A general rotation criterion and its use in orthogonal rotation

Measures of test parsimony and factor parsimony are defined. Minimizing their weighted sum produces a general rotation criterion for either oblique or orthogonal rotation. The quartimax, varimax and equamax criteria are special cases of the expression. Two new criteria are developed. One of these, the parsimax criterion, apparently gives excellent results. It is argued that one of the most important factors bearing on the choice of a rotation criterion for a particular problem is the amount of information available on the number of factors that should be rotated. This research was supported by the National Research Council of Canada research grant 291-13 to Dr. G. A. Ferguson.     

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.     

A simple graphical method for orthogonal rotation of axes

A simple method of making orthogonal rotations by projecting coordinates from one plot to another is presented. An easily constructed apparatus for rapid and accurate work is described in detail. By the technique proposed 12 to 15 rotations of pairs of axes involving 20 test variables can be completed per hour by a practiced technician.     

Hemisphere differences for components of mental rotation

The hemispheric functional lateralization of components of mental rotation performance was investigated. Twenty right-handed males were presented with rotated alphanumerics and unfamiliar characters in the left or right visual field. Subjects decided if the laterally presented stimulus was identical to or a mirror image of a center standard stimulus. Reaction time and errors were measured. Previous mental rotation findings were replicated and the visual field variable produced significant effects for both dependent measures. An overall right visual field advantage was observed in the latency data, suggesting a left hemisphere superiority for at least one component process of the task. A significant interaction in the error data showed that alphanumerics produced less errors in the right visual field than in the left visual field, consistent with a left hemisphere superiority for processing verbal symbolic material. No such hemispheric difference in accuracy was found for unfamiliar characters.     

Varisim: A new machine method for orthogonal rotation

The derivative of Kaiser's Varimax criterion, if set to zero, yields a set of equations which are quite similar to those obtained for a least-squares problem of the Procrustes type. This similarity suggested an iterative technique for orthogonal rotation, dubbed Varisim, which was programmed for the IBM 7094 in FORTRAN. An empirical comparison between Varimax and Varisim, which was based on a number of data sets taken from the literature yielded three major results so far: (i) Varisim is slower than Varimax, roughly by a factor of 3, (ii) Varisim yields factors which in general contribute more evenly to the common test variance than Varimax factors, and which (iii) line up more closely with oblique configurations obtained with Binormamin than Varimax factors.This paper is based on parts of a thesis submitted to the Graduate College of the University of Illinois in partial fulfillment of the requirements for a degree in Psychology. Some of its contents were read at the joint meeting of the Psychonomic and the Psychometric Society, Niagara Falls, Ontario, 1964. Most of the computer-bound work was carried out while the author was employed by the Statistical Service Unit, University of Illinois; it was written up while the author held a Postdoctoral Fellowship at the Psychometric Laboratory of the University of North Carolina. Although space limitations forbid more complete acknowledgments, I am bound to record my deep gratitude to my advisor, Prof. H. F. Kaiser.     

On the equivalence of two oblique congruence rotation methods,and orthogonal approximations

Tucker's method of oblique congruence rotation is shown to be equivalent to a procedure by Meredith. This implies that Monte Carlo studies on congruence by Nesselroade, Baltes and Labouvie and by Korth and Tucker are highly comparable. The problem of rotating two matrices orthogonally to maximal congruence has not yet been solved. An approximate solution to this problem can be derived from Tucker's method. Even better results can be obtained from a Procrustes rotation followed by rotation to simple structure.     

Mental rotation, physical rotation, and surface media.

Subjects made mirror-normal discriminations on alphanumeric characters shown in different orientations in the picture plane. Either the characters or the background rotated during stimulus presentation in Experiments 1-3. Character rotation in the direction of mental rotation facilitated mental rotation, whereas rotation in the opposite direction inhibited it. In Experiment 4, characters were presented in different surface media so as to stimulate only one representation at a time. Mental rotation performance was similar whether the stimuli were defined by luminance, color, texture, relative motion, or binocular disparity, suggesting that mental rotation occurs at a level beyond that of the independent analyses of these different media. These results support those of Experiments 1-3 in excluding the participation of low-level motion analysis centers in the mental rotation processes.     

