣܐ ܠܡܨܚܚ ܐܬܪ̈ܘܬܐ ܣܓܝ̈ܐܐ ܓܘܒ̈ܝܐ (convex) ܓܘ ܦܬܝܘܬܐ ܕ-X ܕܐܝܬ ܓܘܝܗܝ ܡܘܕ̈ܐ (modes)" ܕܦܘܠܓܐ، ܘܡܛܠ ܗܕܐ ܟܐܡܪ ܐܢ ܡܨܝܚ ܠܡܚܘܝܐ ܠܗ "ܐܝܟ ܚܘܠܛܐ ܕܐܬܪ̈ܘܬܐ ܦܫܝ̈ܛܐ ܒܢܘܬ ܕܟܡܚܘܝܐ ܛܘ̈ܦܣܐ ܝܢ ܓܢ̈ܣܐ ܦܪ̈ܝܫܐ ܕܚܙ̈ܝܬܐ" (ܦܐܬܐ 486). Truong ܟܟܬܒ ܗܝ ܦܘܠܓܘܬܐ ܓܘ ܠܘܚܐ (table): "ܡܨܥܬܐ ܕܦܘܫܩܐ" ܕܦܘܫܩܐ ܕܡܢܘ̈ܬܐ ܝܠܗ ܡܫܬܚܠܦ̈ܢܐ، ܕܦܘܫܩܐ ܕܥܢܩܘ̈ܕܐ – ܚܙ̈ܝܬܐ؛ ܩܕܡܝܐ ܟܡܦܩ ܡܢܘ̈ܬܐ ܕܟܡܟܢܫܝ، ܬܪܝܢܐ ܟܡܦܩ ܓܘ̈ܕܐ (Truong, 2026, Table 13.1, p. 459). ܐܗܐ ܝܠܗ ܦܘܪܫܐ ܕܒܨܘܝܐ ܕܟܢܫܐ ܟܝܕܥ ܡܢ ܡܐܛܪܝܟܣ ܕܝܠܗ: ܠܡܒܨܪ ܥܡܘ̈ܕܐ (ܦܣ̈ܩܐ 2-3) ܝܢ ܠܡܟܢܫ ܣܕܪ̈ܐ (ܦܣ̈ܩܐ 4-6). Géron ܟܡܘܣܦ ܬܪܬܝܢ ܥܒܕ̈ܐ ܚܬ̈ܐ – ܦܘܪܫܐ ܕܛܝ̈ܡܐ ܢܘܟܪ̈ܝܐ (outliers)، "ܠܡܝܠܦ ܐܝܟܢܐ ܟܡܬܚܙܝ ܝܗܝ̈ܒܬܐ 'ܢܘܪܡܠܝ̈ܐ' ܘܒܬܪ ܗܕܐ ܠܡܦܠܚ ܒܗܕܐ ܩܐ ܡܦܪܫܐ ܚܙ̈ܝܬܐ ܢܘܟܪ̈ܝܬܐ"، ܘܫܘܥܪܐ ܕܐܬܪܘܬܐ (density estimation) (ܨܚܚܐ 9، ܦܬܚܐ) – ܘܬܪܘܝܗܝ ܟܕܥܪܝ ܓܘ ܦܣܩܐ 4 ܐܝܟ ܦܐܪܐ ܕܓܢܒܐ ܕܗܢܝ̈ ܐܠܓܘܪܝܬܡ̈ܐ.

2. ܒܨܪܐ ܕܡܫܘܚ̈ܬܐ: ܦܘܫܩܐ ܕܡܢܘ̈ܬܐ ܪ̈ܫܝܬܐ

ܥܘܩܬܐ ܕܐܘܪܚܐ ܟܫܪܝܐ ܝܠܗ̇ ܠܘܛܬܐ ܕܡܫܘܚ̈ܬܐ (curse of dimensionality) (ܦܐܬܐ ܕܝܘܠܦܢܐ ܕܡܟܝܢܐ، ܦܣܩܐ 2): Géron ܟܡܥܗܕ ܕܬܪܬܝܢ ܢܘܩ̈ܙܐ ܓܕܫܢ̈ܝܐ ܓܘ ܡܪܒܥܐ ܕܚܕܝܘܬܐ ܪ̈ܚܝܩܐ ܝܢܐ ܚܕܐ ܡܢ ܚܕܐ ܒܡܨܥܝܬܐ 0.52، ܘܓܘ ܟܥܒܐ ܕܡܠܝܘܢ ܡܫܘܚ̈ܬܐ – ܡܩܪܒ 408؛ "ܒܣ ܐܝܬ ܣܓܝ ܕܘܟܐ ܓܘ ܡܫܘܚ̈ܬܐ ܪ̈ܡܐ"، ܘܡܛܠ ܗܕܐ "ܣܕܪ̈ܐ ܕܝܗܝ̈ܒܬܐ ܕܣܓܝ ܡܫܘܚ̈ܬܐ ܓܘ ܣܘܟܢܐ ܝܢܐ ܕܗܘܝܐ ܣܓܝ ܩܠܝ̈ܠܐ" (Géron, 2022, Chapter 8, "The Curse of Dimensionality"). ܦܘܫܩܐ ܕܡܢܘ̈ܬܐ ܪ̈ܫܝܬܐ (PCA) ܟܡܛܠܠ ܝܗܝ̈ܒܬܐ ܥܠ ܦܬܝܘܬܐ ܙܥܘܪܬܐ ܒܢܘܬ. ܠ-Bishop ܐܝܬ ܬܪܬܝܢ ܬܚܘܡܝ̈ܬܐ "ܕܟܝܗܒܝ ܗܘ ܐܠܓܘܪܝܬܡ": ܛܠܠܐ (projection) ܐܘܪܬܘܓܘܢܠܝܐ (orthogonal) "ܗܟܢܐ ܕܫܘܚܠܦܐ ܕܝܗܝ̈ܒܬܐ ܡܛܠ̈ܠܬܐ ܗܘܐ ܥܠܝܐ ܡܢ ܟܠܗ" (Hotelling, 1933)، ܝܢ ܫܘܝܐ ܠܗܕܐ – ܛܠܠܐ ܣܪܛܢܝܐ "ܕܟܒܨܪ ܛܝܡܐ ܡܨܥܝܐ ܕܛܠܠܐ، ܬܚܝܡܐ ܐܝܟ ܡܪܒܥܐ ܕܪܘܚܩܐ ܡܨܥܝܐ ܒܝܢܝ ܢܘܩ̈ܙܐ ܕܝܗܝ̈ܒܬܐ ܘܛܠ̈ܠܐ ܕܝܠܗܝ" (Pearson, 1901) (Bishop, 2006, p. 561)؛ ܒܢܘܣܚܐ ܕ-Hastie ܘܚܒܪ̈ܘܗܝ، ܡܢܘ̈ܬܐ ܪ̈ܫܝܬܐ "ܟܝܗܒܝ ܣܕܪܐ ܕܩܘܪ̈ܒܐ ܣܪ̈ܛܢܝܐ ܛܒ̈ܐ ܡܢ ܟܠ ܠܝܗܝ̈ܒܬܐ، ܡܢ ܟܠ ܕܪ̈ܓܐ" (Hastie et al., 2009, p. 534). Géron ܟܡܦܫܩ ܐܗܐ ܒܨܘܪܬܐ: ܡܢ ܒܝܢܝ ܣܪ̈ܛܐ (axes) ܕܡܨܝܐ ܝܢܐ، PCA ܟܓܒܐ "ܣܪܛܐ ܕܟܢܛܪ ܟܡܝܘܬܐ ܥܠܝܬܐ ܕܫܘܚܠܦܐ"، ܘܒܬܪ ܗܕܐ ܣܪܛܐ ܬܪܝܢܐ، ܙܩܝܦܐ ܥܠ ܩܕܡܝܐ، ܕܟܢܛܪ ܒܢܘܬ ܡܢ ܟܠ ܡܢ ܫܘܚܠܦܐ ܕܦܝܫܐ، ܘܗܟܢܐ ܠܩܕܡ؛ "ܣܪܛܐ ܕ-i ܟܡܬܩܪܐ ܡܢܬܐ ܪܫܝܬܐ ܕ-i" (Chapter 8, "Preserving the Variance", "Principal Components"). ܡܐ ܕܐܝܬ ܒܐܝܕܐ ܒܬܪ ܛܠܠܐ ܝܠܗ ܠܘܚܐ ܕܡܢܘ̈ܬܐ ܕܫܘܚܠܦܐ ܡܦܫܩܐ (explained variance ratio): ܓܘ ܡܬܠܐ ܕ-Géron 76% ܥܠ ܡܢܬܐ ܩܕܡܝܬܐ، 15% ܥܠ ܬܪܝܢܝܬܐ، 9% ܥܠ ܬܠܝܬܝܬܐ – "ܘܡܛܠ ܗܕܐ ܡܣܬܒܪܐ ܝܠܗ ܕܡܢܬܐ ܬܠܝܬܝܬܐ ܟܛܥܢܐ ܩܠܝܠ ܡܘܕܥܢܘܬܐ" (Chapter 8, "Explained Variance Ratio").

ܟܡܐ ܡܢܘ̈ܬܐ. ܬܠܝܐ ܝܠܗ ܩܐ ܡܘܢܐ: "ܐܢ ܟܒܨܪܝܬܘܢ ܡܫܘܚ̈ܬܐ ܩܐ ܚܙܝܐ، ܒܕ ܒܥܝܬܘܢ ܠܡܒܨܪ ܠ-2 ܝܢ 3" – ܐܢ ܠܐ، Géron ܟܡܩܪܒ ܠܡܓܒܐ ܡܢܝܢܐ ܕܡܫܘܚ̈ܬܐ "ܕܟܡܛܝܝ ܠܡܢܬܐ ܪܒܬܐ ܕܫܘܚܠܦܐ – ܐܡܘܪ 95%"، ܝܢ ܠܡܨܪ ܫܘܚܠܦܐ ܟܢܝܫܐ ܐܝܟ ܦܘܢܩܣܝܘܢ ܕܡܢܝܢܐ ܕܡܫܘܚ̈ܬܐ ܘܠܡܒܨܐ ܥܠ "elbow، ܕܒܗ ܫܘܚܠܦܐ ܡܦܫܩܐ ܟܦܣܩ ܠܡܪܒܐ ܒܥܓܠ"؛ ܘܐܢ PCA ܝܠܗ ܕܪܓܐ ܕܛܝܒܘܬܐ ܩܐ ܡܘܕܝܠ ܡܕܒܪܢܐ، "ܬܩܢܘܢ ܡܢܝܢܐ ܕܡܫܘܚ̈ܬܐ ܐܝܟ ܟܠ hyperparameter ܐܚܪܢܐ" – ܒܫܘܪܪܐ ܡܚܠܦܢܐ (cross-validation)، ܠܦܘܬ ܥܒܕܐ ܕܡܘܕܝܠ ܕܒܬܪܗ (Chapter 8, "Choosing the Right Number of Dimensions"). ܘܙܘܗܪܐ ܕܩܕܡ ܟܠ ܡܕܡ: ܒܨܪܐ ܕܡܫܘܚ̈ܬܐ "ܫܪܝܪܐܝܬ ܟܥܒܕ ܚܣܪܢܐ ܕܡܘܕܥܢܘܬܐ… ܘܡܛܠ ܗܕܐ ܟܡܦܩܕܢ ܕܩܕܡܝܐ ܢܣܝܬܘܢ ܠܡܠܦܐ ܠ-system ܥܠ ܝܗܝ̈ܒܬܐ ܫܪ̈ܫܝܐ ܩܕܡ ܕܚܫܒܬܘܢ ܥܠ ܒܨܪܐ ܕܡܫܘܚ̈ܬܐ" (ܨܚܚܐ 8، ܦܬܚܐ). ܬܪܝܢ ܦܘܪ̈ܫܐ ܛܟܢܝܩ̈ܝܐ ܕ-software ܠܐ ܒܕ ܥܒܕ ܩܐ ܕܝܘܟܘܢ: ܡܬܩܢܘܬܐ (standardization) – PCA ܥܠ ܡܫܬܚܠܦ̈ܢܐ ܓܘ ܚܕܝ̈ܘܬܐ ܦܪ̈ܝܫܐ ܝܠܗ PCA ܥܠ ܡܫܬܚܠܦܢܐ ܕܐܝܬ ܠܗ ܫܘܚܠܦܐ ܪܒܐ ܡܢ ܟܠ؛ Gutierrez ܟܡܚܘܐ ܥܠ ܝܗܝ̈ܒܬܐ ܕܥܘܠ̈ܐ ܕܐܬܪ̈ܘܬܐ ܡܚܝܕ̈ܐ ܕܐܡܪܝܟܐ، ܕܓܘܝܗܝ ܫܘܚܠܦܐ ܕܡܚ̈ܘܬܐ ܝܠܗ 6,945 ܘܫܘܚܠܦܐ ܕܩܛܠ̈ܐ 19، ܘܟܦܩܕ: "ܐܠܨܐ ܝܠܗ، ܥܡ ܗܕܐ، ܕܟܬܒܝܬܘܢ scale=TRUE" (Gutierrez, 2015, Chapter 8, "Principal Component Analysis")؛ ܘܐܘܪܚܐ – ܪܘܫܡܐ ܕܛܥ̈ܢܐ (loadings) ܕܡܢܬܐ ܡܢ ܨܒܝܢܐ ܝܠܗ، ܘܡܢܬܐ "ܕܒܨܪܐ" ܠܐ ܝܠܗ ܡܫܟܚܢܘܬܐ. ܘܡܐ ܕܠܐ ܝܠܗ ܕܟܐ: t-SNE، ܕ-Géron ܟܪܫܡ ܕ"ܟܡܬܦܠܚ ܒܪܫܐ ܩܐ ܚܙܝܐ، ܝܚܝܕܐܝܬ ܩܐ ܡܚܘܝܐ ܥܢܩܘ̈ܕܐ ܕܚܙ̈ܝܬܐ ܓܘ ܦܬܝܘܬܐ ܕܣܓܝ ܡܫܘܚ̈ܬܐ"، ܘ-UMAP ܐܚܘܢܗ – ܡܐܢ̈ܐ ܕܨܘܪܬܐ، ܕܪ̈ܘܚܩܐ ܓܘ ܨܘܪܬܐ ܕܝܠܗܘܢ ܠܐ ܝܢܐ ܪ̈ܘܚܩܐ (Chapter 8, "Other Dimensionality Reduction Techniques"). ܘܠܘܚܐ ܕܛܥ̈ܢܐ ܕܡܢܬܐ ܩܕܡܝܬܐ، ܕܟܡܬܚܙܐ ܐܝܟ ܠܘܚܐ ܕܦܘܫܩܐ ܕܥܠ̈ܠܬܐ – ܦܣܩܐ 3.

