PCA Weighting: Principal Component Analysis based objective weighting

Overview

Weight_Objective (PCA variance explained, eigenvector-based). Output typically weight (higher value = preferred).

Strengths

• •Method-specific: Weight_Objective (PCA variance explained, eigenvector-based)

Limitations

• •Assumes: Decision matrix exists with measurable criteria
• •Assumes: Sufficient inter-alternative variation per criterion

Assumptions to verify

• •Decision matrix exists with measurable criteria
• •Sufficient inter-alternative variation per criterion

When not to use

• •No data variation (constant criterion) → weight degenerates
• •Expert judgment is the actual driver → use subjective weighting

Edge cases

• •See F.steps and D.parameters for PCA-WEIGHT-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Hatalı: 'PCA-WEIGHT bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix exists with measurable criteria
• •Hatalı: 'PCA-WEIGHT bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Sufficient inter-alternative variation per criterion
• •Hatalı: PCA-WEIGHT'yi 'No data variation (constant criterion) → weight degenerates' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: PCA-WEIGHT'yi 'Expert judgment is the actual driver → use subjective weighting' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 1: Z-score standardise each criterion column (mean=0, σ=1). Formül: z_{ij}=(x_{ij}-\bar{x}_j)/\sigma_j Anchor: Hotelling 1933, p.419 (standardisation; pending PDF page verification)
2. 2.Adım 2 (F2): Step 2: Compute eigenvalue decomposition of the correlation matrix R = Z^T Z / (m−1). Retain q components (Kaiser: eigenvalue ≥ 1). Formül: R\,v_k = \lambda_k\,v_k,\quad k=1,\ldots,n;\quad q=|\{k:\lambda_k\ge 1\}| Anchor: Pearson 1901, p.562 (principal component extraction; pending PDF page verification)
3. 3.Adım 3 (F3): Step 3: Weight criterion j by its total contribution across retained components: w_j = Σ_{k=1}^{q} (λ_k/Σλ) · v_{kj}². Normalise Σ w_j = 1. Formül: w_j = \frac{\sum_{k=1}^{q}\frac{\lambda_k}{\sum_l\lambda_l}v_{kj}^2}{\sum_{j'}\sum_{k=1}^{q}\frac{\lambda_k}{\sum_l\lambda_l}v_{kj'}^2} Anchor: PCA-MCDM weight adaptation (pending PDF page verification)

Commonly paired with

• •PCA-WEIGHT + TOPSIS (high)
• •PCA-WEIGHT + VIKOR (high)
• •PCA-WEIGHT + EDAS (high)
• •PCA-WEIGHT + WASPAS (high)
• •PCA-WEIGHT + MARCOS (high)

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