ICA-TOPSIS Variants Analysis

• ICA-TOPSIS variants integrate Independent Component Analysis with TOPSIS to address bias from correlated decision criteria.
• They first recover latent, statistically independent sources via blind source separation before applying modified TOPSIS ranking using Euclidean or Mahalanobis distances.
• Empirical results on synthetic and real-world data demonstrate that these methods outperform classical TOPSIS by reducing dependency-induced ranking bias.

ICA-TOPSIS variants comprise multicriteria decision-making methodologies that integrate Independent Component Analysis (ICA) with Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), specifically addressing the challenges posed by dependent or correlated criteria in decision matrices. These approaches reformulate the ranking process by first estimating independent latent criteria through blind source separation, thereby providing robust, unbiased alternative rankings even when criteria are statistically dependent.

1. Problem Context: Dependence in Multicriteria Decision Making

Classical multicriteria decision-making (MCDM) methods, including TOPSIS, commonly assume that evaluation criteria are statistically independent. This independence is often unrealistic; in practical applications, observed criteria exhibit nontrivial correlations—such as socioeconomic indicators that covary—resulting in biased or unbalanced rankings. ICA-TOPSIS variants directly confront this issue by modeling observed criteria as linear mixtures of latent, independent sources and employing ICA to recover these sources prior to ranking alternatives.

2. Mathematical Foundation and Blind Source Separation

Let $\mathcal{A} = [\mathcal{A}_1, ..., \mathcal{A}_K]$ denote the set of alternatives and $\mathcal{C} = [\mathcal{C}_1, ..., \mathcal{C}_M]$ the observed criteria. The evaluations form the matrix:

$\mathbf{V} = [v_{m,k}] \in \mathbb{R}^{M \times K}$

The observed decision matrix is modeled as a noisy linear mixture:

$\mathbf{V} = \mathbf{A}\mathbf{L} + \mathbf{G}$

where $\mathbf{A}$ is an unknown mixing matrix, $\mathbf{L}$ contains the independent latent criteria, and $\mathbf{G}$ denotes additive noise. The goal is to estimate $\mathbf{L}$ given $\mathbf{V}$, analogously to a blind source separation (BSS) task. ICA is employed for this estimation, producing an unmixing (separating) matrix $\mathbf{B}$ such that:

$\hat{\mathbf{L}} = \mathbf{B} \mathbf{V}$

Empirical ICA algorithms include FastICA (maximizing non-Gaussianity) and JADE (joint diagonalization of cumulants), each susceptible to permutation and scaling ambiguities. Systematic ambiguity adjustment is performed by permuting and scaling components to maximize correspondence between estimated sources and observed criteria, typically making the diagonal of the adjusted mixing matrix dominant and nonnegative.

3. TOPSIS on Independent Latent Criteria

After ICA-based recovery and adjustment, the latent criteria form the basis for ranking. The standard TOPSIS procedure is modified to operate in the independent criteria space:

1. Normalization: $u_{m,k} = l_{m,k} / \sqrt{\sum_{k=1}^K l_{m,k}^2}$
2. Weighted aggregation: $p_{m,k} = w_m u_{m,k}$, with $\mathbf{w}$ as criteria weights.
3. Identification of Positive Ideal Alternative (PIA) and Negative Ideal Alternative (NIA): $p_m^+ = \max_k p_{m,k}$, $p_m^- = \min_k p_{m,k}$.
4. Distance calculation: Euclidean distances to PIA and NIA for each alternative.
5. Closeness coefficient computation:

$r_k = \frac{D_k^-}{D_k^+ + D_k^-}$

where $D_k^+, D_k^-$ are respective Euclidean distances. Alternatives are ranked by $r_k$ in descending order.

4. ICA-TOPSIS and ICA-TOPSIS-M: Variant Descriptions

ICA-TOPSIS

This variant performs ICA, ambiguity adjustment, and standard TOPSIS on the latent, independent criteria. All normalization, weighting, and ideal solution determination are executed in the transformed (latent) space, yielding rankings that reflect the independent contribution of each criterion.

ICA-TOPSIS-M

An extension that incorporates Mahalanobis distance to account for variance and covariance among criteria, ICA-TOPSIS-M first computes PIA/NIA directly from the latent criteria without prior normalization. These ideals are then transformed back into the original observed criterion space via the mixing matrix. The alternative rankings are computed using Mahalanobis distance:

$$DM_k^+ = \sqrt{ (\mathbf{u}_k - \mathbf{u}^+)^T \Delta^T \Sigma_U^{-1} \Delta (\mathbf{u}_k - \mathbf{u}^+) }$$

where $\Delta$ is a diagonal matrix of criterion weights and $\Sigma_U$ is the covariance of the observed data. This process maintains the integrity of the ideal solutions derived from independent latent sources, while appropriately evaluating alternatives in the presence of correlation among observed criteria.

Variant	Criterion Space	Distance Metric
ICA-TOPSIS	Latent	Euclidean
ICA-TOPSIS-M	Original	Mahalanobis

5. Experimental Findings: Synthetic and Real Data

Empirical validation was performed using both synthetic data—generated under controlled latent-source mixtures—and real-world data (e.g., countries ranked by forest area, GNI per capita, and life expectancy). Key results include:

• ICA-TOPSIS and ICA-TOPSIS-M consistently outperform classic TOPSIS and TOPSIS-M on correlated criteria, especially as the number of alternatives increases and noise remains moderate or low.
• Performance metrics: Kendall's Tau ($\tau$), Pearson correlation ($\rho$) of rankings, and mean absolute error ($\varepsilon$) in top-20% rankings.
• ICA-TOPSIS-M (especially with JADE) exhibits best approximation to the true latent ranking, given sufficient sample size and moderate signal-to-noise ratio (SNR).
• When SNR is low or alternatives are few, ICA source separation degrades, resulting in reduced ranking accuracy.
• Real-world data reveal that classical TOPSIS disproportionately favors alternatives strong in correlated criteria; ICA-TOPSIS variants mitigate this bias, yielding balanced rankings.

6. Scope, Advantages, and Limitations

ICA-TOPSIS variants offer a signal-processing-informed methodology for handling dependent criteria within MCDM. Notable properties include:

Advantages:

• Reduction in ranking bias due to criterion dependence by revealing latent independent factors.
• Applicability to multidimensional scenarios provided ICA is feasible (sufficient sample size, moderate SNR).
• Systematic ambiguity adjustment process for ICA outputs.
• Flexibility to utilize different ICA algorithms (e.g., FastICA, JADE).

Limitations:

• Assumption of linear dependence and mixture model; real-world dependencies may be nonlinear.
• Effectiveness of ICA contingent on sample size and SNR; poor performance with too few alternatives or high noise.
• Latent criteria may lack direct interpretability.
• Necessity for adjustment of ICA’s intrinsic scaling and permutation ambiguities for each dataset or application.
• ICA-TOPSIS variants only address linear dependence; nonlinear relationships remain unhandled.

7. Implications and Application Domains

The ICA-TOPSIS framework is tailored for decision problems where statistical dependence among criteria is suspected or confirmed—typified by socioeconomic, environmental, or composite performance indicators. A plausible implication is enhanced fairness and robustness of rankings in contexts such as country comparison, institutional benchmarking, or multi-objective engineering design, particularly where criteria are empirically or structurally correlated. Domains requiring explicit accounting for criterion interdependence, such as complex systems evaluation or multi-indicator performance analysis, directly benefit from ICA-TOPSIS variants by mitigating the bias that classical procedures introduce.