3. PCA ܠܐ ܝܠܗ ܦܘܫܩܐ ܕܥܠ̈ܠܬܐ – ܘܩܐ ܡܘܢܐ SPSS ܟܥܒܕ ܕܚܫܒܚ ܕܐܝܢ

ܩܐ ܒܨܘܝܐ ܕܟܢܫܐ ܐܗܐ ܝܠܗ ܦܣܩܐ ܚܫܝܚܐ ܓܘ ܦܐܬܐ. Hastie ܘܚܒܪ̈ܘܗܝ ܟܡܚܘܝܝ ܕܡܨܝܚ ܠܡܩܪܐ PCA ܐܝܟ ܡܘܕܝܠ ܕܡܫܬܚܠܦ̈ܢܐ ܟܣ̈ܝܐ (latent variables) – ܟܠ ܡܫܬܚܠܦܢܐ ܚܙܝܐ ܝܠܗ ܚܘܠܛܐ ܣܪܛܢܝܐ ܕ-p ܡܫܬܚܠܦ̈ܢܐ ܕܠܐ ܐܣܘܪܐ ܕܚܕ̈ܕܐ – "ܐܠܐ ܐܗܐ ܠܐ ܝܠܗ ܣܓܝ ܡܣܦܩܢܐ"، ܡܛܠ ܕܟܠ ܚܘܕܪܐ (rotation) ܐܘܪܬܘܓܘܢܠܝܐ ܟܝܗܒ ܦܘܪܫܢܐ ܐܚܪܢܐ، "ܘܡܛܠ ܗܕܐ ܠܐ ܡܨܝܚ ܠܡܦܪܫܐ ܡܫܬܚܠܦ̈ܢܐ ܟܣ̈ܝܐ ܦܪ̈ܝܫܐ ܐܝܟ ܡܒܘ̈ܥܐ ܫܪ̈ܫܝܐ ܝܚܝܕ̈ܝܐ". "ܡܘܕܝܠ ܩܠܐܣܝܩܝܐ ܕܦܘܫܩܐ ܕܥܠ̈ܠܬܐ، ܕܒܢܝܐ ܝܠܗ ܒܪܫܐ ܒܐܝܕܐ ܕܒܨܘ̈ܝܐ ܓܘ ܦܣܝܟܘܡܝܛܪܝܐ، ܟܡܩܠܠ ܗܠܝܢ ܥܘܩ̈ܬܐ ܡܢܬܐ ܡܢܬܐ": q < p ܥܠ̈ܠܬܐ ܟܣ̈ܝܬܐ، ܡܐܛܪܝܟܣ ܕܛܥ̈ܢܐ، ܘ"ܫܓܘ̈ܫܝܐ" ε ܕܠܐ ܐܣܘܪܐ ܕܚܕ̈ܕܐ – "ܪܥܝܢܐ ܝܠܗ ܕܡܫܬܚܠܦ̈ܢܐ ܟܣ̈ܝܐ ܝܢܐ ܡܒܘ̈ܥܐ ܡܫܘܬ̈ܦܐ ܕܫܘܚܠܦܐ ܒܝܢܝ ܡܫܬܚܠܦ̈ܢܐ ܚܙ̈ܝܐ، ܘܟܡܦܫܩܝ ܒܢܝܢܐ ܕܐܣܘܪ̈ܐ ܕܚܕ̈ܕܐ ܒܝܢܝܗܘܢ، ܘܫܓܘ̈ܫܝܐ ܕܠܐ ܐܣܘܪܐ ܝܚܝܕ̈ܝܐ ܝܢܐ ܩܐ ܟܠ ܡܫܬܚܠܦܢܐ ܘܟܟܢܫܝ ܫܘܚܠܦܐ ܕܦܝܫܐ" (Hastie et al., 2009, pp. 558-559). ܐܗܐ ܝܠܗ ܦܘܪܫܢܐ ܟܠܗ، ܘ-Bishop ܟܟܬܒ ܠܗ ܒܦܬܓܡܐ ܚܕ: ܦܘܫܩܐ ܕܥܠ̈ܠܬܐ ܝܠܗ PCA ܕܡܣܬܒܪܢܘܬܐ (probabilistic PCA) ܕܒܗ ܩܠܐ ܕܡܫܘܚܬܐ ܕܟܠ ܡܫܬܚܠܦܢܐ ܦܪܝܫܐ ܝܠܗ – "ܡܘܕܝܠ ܟܡܦܫܩ ܒܢܝܢܐ ܚܙܝܐ ܕܫܘܚܠܦܐ ܡܫܘܬܦܐ (covariance) ܕܝܗܝ̈ܒܬܐ ܒܡܚܘܝܢܘܬܐ ܕܫܘܚܠܦܐ ܠܐ ܬܠܝܐ ܕܐܣܝܪܐ ܝܠܗ ܥܡ ܟܠ ܣܪܛܐ ܓܘ ܡܐܛܪܝܟܣ Ψ ܘܒܐܚܕܐ ܕܫܘܚܠܦܐ ܡܫܘܬܦܐ ܒܝܢܝ ܡܫܬܚܠܦ̈ܢܐ ܓܘ ܡܐܛܪܝܟܣ W"؛ ܥܡܘ̈ܕܐ ܕ-W ܝܢܐ "ܛܥ̈ܢܐ ܕܥܠ̈ܠܬܐ"، ܘܗܕ̈ܡܐ ܓܘ ܣܪܛܐ ܡܨܠܝܢܐ ܕ-Ψ – "ܝܚܝܕ̈ܝܘܬܐ" (uniquenesses) (Bishop, 2006, pp. 583-584). ܘܡܢ ܟܐ ܦܐܪܐ: "ܦܘܫܩܐ ܕܥܠ̈ܠܬܐ ܟܥܒܕ ܡܘܕܝܠ ܠܒܢܝܢܐ ܕܐܣܘܪ̈ܐ ܕܚܕ̈ܕܐ ܕܡܫܬܚܠܦ̈ܢܐ ܘܠܐ ܠܒܢܝܢܐ ܕܫܘܚܠܦܐ ܡܫܘܬܦܐ. ܐܗܐ ܝܠܗ ܦܘܪܫܐ ܚܫܝܚܐ ܒܝܢܝ ܦܘܫܩܐ ܕܥܠ̈ܠܬܐ ܠ-PCA" (Hastie et al., 2009, p. 560). PCA ܟܡܦܫܩ ܟܠ ܫܘܚܠܦܐ ܕܦܣܘ̈ܩܐ – ܗܢܝ̈ ܦܣܘ̈ܩܐ ܕܒܨܘܝܐ ܟܒܚܪ ܒܐܠܦܐ ܕ-Cronbach – ܥܡ ܛܥܝܘܬܐ ܕܡܫܘܚܬܐ ܕܟܠ ܦܣܘܩܐ؛ ܦܘܫܩܐ ܕܥܠ̈ܠܬܐ ܟܡܦܫܩ ܒܣ ܡܐ ܕܦܣܘ̈ܩܐ ܟܡܫܬܘܬܦܝ ܒܗ. ܟܢܘܫܬܐ ܕܚܝܐ ܝܠܗ̇ ܡܢ ܦܢܩܝ̈ܬܐ ܕܫܘܐ̈ܠܐ ܡܛܝܐ ܝܠܗ̇ ܠܗܝ ܦܐܪܐ ܡܢ ܓܢܒܐ ܐܚܪܢܐ: "PCA ܠܐ ܝܠܗ ܐܘܪܚܐ ܫܪܝܪܬܐ ܕܦܘܫܩܐ ܕܥܠ̈ܠܬܐ، ܘܐܝܬ ܚܪܝܢܐ ܒܝܢܝ ܬܐܘܪܝܩ̈ܝܐ ܕܣܛܐܛܝܣܛܝܩܐ ܐܡܬܝ ܐܠܨܐ ܝܠܗ ܠܡܦܠܚܐ ܒܗ، ܐܢ ܒܟܠܗ"؛ "ܦܘܫܩܐ ܕܡܢܘ̈ܬܐ ܝܠܗ ܒܣ ܐܘܪܚܐ ܕܒܨܪܐ ܕܝܗܝ̈ܒܬܐ"؛ "ܦܘܫܩܐ ܕܡܢܘ̈ܬܐ ܪ̈ܫܝܬܐ ܠܐ ܟܦܪܫ ܒܝܢܝ ܫܘܚܠܦܐ ܡܫܘܬܦܐ ܠܫܘܚܠܦܐ ܝܚܝܕܝܐ" – ܘܥܡ ܗܕܐ "PCA ܝܠܗ ܐܘܪܚܐ ܕܦܠܛܬܐ (extraction) ܓܘ ܓܘܒܝܐ ܡܩܕܡܐ ܕܣܓܝܐܐ ܡܢ ܚܒ̈ܫܐ ܣܛܐܛܝܣܛܝܩ̈ܝܐ ܡܦܪ̈ܣܐ، ܘܡܢܝܗܝ SPSS ܘ-SAS" (Costello & Osborne, 2005, p. 2). ܐܝܢܝ ܕܟܕܚܦܐ ܓܘ SPSS ܥܠ Analyze، Dimension Reduction، Factor ܘܠܐ ܟܡܩܪܒܐ ܠܡܕܡ – ܟܩܒܠܐ ܡܢܘ̈ܬܐ، ܟܩܪܝܐ ܠܗܘܢ ܥܠ̈ܠܬܐ ܘܟܡܘܕܥܐ "ܦܘܫܩܐ ܕܥܠ̈ܠܬܐ". ܐܦ Modeler: ܩܛܪܐ (node) ܟܡܬܩܪܐ PCA/Factor، ܘ"ܟܬܚܡ ܡܢ ܩܕܡ ܕܪ̈ܓܐ ܕ-PCA… ܡܐ ܕܟܡܩܠܠ ܥܒܕܐ ܐܠܐ ܐܦ ܟܒܨܪ ܡܫܘܚܬܐ ܕܫܘܠܛܢܐ ܕܝܠܢ" (Truong, 2026, p. 442). ܘ-Pallant، ܕܟܬܒܐ ܕܝܠܗ̇ ܝܠܗ ܐܘܪܚܐ ܕܒܗ̇ ܣܓܝܐܐ ܡܢ ܒܨܘ̈ܝܐ ܓܘ ܐܝܣܪܐܝܠ ܝܠܦܠܗܘܢ SPSS، ܟܐܡܪܐ ܐܗܐ ܒܓܠܝܘܬܐ: ܓܘ PCA "ܟܠ ܫܘܚܠܦܐ ܓܘ ܡܫܬܚܠܦ̈ܢܐ ܟܡܬܦܠܚ"، ܓܘ ܦܘܫܩܐ ܕܥܠ̈ܠܬܐ "ܒܣ ܫܘܚܠܦܐ ܡܫܘܬܦܐ ܟܡܬܦܫܩ"؛ "ܐܦ ܐܢ PCA ܛܟܢܝܩܐܝܬ ܟܝܗܒ ܡܢܘ̈ܬܐ، ܣܓܝܐܐ ܡܢ ܟܬܘ̈ܒܐ ܟܡܦܠܚܝ ܒܫܡܐ 'ܥܠܬܐ' ܩܐ ܡܦܩܢܐ ܕ-PCA ܘܕ-FA ܬܪܘܝܗܝ. ܠܐ ܣܒܪܝܬܘܢ، ܐܢ ܟܚܙܝܬܘܢ ܫܡܐ 'ܥܠܬܐ' ܐܡܬܝ ܕܟܩܪܝܬܘܢ ܡܐܡܪ̈ܐ، ܕܟܬܘܒܐ ܡܘܦܠܚܠܗ FA" – ܘܗܝ ܓܢܒܗ̇ ܟܓܒܝܐ ܠܡܚܘܝܐ PCA، ܘܟܡܝܬܝܐ ܢܡܘܣܐ ܕ-Tabachnick ܘ-Fidell: "ܐܢ ܟܒܥܝܬܘܢ ܦܘܫܪܐ ܬܐܘܪܝܩܝܐ ܕܠܐ ܛܢܦܐ ܝܠܗ ܒܫܘܚܠܦܐ ܝܚܝܕܝܐ ܘܒܫܘܚܠܦܐ ܕܛܥܝܘܬܐ… FA ܝܠܗ ܓܘܒܝܐ ܕܝܘܟܘܢ. ܐܢ، ܡܢ ܓܢܒܐ ܐܚܪܢܐ، ܒܣ ܟܒܥܝܬܘܢ ܟܢܘܫܝܐ ܐܡܦܝܪܝܩܝܐ ܕܣܕܪܐ ܕܝܗܝ̈ܒܬܐ، PCA ܝܠܗ ܓܘܒܝܐ ܛܒܐ ܒܢܘܬ" (Tabachnick & Fidell, 2013، ܦܐܬܐ 640، ܐܝܟ ܡܝܬܝܐ ܓܘ Pallant, 2020، ܨܚܚܐ 15، ܦܬܚܐ). ܐܗܐ ܝܠܗ ܢܡܘܣܐ ܟܠܗ ܒܦܬܓܡܐ: ܣܒܠܬܐ (scale) ܕܟܡܫܚܐ ܒܢܝܢܐ (construct) – FA؛ ܒܨܪܐ ܕܡܫܬܚܠܦ̈ܢܐ ܩܕܡ ܪܝܓܪܫܢ (regression) – PCA. ܘ-Field ܟܡܦܫܩ ܩܐ ܡܘܢܐ ܗܟܢܐ ܝܠܗ، ܒܐܘܪܚܐ ܕ-communality: ܩܐ ܡܦܠܛܐ ܥܠ̈ܠܬܐ ܐܠܨܐ ܝܠܗ ܠܡܝܕܥ ܟܡܐ ܡܢ ܫܘܚܠܦܐ ܡܫܘܬܦܐ ܝܠܗ، ܘܐܗܐ ܟܡܬܝܕܥ ܒܣ ܒܬܪ ܦܠܛܬܐ؛ "ܐܝܬ ܬܪܝܢ ܦܘܫܪ̈ܐ. ܩܕܡܝܐ ܝܠܗ ܠܡܣܒܪ ܕܟܠ ܫܘܚܠܦܐ ܡܫܘܬܦܐ ܝܠܗ، ܒܣܒܪܐ ܕ-communality ܕܟܠ ܡܫܬܚܠܦܢܐ ܝܠܗ̇ 1… ܐܗܐ ܥܒܕܐ ܝܠܗ PCA" – ܘܡܛܠ ܗܕܐ "PCA ܟܣܒܪ ܕܠܝܬ ܛܥܝܘܬܐ ܕܡܫܘܚܬܐ"، ܘ"ܗܕܐ ܕ-PCA ܟܣܒܪ ܕܠܝܬ ܛܥܝܘܬܐ ܕܡܫܘܚܬܐ ܟܡܪܓܙܐ ܣܓܝܐܐ ܐܢܫ̈ܐ ܕܟܡܦܠܚܝ ܒܦܘܫܩܐ ܕܥܠ̈ܠܬܐ" (Field, 2024, Sections 18.4.2, 18.3.2). ܗܘ ܟܡܝܬܐ ܠ-Cliff: ܣܡܘܟ̈ܐ ܕܦܘܫܩܐ ܕܥܠ̈ܠܬܐ "ܟܡܚܝܠܝ ܕܦܘܫܩܐ ܕܡܢܘ̈ܬܐ ܓܘ ܚܠܬܐ ܛܒܬܐ ܝܠܗ ܦܘܫܩܐ ܕܥܠ̈ܠܬܐ ܡܫܘܬ̈ܦܬܐ ܥܡ ܩܠܝܠ ܛܥܝܘܬܐ ܕܡܘܣܦܬܐ ܝܠܗ̇ ܥܠܗ، ܘܓܘ ܚܠܬܐ ܒܝܫܬܐ ܚܘܠܛܐ ܕܠܐ ܡܬܦܪܫܢܐ ܕܡܕܡ̈ܐ ܕܠܐ ܡܨܝܚ ܠܡܦܣܩܐ ܡܢܗ ܡܕܡ" – ܘܟܡܘܕܐ: "ܐܝܬ ܐܢܝ̈ ܕܟܐܡܪܝ ܕܐܡܬܝ ܕܟܡܦܠܚܚ ܒ-PCA ܠܐ ܐܠܨܐ ܝܠܗ ܠܡܟܬܒ ܠܗ ܐܝܟ ܦܘܫܩܐ ܕܥܠ̈ܠܬܐ (ܐܘܦܣ!)" (ܦܣܩܐ 18.4.3). ܘܓܢܒܐ ܐܚܪܢܐ: ܠܦܘܬ ܒܨܝܪܐ ܕ-Guadagnoli ܘ-Velicer "ܦܘܫܪ̈ܐ ܕ-PCA ܩܠܝܠ ܦܪ̈ܝܫܐ ܝܢܐ ܡܢ ܗܢܝ̈ ܕܐܘܪ̈ܚܬܐ ܕܦܘܫܩܐ ܕܥܠ̈ܠܬܐ" – ܥܡ 30 ܡܫܬܚܠܦ̈ܢܐ ܘܠܥܠ ܘ-communality ܥܠ ܡܢ .7 "ܦܘܫܪ̈ܐ ܦܪ̈ܝܫܐ ܠܐ ܝܢܐ ܡܣܬܒܪ̈ܐ"، ܐܠܐ "ܥܡ ܒܨܝܪ ܡܢ 20 ܡܫܬܚܠܦ̈ܢܐ ܘ-communalities ܢܚܝ̈ܬܐ ܡܕܡ (ܬܚܬ ܡܢ .4) ܡܨܝܐ ܝܢܐ ܕܗܘܝܐ ܦܘܪ̈ܫܢܐ" (ܬܡܢ) – ܘܐܗܐ ܝܠܗ ܚܬܝܬܐܝܬ ܚܠܬܐ ܕܦܢܩܝܬܐ ܕܫܘܐ̈ܠܐ ܟܪܝܬܐ ܥܡ ܦܣܘ̈ܩܐ ܟܪ̈ܝܗܐ. ܬܪܝܢ ܟܬܒ̈ܐ ܕ-SPSS ܟܡܚܘܝܝ ܒܥܒܕܐ ܬܪܘܝܗܝ ܓܘܒ̈ܝܐ: Pallant – PCA؛ Field، ܓܘ ܡܦܩܬܐ ܫܬܝܬܝܬܐ – principal axis factoring (ܦܣܩܐ 18.6.1). ܘܓܘ ܟܬܒܐ ܕܝܘܠܦܢܐ ܕ-Modeler ܗܘ ܦܘܪܫܐ، ܒܢܘܣܚܐ ܕ-Backhaus: ܥܠ̈ܠܬܐ ܕܦܘܫܩܐ ܕܥܠ̈ܠܬܐ "ܡܨܝܚ ܠܡܦܫܩܐ ܐܝܟ 'ܥܠܬܐ ܕܫܘܚܠܦܐ ܡܫܘܬܦܐ'"، ܘܡܢܘ̈ܬܐ ܕ-PCA "ܡܨܝܚ ܠܡܬܚܘܝܐ ܐܝܟ 'ܬܚܘܝܬܐ ܟܠܢܝܬܐ [ܠܐ ܥܠܬܐ!] ܕܫܘܚܠܦܐ ܡܫܘܬܦܐ'" – ܘ"PCA ܟܡܬܦܠܚ ܣܓܝ ܒܢܘܬ ܡܢ PFA" (Wendler & Gröttrup, 2021, pp. 549, 552).

ܐܦ ܐܗܐ ܘܐܦ ܐܗܐ. ܓܢܒܐ ܐܚܪܢܐ ܟܐܬܐ ܡܢ ܗܝ ܦܐܬܐ ܕ-Hastie ܘܚܒܪ̈ܘܗܝ: ܡܛܠ ܕܐܦ ܓܘ ܡܘܕܝܠ ܕܥܠ̈ܠܬܐ ܟܠ ܚܘܕܪܐ ܕܛܥ̈ܢܐ ܟܝܗܒ ܦܘܫܪܐ ܫܘܝܐ، "ܦܝܫܐ ܩܠܝܠ ܡܢ ܪܥܝܢܐ ܕܓܢܐ (subjectivity) ܓܘ ܦܘܠܚܢܐ ܕܦܘܫܩܐ ܕܥܠ̈ܠܬܐ، ܡܛܠ ܕܡܦܠܚܢܐ ܡܨܐ ܠܡܒܨܝܐ ܢܘܣ̈ܚܐ ܡܚܘܕܪ̈ܐ ܕܥܠ̈ܠܬܐ ܕܦܫܝܩ̈ܐ ܝܢܐ ܠܦܘܫܩܐ. ܐܗܐ ܓܢܒܐ ܥܒܕܠܗ ܣܓܝܐܐ ܡܢ ܦܫܘܩ̈ܐ ܕܗܘܝܐ ܡܦܠܓ̈ܐ ܒܪܥܝܢܐ ܥܠ ܦܘܫܩܐ ܕܥܠ̈ܠܬܐ، ܘܒܠܟܝ ܟܡܦܫܩ ܚܣܝܪܘܬܐ ܕܚܘܒܐ ܕܝܠܗ ܓܘ ܣܛܐܛܝܣܛܝܩܐ ܕܐܕܝܘܡ" (ܦܐܬܐ 560). ܦܣܝܟܘܡܝܛܪܝܐ ܟܦܢܝܐ ܕܚܘܕܪܐ ܚܬܝܬܐܝܬ ܐܗܐ ܡܐܢܐ ܝܠܗ: ܥܠ̈ܠܬܐ ܟܣ̈ܝܬܐ ܠܐ ܝܢܐ ܬܚܝܡ̈ܐ ܒܐܝܕܐ ܕܡܬܡܛܝܩܐ ܐܠܐ ܒܐܝܕܐ ܕܦܘܫܩܐ، ܘܚܘܕܪܐ ܟܒܨܐ "ܒܢܝܢܐ ܦܫܝܛܐ" (simple structure) ܕܡܨܝܚ ܠܡܩܪܐ. ܬܪܘܝܗܝ ܩܪ̈ܝܢܐ ܬܪ̈ܝܨܐ ܝܢܐ، ܘܦܐܪܐ ܥܒܕܢܝܐ ܚܕ ܝܠܗ: ܠܡܘܕܥܐ ܐܝܢܐ ܡܘܕܝܠ (ܡܢܘ̈ܬܐ ܝܢ ܥܠ̈ܠܬܐ)، ܐܝܢܐ ܚܘܕܪܐ، ܘܩܐ ܡܘܢܐ. ܘܥܠ ܚܘܕܪܐ ܓܢܒܗ: ܐܢ ܥܠ̈ܠܬܐ ܐܝܬ ܠܗܝܢ ܐܣܘܪܐ ܕܚܕ̈ܕܐ – ܘܒܢܝ̈ܢܐ ܦܣܝܟܘܠܘܓܝ̈ܐ ܩܪܝܒ ܒܟܠ ܙܒܢܐ ܐܝܬ ܠܗܘܢ – ܚܘܕܪܐ ܡܨܠܝܢܐ (oblique rotation) (oblimin، promax) "ܬܐܘܪܝܩܐܝܬ ܐܠܨܐ ܝܠܗ ܕܝܗܒ ܦܘܫܪܐ ܚܬܝܬܐ ܒܢܘܬ، ܘܒܠܟܝ ܐܦ ܡܨܝܚ ܒܢܘܬ ܠܡܬܢܝܐ"؛ ܘ"ܐܢ ܥܠ̈ܠܬܐ ܫܪܝܪܐܝܬ ܠܝܬ ܠܗܝܢ ܐܣܘܪܐ ܕܚܕ̈ܕܐ، ܚܘܕܪܐ ܐܘܪܬܘܓܘܢܠܝܐ ܘܡܨܠܝܢܐ ܟܝܗܒܝ ܦܐܪ̈ܐ ܩܪܝܒ ܚܕ ܐܝܟ ܚܕ" (Costello & Osborne, 2005, p. 4) – ܗܢܘ ܕܠܝܬ ܡܐ ܕܡܚܣܪܚ ܡܢ ܡܨܠܝܢܐ؛ ܘ-Pallant: "ܐܢܐ ܒܟܠ ܙܒܢܐ ܟܡܦܩܕܢ ܠܡܫܪܝܐ ܒܚܘܕܪܐ ܡܨܠܝܢܐ ܩܐ ܡܒܚܪܐ ܡܫܘܚܬܐ ܕܐܣܘܪܐ ܕܚܕ̈ܕܐ ܒܝܢܝ ܥܠ̈ܠܬܐ" (Pallant, 2020, Chapter 15, "Step 3: Rotate and interpret the factors"). Varimax ܐܝܟ ܓܘܒܝܐ ܡܩܕܡܐ ܝܠܗ ܣܒܪܐ، ܠܐ ܓܘܒܝܐ. ܘܩܕܡ ܟܠ ܡܕܡ – ܐܢ ܒܟܠܗ ܐܝܬ ܡܐ ܕܡܦܠܛܚ: Pallant ܟܬܒܥܐ ܐܣܘܪ̈ܐ ܕܚܕ̈ܕܐ ܕ-.3 ܘܠܥܠ ܓܘ ܡܐܛܪܝܟܣ، ܡܚܘܝܢܐ KMO ܕ-.6 ܠܦܚܘܬ ܘܒܘܚܪܢܐ ܕ-Bartlett ܪܫܝܡܐ ܣܛܐܛܝܣܛܝܩܐܝܬ (ibid., "Step 1") – ܬܠܬܝܗܝ ܓܘ ܡܦܩܢܐ ܕ-SPSS، ܘܬܠܬܝܗܝ ܟܒܚܪܝ ܒܣ ܕܐܝܬ ܒܢܝܢܐ ܡܫܘܬܦܐ، ܠܐ ܐܝܢܐ.

