An approach to geometry of visual space with no a priori mapping functions: Multidimensional mapping according to Riemannian metrics

A method of multidimensional mapping is described which constructs a configuration of points {P_i} in a Euclidean map of Riemannian space of constant curvature (hyperbolic, Euclidean, and elliptic) from the dissimilarity matrix (d_ij). The method was applied to the distance matrix in visual space where stimulus points Q_i were either small light points in the dark or small black points in the illuminated field surrounded by white curtains and d_ij represent scaled values of perceptual distances. Configuration of points {Q_i} were at intersections of parallel or distance alleys and horopters for the subject in the horizontal plane of the eye level. In contrast to the theoretical equations for {Q_i} by Luneburg and Blank, no a priori assumption on mapping functions between {Q_i} and {P_i} is necessary in this procedure to fit theoretical curves to {P_i} in the Euclidean map. The data were accounted for better by equations in the hyperbolic plane than by ones in the Euclidean plane. Discussions are made on robustness of Euclidean representation and on how to approach geometry of visual space as a dynamic entity under more natural conditions than the traditional frameless condition for alley and horopter experiments.     

Assessing sensitivity in a multidimensional space: some problems and a definition of a general d'

This article provides a formal definition for a sensivity measure,d′ _g , between two multivariate stimuli. In recent attempts to assess perceptual representations using qualitative tests on response probabilities, the concept of ad′ between two multidimensional stimuli has played a central role. For example, Kadlec and Townsend (1992a, 1992b) proposed several tests based on multidimensional signal detection theory that allow conclusions concerning the perceptual and/or decisional interactions of stimulus dimensions. One proposition, referred to as thediagonal d′test, relies on specific stimulus subsets of a feature-complete factorial identification task to infer perceptual separability. Also, Ashby and Townsend (1986), in a similar manner, attempted to relate perceptual independence to dimensional orthogonality in Tanner’s (1956) model, which also involvesd′ between two multivariate signals. An analysis of the proposedd′ _g reveals shortcomings in the diagonald′ test and also demonstrates that the assumptions behind equating perceptual independence to dimensional orthogonality are too weak. Thisd′ _g can be related to a common measure of statistical distance, Mahalanobis distance, in the special case of equal covariance matrices.     

An analysis of visual detection by temporal probability summation

The probability of detection by temporal probability summation (TPS) is derived assuming a non-white Gaussian noise process, characterised by a finite second spectral moment λ₂; the assumption of stationary noise is discussed with respect to neurophysiological findings suggesting multiplicative noise for individual neurons. The usual white-noise assumption is shown to imply inconsistent detection models. d^′-estimates for 2AFC-experiments do not depend on the value of λ₂ and do not differ from those derived from the standard SDT-assumptions, in contrast to d^′-estimates for yes-no-experiments, which depend on the probability of false alarms. Estimates of properties of a visual channel depend on whether detection is by TPS or temporal peak detection (TPD); it is argued that TPD is not a special case of TPS, and that subjects chose a TPS- or a TPD-strategy depending on experimental conditions.     

Dual scaling of comparison and reference stimuli in multi-dimensional psychological space

Dzhafarov and Colonius (Psychol. Bull. Rev. 6 (1999)239; J. Math. Psychol. 45(2001)670) proposed a theory of Fechnerian scaling of the stimulus space based on the psychometric (discrimination probability) function of a human subject in a same-different comparison task. Here, we investigate a related but different paradigm, namely, referent-probe comparison task, in which the pair of stimuli (x and y) under comparison assumes substantively different psychological status, one serving as a referent and the other as a probe. The duality between a pair of psychometric functions, arising from assigning either x or y to be the fixed reference stimulus and the other to be the varying comparison stimulus, and the 1-to-1 mapping between the two stimulus spaces X and Y under either assignment are analyzed. Following Dzhafarov and Colonius, we investigate two properties characteristic of a referent-probe comparison task, namely, (i) Regular cross-minimality—for the pair of stimulus values involved in referent-probe comparison, each minimizes a discrimination probability function where the other is treated as the fixed reference stimulus; (ii) Nonconstant self-similarity—the value of the discrimination probability function at its minima is a nonconstant function of the reference stimulus value. For the particular form of psychometric functions investigated, it is shown that imposing the condition of regular cross-minimality on the pair of psychometric functions forces a consistent (but otherwise still arbitrary) mapping between X and Y, such that it is independent of the assignment of reference/comparison status to x and to y. The resulting psychometric differentials under both assignments are equal, and take an asymmetric, dualistic form reminiscent of the so-called divergence measure that appeared in the context of differential geometry of the probability manifold with dually flat connections (Differential Geometric Methods in Statistics, Lecture Notes in Statistics, Vol. 28, Springer, New York, 1985). The pair of divergence functions on X and on Y, respectively, induce a Riemannian metric in the small, with psychometric order (defined in Dzhafarov & Colonius, 1999) equal to 2. The difference between the Finsler-Riemann geometric approach to the stimulus space (Dzhafarov & Colonius, 1999) and this dually-affine Riemannian geometric approach to the dual scaling of the comparison and the reference stimuli is discussed.     

