A solution to the weighted procrustes problem in which the transformation is in agreement with the loss function

This paper provides a generalization of the Procrustes problem in which the errors are weighted from the right, or the left, or both. The solution is achieved by having the orthogonality constraint on the transformation be in agreement with the norm of the least squares criterion. This general principle is discussed and illustrated by the mathematics of the weighted orthogonal Procrustes problem.     

A general solution to Mosier's oblique procrustes problem

Browne provided a method for finding a solution to the normal equations derived by Mosier for rotating a factor matrix to a best least squares fit with a specified structure. Cramer showed that Browne's solution is not always valid, and proposed a modified algorithm. Both Browne and Cramer assumed the factor matrix to be of full rank. In this paper a general solution is derived, which takes care of rank deficient factor matrices as well. A new algorithm is offered.     

A weighted procrustes criterion

The Procrustes criterion is a common measure for the distance between two matricesX andY, and can be interpreted as the sum of squares of the Euclidean distances between their respective column vectors. Often a weighted Procrustes criterion, using, for example, a weighted sum of the squared distances between the column vectors, is called for. This paper describes and analyzes the performance of an algorithm for rotating a matrixX such that the column-weighted Procrustes distance toY is minimized. The problem of rotatingX intoY such that an aggregate measure of Tucker's coefficient of congruence is maximized is also discussed.We wish to thank Richard A. Harshman and C. F. Jeff Wu for valuable discussions in the early stages of this work. We would also like to thank Jos ten Berge, John Gower, and the Editor, Associate Editor, and referees whose comments and suggestions greatly improved this paper.     

Variables for designing cube folding tasks influencing sixth-graders' performance

This study investigated whether the three variables of task form, squares carried, and figural complexity, for designing cube folding tasks, affect sixth graders' cube folding performance. Two task forms were used to develop two versions of “cube folding test.” Each version was designed based on two levels of squares carried and three levels of figural complexity, and contained 18 multiple-choice items. The sample included 107 boys and 92 girls with an average age of 12.04 years who were randomly assigned to complete one of the two versions. A linear mixed model was carried out with three factors: task forms, squares carried, and figural complexity, in which squares carried and figural complexity were “within-subjects” factors, and task form was the “between-subjects” factor. The results showed the interaction effect of squares carried and task form, and the main effect of figural complexity, which differentiated the difficulty in cube folding tasks.     

可旋转地图和固定地图的定位效率研究

本研究采用两个分实验,分别比较了在空白背景与复杂背景上地图旋转与地图固定两种导航方式下被试的方位判断反应时之间的差异,然后考察两种背景在两种导航方式差异上的一致性问题。结果发现,在运动点的运动方向为朝上、朝左和朝右时,对于空白背景,地图旋转方式下的反应时显著低于地图固定方式;而对于复杂背景,两种导航方式下的反应时之间不存在显著差异。在运动点的运动方向为朝下时,对于两种背景,地图旋转方式的反应时均显著低于地图固定方式。这一结果基本证实了本研究的假设,即被试利用地图进行空间定向的认知过程包括心理旋转与目标搜索两个阶段。     

Teaching Principal Components Using Correlations

Introducing principal components (PCs) to students is difficult. First, the matrix algebra and mathematical maximization lemmas are daunting, especially for students in the social and behavioral sciences. Second, the standard motivation involving variance maximization subject to unit length constraint does not directly connect to the “variance explained” interpretation. Third, the unit length and uncorrelatedness constraints of the standard motivation do not allow re-scaling or oblique rotations, which are common in practice. Instead, we propose to motivate the subject in terms of optimizing (weighted) average proportions of variance explained in the original variables; this approach may be more intuitive, and hence easier to understand because it links directly to the familiar “R-squared” statistic. It also removes the need for unit length and uncorrelatedness constraints, provides a direct interpretation of “variance explained,” and provides a direct answer to the question of whether to use covariance-based or correlation-based PCs. Furthermore, the presentation can be made without matrix algebra or optimization proofs. Modern tools from data science, including heat maps and text mining, provide further help in the interpretation and application of PCs; examples are given. Together, these techniques may be used to revise currently used methods for teaching and learning PCs in the behavioral sciences.     

Mental rotation with egocentric and object-based transformations

Two experiments were conducted to address methodological issues with past studies investigating the influence of egocentric and object-based transformations on performance and sex differences in mental rotation. In previous work, the egocentric and object-based mental rotation tasks confounded the stimulus type (embodied vs. non-embodied) and transformation task (egocentric vs. object-based). In both experiments presented here, the same stimuli were used regardless of the type of transformation but task instructions were modified to induce either egocentric (left–right judgment) or object-based (same–different judgment) processing. Experiment 1 used pairs of letters whereas Experiment 2 presented pairs of line-drawings of human hands. For both experiments, it was hypothesized that the mental rotation slope for response time would be steeper for object-based than for egocentric transformations. This hypothesis was verified in both experiments. Furthermore, Experiment 2 showed a reduced male advantage for egocentric compared to object-based rotations, whereas this pattern was reversed for Experiment 1. In conclusion, the present study showed that the influence of the type of transformation involved in mental rotation can be examined with the same set of stimuli simply by modifying task instructions.     

Stimulus size matters: do life-sized stimuli induce stronger embodiment effects in mental rotation?

Against the background of the embodied cognition approach this experiment investigated the influence of motor expertise on object-based vs. egocentric transformations in a chronometric mental rotation (MR) task using images of either the own or another person’s body as stimulus material. The present study aimed to clarify two issues: (1) whether stimulus size (life size vs. small) is able to induce embodiment effects and (2) which role self-awareness processes play when using stimuli of the own body. The same design was conducted twice using both small stimuli (Study 1) and life-size human figures (Study 2). Using life-sized figures in Study 2 resulted in an explicit advantage of self-related stimuli and improved performance for motor experts compared to non-motor experts in both object-based and egocentric transformations. In conclusion, these results suggest that life-sized figures do indeed induce stronger embodiment effects in MR.     

物体识别的视角依赖与心理旋转

物体识别的绩效随物体的视角变化而变化,这一物体识别的视角依赖现象引发了研究者对物体识别的机制的广泛讨论。有研究者认为,心理旋转是导致物体识别视角依赖的原因,而另一种观点认为物体识别中不包含心理旋转过程。两种观点都有来自于行为和神经机制两方面研究的证据。然而,现有的行为证据都是间接的证据,缺乏说服力。进一步的研究应注重直接操纵影响心理旋转与物体识别过程的因素,并把行为研究与能进行实时监测的眼动、脑成像等研究结合起来     

Cue effects on memory for location when navigating spatial displays

Participants maneuvered a rat image through a circular region on the computer screen to find a hidden target platform, blending aspects of two well-known spatial tasks. Like the Morris water maze task, participants first experienced a series of learning trials before having to navigate to the hidden target platform from different locations and orientations. Like the dot-location task, they determined the location of a position within a two-dimensional circular region. This procedure provided a way to examine how the number of surrounding cues (1, 2, or 3) affects the memory for spatial location in navigation. Memory performance was better when there were more cues and when targets were close to cues, consistent with the idea that cues bolster fine-grain memory, especially in proximal regions. Early and late measures of bias in memory reflected biases in a direction toward the nearest cue, implicating a cue-based category structure of the navigational space. Collectively, results suggest cue-based spatial memory representations that have been inferred from the dot-location task generalize to a navigation task within a simple, computer-based environment, as demonstrated by the good fits of the spatial model developed for the dot-location task ( Fitting, Wedell, & Allen, 2005, 2007 ).