PCA Weighting - Principal Component Analysis based objective weighting

Overview

PCA weighting extracts weights from the structure of the data itself - criteria that explain more variance (are less redundant) get higher weight. Requires m ≥ n for a non-singular covariance matrix. Results can be sensitive to dataset changes (new alternatives shift weights).

Output

Weight, higher is better

Data

Crisp, complete numeric matrix

Weights

Derived internally, no weight source needed

Size

2+ alternatives, 3-12 criteria works best

Used for

Any (objective weighting)

How it works

1. 1

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

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

   Hotelling 1933, p.419 (standardisation; pending PDF page verification)

2. 2

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

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

   Pearson 1901, p.562 (principal component extraction; pending PDF page verification)

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.

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

   PCA-MCDM weight adaptation (pending PDF page verification)

Look elsewhere when

Assumptions to verify

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

Edge cases and pitfalls

• Singular covariance matrix (m < n or perfectly correlated criteria): eigendecomposition fails. Regularise or use Moore-Penrose pseudo-inverse.

Works with

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