Tensor Mixed Effects Model With Application to Nanomanufacturing Inspection

Raman mapping technique has been used to perform in-line quality inspections of nanomanufacturing processes. In such an application, massive high-dimensional Raman mapping data with mixed effects is 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, and defects. The existing tensor decomposition methods cannot separate mixed effects, and existing mixed effects model can only handle matrix data but not high-dimensional multi-array data. In this article, 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) explore the correlations along different dimensions; and (iii) realize efficient parameter estimation by a proposed iterative double Flip-Flop algorithm. We also investigate the properties of the TME model, existence and identifiability of parameter estimation. 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 case study shows an application of the TME model in quantifying the influence of alignment on carbon nanotubes buckypaper. Moreover, the TME model can be applied to provide potential solutions for a family of tensor data analytics problems with mixed effects.