Observations in various applications are frequently represented as a time series of multidimensional arrays, called tensor time series, preserving the inherent multidimensional structure. In this paper, we present a factor model approach, in a form similar to tensor CANDECOMP/PARAFAC (CP) decomposition, to the analysis of high-dimensional dynamic tensor time series. As the loading vectors are uniquely defined but not necessarily orthogonal, it is significantly different from the existing tensor factor models based on Tucker-type tensor decomposition. The model structure allows for a set of uncorrelated one-dimensional latent dynamic factor processes, making it much more convenient to study the underlying dynamics of the time series. A new high-order projection estimator is proposed for such a factor model, utilizing the special structure and the idea of the higher order orthogonal iteration procedures commonly used in Tucker-type tensor factor model and general tensor CP decomposition procedures. Theoretical investigation provides statistical error bounds for the proposed methods, which shows the significant advantage of utilizing the special model structure. Simulation study is conducted to further demonstrate the finite sample properties of the estimators. Real data application is used to illustrate the model and its interpretations.
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Abstract -
Here, we synthesized and characterized a novel two-dimensional (2D) conjugated electron donor–acceptor (D-A) copolymer (PBDB-T-Ge), wherein the substituent of triethyl germanium was added to the electron donor unit of the polymer. The Turbo–Grignard reaction was used to implement the group IV element into the polymer, resulting in a yield of 86%. This corresponding polymer, PBDB-T-Ge, exhibited a down-shift in the highest occupied molecular orbital (HOMO) level to −5.45 eV while the lowest unoccupied molecular orbital (LUMO) level was −3.64 eV. The peaks in UV-Vis absorption and the PL emission of PBDB-T-Ge were observed at 484 nm and 615 nm, respectively.