The simultaneous maximization of congruence for two or more matrices under orthogonal rotation

While a rotation procedure currently exists to maximize simultaneously Tucker's coefficient of congruence between corresponding factors of two factor matrices under orthogonal rotation of one factor matrix, only approximate solutions are known for the generalized case where two or more matrices are rotated. A generalization and modification of the existing rotation procedure to simultaneously maximize the congruence is described. An example using four data matrices, comparing the generalized congruence maximization procedure with alternative rotation procedures, is presented. The results show a marked improvement of the obtained congruence using the generalized congruence maximization procedure compared to other procedures, without a significant loss of success with respect to the least squares criterion. A computer program written by the author to perform the rotations is briefly discussed.     

Pavlovian conditioning of distinct components of Hermissenda's responses to rotation

In two experiments with the nudibranch mollusk Hermissenda, distinct characteristics of conditioned and unconditioned responses to high-speed orbital rotation were examined. In Experiment 1, two principle unconditioned responses to rotation were identified, namely, reduced rate of locomotion and contraction of the foot. The magnitude of the foot contraction increased throughout a 20-s period of rotation, whereas locomotion was reduced immediately after the onset of the rotation and was maintained at this constant low rate throughout the stimulus presentation. These divergent response patterns suggest that the two responses emerge independently. In Experiment 2, a classical conditioning procedure was employed in which a light (CS) was paired with the rotation (US) employed in Experiment 1. In a subsequent test, it was found that the light had acquired the capability to evoke both foot contraction and decreased locomotion. Although the magnitude of these conditioned responses was reduced relative to the corresponding unconditioned response, the patterns of responding were virtually identical; that is, foot contraction developed gradually whereas locomotion decreased immediately. In contrast, animals that received unpaired presentations of the light and rotation, light alone, or no prior exposure to those stimuli exhibited foot extension in response to the light. These results illustrate a transfer of some of the response-evoking properties of the US to the CS as a result of conditioning, as well as the emergence of two independent conditioned responses. Moreover, these results suggest modulation of at least two distinct motor pathways as a function of learning.     

Orientation disparity, deformation, and stereoscopic slant perception.

Koenderink and van Doorn's theory, that the basis of stereoscopic slant perception is the deformation component of the disparity, field, was tested for slant around a horizontal axis, which produces images with a vertical ramp of horizontal disparity (horizontal shear) characterised by a global orientation disparity at the vertical meridian. The disparity field in this case can be parsed into two components, deformation and curl, which each contribute half of the orientation disparity. This case was compared with similar random-dot stimuli in which the deformation component was doubled and the curl component eliminated or vice versa. All three types of stimuli had identical orientation disparity at the vertical meridian. A condition in which there was no such orientation disparity, but deformation was present, was also included. It was found that perceived slant was not related to the deformation present, as Koenderink and van Doorn's theory would predict, but was predictable from the orientation disparity at the vertical meridian per se.     

Achieving visual object constancy across plane rotation and depth rotation.

Visual object constancy is the ability to recognise an object from its image despite variation in the image when the object is viewed from different angles. I describe research which probes the human visual system's ability to achieve object constancy across plane rotation and depth rotation. I focus on the ecologically important case of recognising familiar objects, although the recognition of novel objects is also discussed. Cognitive neuropsychological studies of patients with specific deficits in achieving object constancy are reviewed, in addition to studies which test neurally intact subjects. In certain cases, the recognition of invariant features allows objects to be recognised irrespective of the view depicted, particularly if small, distinctive sets of objects are presented repeatedly. In contrast, in most situations, recognition is sensitive to both the view in-plane and in-depth from which an object is depicted. This result suggests that multiple, view-specific, stored representations of familiar objects are accessed in everyday, entry-level visual recognition, or that transformations such as mental rotation or interpolation are used to transform between retinal images of objects and view-specific, stored representations.     

Contribution of ankle, knee, and hip joints to the perception threshold for support surface rotation.