ܟܡܐ ܥܠ̈ܠܬܐ – ܢܡܘܣܐ ܕܓܘܒܝܐ ܡܩܕܡܐ ܝܠܗ ܒܝܫܐ ܡܢ ܟܠ. SPSS ܘ-Modeler ܟܫܒܩܝ ܟܠ ܥܠܬܐ ܕܛܝܡܐ ܕܝܠܢܝܐ ܕܝܠܗ̇ ܪܒܐ ܝܠܗ ܡܢ 1 (Kaiser). "ܐܝܬ ܫܠܡܘܬܐ ܦܬܝܬܐ ܓܘ ܟܬܒ̈ܐ ܕܐܗܐ ܝܠܗ ܡܢ ܐܘܪ̈ܚܬܐ ܕܒܨܝܪ ܚܬܝܬ̈ܐ ܡܢ ܟܠ ܩܐ ܓܒܝܐ ܕܡܢܝܢܐ ܕܥܠ̈ܠܬܐ"؛ ܓܘ ܢܣܝܘ̈ܢܐ ܕܕܡܝܘܬܐ (simulations) ܕ-Costello ܘ-Osborne "36% ܡܢ ܕܘ̈ܡܝܐ ܕܝܠܢ ܫܒܩܠܗܘܢ ܣܓܝ ܥܠ̈ܠܬܐ ܠܦܘܬ ܐܗܐ ܢܡܘܣܐ" (ܦܐܬܐ 2-3). Truong ܚܙܐ ܠܗܕܐ ܓܘ ܙܒܢܐ ܫܪܝܪܐ: Modeler ܦܠܛܠܗ ܩܐ ܕܝܠܗ ܚܡܫܐ ܡܢܘ̈ܬܐ "ܥܠ ܫܬܐܣܬܐ ܕܫܪܛܐ ܕܛܝ̈ܡܐ ܕܝܠܢ̈ܝܐ ܪ̈ܒܐ ܝܢܐ ܡܢ 1"، ܘܗܘ ܟܫܐܠ "ܐܗܐ ܝܠܗ ܦܘܣܩܢܐ ܛܒܐ ܡܢ ܟܠ؟"، ܟܡܨܚܚ ܕ"SPSS Modeler ܠܐ ܟܝܗܒ ܨܘܪܬܐ scree ܓܘ ܡܦܩܢܐ ܕܩܛܪܐ PCA/Factor"، ܟܨܪܐ ܚܕܐ ܓܘ ܐܩܣܠ – ܘܟܡܘܕܐ ܕ"ܦܘܪܫܐ ܕ-elbow ܚܬܝܬܐ ܓܘ ܐܗܐ ܨܘܪܬܐ ܩܠܝܠ ܥܣܩܐ ܝܠܗ" (Truong, 2026, pp. 445-446). ܐܗܐ ܝܠܗ ܚܣܝܪܘܬܐ ܕܨܘܪܬܐ scree: ܝܠܗ̇ ܕܝܢܐ ܒܥܝܢܐ. ܢܡܘܣܐ ܕܐܗܐ ܦܐܬܐ ܟܡܦܩܕܐ ܥܠܗ ܝܠܗ ܦܘܫܩܐ ܒܓܢܒ ܚܕ̈ܕܐ (parallel analysis; Horn, 1965): ܟܥܒܕܚ ܝܗܝ̈ܒܬܐ ܓܕܫܢ̈ܝܬܐ ܕܗܝ ܪܒܘܬܐ – ܗܘ N، ܗܘ ܡܢܝܢܐ ܕܦܣܘ̈ܩܐ – ܟܚܫܒܚ ܛܝ̈ܡܐ ܕܝܠܢ̈ܝܐ ܕܝܠܗܝ̈ܢ، ܘܟܫܒܩܚ ܒܣ ܥܠ̈ܠܬܐ ܕܛܝܡܐ ܕܝܠܢܝܐ ܕܝܠܗ̇ ܪܒܐ ܝܠܗ ܡܢ ܡܐ ܕܩܠܐ ܣܪܝܩܐ ܟܝܗܒ. ܐܗܐ ܝܠܗ ܚܬܝܬܐܝܬ ܪܥܝܢܐ ܕܓܘܕܐ ܕܒܘܚܪܢܐ ܡܢ ܦܐܬ̈ܐ ܐܚ̈ܐ، ܡܦܠܚܐ ܥܠ ܫܘܐܠܐ ܝܚܝܕܝܐ ܕܡܨܝܚ: ܠܐ "ܐܢ ܥܠܬܐ ܬܪܝܨܬܐ ܝܠܗ̇" ܐܠܐ "ܐܢ ܝܠܗ̇ ܒܢܘܬ ܡܢ ܩܠܐ ܣܪܝܩܐ". Pallant ܟܟܢܫܐ ܟܬܒ̈ܐ: ܦܘܫܩܐ ܒܓܢܒ ܚܕ̈ܕܐ "ܡܚܘܝܐ ܝܠܗ ܐܝܟ ܚܬܝܬܐ ܡܢ ܟܠ، ܐܡܬܝ ܕܐܦ ܢܡܘܣܐ ܕ-Kaiser ܘܐܦ ܒܘܚܪܢܐ scree ܕ-Cattell ܟܡܝܠܝ ܠܡܫܥܪܐ ܒܢܘܬ ܡܢ ܡܢܝܢܐ ܕܡܢܘ̈ܬܐ"، ܘ"ܐܢ ܟܒܥܝܬܘܢ ܠܡܦܪܣܐ… ܓܘ ܦܣܝܟܘܠܘܓܝܐ ܝܢ ܝܘܠܦܢܐ، ܒܕ ܐܠܨܐ ܗܘܐ ܠܡܦܠܚܐ ܒܦܐܪ̈ܐ ܕܦܘܫܩܐ ܒܓܢܒ ܚܕ̈ܕܐ ܘܠܡܘܕܥܐ ܠܗܘܢ" (Pallant, 2020, Chapter 15, "Parallel analysis"). ܓܘ Modeler ܚܠܬܐ ܒܝܫܬܐ ܝܠܗ̇ ܒܢܘܬ: Wendler ܘ-Gröttrup ܟܡܢܝ ܡܚܘ̈ܝܢܐ ܕܦܐܝܘܬܐ ܕܡܐܛܪܝܟܣ – Bartlett، ܡܐܛܪܝܟܣ ܗܦܝܟܐ، anti-image، KMO – ܘܟܟܢܫܝ: "ܠܥܩܬܐ، SPSS Modeler ܠܐ ܟܝܗܒ ܐܗܐ ܡܚܘܝܢܐ ܘܐܦ ܠܐ ܚܕ ܡܢ ܡܚܘ̈ܝܢܐ ܐܚܪ̈ܢܐ ܕܬܚܘܝ̈ܐ ܝܢܐ ܠܥܠ"؛ ܢܡܘܣܐ ܕ-Kaiser، ܡܢ ܓܢܒܐ ܐܚܪܢܐ، "ܡܦܠܚܐ ܝܠܗ ܘܡܪܗܛܐ ܓܘ SPSS Modeler"، ܘܨܘܪܬܐ scree – "SPSS Modeler، ܠܥܩܬܐ، ܠܐ" (Wendler & Gröttrup, 2021, pp. 551-552). ܡܬܠܐ ܕ-Pallant ܝܠܗ ܣܗܕܘܬܐ: 20 ܦܣܘ̈ܩܐ ܕ-PANAS ܥܠ 435 ܡܦܢܝ̈ܢܐ – Kaiser ܟܫܒܩ ܐܪܒܥܐ ܡܢܘ̈ܬܐ (ܛܝ̈ܡܐ ܕܝܠܢ̈ܝܐ 6.25، 3.40، 1.22، 1.16)، scree ܟܡܚܘܐ ܬܒܪܐ ܒܬܪ ܬܪܝܢܝܬܐ، ܘܦܘܫܩܐ ܒܓܢܒ ܚܕ̈ܕܐ، ܥܡ ܛܝ̈ܡܐ ܕܢܡܘܣܐ ܕ-1.40، 1.33، 1.27 ܘ-1.22 ܡܢ 100 ܣܕܪ̈ܐ ܕܝܗܝ̈ܒܬܐ ܓܕܫܢ̈ܝܐ، ܟܫܒܩ ܬܪܬܝܢ – ܚܬܝܬܐܝܬ ܬܪܬܝܢ ܥܠ̈ܠܬܐ ܕܣܒܠܬܐ ܒܢܝܬܐ ܝܠܗ̇ ܥܠܝܗܝ̈ܢ (ܬܡܢ، "Interpretation of Output from Factor Analysis: Part 1"، ܕܪ̈ܓܐ 2-4). ܓܘܒܝܐ ܡܩܕܡܐ ܕ-SPSS ܛܥܐ ܒܬܪܬܝܢ. Field ܟܡܘܣܦ ܐܡܬܝ Kaiser ܒܟܠ ܐܘܪܚܐ ܚܬܝܬܐ ܝܠܗ: "ܐܡܬܝ ܕܡܢܝܢܐ ܕܡܫܬܚܠܦ̈ܢܐ ܒܨܝܪ ܝܠܗ ܡܢ 30 ܘ-communalities ܕܩܒܝܠ̈ܐ ܝܢܐ (ܒܬܪ ܦܠܛܬܐ) ܟܠܗܝ ܪ̈ܒܐ ܝܢܐ ܡܢ .7؛ ܡܨܐ ܕܗܘܐ ܚܬܝܬܐ ܐܦ ܐܡܬܝ ܕܪܒܘܬܐ ܕܕܘܡܝܐ ܥܠܝܐ ܝܠܗ̇ ܡܢ 250 ܘ-communality ܡܨܥܝܬܐ ܝܠܗ̇ .6 ܘܠܥܠ" (Field, 2024, Section 18.4.5) – ܘܟܟܢܫ ܢܣܝܘܢܐ ܕܕܡܝܘܬܐ ܕ-Auerswald ܘ-Moshagen (2019): "ܐܢ ܥܠ̈ܠܬܐ ܕܝܘܟܘܢ ܐܝܬ ܠܗܝܢ ܐܣܘܪܐ ܕܚܕ̈ܕܐ ܪܒܐ، ܟܠ ܐܘܪ̈ܚܬܐ ܒܕ ܗܘܝܐ ܒܝܫ̈ܐ"؛ "ܥܡ ܬܡܢܝܐ ܦܣܘ̈ܩܐ ܘܠܥܠ ܘܕܘܡܝܐ ܕ-200+، ܦܘܫܩܐ ܒܓܢܒ ܚܕ̈ܕܐ… ܒܕ ܥܒܕ ܥܒܕܐ ܛܒܐ ܡܕܡ"؛ ܘܙܒܢܐ ܝܚܝܕܝܐ ܕ-Kaiser ܛܒܐ ܝܠܗ ܒܢܘܬ ܝܠܗ "ܐܡܬܝ ܕܐܝܬ ܠܘܟܘܢ ܡܢܝܢܐ ܙܥܘܪܐ ܕܦܣܘ̈ܩܐ (ܡܬܠܐ ܐܪܒܥܐ) ܘܥܠ̈ܠܬܐ ܕܐܝܬ ܠܗܝܢ ܐܣܘܪܐ ܕܚܕ̈ܕܐ، ܘܐܦ ܗܝܕܝܟ ܒܣ ܓܘ ܕܘ̈ܡܝܐ ܪ̈ܘܪܒܐ ܘܒܐܘܪܚܐ ܕ-PCA" (ܬܡܢ). ܓܘ R ܦܘܫܩܐ ܒܓܢܒ ܚܕ̈ܕܐ ܝܠܗ ܣܕܪܐ ܚܕ (fa.parallel, Section 8)؛ ܓܘ ܪ̈ܫܝܡܬܐ ܕ-SPSS ܠܝܬ ܠܗ، ܘܬܪܝܢ ܟܬܒ̈ܐ ܟܥܒܪܝ ܥܠܗ ܒܐܘܪ̈ܚܬܐ ܦܪ̈ܝܫܐ – Pallant ܟܡܫܕܪܐ ܠܬܪܥܝܬܐ ܙܥܘܪܬܐ ܦܪܝܫܬܐ، Monte Carlo PCA for Parallel Analysis ܕ-Watkins، ܡܢ ܣܐܝܛ ܕܟܬܒܐ، ܕܟܛܒܥܐ ܛܝ̈ܡܐ ܕܝܠܢ̈ܝܐ ܓܕܫܢ̈ܝܐ ܘܟܦܚܡܚ ܒܐܝܕܐ (ܬܡܢ، ܕܪܓܐ 4)؛ Field ܟܪܗܛ psych ܕ-R ܡܢ ܓܘ ܣܝܢܬܐܟܣ (syntax) ܕ-SPSS (BEGIN PROGRAM R، ܦܐܝܠ ܡܛܝܒܐ ܓܘ ܣܐܝܛ ܕܟܬܒܐ؛ Field, 2024، SPSS Tip 18.1) – ܕܐܗܐ ܝܠܗ، ܒܫܪܪܐ، ܣܕܪܐ ܕ-R ܕܦܣܩܐ 8. ܘܒܚܪܬܐ، N: ܦܘܫܩܐ ܕܥܠ̈ܠܬܐ "ܝܠܗ ܥܒܕܐ ܕ'ܕܘܡܝܐ ܪܒܐ'"؛ ܓܘ ܢܣܝܘܢܐ ܕܕܡܝܘܬܐ ܕ-Costello ܘ-Osborne، ܒܦܘܚܡܐ ܕܬܪܝܢ ܡܫܬܘܬܦ̈ܢܐ ܩܐ ܦܣܘܩܐ ܒܣ 10% ܡܢ ܕܘ̈ܡܝܐ ܬܢܝܠܗܘܢ ܒܢܝܢܐ ܫܪܝܪܐ، ܒܦܘܚܡܐ ܕ-20:1 – 70% (ܦܐܬܐ 4-7). Pallant، ܒܬܪ Tabachnick ܘ-Fidell، ܟܣܦܩܐ ܒ-150 ܡܫܬܘܬܦ̈ܢܐ ܘܒܦܘܚܡܐ ܕ-5:1، ܘܟܡܝܬܝܐ ܐܦ 10:1 ܕ-Nunnally (1978) (Pallant, 2020, Chapter 15, "Step 1"). ܘ-Field ܟܡܥܪܥܪ ܥܠ ܦܘܚܡܐ ܓܢܒܗ: ܥܠ 10-15 ܩܐ ܦܣܘܩܐ "ܫܡܥܠܝ ܐܗܐ ܢܡܘܣܐ ܓܘ ܣܓܝ ܥܕܢ̈ܐ، ܐܠܐ ܫܬܐܣܬܐ ܐܡܦܝܪܝܩܝܬܐ ܕܝܠܗ ܠܐ ܝܠܗ̇ ܓܠܝܬܐ"؛ "ܡܐ ܕܚܫܝܚܐ ܝܠܗ ܪܒܘܬܐ ܟܠܢܝܬܐ ܕܕܘܡܝܐ، ܛܥ̈ܢܐ ܕܥܠ̈ܠܬܐ ܘ-communalities" – Comrey ܘ-Lee ܟܡܕܪܓܝ 100 ܐܝܟ ܕܘܡܝܐ ܒܝܫܐ، 300 ܐܝܟ ܛܒܐ ܘ-1,000 ܐܝܟ ܛܒܐ ܣܓܝ؛ ܥܠܬܐ ܥܡ ܐܪܒܥܐ ܛܥ̈ܢܐ ܘܠܥܠ ܥܠ ܡܢ .6 "ܫܪܝܪܬܐ ܝܠܗ̇ ܕܠܐ ܐܣܘܪܐ ܥܡ ܪܒܘܬܐ ܕܕܘܡܝܐ"؛ ܘܐܡܬܝ ܕ-communalities ܢܚܝ̈ܬܐ ܝܢܐ (ܣܓܝ ܬܚܬ ܡܢ .5) ܘܥܠ̈ܠܬܐ ܣܓܝ̈ܐܐ ܝܢܐ، ܥܠ ܡܢ 500؛ "ܡܐ ܕܓܠܝܐ ܝܠܗ ܡܢ ܐܗܐ ܥܒܕܐ ܝܠܗ ܕܕܘܡܝܐ ܕ-300 ܝܢ ܒܢܘܬ ܒܠܟܝ ܒܕ ܝܗܒ ܦܘܫܪܐ ܕܥܠ̈ܠܬܐ ܩܝܡܐ" (Field, 2024, Section 18.5.3). ܩܪܝܢܐ ܡܚܝܕܐ: ܦܘܚ̈ܡܐ ܝܢܐ ܢܡܘ̈ܣܐ ܕܨܒܥܐ (rules of thumb)، ܡܢܝܢܐ ܕܟܕܥܪ ܓܘ ܟܠ ܡܒܘ̈ܥܐ ܝܠܗ 300 – ܘܡܐ ܕܟܫܒܩ ܠܡܢܚܬ ܡܢܗ ܝܠܗ ܦܣܘ̈ܩܐ ܚܝܠܬ̈ܢܐ، ܠܐ ܡܫܬܘܬܦ̈ܢܐ ܩܠܝ̈ܠܐ. ܦܢܩܝܬܐ ܕܫܘܐ̈ܠܐ ܕ-30 ܦܣܘ̈ܩܐ ܥܠ 90 ܡܫܬܘܬܦ̈ܢܐ ܒܕ ܝܗܒܐ ܥܠ̈ܠܬܐ؛ ܐܢܝ̈ ܠܐ ܒܕ ܕܥܪܝ ܓܘ ܕܘܡܝܐ ܕܐܬܐ.