Binary choice, subset choice, random utility, and ranking: A unified perspective using the permutahedron

The d-permutahedron Π_d−1⊂R^d is defined as the convex hull of all d-dimensional permutation vectors, namely, vectors whose components are distinct values of a d-element set of integers [d]≡{1,2,…,d}. By construction, Π_d−1 is a convex polytope with d! vertices, each representing a linear order (ranking) on [d], and has dimension dim(Π_d−1)=d−1. This paper provides a review of some well-known properties of a permutahedron, applies the geometric-combinatoric insights to the investigation of the various popular choice paradigms and models by emphasizing their inter-connections, and presents a few new results along this line.Permutahedron provides a natural representation of ranking probability; in fact it is shown here to be the space of all Borda scores on ranking probabilities (also called “voters profiles” in the social choice literature). The following relations are immediate consequences of this identification. First, as all d! vertices of Π_d−1 are equidistant to its barycenter, Π_d−1 is circumscribed by a sphere S^d−2 in a (d−1)-dimensional space, with each spherical point representing an equivalent class of vectors whose components are defined on an interval scale. This property provides a natural expression of the random utility model of ranking probabilities, including the condition of Block and Marschak. Second, Π_d−1 can be realized as the image of an affine projection from the unit cube of dimension d(d−1)/2. As the latter is the space of all binary choice vectors describing probabilities of pairwise comparisons within d objects, Borda scores can be defined on binary choice probabilities through this projective mapping. The result is the Young's formula, now applicable to any arbitrary binary choice vector. Third, Π_d−1 can be realized as a “monotone path polytope” as induced from the lift-up of the projection of the cube onto the line segment [0,d]⊂R¹. As the 2^d vertices of the d-cube are in one-to-one correspondence to all subsets of [d], a connection between the subset choice paradigm and ranking probability is established. Specifically, it is shown here that, in the case of approval voting (AV) with the standard tally procedure (Amer. Pol. Sci. Rev. 72 (1978) 831), under the assumption that the choice of a subset indicates an approval (with equal probability) of all linear orders consistent with that chosen subset, the Brams-Fishburn score is then equivalent to the Borda score on the induced profile. Requiring this induced profile (ranking probability) to be also consistent with the size-independent model of subset choice (J. Math. Psychol. 40 (1996) 15) defines the “core” of the AV Polytope. Finally, Π_d−1 can be realized as a canonical projection from the so-called Birkhoff polytope, the space of rank-position probabilities arising out of the rank-matching paradigm; thus Borda scores can be defined on rank-position probabilities. To summarize, the many realizations of a permutahedron afford a unified framework for describing and relating various ranking and choice paradigms.     

Some counterexamples in multidimensional scaling

Given a set X with elements x, y,… which has a partial order < on the pairs of the Cartesian product X², one may seek a distance function ? on such pairs (x, y) which satisfies ?(x₁, y₁) < ?(x₂, y₂) precisely when (x₁, y₁) < (x₂, y₂), and even demand a metric space (X, ?) with some such compatible ? which has an isometric imbedding into a finite-dimensional Euclidean space or a separable Hilbert space. We exhibit here systems (X, <) which cannot meet the latter demand. The space of real m-tuples (ξ₁,…,ξ_m) with either the “city-block” norm Σ_i ∥ξ_i∥ or the “dominance” norm max_i, ∥ξ_i∥ cannot possibly become a subset of any finite-dimensional Euclidean space. The set of real sequences (ξ₁, ξ₂,…) with finitely many nonzero elements and the supremum norm sup_i, ∥ξ_i∥ cannot even become a subset of any separable Hilbert space.     

