Uniqueness proof for a family of models sharing features of Tucker's three-mode factor analysis and PARAFAC/candecomp

Some existing three-way factor analysis and MDS models incorporate Cattell's “Principle of Parallel Proportional Profiles”. These models can—with appropriate data—empirically determine a unique best fitting axis orientation without the need for a separate factor rotation stage, but they have not been general enough to deal with what Tucker has called “interactions” among dimensions. This article presents a proof of unique axis orientation for a considerably more general parallel profiles model which incorporates interacting dimensions. The model, X_k=A^AD_k H^BD_k B', does not assume symmetry in the data or in the interactions among factors. A second proof is presented for the symmetrically weighted case (i.e., where^AD_k=^BD_k). The generality of these models allows one to impose successive restrictions to obtain several useful special cases, including PARAFAC2 and three-way DEDICOM. We want to express appreciation for the contributions of several colleagues: Jos M. F. ten Berge and Henk A. L. Kiers carefully went through more than one version of this article, found an important error, and contributed many improvements. J. Douglas Carroll and Shizuhiko Nishisato acted with unusual editorial preserverance and flexibility, thereby making possible the successful completion of a difficult assessment and revision process. Joseph B. Kruskal has long provided crucial mathematical insights and inspiration to those working in this area, but this is particularly true for us. His contributions to this specific article include early discussion of basic questions and careful examination of some earlier attempted proofs, finding them to be invalid. The anonymous reviewers also made useful suggestions. Some portions of this work were supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.     

Principal component analysis of three-mode data by means of alternating least squares algorithms

A new method to estimate the parameters of Tucker's three-mode principal component model is discussed, and the convergence properties of the alternating least squares algorithm to solve the estimation problem are considered. A special case of the general Tucker model, in which the principal component analysis is only performed over two of the three modes is briefly outlined as well. The Miller & Nicely data on the confusion of English consonants are used to illustrate the programs TUCKALS3 and TUCKALS2 which incorporate the algorithms for the two models described.     

Scale freeness in factor analysis

The notion of scale freeness does not seem to have been well understood in the factor analytic literature. It has been believed that if the loss function that is minimized to obtain estimates of the parameters in the factor model is scale invariant, then the estimates are scale free. It is shown that scale invariance of the loss function is neither a necessary nor a sufficient condition for scale freeness. A theorem that ensures scale freeness in the orthogonal factor model is given in this paper.The authors are grateful for the suggestions of the referees.     

Redundancy analysis an alternative for canonical correlation analysis

A component method is presented maximizing Stewart and Love's redundancy index. Relationships with multiple correlation and principal component analysis are pointed out and a rotational procedure for obtaining bi-orthogonal variates is given. An elaborate example comparing canonical correlation analysis and redundancy analysis on artificial data is presented.A Fortran IV program for the method of redundancy analysis described in this paper can be obtained from the author upon request.     

The indirect effect of cognitive reserve on the relationship between age and cognition in pathological ageing: A cross-sectional retrospective study in an unselected and consecutively enrolled sample

Cognitive reserve (CR) allows individuals to maintain cognitive functionality even in the presence of pathologies. The compensation hypothesis suggests that CR plays an indirect role between age and cognitive decline, contrasting the negative effect of ageing on cognition. We test this hypothesis in an unselected and consecutively enrolled sample of memory clinic attendees (n = 134) who completed the CR Index questionnaire and three neuropsychological tests assessing global cognition (MMSE, FAB, CDT). Participants were divided into two groups based on standard diagnostic criteria (DSM-5): those who were cognitively impaired (n = 92) and those who were preserved (n = 42). A principal component analysis was used to extract a composite measure of global cognitive functioning from the three neuropsychological tests, and mediation analysis was used to examine the relationship between CR, age and global cognitive functioning in the two groups. Results revealed that: (i) age had a significant direct negative effect on the global cognitive score in both groups; (ii) the three socio-behavioural proxies of CR together suppress the direct negative relationship between age and global cognitive score in cognitively impaired patients but not in cognitively preserved participants. This study confirms the association between CR, age and cognition and allows us to validate its role in a population with cognitive impairment and extend findings to a low-to-middle educated population. These results hold important implications for public health and wellness promotion, emphasising the beneficial role of maintaining healthy and active physical, cognitive and social lifestyles.     

个体化降压的临床意义

高血压是由多种病因和发病机制引起的一种慢性疾病,可以导致多种靶器官功能损害,并最终导致严重的心、脑血管并发症,危及人类的健康和生命。如何做到有效的预防和治疗是目前面,临的问题。本文分析了高血压控制的现状,以矛盾的特殊性原理为指导,从高血压的病因和发病机制、临床特点以及药物对个体的敏感性等诸多方面出发,阐述了高血压患者个体差异性。高血压患者个体差异性决定每一个个体综合控制策略的不同,运用“具体问题具体分析”的方法,为每一个个体制定个体化的血压控制方案,以期达到理想的高血压整体控制水平。     

京氏易学中的阴阳对待与流行——兼论京易纳甲、建候、积算的建构原则

本文从《京氏易传》入手,对京氏易学中阴阳二气的对待与流行进行了阐发。文章指出,京氏通过纳甲、卦变、飞伏、游魂、归魂、建候、积算诸说将“一阴一阳”之《易》道具体到实处,构建了一幅生生不息的、立体的阴阳对待流行图;其纳支、建候、积算则依历法及音律之隔八生律法纳入。     

Newton Algorithms for Analytic Rotation: an Implicit Function Approach

In this paper implicit function-based parameterizations for orthogonal and oblique rotation matrices are proposed. The parameterizations are used to construct Newton algorithms for minimizing differentiable rotation criteria applied to m factors and p variables. The speed of the new algorithms is compared to that of existing algorithms and to that of Newton algorithms based on alternative parameterizations. Several rotation criteria were examined and the algorithms were evaluated over a range of values for m. Initial guesses for Newton algorithms were improved by subconvergence iterations of the gradient projection algorithm. Simulation results suggest that no one algorithm is fastest for minimizing all criteria for all values of m. Among competing algorithms, the gradient projection algorithm alone was faster than the implicit function algorithm for minimizing a quartic criterion over oblique rotation matrices when m is large. In all other conditions, however, the implicit function algorithms were competitive with or faster than the fastest existing algorithms. The new algorithms showed the greatest advantage over other algorithms when minimizing a nonquartic component loss criterion.     

几何模块论的局限性——来自梯形实验的证据

以往研究表明幼儿主要依赖几何信息进行再定向,但这些研究大多使用矩形空间。在本研究中,幼儿所处的空间为几何信息更丰富,目标物体所在位置确定的梯形。结果发现,幼儿并不会利用新增的几何信息确定目标物体,仍然去正确位置的对角寻找,表现出的寻找模式与矩形空间一样,这说明幼儿在空间再定向任务中对几何信息的利用很有限。此外,录像分析的结果发现,幼儿在面向梯形两底边时会更多地且径直地走向正确位置或对角,这种寻找过程上的特点表明他们可能基于空间轴朝向结合左右方位感表征物体位置