  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, defects, et al. 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
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) 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.
