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.     

A simple general method for oblique rotation

A simple and very general algorithm for oblique rotation is identified. While motivated by the rotation problem in factor analysis, it may be used to minimize almost any function of a not necessarily square matrix whose columns are restricted to have unit length. The algorithm has two steps. The first is to compute the gradient of the rotation criterion and the second is to project this onto a manifold of matrices with unit length columns. For this reason it is called a gradient projection algorithm. Because the projection step is very simple, implementation of the algorithm involves little more than computing the gradient of the rotation criterion which for many applications is very simple. It is proven that the algorithm is strictly monotone, that is as long as it is not already at a stationary point, each step will decrease the value of the criterion. Examples from a variety of areas are used to demonstrate the algorithm, including oblimin rotation, target rotation, simplimax rotation, and rotation to similarity and simplicity simultaneously. While it may be, the algorithm is not intended for use as a standard algorithm for well established problems, but rather as a tool for investigating new methods where its generality and simplicity may save an investigator substantial effort.The author would like to thank the review team for their insights and recommendations.     

Riemannian Newton and trust-region algorithms for analytic rotation in exploratory factor analysis

In exploratory factor analysis, latent factors and factor loadings are seldom interpretable until analytic rotation is performed. Typically, the rotation problem is solved by numerically searching for an element in the manifold of orthogonal or oblique rotation matrices such that the rotated factor loadings minimize a pre-specified complexity function. The widely used gradient projection (GP) algorithm, although simple to program and able to deal with both orthogonal and oblique rotation, is found to suffer from slow convergence when the number of manifest variables and/or the number of latent factors is large. The present work examines the effectiveness of two Riemannian second-order algorithms, which respectively generalize the well-established truncated Newton and trust-region strategies for unconstrained optimization in Euclidean spaces, in solving the rotation problem. When approaching a local minimum, the second-order algorithms usually converge superlinearly or even quadratically, better than first-order algorithms that only converge linearly. It is further observed in Monte Carlo studies that, compared to the GP algorithm, the Riemannian truncated Newton and trust-region algorithms require not only much fewer iterations but also much less processing time to meet the same convergence criterion, especially in the case of oblique rotation.     

How 9-month-old crawling infants profit from visual-manual rotations in a mental rotation task

Studies show that visual-manual object exploration influences spatial cognition, and specifically mental rotation performance in infancy. The current work with 9-month-old infants investigated which specific exploration procedures (related to crawling experience) support mental rotation performance. In two studies, we examined the effects of two different exploration procedures, manual rotation (Study 1) and haptic scanning (Study 2), on subsequent mental rotation performance. To this end, we constrained infants’ exploration possibilities to only one of the respective procedures, and then tested mental rotation performance using a live experimental set-up based on the task used by Moore and Johnson (2008). Results show that, after manual rotation experience with a target object, crawling infants were able to distinguish between exploration objects and their mirror objects, while non-crawling infants were not (Study 1). Infants who were given prior experience with objects through haptic scans (Study 2) did not discriminate between objects, regardless of their crawling experience. Results indicated that a combination of manual rotations and crawling experience are valuable for building up the internal spatial representation of an object.     

Orthogonal rotations to maximal agreement for two or more matrices of different column orders

Methods for orthogonal Procrustes rotation and orthogonal rotation to a maximal sum of inner products are examined for the case when the matrices involved have different numbers of columns. An inner product solution offered by Cliff is generalized to the case of more than two matrices. A nonrandom start for a Procrustes solution suggested by Green and Gower is shown to give better results than a random start. The Green-Gower Procrustes solution (with nonrandom start) is generalized to the case of more than two matrices. Simulation studies indicate that both the generalized inner product solution and the generalized Procrustes solution tend to attain their global optima within acceptable computation times. A simple procedure is offered for approximating simple structure for the rotated matrices without affecting either the Procrustes or the inner product criterion.The authors are obliged to Charles Lewis for helpful comments on a previous draft of this paper and to Frank Brokken for preparing a computer program that was used in this study.     