4. ܥܢܩܘ̈ܕܐ: ܐܘܪ̈ܚܬܐ ܘܡܐ ܕܟܠ ܚܕܐ ܟܣܒܪܐ

"ܦܘܫܩܐ ܕܥܢܩܘ̈ܕܐ (cluster analysis)، ܕܟܡܬܩܪܐ ܐܦ ܦܘܠܓܐ ܕܝܗܝ̈ܒܬܐ (data segmentation)… ܟܠܗ ܥܣܝܩܐ ܝܠܗ ܒܟܢܫܐ ܝܢ ܒܦܘܠܓܐ ܕܟܢܘܫܝܐ ܕܡܕܡ̈ܐ ܠܓܘ̈ܕܐ ܡܢܬܢ̈ܝܐ ܝܢ 'ܥܢܩܘ̈ܕܐ'، ܗܟܢܐ ܕܐܢܝ̈ ܕܓܘ ܟܠ ܥܢܩܘܕܐ ܐܣܝܪ̈ܐ ܝܢܐ ܚܕ ܥܡ ܚܕ ܒܢܘܬ ܡܢ ܡܕܡ̈ܐ ܕܦܠܝܓ̈ܐ ܝܢܐ ܠܥܢܩܘ̈ܕܐ ܦܪ̈ܝܫܐ" (Hastie et al., 2009, p. 501). Géron: "ܐܗܐ ܝܠܗ ܥܒܕܐ ܕܦܘܪܫܐ ܕܚܙ̈ܝܬܐ ܕܡܝ̈ܬܐ ܘܦܘܠܓܐ ܕܝܠܗܝ̈ܢ ܠܥܢܩܘ̈ܕܐ"؛ "ܚܬܝܬܐܝܬ ܐܝܟ ܓܘ ܦܘܪܫܐ ܠܓܢ̈ܣܐ (classification)، ܟܠ ܚܙܝܬܐ ܟܦܠܓܐ ܠܓܘܕܐ. ܐܠܐ ܠܐ ܐܝܟ ܦܘܪܫܐ ܠܓܢ̈ܣܐ، ܟܢܫܐ ܠܥܢܩܘ̈ܕܐ ܥܒܕܐ ܕܠܐ ܡܕܒܪܢܐ ܝܠܗ" (Géron, 2022, Chapter 9, "Clustering Algorithms"). ܘܡܠܬܐ ܕܟܠ ܡܕܡ ܬܠܝܐ ܝܠܗ ܒܗ̇ ܝܠܗ̇ "ܕܡ̈ܝܐ": Gutierrez ܟܡܢܐ ܐܪܒܥܐ ܫܘܐ̈ܠܐ – ܐܝܟܢܐ ܟܬܚܡܚ "ܩܪܝܒܐ"، ܐܝܟܢܐ ܟܟܢܫܚ، ܐܝܟܢܐ ܟܡܚܘܚ، ܘ"ܐܝܟܢܐ ܟܦܫܩܚ ܟܢܘ̈ܫܝܐ… ܕܒܠܟܝ ܥܣܩܐ ܝܠܗ ܠܡܗܝܡܢܐ ܒܗܘܢ ܝܢ ܕܗܘܝ̈ܐ ܝܢܐ ܒܣ ܡܛܠ ܩܠܐ ܣܪܝܩܐ" – ܘܟܦܣܩ ܕ"ܬܚܡܐ ܕܩܘܪܒܐ ܝܠܗ ܕܪܓܐ ܚܫܝܚܐ ܡܢ ܟܠ ܓܘ ܐܠܓܘܪܝܬܡ ܕܟܢܫܐ" (Gutierrez, 2015, Chapter 8, "Clustering"). ܪܘܚܩܐ – ܕ-Euclid، Manhattan، cosine ܩܐ ܟܬܒ̈ܐ، Jaccard ܩܐ ܪ̈ܫܝܡܬܐ – ܝܠܗ ܦܘܣܩܢܐ ܕܒܨܘܝܐ، ܠܐ ܕܐܠܓܘܪܝܬܡ، ܘ-Truong ܟܡܩܕܫ ܩܐ ܕܝܠܗ ܕܪܓܐ ܬܠܝܬܝܐ ܡܢ ܒܝܢܝ ܬܡܢܝܐ ܕܪ̈ܓܐ ܕܦܘܫܩܐ ܕܝܠܗ (Truong, 2026, p. 462). ܘܩܕܡܗ، ܐܝܟ ܓܘ PCA، ܡܬܩܢܘܬܐ (standardization): "ܚܫܝܚܐ ܝܠܗ ܠܡܬܩܢܐ ܬܚܘܝ̈ܬܐ ܕܡܥܠܬܐ… ܩܕܡ ܕܡܪܗܛܚ k-ܡܨܥܝ̈ܬܐ، ܐܢ ܠܐ ܥܢܩܘ̈ܕܐ ܡܨܝܐ ܝܢܐ ܕܗܘܝܐ ܣܓܝ ܡܬܝ̈ܚܐ ܘ-k-ܡܨܥܝ̈ܬܐ ܒܕ ܥܒܕ ܒܝܫܐܝܬ" (Géron, 2022, Chapter 9, "Limits of k-means").

k-ܡܨܥܝ̈ܬܐ (k-means) ܝܠܗ ܢܘܩܙܐ ܕܡܫܪܝܬܐ، ܘ-Géron ܟܡܦܫܩ ܪܥܝܢܐ ܒܬܪܬܝܢ ܫܘܝ̈ܘܬܐ: ܐܢ ܗܘܚ ܝܕܥܐ ܡܨܥ̈ܬܐ ܕܥܢܩܘ̈ܕܐ، ܗܘܚ ܦܠܓܐ ܟܠ ܚܙܝܬܐ ܠܡܨܥܬܐ ܕܩܪܝܒܐ ܝܠܗ̇ ܡܢܗ̇؛ ܐܢ ܗܘܚ ܝܕܥܐ ܦܘܠ̈ܓܐ، ܗܘܚ ܚܫܒܐ ܟܠ ܡܨܥܬܐ ܐܝܟ ܡܨܥܝܬܐ؛ ܠܝܬ ܠܢ ܠܐ ܐܗܐ ܘܠܐ ܐܗܐ – "ܟܡܫܪܚ ܒܕܘܟܐ ܓܕܫܢܝܬܐ ܕܡܨܥ̈ܬܐ… ܘܒܬܪ ܗܕܐ ܟܪܫܡܚ ܚܙ̈ܝܬܐ ܒܪܘܫܡܐ، ܟܡܚܕܬܚ ܡܨܥ̈ܬܐ، ܟܪܫܡܚ، ܟܡܚܕܬܚ، ܘܗܟܢܐ ܠܩܕܡ ܗܠ ܕܡܨܥ̈ܬܐ ܦܣܩܝ ܡܢ ܙܘܥܐ" (Chapter 9, "The k-means algorithm"). Bishop ܟܡܚܘܐ ܕܬܪܝܢ ܐܢܝ̈ ܕܪ̈ܓܐ ܝܢܐ ܕܪ̈ܓܐ E ܘ-M ܕܐܠܓܘܪܝܬܡ EM (Bishop, 2006, pp. 424-425). ܬܪܬܝܢ ܦܚ̈ܐ: ܐܠܓܘܪܝܬܡ "ܫܪܝܪܐ ܝܠܗ ܕܒܕ ܡܬܟܢܫ، ܐܠܐ ܠܐ ܒܐܠܨܘܬܐ ܠܦܘܫܪܐ ܬܪܝܨܐ" – ܬܠܝܐ ܝܠܗ ܒܡܫܪܝܬܐ، ܘܡܛܠ ܗܕܐ ܟܪܗܛܚ ܟܡܐ ܓܗ̈ܐ ܡܢ ܡܫܪܝܬ̈ܐ ܦܪ̈ܝܫܬܐ ܘܟܢܛܪܚ ܦܘܫܪܐ ܥܡ inertia ܢܚܝܬܐ ܡܢ ܟܠ، ܓܘܡܪܐ ܕܡܪ̈ܒܥܐ ܕܪ̈ܘܚܩܐ ܡܢ ܡܨܥ̈ܬܐ (Géron، ܨܚܚܐ 9، "Centroid initialization methods"؛ ܓܘ R – nstart، Gutierrez، ܨܚܚܐ 8، "K-Means Clustering"؛ ܘܡܫܪܝܬܐ k-means++ ܟܡܒܕܪܐ ܡܨܥ̈ܬܐ ܡܢ ܩܕܡ)؛ ܘ"k-ܡܨܥܝ̈ܬܐ ܠܐ ܟܥܒܕ ܛܒܐ ܐܡܬܝ ܕܥܢܩܘ̈ܕܐ ܐܝܬ ܠܗܘܢ ܪ̈ܒܘܬܐ ܦܪ̈ܝܫܐ، ܐܬܪ̈ܘܬܐ ܦܪ̈ܝܫܐ ܝܢ ܐܣܟܝ̈ܡܐ ܕܠܐ ܐܝܟ ܟܘܪ̈ܐ" – ܓܘ ܡܬܠܐ ܕ-Géron ܗܘ ܟܩܛܥ ܪܘܒܥܐ ܡܢ ܥܢܩܘܕܐ ܡܓܪܓܐ (elliptical) ܘܟܡܕܒܩ ܠܗ ܥܡ ܫܒܒܐ، "ܐܦ ܐܢ inertia ܕܝܠܗ ܢܚܝܬܐ ܝܠܗ̇ ܒܢܘܬ" (Chapter 9, "Limits of k-means"). ܘܣܒܪܐ ܫܠܝܐ ܒܢܘܬ ܡܢ ܟܠ: ܟܠ ܚܙܝܬܐ ܕܥܢܩܘܕܐ ܚܕ ܝܠܗ̇. Bishop: "ܡܨܝܐ ܝܢܐ ܕܗܘܝܐ ܢܘܩ̈ܙܐ ܕܝܗܝ̈ܒܬܐ ܕܐܝܬܝܗܝ ܩܪܝܒ ܒܦܠܓܐ ܕܐܘܪܚܐ ܒܝܢܝ ܡܨܥ̈ܬܐ ܕܥܢܩܘ̈ܕܐ. ܓܘ ܚܠܬܐ ܐܝܟ ܗܕܐ ܠܐ ܝܠܗ ܓܠܝܐ ܕܦܘܠܓܐ ܩܫܝܐ ܠܥܢܩܘܕܐ ܩܪܝܒܐ ܡܢ ܟܠ ܦܐܐܫ ܝܠܗ ܒܢܘܬ ܡܢ ܟܠ" (ܦܐܬܐ 428) – ܘܚܠܦܐ ܝܠܗ ܚܘܠܛܐ ܕ-Gauss (Gaussian mixture): ܟܠ ܥܢܩܘܕܐ ܝܠܗ ܦܘܠܓܐ ܢܘܪܡܠܝܐ ܕܣܓܝ ܡܫܘܚ̈ܬܐ ܕܝܠܗ، "ܥܡ ܐܣܟܝܡܐ ܡܓܪܓܐ، ܪܒܘܬܐ، ܐܬܪܘܬܐ ܘܐܘܪܚܐ ܦܪ̈ܝܫܐ"، ܘܡܘܕܝܠ ܟܝܗܒ ܠܟܠ ܚܙܝܬܐ ܡܣܬܒܪܢܘܬܐ (probability) ܕܗܘܝܐ ܡܢ ܟܠ ܥܢܩܘܕܐ – "ܦܘܠܓܐ ܪܟܝܟܐ، ܥܠ ܫܬܐܣܬܐ ܕܡܣܬܒܪ̈ܢܘܝܬܐ ܕܒܬܪ (posterior)" (Bishop, 2006, p. 443; Géron, 2022, Chapter 9, "Gaussian Mixtures"). k-ܡܨܥܝ̈ܬܐ ܝܠܗ ܬܚܘܡܐ ܕܐܗܐ ܡܘܕܝܠ ܐܡܬܝ ܕܫܘܚܠܦܐ ܕܟܠ ܥܢܩܘ̈ܕܐ ܚܕ ܝܠܗ ܘܟܐܙܠ ܠܨܦܪܐ (Bishop, p. 443). ܩܐ ܒܨܘܝܐ: ܐܢ ܐܝܬ ܡܫܬܘܬܦ̈ܢܐ "ܒܝܢܝ" ܛܘ̈ܦܣܐ – ܘܒܟܠ ܙܒܢܐ ܐܝܬ – ܡܣܬܒܪܢܘܬܐ ܕܗܘܝܐ ܡܢ ܥܢܩܘܕܐ ܝܠܗ̇ ܡܫܟܚܢܘܬܐ، ܠܐ ܪܘܫܡܐ.

ܥܢܩܘ̈ܕܐ ܕܕܪ̈ܓܐ (hierarchical clustering) ܠܐ ܟܬܒܥܝ k ܡܢ ܩܕܡ: "ܒܕܘܟܬܐ ܕܗܕܐ، ܟܬܒܥܝ ܡܢ ܡܦܠܚܢܐ ܕܬܚܡ ܡܫܘܚܬܐ ܕܠܐ ܕܡܝܘܬܐ ܒܝܢܝ ܓܘ̈ܕܐ (ܦܪ̈ܝܫܐ) ܕܚܙ̈ܝܬܐ" – ܘܒܬܪ ܗܕܐ، ܒܐܘܪܚܐ ܕܡܚܝܕܢܘܬܐ، ܟܡܫܪܚ ܡܢ ܟܠ ܚܙܝܬܐ ܐܝܟ ܥܢܩܘܕܐ ܘܓܘ ܟܠ ܕܪܓܐ ܟܡܚܝܕܚ ܬܪܝܢ ܩܪ̈ܝܒܐ ܡܢ ܟܠ، ܗܠ ܥܢܩܘܕܐ ܚܕ؛ ܦܐܪܐ ܝܠܗ dendrogram، ܐܝܠܢܐ ܕܪܘܡܐ ܕܟܠ ܩܛܪܐ ܓܘܗ ܝܠܗ ܠܐ ܕܡܝܘܬܐ ܒܝܢܝ ܬܪܝܢ ܒܢ̈ܘܗܝ، ܘܩܛܥܐ ܦܫܝܛܐ ܕܝܠܗ ܟܝܗܒ ܦܘܠܓܐ (Hastie et al., 2009, pp. 520-521). ܡܐ ܝܠܗ "ܩܪܝܒܐ" ܒܝܢܝ ܓܘ̈ܕܐ – ܐܣܘܪܐ (linkage): ܝܚܝܕܝܐ، ܓܡܝܪܐ، ܡܨܥܝܐ، ܝܢ Ward (ܒܨܪܐ ܕܫܘܚܠܦܐ ܓܘ ܥܢܩܘ̈ܕܐ) – ܟܡܫܢܐ ܐܝܠܢܐ. ܘܙܘܗܪܐ ܕ-Hastie ܘܚܒܪ̈ܘܗܝ، ܕܐܠܨܐ ܝܠܗ ܠܡܝܬܝܐ ܠܗ ܓܡܝܪܐܝܬ: "dendrogram ܐܝܟ ܗܕܐ ܣܓܝ ܓܗ̈ܐ ܟܡܬܚܫܒ ܐܝܟ ܟܢܘܫܝܐ ܕܨܘܪܬܐ ܕܝܗܝ̈ܒܬܐ ܓܢܒܗܝ̈ܢ، ܘܠܐ ܐܝܟ ܬܚܘܝܬܐ ܕܦܐܪ̈ܐ ܕܐܠܓܘܪܝܬܡ. ܐܠܐ ܐܠܨܐ ܝܠܗ ܠܡܚܫܒܐ ܠܦܘܫ̈ܩܐ ܐܝܟ ܗܢܝ̈ ܒܙܗܝܪܘܬܐ. ܩܕܡܝܐ، ܐܘܪ̈ܚܬܐ ܕܕܪ̈ܓܐ ܦܪ̈ܝܫܐ…، ܘܐܦ ܫܘܢܝ̈ܐ ܙܥܘܪ̈ܐ ܓܘ ܝܗܝ̈ܒܬܐ، ܡܨܝܐ ܝܢܐ ܕܡܝܬܝܐ ܠ-dendrograms ܣܓܝ ܦܪ̈ܝܫܐ. ܐܦ، ܟܢܘܫܝܐ ܐܝܟ ܗܕܐ ܒܕ ܗܘܐ ܬܪܝܨܐ ܒܣ ܐܢ ܠܐ ܕܡܝܘܬܐ ܙܘܓܢܝܬܐ ܒܝܢܝ ܚܙ̈ܝܬܐ ܫܪܝܪܐܝܬ ܟܡܩܝܡܐ ܒܢܝܢܐ ܕܕܪ̈ܓܐ ܕܐܠܓܘܪܝܬܡ ܟܥܒܕ. ܐܘܪ̈ܚܬܐ ܕܕܪ̈ܓܐ ܟܡܛܠܨܝ ܒܢܝܢܐ ܕܕܪ̈ܓܐ ܐܢ ܒܢܝܢܐ ܐܝܟ ܗܕܐ ܐܝܬ ܓܘ ܝܗܝ̈ܒܬܐ ܘܐܢ ܠܝܬ" (ܦܐܬܐ 521-522). DBSCAN ܟܬܚܡ ܥܢܩܘ̈ܕܐ "ܐܝܟ ܐܬܪ̈ܘܬܐ ܡܬܡܫܚ̈ܢܐ ܕܐܬܪܘܬܐ ܥܒܝܬܐ": ܚܙܝܬܐ ܥܡ ܡܣܦܩܢܘܬܐ ܕܫܒܒ̈ܐ ܓܘ ܦܠܓܘܬ-ܩܛܪܐ ε ܝܠܗ̇ "ܠܒܐ" (core)، ܫܒܒ̈ܐ ܕܠܒ̈ܐ ܟܡܚܝܕܝ ܠܥܢܩܘܕܐ، ܘܐܝܢܐ ܕܠܐ ܝܠܗ ܠܒܐ ܘܠܝܬ ܠܗ ܠܒܐ ܓܘ ܫܒܒܘܬܐ "ܚܫܝܒܐ ܝܠܗ ܢܘܟܪܝܐ (outlier)" – ܘܗܘ "ܟܥܒܕ ܛܒܐ ܐܢ ܟܠ ܥܢܩܘ̈ܕܐ ܦܪ̈ܝܫܐ ܝܢܐ ܛܒܐ ܒܐܝܕܐ ܕܐܬܪ̈ܘܬܐ ܕܐܬܪܘܬܐ ܢܚܝܬܐ" (Géron, 2022, Chapter 9, "DBSCAN")؛ ܠܝܬ k، ܐܝܬ ε، ܘܐܝܬ ܛܝ̈ܡܐ ܢܘܟܪ̈ܝܐ ܐܝܟ ܦܐܪܐ ܕܓܢܒܐ. ܘ-TwoStep ܕ-SPSS ܘ-Modeler: ܥܒܪܐ ܚܕ ܥܠ ܝܗܝ̈ܒܬܐ ܕܟܚܒܫ ܠܗܝ̈ܢ ܠܥܢܩܘ̈ܕܐ ܡܢܬܢ̈ܝܐ، ܘܒܬܪ ܗܕܐ ܥܢܩܘܕܐ ܕܕܪ̈ܓܐ ܥܠܝܗܝ – ܡܛܝܒܐ ܝܠܗ ܩܐ ܝܗܝ̈ܒܬܐ ܪ̈ܘܪܒܐ، ܘܥܡ "ܦܘܣܩܢܐ ܡܢ ܓܢܗ ܕܡܢܝܢܐ ܕܥܢܩܘ̈ܕܐ ܥܠ ܫܬܐܣܬܐ ܕܢܡܘ̈ܣܐ ܣܛܐܛܝܣܛܝܩ̈ܝܐ" (Truong, 2026, pp. 470, 473) – ܕܝܠܗ BIC، ܗܢܘ ܦܣܩܐ 5.

