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.     

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.     

The effect of punishment on a factor I stuttering behavior

On each of five days the subject of this experiment read aloud different material during an 18-min A₁ (base rate), B (experimental), and A₂ (extinction) period. No stimulation was delivered during the first three days. On the fourth day, the nonsense word “norg” was presented noncontingently during the experimental period. There was no statistically significant difference between the A₁ and B periods on any of these four days. On day five, “wrong” was made contingent on Factor I prolongations during the experimental period. As a result, there was a statistically significant increase in their frequency. When the response consequence was removed in the A₂ priod, there was a reliable reduction in prolongations and the frequency closely approximated that found in the A₁ period. These data suggest that response-contingent negative stimulation is contraindicated as a clinical tactic for use with Factor I prolongations. It not only failed to suppress them, it increased their frequency.     

An alternative scaling method for priorities in hierarchical structures

The early contributions of Saaty have spawned a multitude of applications of principal right (PR) eigenvector “scaling” of a dominance matrix [R]. Prior to Saaty's work (1977–1984) scaling of dominance matrices received little attention in multidimensional scaling, e.g., see Shepard (1972, pp. 26–27). This eigenvector method (EM) of scaling [R] yields u_i scores (weights) popularly used at each branching of the Analytic Hierarchy Process (AHP) technique that has been increasingly applied in multiple criterion analysis of utility, preference, probability, and performance. In this paper, it is proposed that an alternate least squares method (LSM) scaling technique yielding least squares optimal scores (weights) provides $\text{w}{}_{\mathrm{i}}{}^1$ values having a number of important advantages over u_i scores popularly utilized to date.     

An algorithm for clustering relational data with applications to social network analysis and comparison with multidimensional scaling

A method of hierarchical clustering for relational data is presented, which begins by forming a new square matrix of product-moment correlations between the columns (or rows) of the original data (represented as an n × m matrix). Iterative application of this simple procedure will in general converge to a matrix that may be permuted into the blocked form [_?1¹₁^?1]. This convergence property may be used as the basis of an algorithm (CONCOR) for hierarchical clustering. The CONCOR procedure is applied to several illustrative sets of social network data and is found to give results that are highly compatible with analyses and interpretations of the same data using the blockmodel approach of White (White, Boorman & Breiger, 1976). The results using CONCOR are then compared with results obtained using alternative methods of clustering and scaling (MDSCAL, INDSCAL, HICLUS, ADCLUS) on the same data sets.     

Analysis of the Talmudic Argumentum A Fortiori Inference Rule (Kal Vachomer) using Matrix Abduction

We motivate and introduce a new method of abduction, Matrix Abduction, and apply it to modelling the use of non-deductive inferences in the Talmud such as Analogy and the rule of Argumentum A Fortiori. Given a matrix \({\mathbb {A}}\) with entries in {0, 1}, we allow for one or more blank squares in the matrix, say a _i,j =?. The method allows us to decide whether to declare a _i,j = 0 or a _i,j = 1 or a _i,j =? undecided. This algorithmic method is then applied to modelling several legal and practical reasoning situations including the Talmudic rule of Kal-Vachomer. We add an Appendix showing that this new rule of Matrix Abduction, arising from the Talmud, can also be applied to the analysis of paradoxes in voting and judgement aggregation. In fact we have here a general method for executing non-deductive inferences.     

On Some Generalizations of the Concept of Partition

There are well-known isomorphisms between the complete lattice of all partitions of a given set A and the lattice of all equivalence relations on A. In the paper the notion of partition is generalized in order to work correctly for wider classes of binary relations than equivalence ones such as quasiorders or tolerance relations. Some others classes of binary relations and corresponding counterparts of partitions are considered.     

How one gambles if one must: Effects of differing return rates on multistage betting decisions

In the multistage betting game (MBG), a decision maker (DM) is provided with some capital x which he is required to bet over m (m > 1) mutually exclusive and collectively exhaustive alternatives, each of which occurs with probability p_i (p_i > 0, i = 1,…, m; Σ_{i = 1}^mp_i = 1). If y_i is bet on alternative i (y_i ≥ 0, Σ_{i = 1}^my_i = x) and alternative i obtains, the DM's capital for the next stage is y_ir_i, (r_i > 0). The MBG lasts until either the DM loses his capital or N stages elapse, whichever comes first. Each of six subjects participated in six sessions consisting of several hundred 3-alternative MBG stages. A within-subject design assigned negative expected value (EV) bets to the first three sessions and positive EV bets to three more sessions. Significant effects were found due to return rate, capital size, homogeneous runs of either wins or losses, and individual differences. Four maximization of expected utility and two minimization of risk models were presented and tested. A modified logarithmic utility model is proposed, which provides the best fit to the data. The implications of the results and directions for further research are briefly discussed.     