A quasi-nonmetric method for multidimensional scaling VIA an extended euclidean model

An Extended Two-Way Euclidean Multidimensional Scaling (MDS) model which assumes both common and specific dimensions is described and contrasted with the standard (Two-Way) MDS model. In this Extended Two-Way Euclidean model then stimuli (or other objects) are assumed to be characterized by coordinates onR common dimensions. In addition each stimulus is assumed to have a dimension (or dimensions) specific to it alone. The overall distance between objecti and objectj then is defined as the square root of the ordinary squared Euclidean distance plus terms denoting the specificity of each object. The specificity,s _j , can be thought of as the sum of squares of coordinates on those dimensions specific to objecti, all of which have nonzero coordinatesonly for objecti. (In practice, we may think of there being just one such specific dimension for each object, as this situation is mathematically indistinguishable from the case in which there are more than one.)We further assume that _ij =F(d _ij ) +e _ij where _ij is the proximity value (e.g., similarity or dissimilarity) of objectsi andj,d _ij is the extended Euclidean distance defined above, whilee _ij is an error term assumed i.i.d.N(0, ²).F is assumed either a linear function (in the metric case) or a monotone spline of specified form (in the quasi-nonmetric case). A numerical procedure alternating a modified Newton-Raphson algorithm with an algorithm for fitting an optimal monotone spline (or linear function) is used to secure maximum likelihood estimates of the paramstatistics) can be used to test hypotheses about the number of common dimensions, and/or the existence of specific (in addition toR common) dimensions.This approach is illustrated with applications to both artificial data and real data on judged similarity of nations.     

Establishing an association between the Flynn effect and ability differentiation

The relationship between the Flynn effect and ability differentiation is investigated in a reanalysis of published data on Estonian student cohorts tested in 1933/36, 1997/98 and 2006 on the National Intelligence Test (Must, te Nijenhuis, Must, & van Vianen, 2009). To determine whether there was a relationship we computed the vector correlation between the Flynn effects (d) and the change in the g loading (Δg) between measurement occasions for each of the 10 NIT subtests and for each of the seven cohort comparisons, giving a total N of 70 effect sizes. The association between d and Δg was robustly negative (indicating that the Flynn effects were negatively associated with changes in the g loading of subtests) for all cohort comparisons, with values of r ranging from −.100 to −.461 (N = 10). When all effect sizes were analyzed together, the vector correlation was found to be −.281 (p ? .05, N = 70). This indicates a significant association between the Flynn effect and ability differentiation. Possible causes of this association are discussed.     

A locus problem involving the three-dimensional city-block metric

In connection with multidimensional scaling, representations have been considered of the form abDcd ← ?(f(a), f(b)) ≦ ?(f(c), f(d)), for all a, b, c, d ∈ A, where A is a nonvoid finite set, D is a four-place relation on A, f is a function from A into Euclidean n-space, $\text{R}{}^{\mathrm{n}}$, and ? is a metric in $\text{R}{}^{\mathrm{n}}$. For particular metrics there exist finite universal axiomatizations which are necessary and sufficient for the above representation. On the other hand, it is known that no such axiomatizations can be given for either the supremum metric or the ordinary Euclidean metric. Methods for showing this apply easily to the city-block metrics in $\text{R}{}^1$ and $\text{R}{}^2$. This article describes a computer-aided verification of a locus result which shows the impossibility of finite universal axiomatizability for the case of the city-block metric in $\text{R}{}^3$. The result was obtained by dealing with 21,780 cases, each of which involved a set of 10 equations in 12 unknowns along with a related set of inequalities.     