Rotation to Simple Loadings Using Component Loss Functions: The Oblique Case

Component loss functions (CLFs) similar to those used in orthogonal rotation are introduced to define criteria for oblique rotation in factor analysis. It is shown how the shape of the CLF affects the performance of the criterion it defines. For example, it is shown that monotone concave CLFs give criteria that are minimized by loadings with perfect simple structure when such loadings exist. Moreover, if the CLFs are strictly concave, minimizing must produce perfect simple structure whenever it exists. Examples show that methods defined by concave CLFs perform well much more generally. While it appears important to use a concave CLF, the specific CLF used is less important. For example, the very simple linear CLF gives a rotation method that can easily outperform the most popular oblique rotation methods promax and quartimin and is competitive with the more complex simplimax and geomin methods. The author would like to thank the editor and three reviewers for helpful suggestions and for identifying numerous errors.     

Children's left parietal brain activation during mental rotation is reliable as well as specific

Some recent evidence suggests that mental rotation of characters in children aged 7 or 8 years might be lateralized to the left parietal hemisphere. An alternative statement exists, however, the finding might be completely unspecific for mental rotation but either be simply a function of task difficulty or a consequence of the use of characters as stimuli. To test these alternatives, ERPs of 24 second graders were measured twice: (a) during mental rotation with characters as stimuli and orientations of 30°, 90°, or 150° and (b) during memory scanning with characters as stimuli and set sizes of 1, 2, or 3 letters. In both cases, an amplitude modulation was found. The effect of mental rotation as a function of character orientation turned out to be lateralized to the left parietal hemisphere. The effect of memory scanning as a function of set size, however, turned out to be completely non-lateralized. Thus, children's left hemisphere activation during mental rotation is reliable as well as specific.     

Multidimensional rotation and scaling of configurations to optimal agreement

An integrated method for rotating and rescaling a set of configurations to optimal agreement in subspaces of varying dimensionalities is developed. The approach relates existing orthogonal rotation techniques as special cases within a general framework based on a partition of variation which provides convenient measures of agreement. In addition to the well-known Procrustes and inner product optimality criteria, a criterion which maximizes the consensus among subspaces of the configurations is suggested. Since agreement of subspaces of the configurations can be examined and compared, rotation and rescaling is extended from a data transformation technique to an analytical method.     

A monotonically convergent algorithm for orthogonal congruence rotation

Brokken has proposed a method for orthogonal rotation of one matrix such that its columns have a maximal sum of congruences with the columns of a target matrix. This method employs an algorithm for which convergence from every starting point is not guaranteed. In the present paper, an iterative majorization algorithm is proposed which is guaranteed to converge from every starting point. Specifically, it is proven that the function value converges monotonically, and that the difference between subsequent iterates converges to zero. In addition to the better convergence properties, another advantage of the present algorithm over Brokken's one is that it is easier to program. The algorithms are compared on 80 simulated data sets, and it turned out that the new algorithm performed well in all cases, whereas Brokken's algorithm failed in almost half the cases. The derivation of the algorithm is given in full detail because it involves a series of inequalities that can be of use to derive similar algorithms in different contexts.This research has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences to the first author. The authors are obliged to Willem J. Heiser and Jos M. F. ten Berge for useful comments on an earlier version of this paper.     

Iteration of Partially Specified Target Matrices: Application to the Bi-Factor Case

The current study proposes a new bi-factor rotation method, Schmid-Leiman with iterative target rotation (SLi), based on the iteration of partially specified target matrices and an initial target constructed from a Schmid-Leiman (SL) orthogonalization. SLi was expected to ameliorate some of the limitations of the previously presented SL bi-factor rotations, SL and SL with target rotation (SLt), when the factor structure either includes cross-loadings, near-zero loadings, or both. A Monte Carlo simulation was carried out to test the performance of SLi, SL, SLt, and the two analytic bi-factor rotations, bi-quartimin and bi-geomin. The results revealed that SLi accurately recovered the bi-factor structures across the majority of the conditions, and generally outperformed the other rotation methods. SLi provided the biggest improvements over SL and SLt when the bi-factor structures contained cross-loadings and pure indicators of the general factor. Additionally, SLi was superior to bi-quartimin and bi-geomin, which performed inconsistently across the types of factor structures evaluated. No method produced a good recovery of the bi-factor structures when small samples (N = 200) were combined with low factor loadings (0.30–0.50) in the specific factors. Thus, it is recommended that larger samples of at least 500 observations be obtained.     