5. ܟܡܐ ܥܢܩܘ̈ܕܐ – ܘܐܢ ܒܟܠܗ

Hastie ܘܚܒܪ̈ܘܗܝ ܟܦܪܫܝ ܒܝܢܝ ܬܪܬܝܢ ܚܠ̈ܬܐ. ܐܡܬܝ ܕܢܝܫܐ ܝܠܗ ܦܘܠܓܐ (segmentation)، "k ܒܥܕܬܐ ܬܚܝܡܐ ܝܠܗ ܐܝܟ ܡܢܬܐ ܡܢ ܥܘܩܬܐ" – ܫܪܟܬܐ ܥܡ k ܙܒܘܢ̈ܐ ܒܕ ܦܠܓܐ ܙܒܘܢ̈ܐ ܕܝܠܗ̇ ܠ-k؛ "ܐܠܐ ܣܓܝ ܓܗ̈ܐ ܦܘܫܩܐ ܕܥܢܩܘ̈ܕܐ ܟܡܬܦܠܚ ܐܝܟ ܣܛܐܛܝܣܛܝܩܐ ܬܚܘܝܝܬܐ ܩܐ ܡܒܚܪܐ ܡܫܘܚܬܐ ܕܒܗ̇ ܚܙ̈ܝܬܐ… ܟܢܦܠܝ ܠܓܘ̈ܕܐ ܛܒܥ̈ܝܐ ܦܪ̈ܝܫܐ. ܟܐ ܡܢܝܢܐ ܕܓܘ̈ܕܐ k* ܠܐ ܝܠܗ ܝܕܝܥܐ" ܘܐܠܨܐ ܝܠܗ ܠܡܫܥܪܐ ܠܗ ܡܢ ܝܗܝ̈ܒܬܐ (Hastie et al., 2009, p. 518). ܘܥܘܩܬܐ: ܠܐ ܕܡܝܘܬܐ ܓܘ ܥܢܩܘ̈ܕܐ، W_k، "ܒܥܕܬܐ ܟܢܚܬܐ ܐܝܟ ܕ-k ܟܪܒܐ. ܗܟܢܐ ܒܕ ܗܘܐ ܐܦ ܐܡܬܝ ܕܢܡܘܣܐ ܡܫܘܥܪܐ ܝܠܗ ܥܠ ܓܘܕܐ ܕܒܘܚܪܢܐ (test set) ܠܐ ܬܠܝܐ، ܡܛܠ ܕܡܢܝܢܐ ܪܒܐ ܕܡܨܥ̈ܬܐ ܕܥܢܩܘ̈ܕܐ ܟܡܝܠ ܠܡܡܠܝܐ ܦܬܝܘܬܐ ܕܬܚܘܝ̈ܬܐ (features) ܒܐܬܪܘܬܐ ܘܡܛܠ ܗܕܐ ܠܗܘܝܐ ܩܪܝܒܐ ܠܟܠ ܢܘܩ̈ܙܐ ܕܝܗܝ̈ܒܬܐ. ܡܛܠ ܗܕܐ ܐܘܪ̈ܚܬܐ ܕܫܘܪܪܐ ܡܚܠܦܢܐ (cross-validation)، ܕܣܓܝ ܡܘܬܪ̈ܢܐ ܝܢܐ ܩܐ ܓܒܝܐ ܕܡܘܕܝܠ ܓܘ ܝܘܠܦܢܐ ܡܕܒܪܢܐ، ܠܐ ܡܨܝܐ ܝܢܐ ܠܡܬܦܠܚܐ ܓܘ ܐܗܐ ܣܝܩܐ" (ܦܐܬܐ 519). ܐܗܐ ܝܠܗ ܥܠܬܐ ܕܠܝܬ ܟܐ "ܡܚܘ̈ܝܢܐ ܕܥܒܕܐ": ܟܠ ܡܐ ܕܦܝܫܐ ܝܠܗ ܐܘܪ̈ܚܬܐ ܕܡܨܝܕܢܘܬܐ (heuristics)، ܘܦܐܬܐ ܟܡܦܩܕܐ ܥܠ ܬܠܬܐ، ܒܛܟܣܐ ܣܠܩܐ ܕܬܪܝܨܘܬܐ (validity).

Elbow. Géron: inertia ܩܐ k=3 ܝܠܗ̇ 653، ܩܐ k=5 – 212، ܩܐ k=8 – 119؛ "inertia ܠܐ ܝܠܗ̇ ܡܚܘܝܢܐ ܛܒܐ ܕܥܒܕܐ ܩܐ ܓܒܝܐ ܕ-k، ܡܛܠ ܕܟܡܦܫܐ ܠܡܢܚܬ ܐܝܟ ܕܟܡܪܒܚ k". ܟܨܪܚ ܠܗ̇ ܐܝܟ ܦܘܢܩܣܝܘܢ ܕ-k ܘܟܒܨܚ "ܢܘܩܙܐ ܕܟܦܦܐ ܕܟܡܬܩܪܐ elbow" – "ܐܗܐ ܐܘܪܚܐ… ܣܓܝ ܓܣܝܬܐ ܝܠܗ̇" (Géron, 2022, Chapter 9, "Finding the optimal number of clusters")؛ ܓܘ Hastie ܘܚܒܪ̈ܘܗܝ ܐܗܐ ܝܠܗ "kink" ܓܘ W_k، "ܘܐܝܟ ܓܢ̈ܒܐ ܐܚܪ̈ܢܐ ܕܥܒ̈ܕܐ ܕܥܢܩܘ̈ܕܐ، ܐܗܐ ܐܘܪܚܐ ܩܠܝܠ ܕܡܨܝܕܢܘܬܐ (heuristic) ܝܠܗ̇" (ܦܐܬܐ 519). Silhouette (silhouette; Rousseeuw, 1987): ܩܐ ܟܠ ܚܙܝܬܐ، (b − a) / max(a, b)، ܐܡܬܝ ܕ-a ܝܠܗ ܪܘܚܩܐ ܡܨܥܝܐ ܠܗܕ̈ܡܐ ܕܥܢܩܘܕܐ ܕܝܠܗ̇ ܘ-b ܪܘܚܩܐ ܡܨܥܝܐ ܠܥܢܩܘܕܐ ܩܪܝܒܐ ܕܒܬܪܗ؛ "ܡܢܝܢܐ ܡܫܘܬܦܐ ܩܪܝܒܐ ܠ-+1 ܟܐܡܪ ܕܚܙܝܬܐ ܛܒܐ ܝܠܗ̇ ܓܘ ܥܢܩܘܕܐ ܕܝܠܗ̇ ܘܪܚܝܩܬܐ ܡܢ ܐܚܪ̈ܢܐ… ܩܪܝܒܐ ܠ-0 – ܕܩܪܝܒܬܐ ܝܠܗ̇ ܠܬܚܘܡܐ… ܘܩܪܝܒܐ ܠ-−1 – ܕܒܠܟܝ ܦܠܝܓܬܐ ܝܠܗ̇ ܠܥܢܩܘܕܐ ܠܐ ܬܪܝܨܐ"؛ ܡܢܝܢܐ ܕ-silhouette ܝܠܗ ܡܨܥܝܬܐ ܥܠ ܟܠ ܚܙ̈ܝܬܐ، ܘܨܘܪܬܐ ܕ-silhouette – "ܐܣܟܝܡܐ ܕܣܟܝܢܐ" ܩܐ ܟܠ ܥܢܩܘܕܐ – ܟܡܚܘܝܐ ܐܦ ܥܢܩܘ̈ܕܐ ܕܣܓܝܐܘܬܐ ܕܗܕ̈ܡܐ ܕܝܠܗܘܢ ܬܚܬ ܡܢ ܡܨܥܝܬܐ ܝܢܐ (ܨܚܚܐ 9، ܬܡܢ). ܓܘ ܡܬܠܐ ܕ-Géron silhouette ܟܡܫܪܪ k=4 ܐܠܐ ܟܡܚܘܐ ܕ-k=5 "ܐܦ ܛܒܐ ܡܕܡ ܝܠܗ، ܘܣܓܝ ܛܒܐ ܒܢܘܬ ܡܢ 6 ܝܢ 7. ܐܗܐ ܠܐ ܗܘܐ ܡܬܚܙܝܐ ܒܦܘܚܡܐ ܕ-inertias" – ܘܗܘ ܟܓܒܐ 5، ܩܐ ܥܢܩܘ̈ܕܐ ܕܪ̈ܒܘܬܐ ܕܡ̈ܝܬܐ: silhouette ܟܡܥܕܪ، ܕܝܢܐ ܟܦܐܫ. Gap statistic (gap statistic; Tibshirani, Walther & Hastie, 2001) ܝܠܗ ܝܚܝܕܝܐ ܡܢ ܬܠܬܝܗܝ ܕܟܦܢܐ ܐܦ ܥܠ "ܘܐܢ ܒܟܠܗ": ܗܘ "ܟܡܦܚܡ ܣܪܛܐ ܟܦܝܦܐ log W_k ܥܡ ܣܪܛܐ ܟܦܝܦܐ ܕܟܦܐܫ ܡܢ ܝܗܝ̈ܒܬܐ ܡܒܕܪ̈ܬܐ ܫܘܝܐܝܬ ܥܠ ܡܪܒܥܐ ܡܓܪܓܐ ܕܐܝܬ ܓܘܗ ܝܗܝ̈ܒܬܐ"، ܘܟܓܒܐ k ܕܒܗ ܦܘܪܫܢܐ ܒܝܢܝ ܬܪܘܝܗܝ ܪܒܐ ܝܠܗ ܡܢ ܟܠ؛ "ܒܫܪܪܐ ܐܗܐ ܐܘܪܚܐ ܝܠܗ̇ ܕܡܢ ܓܢܗ̇ ܠܡܨܚܚܐ 'kink'… ܐܦ ܟܥܒܕ ܛܒܐ ܡܕܡ ܐܡܬܝ ܕܝܗܝ̈ܒܬܐ ܟܢܦܠܝ ܠܥܢܩܘܕܐ ܝܚܝܕܝܐ، ܘܓܘ ܚܠܬܐ ܐܝܟ ܗܕܐ ܒܕ ܡܝܠ ܠܡܫܥܪܐ ܡܢܝܢܐ ܛܒܐ ܡܢ ܟܠ ܕܥܢܩܘ̈ܕܐ ܐܝܟ ܚܕ. ܐܗܐ ܝܠܗ ܚܠܬܐ ܕܒܗ̇ ܣܓܝܐܐ ܡܢ ܐܘܪ̈ܚܬܐ ܐܚܪ̈ܢܐ ܠܐ ܟܡܨܝ" (Hastie et al., 2009, p. 519). ܩܐ ܚܘ̈ܠܛܐ ܕ-Gauss، ܕܓܘܝܗܝ inertia ܘ-silhouette "ܠܐ ܝܢܐ ܡܗܝܡܢ̈ܐ ܐܡܬܝ ܕܥܢܩܘ̈ܕܐ ܠܐ ܝܢܐ ܐܝܟ ܟܘܪ̈ܐ ܝܢ ܕܪ̈ܒܘܬܐ ܦܪ̈ܝܫܐ"، Géron ܟܥܒܪ ܠ-BIC ܘ-AIC – ܢܡܘ̈ܣܐ ܕܡܘܕܥܢܘܬܐ (information criteria) ܕܟܡܢܟܝ ܥܠ ܦܪܡܛܪ̈ܐ (Chapter 9, "Selecting the Number of Clusters") – ܘܐܗܐ ܝܠܗ ܡܐ ܕ-TwoStep ܟܥܒܕ ܒܬܪ "ܡܢ ܓܢܗ".

ܘܓܘ ܡܚܫܒܢܐ (calculator): ܦܐܬܐ ܦܘܫܩܐ ܕܥܢܩܘ̈ܕܐ ܟܪܗܛ ܐܘܪܚܐ ܕܬܪܝܢ ܕܪ̈ܓܐ ܥܠ ܡܫܬܚܠܦ̈ܢܐ ܕܟܡܝܘܬܐ – Ward، ܘܒܬܪܗ k-ܡܨܥܝ̈ܬܐ ܕܟܡܫܪܝ ܡܢ ܡܨܥ̈ܬܐ ܕܥܢܩܘ̈ܕܐ ܕ-Ward – ܘܟܛܒܥ ܩܐ ܟܠ k ܬܠܬܐ ܢܡܘ̈ܣܐ ܕܐܗܐ ܦܣܩܐ: W_k (elbow)، ܡܨܥܝܬܐ ܕ-silhouette ܘ-gap statistic (ܐܝܟ clusGap، ܥܡ ܡܘܠܕܢܐ ܕ-R). ܡܢ ܬܡܐ – dendrogram، ܬܚܘܝܬܐ ܕܥܢܩܘ̈ܕܐ (cluster profile)، ܨܘܪܬܐ ܕ-silhouette ܘܦܐܝܠ ܥܡ ܥܡܘܕܐ ܕܥܢܩܘܕܐ ܩܐ ܫܘܪܪܐ ܠܦܘܬ ܦܣܩܐ 6. ܘܒܕܚܦܬܐ – ܩܝܡܘܬܐ (stability) ܒܓܘܒܝܐ ܕܕܘܡܝܐ ܡܢ ܪܝܫ (resampling): ܐܢ ܦܘܠܓܐ، ܘܟܠ ܥܢܩܘܕܐ ܓܘܗ، ܟܕܥܪܝ ܓܘ ܕܘ̈ܡܝܐ ܕ-bootstrap – ܫܘܪܪܐ ܕܠܐ ܡܚܘܝܢܐ ܕܠܒܪ ܕܐܝܬ ܓܘ ܦܣܩܐ 6.

6. ܥܢܩܘܕܐ ܐܝܟ ܗܘܦܘܬܣܝܣ: ܡܐ ܟܥܒܕܚ ܒܬܪ ܕܡܨܚܚ ܠܗ

ܢܡܘܣܐ ܕܠܝܬ ܠܗ ܚܠܦܐ: ܐܠܓܘܪܝܬܡ ܒܟܠ ܙܒܢܐ ܟܡܕܥܪ ܥܢܩܘ̈ܕܐ. k-ܡܨܥܝ̈ܬܐ ܥܠ ܬܪܝܢ ܡܫܬܚܠܦ̈ܢܐ ܫܘ̈ܝܐ ܓܡܝܪܐܝܬ، ܕܠܐ ܚܕ ܒܢܝܢܐ، ܒܕ ܡܕܥܪ ܬܠܬܐ ܥܢܩܘ̈ܕܐ ܒܪ̈ܒܘܬܐ ܡܣܬܒܪ̈ܢܝܬܐ ܘܥܡ silhouette ܕܠܐ ܒܗܬܬܐ – ܦܣܩܐ 8 ܟܡܚܘܐ ܐܗܐ ܒܡܢܝܢ̈ܐ. Truong ܟܡܢܐ ܒܝܢܝ ܚܣܝܪ̈ܘܝܬܐ ܕܐܘܪܚܐ ܪܓܝܫܘܬܐ ܠܡܫܪܝܬܐ، ܠܛܝ̈ܡܐ ܢܘܟܪ̈ܝܐ (outliers) ܘܠܡܫܘܚ̈ܬܐ، ܘܦܘܫܩܐ: "ܥܢܩܘ̈ܕܐ ܓܢܒܗܘܢ ܡܨܝܐ ܝܢܐ ܕܗܘܝܐ ܕܠܐ ܪܘ̈ܫܡܐ ܝܢ ܦܘܫ̈ܩܐ ܓܠܝ̈ܐ" (Truong, 2026, p. 462) – ܘܬܪܝܢ ܕܪ̈ܓܐ ܐܚܪ̈ܝܐ ܕܝܠܗ ܝܢܐ ܬܚܘܝܬܐ (profiling: ܡܨܥܝ̈ܬܐ ܘܣܓܝܐܘ̈ܬܐ ܕܟܠ ܥܢܩܘܕܐ) ܘܫܘܪܪܐ، "ܠܡܫܪܪܐ ܕܦܐܪ̈ܐ ܕܥܢܩܘܕܐ ܐܝܬ ܠܗܘܢ ܣܘܟܠܐ، ܡܗܝܡܢ̈ܐ ܝܢܐ ܘܦܐܐܫ̈ܐ ܝܢܐ ܠܝܗܝ̈ܒܬܐ" (ܦܐܬܐ 462). ܐܝܟܢܐ ܟܫܪܪܚ ܡܕܡ ܕܠܝܬ ܠܗ ܡܚܘܝܢܐ؟ ܒܬܠܬܐ ܐܘܪ̈ܚܬܐ، ܡܢ ܟܪܝܗܬܐ ܠܚܝܠܬܢܝܬܐ، ܘܬܠܬܝܗܝ ܥܠ ܝܗܝ̈ܒܬܐ ܕܥܢܩܘܕܐ ܠܐ ܒܢܝܐ ܝܠܗ ܡܢܝܗܝ̈ܢ.

