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Raman mapping technique has been used to perform in-line quality inspectionsof nanomanufacturing processes. In such an application, massivehigh-dimensional Raman mapping data with mixed effects is generated. Ingeneral, fixed effects and random effects in the multi-array Raman data areassociated with different quality characteristics such as fabricationconsistency, uniformity, defects, et al. The existing tensor decompositionmethods cannot separate mixed effects, and existing mixed effects model canonly handle matrix data but not high-dimensional multi-array data. In thispaper, we propose a tensor mixed effects (TME) model to analyze massivehigh-dimensional Raman mapping data with complex structure. The proposed TMEmodel can (i) separate fixed effects and random effects in a tensor domain;(ii) explore the correlations along different dimensions; and (iii) realizeefficient parameter estimation by a proposed iterative double Flip-Flopalgorithm. We also investigate the properties of the TME model, existence andidentifiability of parameter estimation. The numerical analysis demonstratesthe efficiency and accuracy of the parameter estimation in the TME model.Convergence and asymptotic properties are discussed in the simulation andsurrogate data analysis. The case study shows an application of the TME modelin quantifying the influence of alignment on carbon nanotubes buckypaper.Moreover, the TME model can be applied to provide potential solutions for afamily of tensor data analytics problems with mixed effects.

Tensor Mixed Effects Model with Applications in Nanomanufacturing Inspection