PCA Weighting - Principal Component Analysis based objective weighting

1. 1. Z-score standardise each criterion column (mean=0, σ=1).

   $$z_{ij}=(x_{ij}-{\bar{x}}_j)/{\sigma }_j$$

2. 2. Compute eigenvalue decomposition of the correlation matrix R = Z^T Z / (m−1). Retain q components (Kaiser: eigenvalue ≥ 1).

   $$R{v}_k={\lambda }_k{v}_k,k=1,\dots ,n;q=|\{k:{\lambda }_k\ge 1\}|$$

3. 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.

   $$w_j=\frac{{\sum }_{k=1}^q\frac{{\lambda }_k}{{\sum }_l{\lambda }_l}{v}_{kj}^2}{{\sum }_{j^{\prime }}{\sum }_{k=1}^q\frac{{\lambda }_k}{{\sum }_l{\lambda }_l}{v}_{k{j}^{\prime }}^2}$$