The purpose of the present experiment was to investigate the extent to which subjects can perceive, at very slow velocities, an angular rotation of the support surface about the medio-lateral axis of the ankle, knee, hip, or neck joint when visual cues are not available. Subjects were passively displaced on a slowly rotating platform at .01, .03, and .05 deg/sec. The subjects' task was to detect movements of the platform in four different postural conditions allowing body oscillations about the ankle, knee, hip, or neck joint. In Experiment 1, subjects had to detect backward and forward rotation (pitching). In Experiment 2, they had to detect left and right rotations of the platform (rolling). In Experiment 3, subjects had to detect both backward/forward and left/right rotations of the platform, with the body fixed and the head either fixed or free to move. Overall, when the body was free to oscillate about the ankle, knee, or hip joints, a similar threshold for movement perception was observed. This threshold was lower for rolling than for pitching. Interestingly, in these postural conditions, an unconscious compensation in the direction opposite to the platform rotation was observed on most trials. The threshold for movement perception was much higher when the head was the only segment free to oscillate about the neck joint. These results suggest that, in static conditions, the otoliths are poor detectors of the direction of gravity forces. They also suggest that accurate perception of body orientation is improved when proprioceptive information can be dynamically integrated.     

Mental rotation and temporal contingencies.

A task that requires subjects to determine whether two forms of the same shape, but in different orientations, are mirror images or identical except for orientation is called a handedness recognition task. Subjects' reaction times (RT) on this task are consistently related to the angular disparity (termed alpha) between the two presented forms. This pattern of data has been interpreted to indicate that subjects solve the task by imagining that one of the forms rotates into the orientation of the other (termed mental rotation). The speed with which one imagines one of the forms rotating has been widely considered a fixed capability of the individual, and thus immune to the effect of contingencies. We present an experiment that assesses the effects of temporal contingencies in a handedness recognition task on the slope of the function RT = f(alpha). The data indicate that the slope of this function can come under the control of temporal contingencies.     

Mental rotation of letters, pictures, and three-dimensional objects in German dyslexic children.

This study examines mental rotation ability in children with developmental dyslexia. Prior investigations have yielded equivocal results that might be due to differences in stimulus material and testing formats employed. Whereas some investigators found dyslexic readers to be impaired in mental rotation, others did not report any performance differences or even superior spatial performance for dyslexia. Here, we report a comparison of mental rotation for letters, three-dimensional figures sensu Shepard and Metzler, and colored pictures of animals or humans in second-grade German dyslexic readers. Findings indicate that dyslexic readers are impaired in mental rotation for all three kinds of stimuli. Effects of general intelligence were controlled. Furthermore, dyslexic children were deficient in other spatial abilities like identifying letters or forms among distracters. These results are discussed with respect to the hypotheses of a developmental dysfunction of the parietal cortex or a subtle anomaly in cerebellar function in dyslexic readers.     

Perception of three-dimensional angular rotation.

In three experiments, difference thresholds (dLs) and points of subjective equality (PSEs) for three-dimensional (3-D) rotation simulations were examined. In the first experiment, observers compared pairs of simulated spheres that rotated in polar projection and that differed in their structure (points plotted in the volume vs. on the surface), axis of rotation (vertical, y, vs. horizontal, x), and magnitude of rotation (20 degrees-70 degrees). DLs were lowest (7%) when points were on the surface and when at least one sphere rotated around the y-axis and varied with changes in the independent variables. PSEs were closest to objective equality when points were on the surface of both spheres and when both spheres rotated about the x-axis. In the second experiment, subjects provided direct estimates of the rotations of the same spheres. Results suggested a reasonable agreement between PSEs for the indirect-scaling and direct-estimate procedures. The third experiment varied sphere diameter (and therefore mean linear velocity of stimulus elements) and showed that although rotation judgments are biased by mean linear velocity, they are not likely to be made solely on the basis of that information. These and past results suggest a model whereby recovery of structure is conducted by low-level motion-detecting mechanisms, whereas rotation (and other) judgments are based on a higher level representation.