A performance evaluation of integrating machine learning schemes utilizing fluidic lenses

A combination of statistical inference and machine learning (ML) schemes has been utilized to create a thorough understanding of coarse experimental data based on Zernike variables characterizing optical aberrations in fluidic lenses. A classification of surplus-response variables through tolerance manipulation was included to unravel the dimensional aspect of the data. Similarly, the impact of the exclusion of supererogatory variables through the identification of clustering movements of constituents is examined. The method of constructing a spectrum of collaborative results through the application of similar techniques has been tested. To evaluate the suitability of each statistical method before its application on a large dataset, a selection of ML schemes has been proposed. The supervised learning tools principal component analysis (PCA), factor analysis (FA), and hierarchical clustering (HC) were employed to define the elemental characteristics of Zernike variables. PCA enabled to reduce the dimensionality of the system by identifying two principal components which collectively account for 95% of the total variance. The execution of FA indicated that a specific tolerance of independent variability of 0.005 could be used to reduce the dimensionality of the system without losing essential data information. A high cophenetic coefficient value of c=0.9629 validated an accurate clustering division of variables with similar characteristics. The current approach of mutually validating ML and statistical analysis methods will aid in laying the foundation for state-of-the-art (SOTA) analysis. The benefit of our approach can be assessed by considering that the associated SOTA will enhance the predictive accuracy between two comparable methods, in contrast to the SOTA analysis conducted between two arbitrary ML methods.

💡 Research Summary

The paper presents an integrated statistical‑machine‑learning framework for analyzing coarse experimental data obtained from fluidic lenses. The authors measured the first 15 Zernike coefficients of the phase front transmitted by a PDMS double‑membrane fluidic lens for eight different fluid volumes using a Shack‑Hartmann wave‑front sensor. These 15 coefficients, recorded across the eight volume conditions, constitute a modest dataset (120 samples) that the authors treat as a testbed for evaluating several supervised learning tools—principal component analysis (PCA), factor analysis (FA), and hierarchical clustering (HC).

The study begins with exploratory data analysis. Box‑plots reveal that Z₁ exhibits the highest variability while Z₇–Z₁₅ are relatively stable across volumes. An X‑Bar control chart of pairwise correlations shows that Z₂ and Z₅ are positively correlated, Z₁ and Z₇ are negatively correlated, and the remaining coefficients display weak inter‑correlations. These observations guide subsequent dimensionality‑reduction and variable‑selection steps.

In the PCA stage, the authors weight each Zernike variable by the inverse of its variance, thereby emphasizing more stable coefficients. The analysis yields five outputs (loadings, scores, latent values, Hotelling’s T², and explained variance). The scree plot demonstrates that the first two principal components together account for 95 % of the total variance (PC1 ≈ 70 %, PC2 ≈ 25 %). Loadings indicate that Z₂, Z₅, and Z₁ dominate the first two components, while Z₃, Z₄, and Z₈ contribute mainly to PC2. A bi‑plot of loadings and scores visualizes how the coefficients cluster in the reduced space, suggesting that PC1 captures a dominant aberration mode (e.g., defocus or spherical aberration) and PC2 captures a secondary mode (e.g., astigmatism).

Factor analysis is performed using the maximum‑likelihood estimator to extract three common factors (m = 3). The loading matrix (both raw and rotated) shows that Z₁–Z₈ load strongly on one of the three factors, whereas Z₉–Z₁₅ have negligible loadings. Specific variances are reported for each coefficient; notably, Z₈ exhibits a high specific variance (0.2427), indicating it contains largely independent information. The authors adopt a tolerance threshold of 0.005 on specific variance, allowing them to discard variables below this value without appreciable loss of information. The rotated factor plot confirms that each coefficient aligns predominantly with a single factor, supporting the interpretability of the three‑factor model.

Hierarchical clustering is applied to the normalized Zernike data using average linkage and Euclidean distance. The resulting dendrogram, validated by a cophenetic correlation coefficient of 0.9629, reveals three major clusters: (1) Z₁–Z₃, (2) Z₄–Z₇, and (3) Z₈–Z₁₅. The high cophenetic value indicates that the dendrogram faithfully represents the underlying pairwise distances, and the cluster assignments correspond well with the correlation patterns observed earlier.

The authors emphasize that the three techniques are not used in isolation but rather in a mutually validating loop. PCA provides a linear orthogonal basis that captures most variance; FA uncovers latent common factors and quantifies variable‑specific noise; HC reveals non‑linear grouping structure. By cross‑checking results (e.g., variables that dominate PC1 also load heavily on Factor 1 and belong to the same HC cluster), the authors achieve a robust “state‑of‑the‑art” (SOTA) analysis that surpasses naïve comparisons of arbitrary ML models.

In conclusion, the paper demonstrates that even with a relatively small, high‑dimensional optical dataset, a combined statistical‑ML pipeline can effectively reduce dimensionality, identify redundant variables, and uncover meaningful groupings. The methodology is presented as a foundation for future work on larger optical datasets, integration with more complex ML models (e.g., deep neural networks), and real‑time adaptive control of fluidic lenses.