No relevant spectator impact on home advantage in male and female professional volleyball – A longitudinal multilevel logistic model analysis over 25 years

Home advantage (HA) regularly occurs in volleyball (Pollard et al., 2017: men: 56.62%, women: 55.26%). Research to date has investigated primarily small samples of mostly female matches and not looked into the potential impact of spectators on HA. This archival analysis uses multilevel modelling to examine HA in professional German volleyball (men & women) over 25 seasons in all regular and play-off matches (N = 6,833). We analyze how spectators drive HA and whether this projects to the COVID-19 season 2020/21.When intercepts varied between teams (2-level model, ICC = 27%), the winning probability increased when playing at home (men: 57.01%, OR_men = 2.39, d = 0.48; women: 55.39%, OR_women = 2.19, d = 0.43), while controlling for team strength, interaction with gender, and travelling distance. More spectators had a negligible effect on the men’s and women’s chances (|d| < 0.07). Similar trends were observed for the probability of winning sets. Contrary to other team sports (e.g., soccer), there is no HA-development over the last decades.     

Recognizing upper limb movements with wrist worn inertial sensors using k-means clustering classification

In this paper we present a methodology for recognizing three fundamental movements of the human forearm (extension, flexion and rotation) using pattern recognition applied to the data from a single wrist-worn, inertial sensor. We propose that this technique could be used as a clinical tool to assess rehabilitation progress in neurodegenerative pathologies such as stroke or cerebral palsy by tracking the number of times a patient performs specific arm movements (e.g. prescribed exercises) with their paretic arm throughout the day. We demonstrate this with healthy subjects and stroke patients in a simple proof of concept study in which these arm movements are detected during an archetypal activity of daily-living (ADL) – ‘making-a-cup-of-tea’. Data is collected from a tri-axial accelerometer and a tri-axial gyroscope located proximal to the wrist. In a training phase, movements are initially performed in a controlled environment which are represented by a ranked set of 30 time-domain features. Using a sequential forward selection technique, for each set of feature combinations three clusters are formed using k-means clustering followed by 10 runs of 10-fold cross validation on the training data to determine the best feature combinations. For the testing phase, movements performed during the ADL are associated with each cluster label using a minimum distance classifier in a multi-dimensional feature space, comprised of the best ranked features, using Euclidean or Mahalanobis distance as the metric. Experiments were performed with four healthy subjects and four stroke survivors and our results show that the proposed methodology can detect the three movements performed during the ADL with an overall average accuracy of 88% using the accelerometer data and 83% using the gyroscope data across all healthy subjects and arm movement types. The average accuracy across all stroke survivors was 70% using accelerometer data and 66% using gyroscope data. We also use a Linear Discriminant Analysis (LDA) classifier and a Support Vector Machine (SVM) classifier in association with the same set of features to detect the three arm movements and compare the results to demonstrate the effectiveness of our proposed methodology.     

Influence of step-surface visual properties on confidence,anxiety, dynamic stability,and gaze behaviour in young and older adults

BackgroundStep-surface visual properties are often associated with stair falls. However, evidence for decorating stairs typically concerns the application of step-edge highlighters rather than the entire step-surface. Here we examine the influence of step-surface visual properties on stair descent safety, with a view to generating preliminary evidence for safe stair décor.MethodsFourteen young (YA: 23.1 ± 3.7 years), 13 higher (HAOA: 67 ± 3.5) and 14 lower (LAOA: 73.4 ± 5.7) ability older adults descended a seven-step staircase. Older adults were stratified based on physiological/cognitive function. Step-surface décor patterns assessed were: Black and white (Busy); fine grey (Plain); and striped multicolour (Striped); each implemented with/without black edge-highlighters (5.5 cm width) totalling six conditions. Participants descended three times per condition. Confidence was assessed prior to, and anxiety following, the first descent in each condition. 3D kinematics (Vicon) quantified descent speed, margin of stability, and foot clearances with respect to step-edges. Eye tracking (Pupil-labs) recorded gaze. Data from three phases of descent (entry, middle, exit) were analysed. Linear mixed-effects models assessed within-subject effects of décor (×3) and edge highlighters (×2), between-subject effects of age (×3), and interactions between terms (α = p < .05).ResultsDécor: Plain décor reduced anxiety in all ages and abilities (p = .032, effect size: g_av = 0.3), and increased foot clearances in YA and HAOA in the middle phase (p < .001, g_av = 0.53), thus improving safety. In contrast, LAOA exhibited no change in foot clearance with Plain décor. Patterned décor slowed descent (Busy: p < .001, g_av = 0.2), increased margins of stability (Busy: p < .001, g_av = 0.41; Striped: p < .001, gav = 0.25) and reduced steps looked ahead (Busy: p = .053, gav = 0.25; Striped: p = .039, gav = 0.28) in all ages and abilities. This reflects cautious descent, likely due to more challenging conditions for visually extracting information about the spatial characteristics of the steps useful to guide descent.Edge highlighters: Step-edge highlighters increased confidence (p < .001, g_av = 0.53) and reduced anxiety (p < .001, g_av = 0.45) in all ages and abilities and for all décor, whilst removing them slowed descent in HAOA (p = .01, g_av = 0.26) and LAOA (p = .003, g_av = 0.25). Step-edge highlighters also increased foot clearance in YA and HAOA (p = .003, g_av = 0.14), whilst LAOA older adults showed no adaptation. No change in foot clearances with décor or step-edge highlighters in LAOA suggests an inability to adapt to step-surface visual properties.ConclusionPatterned step surfaces can lead to more cautious and demanding stair negotiation from the perspective of visually extracting spatial information about the steps. In contrast, plain décor with step edge highlighters improves safety. We therefore suggest plain décor with edge highlighters is preferable for use on stairs.     

