This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or $N$-way array. Decompositions of higher-order tensors (i.e., $N$-way arrays with $N \geq 3$) have applications in psycho-metrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, and elsewhere. Two particular tensor decompositions can be considered to be higher-order extensions of the matrix singular value decomposition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rank-one tensors, and the Tucker decomposition is a higher-order form of principal component analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The N-way Toolbox, Tensor Toolbox, and Multilinear Engine are examples of software packages for working with tensors.

/pdf/tensor-decompositions-and-applications-46vl2ih5gn.pdf

Tensor Decompositions and Applications

https://www.cs.cmu.edu/~pmuthuku/mlsp_page/lectures/Parafac.pdf

PARAFAC. Tutorial and applications

Nondestructive measurement of fruit and vegetable quality by means of NIR spectroscopy: A review

Tracing dissolved organic matter in aquatic environments using a new approach to fluorescence spectroscopy

A new diagnostic called the core consistency diagnostic (CORCONDIA) is suggested for determining the proper number of components for multiway models. It applies especially to the parallel factor analysis (PARAFAC) model, but also to other models that can be considered as restricted Tucker3 models. It is based on scrutinizing the ‘appropriateness’ of the structural model based on the data and the estimated parameters of gradually augmented models. A PARAFAC model (employing dimension-wise combinations of components for all modes) is called appropriate if adding other combinations of the same components does not improve the fit considerably. It is proposed to choose the largest model that is still sufficiently appropriate. Using examples from a range of different types of data, it is shown that the core consistency diagnostic is an effective tool for determining the appropriate number of components in e.g. PARAFAC models. However, it is also shown, using simulated data, that the theoretical understanding of CORCONDIA is not yet complete. Copyright © 2003 John Wiley & Sons, Ltd.

A new efficient method for determining the number of components in PARAFAC models

Multivariate instrument standardization

The successful application of multivariate calibration in solving industrial analytical problems requires a protocol to standardize calibration models between different instruments, e.g., from a central laboratory instrument to a process instrument. In a previous study, several standardization methods based on the measurement of a small set of strandards have been developed and compared to the recalibration on individual instruments

Improvement of multivariate calibration through instrument standardization

Near-infrared (NIR) spectroscopy has been widely accepted as a quantitative technique in which multivariate calibration plays an important role. The application of NIR to process analysis, however, has been largely limited by a problem identified as calibration transfer, the attempt to transfer a well-established calibration model from one instrument (e.g., located in the central laboratory) to another instrument of the same type (e.g., located on an industrial process). A calibration transfer method called piecewise direct standardization (PDS) is applied to a set of gasoline samples measured on two different NIR spectrometers. On the basis of the measurement of a small set of transfer samples on both instruments, a structured transformation matrix can be determined and applied to transform spectra between two instruments, enabling the transfer of calibration models. The effect of spectrum preprocessing on standardization is studied with the use of a set of gasoline samples. In a separate study, the day-to-day instrument variation as observed from the change in the polystyrene spectrum is related to the prediction of moisture, oil, protein, and starch content in corn samples, and then the possibility of using such generic standards to replace real samples in a transfer set is explored. In all cases, a standard error for prediction comparable to full set cross-validation is obtained through standardization.

Calibration Transfer and Measurement Stability of Near-Infrared Spectrometers

If an analytical instrument or instrumental method gives a response matrix when analyzing a pure analyte, the instrument or instrumental method is called a second-order method. Second-order methods that generate a response matrix for a pure analyte of rank one are called rank-one second-order methods. If the response matrix of a pure analyte is not rank one, essentially two cases exist: medium rank (between two and five) and high rank (greater than five). Subsequently, medium- and high-rank second-order calibration tries to use medium- and high-rank second-order methods to analyze for analytes of interest in a mixture. A particular advantage of second-order methods is the ability to analyze for analytes of interest in a mixture which contains unknown interferences. Keeping this advantage is the challenge on moving away from rank-one second-order calibration methods. In this paper a medium-rank second-order calibration method is proposed based on least-squares restricted Tucker models. With this method the second-order advantage is retained.

Yongdong Wang

Papers

Multivariate instrument standardization

Improvement of multivariate calibration through instrument standardization

Calibration Transfer and Measurement Stability of Near-Infrared Spectrometers

Theory of medium‐rank second‐order calibration with restricted‐Tucker models

Temperature-compensating calibration transfer for near-infrared filter instruments