Alternative measures of fit for the Schönemann-carroll matrix fitting algorithm

In connection with a least-squares solution for fitting one matrix,A, to another,B, under optimal choice of a rigid motion and a dilation, Schönemann and Carroll suggested two measures of fit: a raw measure,e, and a refined similarity measure,e _s , which is symmetric. Both measures share the weakness of depending upon the norm of the target matrix,B,e.g.,e(A,kB) ≠e(A,B) fork ≠ 1. Therefore, both measures are useless for answering questions of the type: “DoesA fitB better thanA fitsC?”. In this note two new measures of fit are suggested which do not depend upon the norms ofA andB, which are (0, 1)-bounded, and which, therefore, provide meaningful answers for comparative analyses.     

An approximation of theK outN reliability of a test,and a scoring procedure for determining which items an examinee knows

Consider any scoring procedure for determining whether an examinee knows the answer to a test item. Letx _i = 1 if a correct decision is made about whether the examinee knows the ith item; otherwisex _i = 0. Thek out ofn reliability of a test isρ _k = Pr (Σx _i ≥k). That is,ρ _k is the probability of making at leastk correct decisions for a typical (randomly sampled) examinee. This paper proposes an approximation ofρ _k that can be estimated with an answer-until-correct test. The paper also suggests a scoring procedure that might be used whenρ _k is judged to be too small under a conventional scoring rule where it is decided an examinee knows if and only if the correct response is given.     

Decision policies minimizing risk in a multistage betting game

Considered in this paper is a decision task which has been employed to study multistage betting behavior. When the task commences, a decision maker (DM) is provided with some capital x (x > 0) which he is required to allocate over m (m > 1) mutually exclusive and collectively exhaustive alternatives, each of which occurs with probability p_i (p_i > 0, i = 1,…, m; Σ_i=1^mp_i = 1). If the amount y_i is allocated to alternative i (y_i ≥ 0, Σ_i=1^my_i = x) and alternative i obtains, DM's capital for the next stage of the game becomes y_ir_i, where r_i (r_i > 0) is the return per unit allocated to alternative i. The task consists of N stages.Defining risk in terms of the mean and variance of DM's bets, and assuming that the minimization of risk is DM's objective, decision policies satisfying this objective are derived in closed form and their testable properties are briefly discussed.     

A strong developmental theory of field dependence-independence

Assume an X-linked gene in two alleles mediates performance on field dependent-independent tests such as the rod-and-frame test. Only the recessive gene with relative frequency q facilitates field independence. Other genotypes lead to field dependence. Under a simple genetic model, field dependence-independence may be viewed as outcomes of a discrete random variable B with field independent and dependent probabilities π_iq and 1 ? π_iq for men, and π_iq² and 1 ? π_iq² for women, respectively. The parameter π_i is a maturational ageindexed parameter, 0 < π_i ≤ 1, monotonically increasing with development until maturity when π_k = 1. Observations of performance are made on a random variable W of the form W = B + N, where N is normal in distribution independent of B; N represents a composite of influences including error. The model implies testable age-related between- and within-sex predictions regarding E(W) and Var(W), predictions which appear to coincide with major empirical findings; it also generates novel predictions. For instance, W is a mixture of normals distribution. The model is briefly evaluates in two data sets.     

Matrices with a given number of violations of Regular Minimality

A row (or column) of an n×n matrix complies with Regular Minimality (RM) if it has a unique minimum entry which is also a unique minimum entry in its column (respectively, row). The number of violations of RM in a matrix is defined as the number of rows (equivalently, columns) that do not comply with RM. We derive a formula for the proportion of n×n matrices with a given number of violations of RM among all n×n matrices with no tied entries. The proportion of matrices with no more than a given number of violations can be treated as the p-value of a permutation test whose null hypothesis states that all permutations of the entries of a matrix without ties are equiprobable, and the alternative hypothesis states that RM violations occur with lower probability than predicted by the null hypothesis. A matrix with ties is treated as being represented by all matrices without ties that have the same set of strict inequalities among their entries.     