How many response options? A study of remember-know testing procedures

The remember-know procedure can be conducted in one or two steps. The one-step procedure does not include a recognition response (old-new) prior to the remember-know response. It is observed consistently that the one-step procedure leads to a more liberal placement of the response criterion, but it is unclear whether recognition accuracy is affected by the number of procedural steps. However, previous studies used bias-dependent measures of accuracy (A′ and d′). We manipulated the number of steps and confirmed the finding that the response criterion is more liberal with the one-step procedure. More importantly, we employed a signal detection theory bias-free accuracy measure (d_a) to show that varying the number of steps does not affect recognition accuracy, and we demonstrated that this pattern of results does not change when the dual process signal detection model (Yonelinas, 1997) is applied.     

Nondecomposable conjoint measurement for bisymmetric structures

This paper discusses two “nondecomposable” conjoint measurement representations for an asymmetric binary relation ? on a product set A × X, namely (a, x) ? (b, y) iff f₁(a) + g₁(a)g₂(x) > f₁(b) + g₁(b)g₂(y), and (a, x) ? (b, y) iff f₁(a) + f₂(x) + g₁(a)g₂(x) > f₁(b) + f₂(y) + g₁(b)g₂(y). Difficulties in developing axioms for ? on A × X which imply these representations in a general formulation have led to their examination from the standpoint of bisymmetric structures based on applications of a binary operation to A × X. Depending on context, the binary operation may refer to concatenation, extensive or intensive averaging, gambles based on an uncertain chance event, or to some other interpretable process. Independence axioms which are necessary and sufficient for the special representations within the context of bisymmetric structures are presented.     

Testing for selectivity in the dependence of random variables on external factors

Random variables A and B, whose joint distribution depends on factors (x,y), are selectively influenced by x and y, respectively, if A and B can be represented as functions of, respectively, (x,S_A,C) and (y,S_B,C), where S_A,S_B,C are stochastically independent and do not depend on (x,y). Selective influence implies selective dependence of marginal distributions on the respective factors: thus no parameter of A may depend on y. But parameters characterizing stochastic interdependence of A and B, such as their mixed moments, are generally functions of both x and y. We derive two simple necessary conditions for selective dependence of (A,B) on (x,y), which can be used to conduct a potential infinity of selectiveness tests. One condition is that, for any factor values x,x^′ and y,y^′,