Techniques for rotating two or more loading matrices to optimal agreement and simple structure: A comparison and some technical details

Matrices of factor loadings are often rotated to simple structure. When more than one loading matrix is available for the same variables, the loading matrices can be compared after rotating them all (separately) to simple structure. An alternative procedure is to rotate them to optimal agreement, and then compare them. In the present paper techniques are described that combine these two procedures. Specifically, five techniques that combine the ideals of rotation to optimal agreement and rotation to simple structure are compared on the basis of contrived and empirical data. For the contrived data, it is assessed to what extent the rotations recover the underlying common structure. For both the contrived and the empirical data it is studied to what extent the techniques give well matching rotated matrices, to what extent these have a simple structure, and to what extent the most prominent parts of the different loading matrices agree. It was found that the simple procedure of combining a Generalized Procrustes Analysis (GPA) with Varimax on the mean of the matched loading matrices performs very well on all criteria, and, for most purposes, offers an attractive compromise of rotation to agreement and simple structure. In addition to this comparison, some technical improvements are proposed for Bloxom's rotation to simple structure and maximum similarity.This research has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences to the author. The author is obliged to René van der Heijden for assistance in programming the procedures in the simulation study reported in this paper, and to Jos ten Berge, three anonymous reviewers and an associate editor for helpful comments on an earlier version of this paper.     

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     

Functionplane—A new approach to simple structure rotation

A new criterion for rotation to an oblique simple structure is proposed. The results obtained are similar to that obtained by Cattell and Muerle's maxplane criterion. Since the proposed criterion is smooth it is possible to locate the local maxima using simple gradient techniques. The results of the application of the Functionplane criterion to three sets of data are given. In each case a better fit to the subjective solution was obtained using the functionplane criterion than was reported for by Hakstian for the oblimax, promax, maxplane, or the Harris-Kaiser methods.This paper is contribution No. 66 from the Program in Ecology and Evolution at the State University of New York, Stony Brook, New York. This work was supported in part by a grant (GB-20496) from the National Science Foundation. The computations were performed on an IBM 360/67 computer at the State University of New York at Stony Brook.     

A Comparative Investigation of Rotation Criteria Within Exploratory Factor Analysis

Exploratory factor analysis (EFA) is a commonly used statistical technique for examining the relationships between variables (e.g., items) and the factors (e.g., latent traits) they depict. There are several decisions that must be made when using EFA, with one of the more important being choice of the rotation criterion. This selection can be arduous given the numerous rotation criteria available and the lack of research/literature that compares their function and utility. Historically, researchers have chosen rotation criteria based on whether or not factors are correlated and have failed to consider other important aspects of their data. This study reviews several rotation criteria, demonstrates how they may perform with different factor pattern structures, and highlights for researchers subtle but important differences between each rotation criterion. The choice of rotation criterion is critical to ensure researchers make informed decisions as to when different rotation criteria may or may not be appropriate. The results suggest that depending on the rotation criterion selected and the complexity of the factor pattern matrix, the interpretation of the interfactor correlations and factor pattern loadings can vary substantially. Implications and future directions are discussed.     

自我和物体为参照系的心理旋转分离：内旋效应

采用虚拟的旋转不同角度左、右手模型,构建“左右手判断(Left and right hand judgment: LR)”任务和“相同－不同判断(same and different judgment: SD)”任务,考察这两种实验任务是否都存在内旋效应和角度效应,以此推论被试采用何种旋转策略。结果发现：（1） 两种实验任务结果均表现出显著的角度效应。（2）在LR任务条件下,存在显著的内旋效应,而在SD任务中不存在内旋效应。从而表明当人手图片作为心理旋转材料时,它具有双重角色。被试心理旋转加工时究竟选用何种参照系的旋转策略,与实验材料和实验任务两者都密不可分     

