Tensor Mixed Effects Model with Applications in Nanomanufacturing Inspection
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Raman mapping technique has been used to do in-line quality inspections of nanomanufacturing process. In such an application, a massive high-dimensional Raman mapping data with mixed effects are generated. In general, fixed effects and random effects in the multi-array Raman data are associated with different quality characteristics such as fabrication consistency, uniformity, defects, et al. The existing tensor decomposition methods cannot separate mixed effects, and existing mixed effects model can only handle matrix data instead of high-dimensional multi-array data. In this paper, we propose a tensor mixed effects (TME) model to analyze massive high-dimensional Raman mapping data with complex structure. The proposed TME model can (i) separate fixed effects and random effects in a tensor domain; (ii) exploit the correlations along different dimensions; and (iii) realize efficient parameter estimation by a proposed iterative double Flip-Flop algorithm. Properties of the TME model, existence and identifiability of parameter estimation are investigated. The numerical analysis demonstrates the efficiency and accuracy of the parameter estimation in the TME model. Convergence and asymptotic properties are discussed in the simulation and surrogate data analysis. The Real case study shows an application of the TME model in quantifying the influence of alignment on carbon nanotubes buckypaper.
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