ܩܝܡܘܬܐ (stability). ܥܢܩܘܕܐ ܕܐܝܬ ܓܘ ܝܗܝ̈ܒܬܐ ܒܕ ܕܥܪ: ܓܘ ܐܠܓܘܪܝܬܡ ܐܚܪܢܐ (k-ܡܨܥܝ̈ܬܐ ܩܘܒܠ Ward ܩܘܒܠ ܚܘܠܛܐ ܕ-Gauss – ܡܚܘܝܢܐ ܕ-Rand ܡܬܘܪܨܐ، ARI، ܒܝܢܝ ܬܪܝܢ ܦܘܠ̈ܓܐ)، ܓܘ ܦܠܓܐ ܐܚܪܢܐ ܕܕܘܡܝܐ (ܟܟܢܫܚ ܦܠܓܐ، ܟܦܠܓܚ ܦܠܓܐ ܬܪܝܢܐ ܠܡܨܥ̈ܬܐ ܕܡܨܚܚ̈ܐ ܝܢܐ، ܘܟܦܚܡܚ ܥܡ ܦܘܠܓܐ ܠܐ ܬܠܝܐ ܕܝܠܗ)، ܘܓܘ ܡܫܪܝܬܐ ܐܚܪܬܐ. ܥܢܩܘܕܐ ܕܟܫܢܐ ܐܡܬܝ ܕܟܡܚܠܦܚ ܐܣܘܪܐ (linkage) ܝܢ ܟܡܦܩܚ ܥܣܪܐ ܡܫܬܘܬܦ̈ܢܐ ܝܠܗ ܦܐܪܐ ܕܐܠܓܘܪܝܬܡ، ܠܐ ܕܟܢܫܐ ܟܠܢܝܐ (population) – ܐܗܐ ܝܠܗ ܙܘܗܪܐ ܕ-Hastie ܘܚܒܪ̈ܘܗܝ ܥܠ dendrogram (ܦܣܩܐ 4)، ܘܗܘ ܟܢܦܠ ܥܠ ܟܠ ܐܘܪܚܐ. ܡܫܬܚܠܦ̈ܢܐ ܕܠܒܪ. ܐܢ ܥܢܩܘ̈ܕܐ ܛܘ̈ܦܣܐ ܫܪ̈ܝܪܐ ܝܢܐ، ܐܢܝ̈ ܒܕ ܦܪܫܝ ܐܦ ܓܘ ܡܫܬܚܠܦܢܐ ܕܠܐ ܡܫܬܘܬܦܠܗ ܓܘ ܒܢܝܢܐ ܕܝܠܗܘܢ – ܦܐܪܐ ܕܡܫܝܚܐ ܝܠܗ ܒܬܪ ܗܕܐ، ܡܫܬܚܠܦܢܐ ܕܬܚܘܝܬܐ (background variable) ܕܫܒܝܩܐ ܝܠܗ ܠܒܪ. ܐܗܐ ܦܘܪܫܢܐ ܝܠܗ ܣܗܕܘܬܐ ܝܚܝܕܝܬܐ ܕܐܝܬ ܠܒܘܨܝܐ ܟܢܫܝܐ ܕܥܢܩܘܕܐ "ܟܐܡܪ ܡܕܡ"، ܘܡܒܘܚܪܐ ܝܠܗ ܒܡܐܢ̈ܐ ܥܕ̈ܝܕܐ (ܟܝ ܡܪܒܥܐ، ANOVA) – ܒܫܪܛܐ ܚܕ: ܡܫܬܚܠܦܢܐ ܫܪܝܪܐܝܬ ܦܝܫܐ ܝܠܗ ܠܒܪ. ܐܝܢܐ ܕܟܟܢܫ ܠܦܘܬ ܚܡܫܐ ܡܫܬܚܠܦ̈ܢܐ ܕܚܕ ܡܢܝܗܝ ܝܠܗ ܦܐܪܐ، ܘܒܬܪ ܗܕܐ ܟܡܘܕܥ ܕܥܢܩܘ̈ܕܐ "ܟܡܢܒܝܝ" ܠܗ، ܟܡܘܕܥ ܬܚܡܐ. ܘܡܐ ܕܥܢܩܘܕܐ ܠܐ ܝܠܗ. ܗܘ ܝܠܗ "ܣܛܐܛܝܣܛܝܩܐ ܬܚܘܝܝܬܐ" (Hastie et al., 2009, p. 518) ܕܐܗܐ ܕܘܡܝܐ، ܥܡ ܐܗܐ ܪܘܚܩܐ ܘܐܗܢܝ̈ ܡܫܬܚܠܦ̈ܢܐ؛ ܫܡܐ ܕܒܨܘܝܐ ܟܝܗܒ ܠܗ – "ܦܪ̈ܝܫܐ"، "ܡܫܬܘܬܦ̈ܢܐ ܒܚܝܠܐ" – ܝܠܗ ܦܘܫܩܐ، ܚܬܝܬܐܝܬ ܐܝܟ ܫܡܐ ܕܥܠܬܐ. ܨܘܪܬܐ ܕܡܐܢ̈ܐ ܟܡܝܬܝܐ ܒܩܪܬܐ ܕ-Drucker ܘ-Albrecht ܥܠ topic modeling، ܕܟܡܬܚܘܐ ܐܝܟ ܕܡܘܬ̈ܒܐ (topics) "ܡܢ ܩܕܡ ܐܝܬܝܗܘܢ ܓܘ ܟܬܒܐ ܘܠܐ ܟܗܘܝܐ ܐܝܟ ܥܒܕܐ ܕܩܪܝܢܐ ܘܦܘܫܩܐ" (ܬܡܢ، ܦܣܩܐ 1) – ܗܝ ܟܢܦܠܐ، ܡܠܬܐ ܒܡܠܬܐ، ܥܠ ܥܢܩܘ̈ܕܐ ܕܡܫܬܘܬܦ̈ܢܐ. ܘܐܢ ܥܢܩܘܕܐ ܟܡܬܦܠܚ ܒܬܪ ܗܕܐ ܐܝܟ ܡܫܬܚܠܦܢܐ ܓܘ ܡܘܕܝܠ ܡܕܒܪܢܐ – ܐܝܟ ܬܚܘܝܬܐ (feature)، ܝܢ ܐܝܟ ܪܘܫܡܐ ܕܟܒܨܪ ܥܒܕܐ ܕܪܫܡܐ ܒܪ̈ܘܫܡܐ: Géron ܟܡܚܘܐ ܕܪܫܡܐ ܕ-50 "ܨܘܪ̈ܬܐ ܕܟܡܚܘܝܐ"، ܚܕܐ ܩܐ ܡܨܥܬܐ ܕܟܠ ܥܢܩܘܕܐ، ܒܕܘܟܬܐ ܕ-50 ܓܕܫܢ̈ܝܬܐ، ܟܡܪܡܐ ܚܬܝܬܘܬܐ (accuracy) ܡܢ 74.8% ܠ-84.9% (Chapter 9, "Using Clustering for Semi-Supervised Learning") – ܟܢܦܠܝ ܥܠܗ ܟܠ ܢܡܘ̈ܣܐ ܕܛܘܦܐ (leakage): ܥܢܩܘܕܐ ܡܚܘܫܒܐ ܝܠܗ ܥܠ ܓܘܕܐ ܕܝܘܠܦܢܐ (training set) ܒܠܚܘܕ، ܐܝܟ ܕܪܓܐ ܓܘ ܩܢܝܐ (pipeline).

7. ܘܐܝܟܢܐ ܟܡܬܩܪܐ ܓܘ ܪܫܝܡܬܐ: SPSS Statistics ܘ-SPSS Modeler

SPSS Statistics. ܐܘܪܚܐ ܕ-Pallant، ܕܪܓܐ ܒܬܪ ܕܪܓܐ (Pallant, 2020, Chapter 15, "Procedure for Factor Analysis: Part 1"): Analyze، Dimension Reduction، Factor؛ ܟܡܫܢܚ ܦܣܘ̈ܩܐ (items) ܠ-Variables؛ ܓܘ button Descriptives ܟܪܫܡܚ Initial Solution، Coefficients ܘ-KMO and Bartlett's test of sphericity؛ ܓܘ button Extraction – "ܫܪܪܘܢ ܕ-Principal components ܡܚܘܝܐ ܝܠܗ، ܝܢ ܓܒܘܢ ܚܕܐ ܡܢ ܐܘܪ̈ܚܬܐ ܐܚܪ̈ܢܐ ܕܦܠܛܬܐ (extraction) (e.g., Maximum likelihood)" – ܐܗܐ ܝܠܗ̇ ܪܫܝܡܬܐ ܕܟܦܬܚܐ ܕܦܣܩܐ 3 ܥܣܝܩܐ ܝܠܗ ܒܗ̇، ܘܩܐ ܦܘܫܩܐ ܕܥܠ̈ܠܬܐ (factor analysis) ܟܓܒܚ ܓܘܗ̇ Principal axis factoring ܝܢ Maximum likelihood؛ ܟܪܫܡܚ Scree plot، ܘܓܘ ܡܢܬܐ ܕ-Extract – Based on Eigenvalue (ܢܡܘܣܐ ܕ-Kaiser، ܓܘܒܝܐ ܡܩܕܡܐ) ܝܢ Fixed number of factors، ܕܓܘܗ ܟܡܥܠܚ ܡܢܝܢܐ ܕܦܘܫܩܐ ܒܓܢܒ ܚܕ̈ܕܐ (parallel analysis) ܝܗܒܠܗ؛ ܓܘ button Rotation – Direct Oblimin (ܓܘܒܝܐ ܡܩܕܡܐ ܝܠܗ None)؛ ܓܘ button Options – Sorted by size ܘ-Suppress small coefficients ܥܡ .3. ܣܝܢܬܐܟܣ (syntax) ܕܟܦܐܫ ܡܢ ܗܕܐ ܝܠܗ ܦܘܩܕܢܐ FACTOR ܥܡ /EXTRACTION PC، /CRITERIA MINEIGEN(1) ܘ-/ROTATION OBLIMIN – ܬܠܬܐ ܣܕܪ̈ܐ ܕܟܓܠܝܝ ܡܐ ܕܥܒܝܕܐ ܝܠܗ، ܘܡܛܠ ܗܕܐ ܐܢܝ̈ ܝܢܐ ܡܐ ܕܟܡܚܝܕܚ ܠܡܘܕܥܢܘܬܐ. ܘܡܐ ܟܡܘܕܥܚ، ܠܦܘܬ Pallant: ܐܘܪܚܐ ܕܦܠܛܬܐ، ܢܡܘ̈ܣܐ ܩܐ ܡܢܝܢܐ ܕܥܠ̈ܠܬܐ – "ܘܐܗܐ ܐܠܨܐ ܝܠܗ ܕܐܝܬ ܓܘܗ ܦܘܫܩܐ ܒܓܢܒ ܚܕ̈ܕܐ" – ܐܘܪܚܐ ܕܚܘܕܪܐ (rotation)، ܓܘܡܪܐ ܕܫܘܚܠܦܐ ܡܦܫܩܐ، ܛܝ̈ܡܐ ܕܝܠܢ̈ܝܐ ܩܕܡ ܚܘܕܪܐ ܘܒܬܪܗ، ܠܘܚܐ ܓܡܝܪܐ ܕܛܥ̈ܢܐ (loadings) "ܕܟܡܚܘܐ ܟܠ ܛܝ̈ܡܐ (ܠܐ ܒܣ ܗܢܝ̈ ܕܥܠ ܡܢ .3)"، ܘܓܘ ܚܘܕܪܐ oblimin – pattern matrix ܘ-structure matrix ܓܡܝܪܐܝܬ، ܥܡ ܐܣܘܪ̈ܐ ܕܚܕ̈ܕܐ ܒܝܢܝ ܥܠ̈ܠܬܐ (ibid., "Presenting the Results from Factor Analysis"). Field، ܓܘ ܗܢܝ̈ ܟܘ̈ܐ، ܟܓܒܐ Principal axis factoring، ܟܦܫܩ ܡܐܛܪܝܟܣ ܕܐܣܘܪ̈ܐ ܕܚܕ̈ܕܐ (correlation matrix) ("ܓܘܒܝܐ ܡܩܕܡܐ ܡܘܬܪܢܐ، ܡܛܠ ܕܟܫܩܠ ܐܣܟܝܡܐ ܡܬܩܢܐ… ܐܢ ܡܫܬܚܠܦ̈ܢܐ ܡܫܝ̈ܚܐ ܝܢܐ ܒܣܒ̈ܠܬܐ ܦܪ̈ܝܫܬܐ ܐܗܐ ܠܐ ܒܕ ܡܥܒܕ ܥܠ ܦܘܫܪܐ")، ܘܟܡܦܩܕ "ܠܡܪܗܛܐ ܦܘܫܩܐ ܩܕܡܝܐ ܥܡ ܛܝ̈ܡܐ ܕܝܠܢ̈ܝܐ ܪ̈ܒܐ ܡܢ 1، ܠܡܓܒܐ ܨܘܪܬܐ scree ܘܠܡܦܚܡܐ… ܛܒܐ ܝܠܗ ܠܡܪܗܛܐ ܐܦ ܦܘܫܩܐ ܒܓܢܒ ܚܕ̈ܕܐ. ܐܢ ܢܡܘ̈ܣܐ ܦܪ̈ܝܫܐ ܟܝܗܒܝ ܦܐܪ̈ܐ ܦܪ̈ܝܫܐ، ܒܚܘܪܘܢ communalities ܘܦܣܘܩܘܢ ܒܓܢܒܘܟܘܢ" (Field, 2024, Section 18.6.1, SPSS Tip 18.2). ܬܘܢܝܐ ܠܦܘܬ Field: ܠܘܚܐ ܕܛܥ̈ܢܐ ܡܚܘܕܪ̈ܐ ܕܟܠ ܦܣܘ̈ܩܐ "ܥܡ ܛܝ̈ܡܐ ܥܠ ܡܢ ܢܡܘܣܐ (ܐܢܐ ܒܕ ܗܘܝܢ ܓܒܝܐ .40) ܡܥܘܒ̈ܝܐ"، ܡܢܬܐ ܒܡܐܐ ܕܫܘܚܠܦܐ ܩܐ ܟܠ ܥܠܬܐ ܘܒܠܟܝ ܐܦ ܛܝܡܐ ܕܝܠܢܝܐ، ܡܚܘ̈ܝܢܐ ܕܦܐܝܘܬܐ ܕܝܗܝ̈ܒܬܐ – KMO ܥܡ ܪ̈ܘܫܡܐ ܕ-Kaiser ܘ-Rice (.90 "ܓܘܪܓܐ"، .80 "ܫܘܒܚܢܝܐ"، .70 "ܡܨܥܝܐ"، .60 "ܬܚܬ ܡܢ ܡܨܥܝܐ"، .50 "ܡܣܟܢܐ"، ܘܬܚܬ ܡܢ .5 – ܠܡܟܢܫܐ ܝܬܝܪ ܝܗܝ̈ܒܬܐ ܝܢ ܠܡܚܫܒܐ ܡܢ ܪܝܫ ܥܠ ܡܫܬܚܠܦ̈ܢܐ؛ ܦܣܩܐ 18.5.2) – "ܘܒܨܘܪܬܐ ܓܡܝܪܬܐ، ܠܡܦܩܕܐ ܠܘܚܐ ܕܐܣܘܪ̈ܐ ܕܚܕ̈ܕܐ ܓܘ ܒܝܬ ܓܢܙܐ ܕܝܕܥܬܐ ܦܬܝܚܬܐ ܐܝܟ OSF ܗܟܢܐ ܕܐܢܫ ܡܨܐ ܠܡܬܢܝܐ ܦܘܫܩܐ" (ܦܣܩܐ 18.8)؛ ܡܬܠܐ ܕܝܠܗ: principal axis، oblimin، KMO .92، ܐܪܒܥܐ ܥܠ̈ܠܬܐ ܒܦܘܫܩܐ ܒܓܢܒ ܚܕ̈ܕܐ ܩܘܒܠ ܕܪܓܐ 95 ܡܢ ܡܐܐ (percentile) ܕ-20 ܣܕܪ̈ܐ ܓܕܫܢ̈ܝܐ. ܘܦܬܓܡܐ ܕܟܢܘܫܝܐ ܕܝܠܗ، ܕܐܠܨܐ ܝܠܗ ܕܗܘܐ ܡܕܒܩܐ ܥܠ ܡܚܙܝܬܐ: "ܣܓܝ ܣܓܝ ܦܘܣ̈ܩܢܐ ܟܦܝܫܝ ܩܐ ܒܨܘܝܐ ܕܟܡܦܠܚ ܒܦܘܫܩܐ ܕܥܠ̈ܠܬܐ، ܘܡܛܠ ܗܕܐ ܕܪ̈ܓܐ ܕܚܐܪܘܬܐ (degrees of freedom) ܕܒܨܘܝܐ… ܟܥܐܠܝ ܠܨܘܪܬܐ ܠܐ ܩܠܝܠ: ܢܣܘܢ ܠܡܫܩܠܐ ܦܘܣ̈ܩܢܐ ܚܟܝ̈ܡܐ ܘܕܠܐ ܡܣܒ ܒܐܦ̈ܐ، ܘܩܘܡܘܢ ܠܩܘܒܠ ܢܣܝܘܢܐ ܕܦܘܣ̈ܩܢܐ ܕܟܝܗܒܝ ܠܘܟܘܢ ܦܐܪ̈ܐ ܕܟܒܥܝܬܘܢ ܠܡܩܒܠܐ" (ܦܣܩܐ 18.7.6). ܥܢܩܘ̈ܕܐ – Analyze، Classify (ܟܐ ܡܢ ܩܕܡ ܕܠܐ Pallant، ܕܠܐ ܥܣܝܩܬܐ ܝܠܗ̇ ܒܥܢܩܘ̈ܕܐ؛ ܫܒܝ̈ܠܐ – ܡܘܕܥܢܘܬܐ ܟܠܢܝܬܐ ܥܠ software): Hierarchical Cluster (ܥܡ dendrogram ܬܚܬ Plots ܘܓܒܝܐ ܕܪܘܚܩܐ ܘܕܐܣܘܪܐ ܬܚܬ Method)، K-Means Cluster (ܡܢܝܢܐ ܕܥܢܩܘ̈ܕܐ ܐܠܨܝܐ ܝܠܗ؛ Iterate ܘ-Classify)، ܘ-TwoStep Cluster (ܡܢܝܢܐ ܡܢ ܓܢܗ ܠܦܘܬ BIC ܝܢ AIC). ܡܬܩܢܘܬܐ (standardization) – Descriptives، Save standardized values، ܝܢ Standardize ܓܘ ܟܘܬܐ ܕܕܪ̈ܓܐ – ܝܠܗ̇ ܕܪܓܐ ܦܪܝܫܐ، ܘܐܝܢܝ ܕܟܡܛܥܝܐ ܠܗ̇ ܟܩܒܠܐ ܥܢܩܘ̈ܕܐ ܕܡܫܬܚܠܦܢܐ ܕܐܝܬ ܠܗ ܦܬܝܘܬܐ (range) ܪܒܬܐ. Silhouette ܘ-gap statistic SPSS ܠܐ ܟܚܫܒ؛ TwoStep ܟܡܚܘܐ "ܡܚܘܝܢܐ ܕ-silhouette ܕܐܚܝܕܘܬܐ ܘܦܘܪܫܐ" ܓܘ ܡܦܩܢܐ ܕ-Model Viewer، ܘܩܐ ܫܪܟܐ – ܦܣܩܐ 8.