s_xy≤s_xy^′+s_x^′y^′+s_x^′y,     

Extraction of relief from visual motion

We quantified the ability of human subjects to discriminate the relative distance of two points from a slanted plane when viewing the projected velocities of this scene (orthographic projection). The relative distance from a plane (called relief) is a 3-D property that is invariant under linear (affine) transformations. As such, relief canin principle be extracted from the instantaneous projected velocity field; a metric representation, which requires the extraction of visual acceleration, is not required. The stimulus consisted of a slanted planeP (specified by three points) and two pointsQ ₁ andQ ₂ that are non-coplanar withP. This configuration of points oscillated rigidly around the vertical axis. We have measured thesystematic error andaccuracy with which human subjects estimate the relative distance of pointsQ ₁ andQ ₂ from planeP as a function of the slant of_P. The systematic error varies with slant: it is low for small slant values, reaches a maximum for medium slant values, and drops again for high slant values. The accuracy covaries with the systematic error and is thus high for small and large slant values and low for medium slant values. These results are successfully modeled by a simple relief-from-motion computation based on local estimates of projected velocities. The data are well predicted by assuming (1) a measurement error in velocity estimation that varies proportionally to velocity (Weber’s law) and (2) an eccentricity-dependent underestimation of velocity.     

Extensions of the significance test for one-parameter signal detection hypotheses

The basic models of signal detection theory involve the parametric measure,d, generally interpreted as a detectability index. Given two observers, one might wish to know whether their detectability indices are equal or unequal. Gourevitch and Galanter (1967) proposed a large sample statistical test that could be used to test the hypothesis of equald values. In this paper, their large two sample test is extended to aK-sample detection test. If the null hypothesisd ₁=d ₂=...=d _K is rejected, one can employ the post hoc confidence interval procedure described in this paper to locate possible statistically significant sources of variance and differences. In addition, it is shown how one can use the Gourevitch and Galanter statistics to testd=0 for a single individual.This paper was written while the author was associated with the Institute of Human Learning at the University of California at Berkeley.     

Gray matter and intelligence factors: Is there a neuro-g?

Heterogeneous results among neuro-imaging studies using psychometric intelligence measures may result from the variety of tests used. The g-factor may provide a common metric across studies. Here we derived a g-factor from a battery of eight cognitive tests completed by 6929 young adults, 40 of whom also completed structural MRI scans. Regional gray matter (GM) was determined using voxel-based-morphometry (VBM) and correlated to g-scores. Results showed correlations distributed throughout the brain, but there was limited overlap with brain areas identified in a similar study that used a different battery of tests to derive g-scores. Comparable spatial scores (with g variance removed) also were derived from both batteries, and there was considerable overlap in brain areas where GM was correlated to the respective spatial scores. The results indicate that g-scores derived from different test batteries do not necessarily have equivalent neuro-anatomical substrates, suggesting that identifying a “neuro-g” will be difficult. The neuro-anatomical substrate of a spatial factor, however, appears more consistent and implicates a distributed network of brain areas that may be involved with spatial ability. Future imaging studies directed at identifying the neural basis of intelligence may benefit from using a psychometric test battery chosen with specific criteria.     

The secular rise in IQs in the Netherlands: Is the Flynn effect on g?

IQ scores have been increasing over the last half century, a phenomenon known as the Flynn effect. In this study, we focused on the question to what extent these secular gains are on the g factor. Two IQ batteries: the Interest-School achievement-Intelligence Test (ISI) and the Groningen Final Examination Primary Education (GALO) yielded small and modest negative correlations between standardized gains and g loadings. As these studies employ large samples this suggests that the combined literature now shows a modest negative relationship between d (the secular change in test score) and g.     

An axiomatization of the ratio/difference representation

If ≥_r and ≥_d are two quaternary relations on an arbitrary set A, a ratio/difference representation for ≥_r and ≥_d is defined to be a function f that represents ≥_r as an ordering of numerical ratios and ≥_d as an ordering of numerical differences. Krantz, Luce, Suppes and Tversky (1971, Foundations of Measurement. New York, Academic Press) proposed an axiomatization of the ratio/difference representation, but their axiomatization contains an error. After describing a counterexample to their axiomatization, Theorem 1 of the present article shows that it actually implies a weaker result: if ≥_r and ≥_d are two quaternary retations satisfying the axiomatization proposed by Krantz et al. (1971), and if ≥_r′ and ≥_d′ are the relations that are inverse to ≥_r and ≥_d, respectively, then either there exists a ratio/difference representation for ≥_r and ≥_d, or there exists a ratio/difference representation for ≥_r′ and ≥_d′, but not both. Theorem 2 identifies a new condition which, when added to the axioms of Krantz et al. (1971), yields the existence of a ratio/difference representation for relations ≥_r and ≥_d.