Evidence for Mixed Processes in Normal/Mirror Discrimination of Rotated Letters: A Bayesian Model Comparison Between Single- and Mixed-Distribution Models1

When determining whether a rotated letter is normal or mirrored, an observer mentally rotates the letter to its canonical orientation. To account for patterns of response times (RTs) for the normal/mirror discrimination of rotated letters, previous research formulated a model that postulated a mixture of trials with and without mental rotation. While this model could explain the curvilinear relationship that has been found between averaged RT and letter orientations, the curved RT function is still open to alternative explanations without assuming mixed processes. To address this issue and test the mixed-process hypothesis more directly, we analyzed trial-by-trial RT data instead of averaged RTs by employing a Bayesian model comparison technique. If rotation and non-rotation trials are mixed, trial-by-trial RTs for letters in a particular orientation should not follow a single distribution but a mixed one formed from the superposition of two separate distributions, one for rotation and one for non-rotation trials. In the present study, we compared single- and mixed-distribution models. Bayes-factor analysis showed decisive support for the mixed-distribution model over the single-distribution model. In addition, using the widely applicable information criterion (WAIC), the predictive accuracy of the mixed-distribution model was found to be as high as that of the single-distribution model. These results indicated the involvement of mixed processes in normal/mirror discrimination of rotated letters. The usefulness of statistical modeling in psychological study and necessary precautions to take in the interpretation of the parameters of unconfirmed models are also discussed.     

Generalization gradients of inhibition following auditory discrimination learning

A more direct method than the usual ones for obtaining inhibitory gradients requires that the dimension of the nonreinforced stimulus selected for testing be orthogonal to the dimensions of the reinforced stimulus. In that case, the test points along the inhibitory gradient are equally distant from the reinforced stimulus. An attempt was made to realize this condition by obtaining inhibitory gradients along the frequency dimension of a pure tone after discrimination training in which the nonreinforced stimulus was a pure tone (or tones), and the reinforced stimulus was either white noise or the absence of a tone. The results showed that some degree of specific inhibitory control was exerted by the frequency of the tone, although the gradients were broad and shallow in slope.

A further experiment was conducted to see whether the modification of an excitatory gradient resulting from training to discriminate neighboring tones could arise from a simple interaction of inhibitory and excitatory gradients. The results indicated that it could not, since discrimination training produced a concentration of responding in the vicinity of the reinforced stimulus which cannot be derived from any plausible gradient of inhibition.

     

Exploratory Bi-Factor Analysis

Bi-factor analysis is a form of confirmatory factor analysis originally introduced by Holzinger. The bi-factor model has a general factor and a number of group factors. The purpose of this paper is to introduce an exploratory form of bi-factor analysis. An advantage of using exploratory bi-factor analysis is that one need not provide a specific bi-factor model a priori. The result of an exploratory bi-factor analysis, however, can be used as an aid in defining a specific bi-factor model. Our exploratory bi-factor analysis is simply exploratory factor analysis using a bi-factor rotation criterion. This is a criterion designed to produce perfect cluster structure in all but the first column of a rotated loading matrix. Examples are given to show how exploratory bi-factor analysis can be used with ideal and real data. The relation of exploratory bi-factor analysis to the Schmid-Leiman method is discussed.     

A cluster‐based factor rotation

A new oblique factor rotation method is proposed, the aim of which is to identify a simple and well‐clustered structure in a factor loading matrix. A criterion consisting of the complexity of a factor loading matrix and a between‐cluster dissimilarity is optimized using the gradient projection algorithm and the k‐means algorithm. It is shown that if there is an oblique rotation of an initial loading matrix that has a perfect simple structure, then the proposed method with Kaiser's normalization will produce the perfect simple structure. Although many rotation methods can also recover a perfect simple structure, they perform poorly when a perfect simple structure is not possible. In this case, the new method tends to perform better because it clusters the loadings without requiring the clusters to be perfect. Artificial and real data analyses demonstrate that the proposed method can give a simple structure, which the other methods cannot produce, and provides a more interpretable result than those of widely known rotation techniques.