SPSS Modeler. ܗܘ ܡܕܡ ܓܘ ܩܛܪ̈ܐ (nodes) (ܢܓܕܐ (stream) – ܦܐܬܐ ܕܝܘܠܦܢܐ ܕܡܟܝܢܐ، ܦܣܩܐ 10)، ܘܐܗܐ ܓܗܐ ܡܢ ܬܚܘܝܬܐ ܕ-IBM ܓܢܒܗ̇. ܩܛܪܐ PCA/Factor ܡܢ ܓܘܕܐ ܕ-Modeling ܟܡܩܪܒ "ܬܪܬܝܢ ܐܘܪ̈ܚܬܐ ܕܡ̈ܝܬܐ ܐܠܐ ܦܪ̈ܝܫܬܐ"، ܘܟܟܬܒ ܦܣܩܐ 3 ܒܡ̈ܠܐ ܕܝܠܗ: "PCA ܟܡܨܥܐ ܥܠ ܟܠ ܫܘܚܠܦܐ، ܥܡ ܫܘܚܠܦܐ ܡܫܘܬܦܐ ܘܝܚܝܕܝܐ… ܦܘܫܩܐ ܕܥܠ̈ܠܬܐ ܟܡܨܥܐ ܥܠ ܫܘܚܠܦܐ ܡܫܘܬܦܐ ܒܠܚܘܕ. ܫܘܚܠܦܐ ܝܚܝܕܝܐ ܩܐ ܚܩܠ̈ܐ (fields) ܡܕܡ ܠܐ ܟܡܬܚܫܒ ܓܘ ܫܘܥܪܐ ܕܡܘܕܝܠ" (IBM Corporation, 2016a, p. 182). ܓܘܒܝܐ ܡܩܕܡܐ ܕ-Extraction Method ܝܠܗ Principal Components؛ ܚܠ̈ܦܐ – Unweighted / Generalized Least Squares، Maximum Likelihood، Principal Axis Factoring ܘܐܚܪ̈ܢܐ – ܓܘ ܚܠܬܐ ܕ-Expert، ܘܬܡܐ ܐܦ Extract factors: Eigenvalues over (ܢܡܘܣܐ ܕ-Kaiser) ܝܢ Maximum number، ܕ"ܓܘܒܝܐ ܡܩܕܡܐ ܕܝܠܗ ܝܠܗ ܚܡܫܐ ܥܠ̈ܠܬܐ/ܡܢܘ̈ܬܐ"؛ ܘ-Rotation، ܕܓܘܒܝܐ ܡܩܕܡܐ ܕܝܠܗ "No rotation" (ܦܐܬܐ 183). ܗܢܘ ܗܢܝ̈ ܬܠܬܐ ܓܘܒ̈ܝܐ ܡܩܕ̈ܡܐ ܕ-SPSS Statistics، ܒܫܡ̈ܗܐ ܐܚܪ̈ܢܐ – ܘܕܠܐ scree: Truong، ܕܩܛܪܐ ܝܗܒܠܗ ܚܡܫܐ ܡܢܘ̈ܬܐ، ܨܪܝܠܗ ܓܘ ܐܩܣܠ (Truong, 2026, pp. 445-446). ܩܐ ܥܢܩܘ̈ܕܐ ܬܠܬܐ ܩܛܪ̈ܐ: K-Means – "ܡܢܝܢܐ ܕܥܢܩܘ̈ܕܐ ܕܟܬܚܡܚ… ܓܘܒܝܐ ܡܩܕܡܐ ܝܠܗ 5"، ܘܙܘܗܪܐ ܕܐܠܨܐ ܝܠܗ ܠܡܩܪܐ ܬܪܝܢ ܓܗ̈ܐ: "ܡܘܕܝܠ ܕܟܦܐܫ ܬܠܝܐ ܝܠܗ ܩܠܝܠ ܒܛܟܣܐ ܕܝܗܝ̈ܒܬܐ ܕܝܘܠܦܢܐ (training data). ܛܟܣܐ ܡܢ ܪܝܫ ܕܝܗܝ̈ܒܬܐ ܘܒܢܝܢܐ ܡܢ ܪܝܫ ܕܡܘܕܝܠ ܡܨܝܐ ܝܢܐ ܕܡܝܬܝܝ ܠܡܘܕܝܠ ܚܪܝܐ ܦܪܝܫܐ ܕܥܢܩܘ̈ܕܐ" (IBM Corporation, 2016a, p. 227) – ܪܗܛܐ ܚܕ، ܕܠܐ nstart ܕܦܣܩܐ 8، ܘ-Wendler ܘ-Gröttrup ܟܡܦܫܩܝ ܩܐ ܡܘܢܐ: ܡܨܥܬܐ ܩܕܡܝܬܐ ܝܠܗ̇ "ܛܝ̈ܡܐ ܕܣܕܪܐ ܩܕܡܝܐ ܓܘ ܣܕܪܐ ܕܝܗܝ̈ܒܬܐ"، ܘܗܢܝ̈ ܕܒܬܪܗ̇ – ܣܕܪ̈ܐ ܪ̈ܚܝܩܐ ܡܢ ܟܠ ܡܢ ܡܨܥ̈ܬܐ ܕܦܣܝܩ̈ܐ ܝܢܐ ܡܢ ܩܕܡ؛ ܡܫܬܚܠܦ̈ܢܐ ܕܡܢܝ̈ܢܐ ܡܬܩܢ̈ܐ ܝܢܐ ܡܢ ܩܕܡ، ܡܢ ܓܘ، ܠܦܬܝܘܬܐ 0-1 (Wendler & Gröttrup, 2021, pp. 676-677)؛ TwoStep – ܟܡܬܩܢ ܚܩܠ̈ܐ ܕܡܢܝ̈ܢܐ ܐܝܟ ܓܘܒܝܐ ܡܩܕܡܐ، ܡܨܐ ܠܡܦܩܐ ܛܝ̈ܡܐ ܢܘܟܪ̈ܝܐ ܠܥܢܩܘܕܐ "ܩܠܐ ܣܪܝܩܐ"، ܘܟܓܒܐ ܡܢܝܢܐ ܕܥܢܩܘ̈ܕܐ ܡܢ ܓܢܗ "ܥܠ ܫܬܐܣܬܐ ܕܫܘܢܝܐ ܓܘ ܢܡܘܣܐ ܕܡܘܕܥܢܘܬܐ ܕ-Bayes (BIC) ܐܝܟ ܕܟܡܘܣܦܚ ܥܢܩܘ̈ܕܐ" ܘܒܬܪ ܗܕܐ ܠܦܘܬ ܫܘܢܝܐ ܪܒܐ ܡܢ ܟܠ ܓܘ ܪܘܚܩܐ ܒܨܝܪܐ ܒܝܢܝ ܥܢܩܘ̈ܕܐ؛ ܪܘܚܩܐ – ܠܘܓ-ܡܬܚܙܝܢܘܬܐ (log-likelihood) (ܕܡܢܝ̈ܢܐ ܘܕܓܢ̈ܣܐ ܚܕ ܚܕܐ؛ ܟܣܒܪ ܡܫܬܚܠܦ̈ܢܐ ܡܬܡܫܚ̈ܢܐ ܢܘܪܡܠܝ̈ܐ ܘܠܐ ܬܠܝ̈ܐ) ܝܢ ܕ-Euclid (ܦܐܬܐ 229-230) – ܘܐܦ ܗܘ، ܠܦܘܬ Wendler ܘ-Gröttrup، "ܬܠܝܐ ܝܠܗ ܒܛܟܣܐ ܕܡܕܡ̈ܐ ܓܘ ܣܕܪܐ ܕܝܗܝ̈ܒܬܐ" ܡܛܠ ܥܢܩܘܕܐ ܩܕܡܝܐ (Wendler & Gröttrup, 2021, p. 651)؛ ܘ-Auto Cluster، ܕܟܪܗܛ TwoStep، K-Means ܘ-Kohonen، ܟܡܕܪܓ ܠܦܘܬ silhouette (ܓܘܒܝܐ ܡܩܕܡܐ؛ "ܡܘܕܝܠ̈ܐ ܥܡ silhouette ܕܒܨܪܐ ܡܨܝܚ ܠܡܫܕܝܐ ܠܗܘܢ ܒܫܠܡܘܬܐ")، ܠܦܘܬ ܡܢܝܢܐ ܕܥܢܩܘ̈ܕܐ ܝܢ ܠܦܘܬ ܪܒܘܬܐ ܕܝܠܗܘܢ – ܘܟܫܒܩ "ܚܩܠ̈ܐ ܕܫܘܥܪܐ": "ܐܦ ܐܢ ܠܐ ܟܡܦܠܚܚ ܒܢܝܫܐ (target)، ܡܨܝܚ ܠܬܚܡܐ ܚܩܠܐ ܚܕ ܝܢ ܒܢܘܬ ܕܫܘܥܪܐ ܩܐ ܦܘܚܡܐ ܕܡܘܕܝܠ̈ܐ. ܡܘܬܪܢܘܬܐ ܕܡܘܕܝܠ ܕܥܢܩܘ̈ܕܐ ܡܨܝܚ ܠܡܫܥܪܐ ܒܡܫܘܚܬܐ ܕܟܡܐ (ܛܒܐ ܝܢ ܒܝܫܐ) ܥܢܩܘ̈ܕܐ ܟܦܪܫܝ ܓܘ ܐܗܢܝ̈ ܚܩܠ̈ܐ" (ܦܐܬܐ 70-71) – ܐܗܐ ܝܠܗ ܡܫܬܚܠܦܢܐ ܕܠܒܪ ܕܦܣܩܐ 6، ܒܢܝܐ ܓܘ software. ܘܡܐ ܕܠܝܬ: ܩܛܪܐ Analysis، ܕܟܡܦܚܡ ܡܢܒܝܢ̈ܘܝܬܐ ܥܡ ܫܪܪܐ، "ܡܘܬܪܢܐ ܝܠܗ ܒܣ ܓܘ ܡܘܕܝܠ̈ܐ ܡܕܒܪ̈ܢܐ… ܩܐ ܡܘܕܝܠ̈ܐ ܕܠܐ ܡܕܒܪ̈ܢܐ ܐܝܟ ܐܠܓܘܪܝܬܡ̈ܐ ܕܥܢܩܘ̈ܕܐ ܠܝܬ ܦܐܪ̈ܐ ܫܪ̈ܝܪܐ ܕܗܘܝܐ ܫܬܐܣܬܐ ܩܐ ܦܘܚܡܐ" (IBM Corporation, 2016b, p. 276) – IBM ܟܐܡܪܐ ܦܣܩܐ 1 ܒܦܬܓܡܐ. ܘܟܬܒܐ ܕܝܘܠܦܢܐ ܕ-Modeler ܟܐܡܪ ܦܣ̈ܩܐ 5-6: "ܥܠ ܫܬܐܣܬܐ ܕܢܣܝܘܢܐ ܥܒܕܢܝܐ، ܚܢܢ ܟܚܫܒܚ ܕܦܘܣܩܢܐ ܕܟܡܐ ܥܢܩܘ̈ܕܐ ܠܡܦܣܩܐ ܐܠܨܐ ܝܠܗ ܕܗܘܐ ܡܩܒܠܐ ܩܕܡܝܐ"، ܥܡ silhouette ܐܝܟ ܡܚܘܝܢܐ ܩܐ ܦܐܝܘܬܐ (ܦܐܬܐ 724)؛ "ܠܐ ܒܨܝܪ ܡܢ ܗܕܐ، ܡܢܝܢܐ ܕܥܢܩܘ̈ܕܐ ܐܠܨܐ ܝܠܗ ܕܗܘܐ ܦܣܝܩܐ ܥܠ ܫܬܐܣܬܐ ܕܝܕܥܬܐ ܥܒܕܢܝܬܐ" (ܦܐܬܐ 750)؛ ܘ"ܐܘܪ̈ܚܬܐ ܕܥܢܩܘ̈ܕܐ ܠܐ ܟܝܗܒܝ ܚܕ ܥܘܕܪܢܐ ܓܘ ܡܨܚܚܬܐ ܕܬܚܘܝܬܐ ܦܐܬܐ ܩܐ ܟܠ ܥܢܩܘܕܐ. ܒܨܘܝܬܐ ܝܢ ܒܨܘܝܐ ܐܠܨܐ ܝܠܗ ܕܦܗܡܝ ܒܓܢܒܗܘܢ ܐܝܟܢܐ ܠܡܬܚܘܝܐ ܟܠ ܥܢܩܘܕܐ ܒܐܘܪܚܐ ܛܒܬܐ ܡܢ ܟܠ" (Wendler & Gröttrup, 2021, p. 650). ܦܘܫܩܐ ܒܓܢܒ ܚܕ̈ܕܐ، gap statistic ܘܩܝܡܘܬܐ ܒܝܢܝ ܐܠܓܘܪܝܬܡ̈ܐ – ܐܝܬ ܓܘ ܣܕܪܐ ܕ-R.

8. ܘܐܝܟܢ ܟܥܒܕܚ ܐܗܐ ܒ-R

ܣܩܪܝܦܬ ܕܒܬܪ ܗܕܐ ܪܗܛܠܗ ܐܝܟ ܕܐܝܬܘܗܝ ܥܠ ܦܐܝܠ ܕܡܬܠܐ (sample data file) ܕܦܐܬܐ – DATA_unsup_example_eng.xlsx (ܘܓܘ ܥܒܪܝܐ _heb.xlsx؛ ܬܪܘܝܗܝ ܓܘ qa/ml/): 300 ܡܦܢܝ̈ܢܐ (respondents) ܓܘ ܒܘܨܝܐ ܕܫܘܐ̈ܠܐ (survey) ܕܥܕܬܐ، 12 ܦܣܘ̈ܩܐ ܕ-Likert (ܕܝܠܝܘܬܐ، ܗܝܡܢܘܬܐ، ܫܠܡܘܬܐ – ܐܪܒܥܐ ܦܣܘ̈ܩܐ ܩܐ ܟܠ ܚܕ)، ܐܪܒܥܐ ܡܫܬܚܠܦ̈ܢܐ ܕܗܘܦܟܐ (ܫܥ̈ܐ ܕܬܫܡܫܬܐ ܡܢ ܨܒܝܢܐ، ܓܕ̈ܫܐ، ܫܒܒ̈ܐ ܝܕܝܥ̈ܐ، ܫܢ̈ܐ ܓܘ ܥܕܬܐ)، ܬܪܝܢ ܡܫܬܚܠܦ̈ܢܐ ܕ"ܩܠܐ ܣܪܝܩܐ" ܫܘ̈ܝܐ، ܘܦܐܪܐ ܕܡܫܝܚܐ ܝܠܗ ܒܬܪ ܗܕܐ – ܫܒܩܬܐ. ܝܗܝ̈ܒܬܐ ܥܒܝܕ̈ܬܐ ܝܢܐ، ܘܒܪܝ̈ܬܐ ܝܢܐ ܡܢ ܬܠܬܐ ܥܠ̈ܠܬܐ ܘܡܢ ܬܠܬܐ ܬܚܘܝ̈ܬܐ (profiles) – ܗܟܢܐ ܕܝܕܥܚ ܡܐ ܝܠܗ ܦܘܢܝܐ ܘܚܙܚ ܐܝܢܐ ܢܡܘܣܐ ܟܡܨܚܚ ܠܗ؛ ܬܚܘܝܬܐ ܫܪܝܪܬܐ ܠܝܬ ܓܘ ܦܐܝܠ. ܚܒ̈ܫܐ: readxl، psych (ܦܘܫܩܐ ܒܓܢܒ ܚܕ̈ܕܐ، ܥܠ̈ܠܬܐ)، cluster (silhouette، gap statistic)، mclust – ܚܘ̈ܠܛܐ ܘܡܚܘܝܢܐ ܕ-Rand (Scrucca et al., 2016). ܫܒܝܠܐ (path) – ܠܡܚܠܦܐ. ܡܥܝܪ̈ܢܘܝܬܐ ܓܘ ܢܓܠܝܙܝܐ ܝܢܐ، ܡܛܠ ܕܓܘܫܡܐ ܡܚܘܝܐ ܝܠܗ ܡܢ ܣܡܠܐ ܠܝܡܝܢܐ.

library(readxl); library(psych); library(cluster); library(mclust)
d <- read_excel("C:/Rdata/DATA_unsup_example_eng.xlsx")
items <- d[, c("belong1","belong2","belong3","belong4","trust1","trust2","trust3","trust4",
               "safe1","safe2","safe3","safe4")]
beh   <- d[, c("volunteer_h","events","neighbours","years")]

# --- Step 1: PCA on the 12 items - standardised (scale = TRUE), because units differ ------
pca <- prcomp(items, scale. = TRUE)
ev  <- pca$sdev^2                                   # eigenvalues of the correlation matrix
round(rbind(eigenvalue = ev, proportion = ev / sum(ev), cumulative = cumsum(ev) / sum(ev)), 3)
cat("Kaiser (eigenvalue > 1) keeps:", sum(ev > 1), "components\n")

# --- Step 2: how many? parallel analysis (Horn) for components AND factors ----------------
set.seed(20260922)
pa <- fa.parallel(items, fa = "both", n.iter = 100, plot = FALSE)
cat("parallel analysis: components =", pa$ncomp, " factors =", pa$nfact, "\n")

# --- Step 3: PCA is not factor analysis: same items, two models ---------------------------
pc3 <- principal(items, nfactors = 3, rotate = "oblimin")   # components: total variance
fa3 <- fa(items, nfactors = 3, fm = "ml", rotate = "oblimin") # common factors: shared variance
print(pc3$loadings, cutoff = .30)
print(fa3$loadings, cutoff = .30)
cat("factor correlations (EFA):\n"); print(round(fa3$Phi, 2))
cat("communalities - PCA vs EFA (first 4 items):\n")
print(round(rbind(PCA = pc3$communality[1:4], EFA = fa3$communality[1:4]), 2))

