Tensor and Multimodal Data Analysis

https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/j.2517-6161.1977.tb01600.x

Maximum likelihood from incomplete data via the EM algorithm

Eighty-three brains obtained at autopsy from nondemented and demented individuals were examined for extracellular amyloid deposits and intraneuronal neurofibrillary changes. The distribution pattern and packing density of amyloid deposits turned out to be of limited significance for differentiation of neuropathological stages. Neurofibrillary changes occurred in the form of neuritic plaques, neurofibrillary tangles and neuropil threads. The distribution of neuritic plaques varied widely not only within architectonic units but also from one individual to another. Neurofibrillary tangles and neuropil threads, in contrast, exhibited a characteristic distribution pattern permitting the differentiation of six stages. The first two stages were characterized by an either mild or severe alteration of the transentorhinal layer Pre-alpha (transentorhinal stages I-II). The two forms of limbic stages (stages III-IV) were marked by a conspicuous affection of layer Pre-alpha in both transentorhinal region and proper entorhinal cortex. In addition, there was mild involvement of the first Ammon's horn sector. The hallmark of the two isocortical stages (stages V-VI) was the destruction of virtually all isocortical association areas. The investigation showed that recognition of the six stages required qualitative evaluation of only a few key preparations.

Neuropathological stageing of Alzheimer-related changes.

Advances in Neural Information Processing Systems 28

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

Concepts of correlation and regression may be applied not only to ordinary one-dimensional variates but also to variates of two or more dimensions. Marksmen side by side firing simultaneous shots at targets, so that the deviations are in part due to independent individual errors and in part to common causes such as wind, provide a familiar introduction to the theory of correlation; but only the correlation of the horizontal components is ordinarily discussed, whereas the complex consisting of horizontal and vertical deviations may be even more interesting. The wind at two places may be compared, using both components of the velocity in each place. A fluctuating vector is thus matched at each moment with another fluctuating vector. The study of individual differences in mental and physical traits calls for a detailed study of the relations between sets of correlated variates. For example the scores on a number of mental tests may be compared with physical measurements on the same persons. The questions then arise of determining the number and nature of the independent relations of mind and body shown by these data to exist, and of extracting from the multiplicity of correlations in the system suitable characterizations of these independent relations. As another example, the inheritance of intelligence in rats might be studied by applying not one but s different mental tests to N mothers and to a daughter of each

Relations Between Two Sets of Variates

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Tensor and Multimodal Data Analysis

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References

Maximum likelihood from incomplete data via the EM algorithm

Neuropathological stageing of Alzheimer-related changes.

Advances in Neural Information Processing Systems 28

Tensor Decompositions and Applications

Relations Between Two Sets of Variates