# --- Step 4: k-means on the 4 behaviour variables - scaled first ---------------------------
X <- scale(beh)
set.seed(20260922)
km <- lapply(2:6, function(k) kmeans(X, centers = k, nstart = 25, iter.max = 100))
inertia <- sapply(km, function(m) m$tot.withinss)              # the "elbow" curve
sil     <- sapply(km, function(m) mean(silhouette(m$cluster, dist(X))[, 3]))
gap     <- clusGap(X, FUN = function(x, k) kmeans(x, k, nstart = 25), K.max = 6, B = 100)
round(cbind(k = 2:6, inertia = inertia, silhouette = sil), 3)
print(gap, method = "Tibs2001SEmax")

# --- Step 5: the chosen solution - and what it looks like in the original units ---------
k3 <- km[[2]]                                                   # k = 3
cat("sizes:", k3$size, "\n")
round(aggregate(beh, by = list(cluster = k3$cluster), FUN = mean), 1)

# --- Step 6: does another algorithm agree? hierarchical (Ward) vs k-means ----------------
hc  <- hclust(dist(X), method = "ward.D2")
hc3 <- cutree(hc, k = 3)
table(kmeans = k3$cluster, ward = hc3)
cat("adjusted Rand index, k-means vs Ward:", round(adjustedRandIndex(k3$cluster, hc3), 3), "\n")

# --- Step 7: stability - split-half: cluster each half, predict the other by nearest centre
set.seed(20260922)
h  <- sample(rep(c(TRUE, FALSE), length.out = nrow(X)))
kA <- kmeans(X[h, ], 3, nstart = 25); kB <- kmeans(X[!h, ], 3, nstart = 25)
nearest <- function(x, centres) apply(x, 1, function(r) which.min(colSums((t(centres) - r)^2)))
predB <- nearest(X[!h, ], kA$centers)                          # A's rule applied to B
cat("split-half ARI (A's centres on B vs B's own clusters):",
    round(adjustedRandIndex(predB, kB$cluster), 3), "\n")

# --- Step 8: the control - two pure-noise variables get clustered too ---------------------
Z <- scale(d[, c("noise_a", "noise_b")])
set.seed(20260922)
kz <- kmeans(Z, 3, nstart = 25)
cat("noise, k = 3: sizes", kz$size, " silhouette",
    round(mean(silhouette(kz$cluster, dist(Z))[, 3]), 3), "\n")
gz <- clusGap(Z, FUN = function(x, k) kmeans(x, k, nstart = 25), K.max = 6, B = 100)
cat("noise, gap statistic chooses k =", maxSE(gz$Tab[, "gap"], gz$Tab[, "SE.sim"], method = "Tibs2001SEmax"), "\n")

# --- Step 9: model-based alternative - Gaussian mixture, number of components by BIC ------
mc <- Mclust(beh, G = 1:6, verbose = FALSE)
cat("Mclust: model", mc$modelName, "with G =", mc$G, "components; ARI vs k-means",
    round(adjustedRandIndex(mc$classification, k3$cluster), 3), "\n")

# --- Step 10: the cluster is a hypothesis, not a finding: an outcome that was NOT clustered on
tab <- table(cluster = k3$cluster, moved_out = d$moved_out)
round(prop.table(tab, 1), 2)
chisq.test(tab)

ܡܐ ܕܦܠܛܠܗ (R 4.3.3، psych 2.4.1، cluster 2.1.6، mclust 6.0.1؛ ܡܦܩܢܐ ܓܡܝܪܐ – qa/ml/unsup_out.txt). ܦܣܘ̈ܩܐ: ܛܝ̈ܡܐ ܕܝܠܢ̈ܝܐ (eigenvalues) ܕ-12 ܦܣܘ̈ܩܐ – 2.88، 2.12، 1.80، ܘܒܬܪ ܗܕܐ 0.85 ܘܠܬܚܬ؛ ܬܠܬܐ ܩܕܡ̈ܝܐ ܟܡܦܫܩܝ 56.7% ܡܢ ܫܘܚܠܦܐ. ܢܡܘܣܐ ܕ-Kaiser ܟܫܒܩ ܬܠܬܐ، ܘܗܟܢܐ ܐܦ ܦܘܫܩܐ ܒܓܢܒ ܚܕ̈ܕܐ – ܬܠܬܐ ܡܢܘ̈ܬܐ ܘܬܠܬܐ ܥܠ̈ܠܬܐ؛ ܟܐ ܐܢܝ̈ ܫܠܡܝ، ܡܛܠ ܕܒܢܝܢܐ ܕܟܝܐ ܝܠܗ، ܘܓܘ ܦܢܩܝܬܐ ܕܫܘܐ̈ܠܐ ܫܪܝܪܬܐ ܠܐ ܒܟܠ ܙܒܢܐ ܒܕ ܫܠܡܝ. ܗܢܝ̈ ܦܣܘ̈ܩܐ ܓܘ ܬܪܝܢ ܡܘܕܝܠ̈ܐ، ܬܪܘܝܗܝ ܒܚܘܕܪܐ oblimin: ܟܠ ܦܣܘܩܐ ܟܛܥܢ ܥܠ ܥܠܬܐ ܕܝܠܗ ܓܘ ܬܪܘܝܗܝ – ܐܠܐ ܛܥ̈ܢܐ ܕܡܢܘ̈ܬܐ ܪ̈ܡܐ ܝܢܐ ܒܢܘܬ (0.80، 0.79، 0.68، 0.75 ܩܐ ܕܝܠܝܘܬܐ، ܩܘܒܠ 0.74، 0.71، 0.55، 0.63)، ܘ-communalities ܕܐܪܒܥܐ ܦܣܘ̈ܩܐ ܕܕܝܠܝܘܬܐ 0.65، 0.63، 0.48، 0.55 ܓܘ PCA ܩܘܒܠ 0.56، 0.51، 0.32، 0.39 ܓܘ ܦܘܫܩܐ ܕܥܠ̈ܠܬܐ؛ ܬܠܬܐ ܡܢܘ̈ܬܐ "ܟܡܦܫܩܝ" 56.7% ܘܬܠܬܐ ܥܠ̈ܠܬܐ 43.0%. ܦܘܪܫܢܐ ܝܠܗ ܫܘܚܠܦܐ ܝܚܝܕܝܐ ܕܦܣܘ̈ܩܐ – ܛܥܝܘܬܐ ܕܡܫܘܚܬܐ – ܕ-PCA ܟܡܥܠܐ ܠܡܢܘ̈ܬܐ ܘܦܘܫܩܐ ܕܥܠ̈ܠܬܐ ܟܫܒܩ ܠܒܪ؛ ܐܗܐ ܝܠܗ ܦܣܩܐ 3 ܒܡܢܝܢ̈ܐ. ܐܣܘܪ̈ܐ ܕܚܕ̈ܕܐ ܕܥܠ̈ܠܬܐ .17، .09، .26 – ܠܐ ܨܦܪܐ، ܘܡܛܠ ܗܕܐ ܡܨܠܝܢܐ. ܗܘܦܟܐ: ܥܠ ܐܪܒܥܐ ܡܫܬܚܠܦ̈ܢܐ ܡܬܩܢ̈ܐ، inertia ܟܢܚܬܐ 498، 274، 245، 227، 200 ܩܐ k ܡܢ 2 ܗܠ 6 – elbow ܓܘ 3؛ silhouette .47، .48، .44، .44، .37 – ܥܠܝܐ ܓܘ 3، ܐܠܐ 2 ܘ-4 ܩܪ̈ܝܒܐ ܝܢܐ؛ gap statistic، ܓܘ 100 ܢܣܝܘ̈ܢܐ ܕܕܡܝܘܬܐ (simulations) – 3. ܪ̈ܒܘܬܐ ܕܥܢܩܘ̈ܕܐ 71، 120 ܘ-109، ܘܡܨܥܝ̈ܬܐ ܕܝܠܗܘܢ ܒܚܕܝ̈ܘܬܐ ܫܪ̈ܫܝܬܐ: 15 ܫܥ̈ܐ ܕܬܫܡܫܬܐ ܡܢ ܨܒܝܢܐ، 22 ܓܕ̈ܫܐ، 25 ܫܒܒ̈ܐ ܘ-20 ܫܢ̈ܐ ܓܘ ܥܢܩܘܕܐ ܚܕ؛ 1.8، 3، 6 ܘ-4.5 ܓܘ ܬܪܝܢܐ؛ 8، 10، 14 ܘ-12.5 ܓܘ ܬܠܝܬܝܐ – ܡܫܬܘܬܦ̈ܢܐ ܒܚܝܠܐ، ܦܪ̈ܝܫܐ، ܒܝܢܝ ܒܝܢܝ؛ ܫܡ̈ܗܐ ܕܝܠܢ ܝܢܐ، ܠܐ ܕܐܠܓܘܪܝܬܡ. ܩܝܡܘܬܐ: Ward ܟܫܠܡ ܥܡ k-ܡܨܥܝ̈ܬܐ ܥܠ 298 ܡܢ 300 ܡܫܬܘܬܦ̈ܢܐ (ARI .98)؛ ܦܘܠܓܐ ܕܝܠܝܦܐ ܝܠܗ ܥܠ ܦܠܓܐ ܚܕ ܘܡܦܠܚܐ ܝܠܗ ܥܠ ܬܪܝܢܐ ܟܫܠܡ ܥܡ ܦܘܠܓܐ ܠܐ ܬܠܝܐ ܕܗܘ ܦܠܓܐ ܒ-ARI .96؛ ܚܘܠܛܐ ܕ-Gauss ܟܓܒܐ ܒ-BIC ܬܠܬܐ ܡܢܘ̈ܬܐ (ܡܘܕܝܠ VEI) ܘܟܫܠܡ ܥܡ k-ܡܨܥܝ̈ܬܐ ܓܡܝܪܐܝܬ (ARI 1). ܩܠܐ ܣܪܝܩܐ: ܥܠ ܬܪܝܢ ܡܫܬܚܠܦ̈ܢܐ ܫܘ̈ܝܐ ܕܠܐ ܚܕ ܒܢܝܢܐ، k-ܡܨܥܝ̈ܬܐ ܟܡܕܥܪ ܬܠܬܐ ܥܢܩܘ̈ܕܐ ܒܪ̈ܒܘܬܐ 108، 104 ܘ-88 ܥܡ silhouette .39 – ܡܢܝܢܐ ܕܓܘ ܦܢܩܝܬܐ ܕܫܘܐ̈ܠܐ ܫܪܝܪܬܐ ܗܘܘ ܡܘܕܥܝ ܠܗ؛ gap statistic ܟܓܒܐ k=1. ܐܗܐ ܝܠܗ ܡܠܬܐ ܟܠܗ̇. ܘܗܘܦܘܬܣܝܣ: ܫܒܩܬܐ، ܕܠܐ ܡܫܬܘܬܦܠܗ̇ ܓܘ ܥܢܩܘܕܐ، ܦܪܝܫܬܐ ܝܠܗ̇ ܒܝܢܝ ܥܢܩܘ̈ܕܐ – 7%، 30% ܘ-21% (χ²(2) = 14.0, p < .001)؛ ܐܗܐ ܝܠܗ̇ ܣܗܕܘܬܐ ܕܥܢܩܘ̈ܕܐ ܟܐܡܪܝ ܡܕܡ، ܘܐܗܐ ܝܠܗ ܕܪܓܐ ܕܒܗ ܟܡܘܕܥܚ ܠܗܘܢ: ܦܘܪܫܢܐ ܓܘ ܡܫܬܚܠܦܢܐ ܕܠܒܪ، ܠܐ "ܬܠܬܐ ܛܘ̈ܦܣܐ ܓܘ ܥܕܬܐ".

ܛܥܝ̈ܘܬܐ ܕܡܬܚܙܝܢ ܣܓܝ (Common mistakes)

ܠܡܘܕܥܐ "ܦܘܫܩܐ ܕܥܠ̈ܠܬܐ" ܥܠ ܡܢܘ̈ܬܐ ܪ̈ܫܝܬܐ. ܓܘܒܝܐ ܡܩܕܡܐ ܕ-SPSS – "ܐܘܦܣ!" (Field). ܡܢܘ̈ܬܐ ܐܝܬ ܓܘܝܗܝ ܛܥܝܘܬܐ ܕܡܫܘܚܬܐ، communalities ܡܢܦܚ̈ܬܐ ܝܢܐ، ܘܠܐ ܡܨܝܚ ܠܡܡܠܠܐ ܥܠ ܒܢܝܢܐ ܟܣܝܐ (ܦܣܩܐ 3). ܩܐ ܦܘܫܩܐ ܕܥܠ̈ܠܬܐ – Principal axis ܝܢ Maximum likelihood، ܚܘܕܪܐ ܡܨܠܝܢܐ.

ܠܡܫܒܩܐ ܠܛܝܡܐ ܕܝܠܢܝܐ 1 ܕܦܣܩ. "ܡܢ ܐܘܪ̈ܚܬܐ ܕܒܨܝܪ ܚܬܝܬ̈ܐ ܡܢ ܟܠ"، ܘܒܟܠ ܐܘܪܚܐ ܓܘܒܝܐ ܡܩܕܡܐ ܓܘ ܬܪܝܢ software؛ ܓܘ ܡܬܠܐ ܕ-PANAS ܕ-Pallant ܗܘ ܟܝܗܒ ܐܪܒܥܐ ܡܢܘ̈ܬܐ ܩܐ ܣܒܠܬܐ ܕܬܪܬܝܢ. ܦܘܫܩܐ ܒܓܢܒ ܚܕ̈ܕܐ – ܘܐܢ ܠܝܬ، ܠܦܚܘܬ scree ܥܡ ܕܝܢܐ ܡܥܠܠܐ (ܠܘܚܐ 2).

ܠܡܟܢܫܐ ܕܠܐ ܡܬܩܢܘܬܐ. ܡܫܬܚܠܦܢܐ ܓܘ ܦܬܝܘܬܐ 0-100 ܓܢܒ ܡܫܬܚܠܦ̈ܢܐ ܓܘ ܦܬܝܘܬܐ 1-5: ܥܢܩܘ̈ܕܐ (clusters) ܕܡܫܬܚܠܦܢܐ ܪܒܐ ܝܢܐ (Géron، ܨܚܚܐ 9؛ Gutierrez، ܨܚܚܐ 8). ܓܘ SPSS ܡܬܩܢܘܬܐ (standardization) ܕܪܓܐ ܦܪܝܫܐ ܝܠܗ̇.

ܠܡܓܒܐ k ܠܦܘܬ inertia – ܝܢ ܠܦܘܬ ܥܝܕܐ. Inertia ܒܟܠ ܙܒܢܐ ܟܢܚܬܐ؛ "ܬܠܬܐ ܥܢܩܘ̈ܕܐ" ܥܝܕܐ ܝܠܗ. Silhouette ܘ-gap statistic، ܘܦܘܢܝܐ ܐܦ ܠ"ܚܕ" (ܦܣܩܐ 5).

ܠܡܩܪܐ ܠ-dendrogram "ܒܢܝܢܐ ܕܝܗܝ̈ܒܬܐ". ܗܝ ܦܐܪܐ ܕܐܠܓܘܪܝܬܡ ܝܠܗ̇؛ ܐܣܘܪܐ (linkage) ܐܚܪܢܐ، ܥܣܪܐ ܡܫܬܘܬܦ̈ܢܐ ܒܨܝܪ – ܐܝܠܢܐ ܐܚܪܢܐ. "ܐܘܪ̈ܚܬܐ ܕܕܪ̈ܓܐ ܟܡܛܠܨܝ ܒܢܝܢܐ ܕܕܪ̈ܓܐ ܐܢ ܐܝܬ ܘܐܢ ܠܝܬ" (Hastie ܘܚܒܪ̈ܘܗܝ، ܦܐܬܐ 522).

ܠܡܫܪܪܐ ܥܢܩܘܕܐ ܓܘ ܡܫܬܚܠܦܢܐ ܕܒܢܐ ܠܗ. "ܥܢܩܘ̈ܕܐ ܦܪ̈ܝܫܐ ܝܢܐ ܪܫܝܡܐܝܬ ܓܘ ܬܫܡܫܬܐ ܡܢ ܨܒܝܢܐ" – ܐܡܬܝ ܕܬܫܡܫܬܐ ܡܢ ܨܒܝܢܐ ܚܕ ܡܢ ܡܫܬܚܠܦ̈ܢܐ ܕܥܢܩܘܕܐ ܝܠܗ̇ – ܝܠܗ ܬܘܢܝܐ ܕܬܚܡܐ. ܫܘܪܪܐ ܒܣ ܓܘ ܡܫܬܚܠܦܢܐ ܕܦܝܫܐ ܝܠܗ ܠܒܪ (ܦܣܩܐ 6).

ܠܡܬܠ ܫܡ̈ܗܐ ܘܠܡܛܥܝܐ ܕܐܢܝ̈ ܕܝܘܟܘܢ ܝܢܐ. "ܦܪ̈ܝܫܐ" ܝܠܗ ܦܘܫܩܐ ܕܡܨܥܝ̈ܬܐ، ܠܐ ܬܚܘܝܬܐ ܕܐܠܓܘܪܝܬܡ ܓܠܐ ܠܗ̇؛ ܘܚܙ̈ܝܬܐ ܕܓܘ ܡܨܥܬܐ – ܕܚܘܠܛܐ ܕ-Gauss ܗܘܐ ܝܗܒܐ ܠܗܝ̈ܢ 50/50 – ܟܓܐܒܝ ܓܘ ܪܘܫܡܐ (label).

ܠܡܦܫܩܐ ܪ̈ܘܚܩܐ ܓܘ t-SNE. ܨܘܪܬܐ ܩܐ ܚܙܝܐ ܝܠܗ̇؛ ܪܒܘܬܐ ܕܥܢܩܘܕܐ ܘܪܘܚܩܐ ܕܝܠܗ ܡܢ ܐܚܪ̈ܢܐ ܓܘ ܨܘܪܬܐ ܬܠܝ̈ܐ ܝܢܐ ܒܦܪܡܛܪ̈ܐ ܘܠܐ ܝܢܐ ܡܫܟܚܢܘܬܐ (ܦܣܩܐ 2).

ܪܫܝܡܬܐ ܕܒܘܚܪܢܐ (checklist) – ܩܕܡ ܕܡܘܕܥܚ ܡܢܘ̈ܬܐ، ܥܠ̈ܠܬܐ ܝܢ ܥܢܩܘ̈ܕܐ

ܡܒܘ̈ܥܐ

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Gutierrez, D. D. (2015). Machine learning and data science: An introduction to statistical learning methods with R. Technics Publications.

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

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