A general nonmetric technique for finding the smallest coordinate space for a configuration of points

LetA ₁,A ₂, ...,A _n be anyn objects, such as variables, categories, people, social groups, ideas, physical objects, or any other. The empirical data to be analyzed are coefficients of similarity or distance within pairs (A _i,A _i ), such as correlation coefficients, conditional probabilities or likelihoods, psychological choice or confusion, etc. It is desired to represent these data parsimoniously in a coordinate space, by calculatingm coordinates {x _ia } for eachA _i for a semi-metricd of preassigned formd _ij =d(|x _i1 -x _j1 |, |x _i2 -x _j2|, ..., |x _im -x _jm |). The dimensionalitym is sought to be as small as possible, yet satisfy the monotonicity condition thatd _ij <d _kl whenever the observed data indicate thatA _i is closer toA _j thanA _k is toA _l . Minkowski and Euclidean spaces are special metric examples ofd. A general coefficient of monotonicity is defined, whose maximization is equivalent to optimal satisfaction of the monotonicity condition, and which allows various options both for treatment of ties and for weighting error-of-fit. A general rationale for algorithm construction is derived for maximizing by gradient-guided iterations; this provides a unified mathematical solution to the basic operational problems of norming the gradient to assure proper convergence, of trading between speed and robustness against undesired stationary values, and of a rational first approximation. Distinction is made between single-phase (quadratic) and two-phase (bilinear) strategies for algorithm construction, and between hard-squeeze and soft-squeeze tactics within these strategies. Special reference is made to the rank-image and related transformational principles, as executed by current Guttman-Lingoes families of computer programs.This research was supported in part by the National Science Foundation of the United States Government, through Grant No. GS 929 to the University of Michigan.I am deeply indebted for many helpful comments on earlier drafts of this paper received from Joseph Kruskal, James Lingoes, and the managing editor.     

Problem syntezy sieci w logice progowej. I

Tre??: Wst?p. Cz??? pierwsza: Synteza sieci zbudowanych na elementach doprogowych. Rozdzia? I. Synteza sieci zbudowanych na elementach doprogowych w algebrze Boole'a: 1. Klasa elementów doprogowych. 2. Metoda Lindemana syntezy sieci. Rozdzia? II. Synteza sieci w logice doprogowej. 3. Aksjomatyka logiki doprogowej. 4. Podstawowe twierdzenia logiki doprogowej. 5. Zasadnicze twierdzenie logiki doprogowej. 6. Metody syntezy sieci w logice doprogowej. Cz??? druga: Synteza sieci zbudowanych na ogólnych elementach progowych. Rozdzia? I. Interpretacja geometryczna elementów progowych i funkcji algebry logiki: 1. Definicje i twierdzenia pomocnicze. 2. Interpretacja geometryczna ogólnych elementów progowych. 3. Definicje elementówn-wymiarowej kostki. 4. Interpretacja geometryczna funkcji algebry logiki. Rozdzia? II. Synteza siec_i jednego elementu progowego z pobudzaj?cymi wej?ciami: 5. W?asno?ci zbiorówA iB oraz ich zbiorów oporowychA _o iB _o. 6. Równanie hiperp?aszczyzny ?ci?le oddzielaj?cej zbiórB _o od zbioruA _o. 7. Warunki realizacji funkcji na progowych elementach z pobudzaj?cymi wej?ciami. 8. Metody syntezy progowego elementu z pobudzaj?cymi wej?ciami. Rozdzia? III. Synteza sieci jednego ogólnego elementu progowego: 9. Uogólnienie twierdzeń dla elementów progowych z pobudzaj?cymi i hamuj?cymi wej?ciami. 10. Warunki realizacji funkcji jednym ogólnym elementem progowym. 11. Metody syntezy jednym ogólnym elementem progowym. Rozdzia? IV. Synteza sieci zbudowanych na ogólnych elementach progowych: 12. Synteza sieci rz?du pierwszego i drugiego. 13. Synteza sieci rz?du trzeciego i sieci wy?szych rz?dów. Zakończenie. Dodatek: Progowe funkcjen≤4 zmiennych. Bibliografia.     

Correspondence and mediational properties of the Fishbein model: An application to adolescent alcohol use

The generalizability of the Fishbein model for behavior prediction was extended to a new field behavior, alcohol drinking by adolescents. The model's two components (i.e., attitudes and normative beliefs) varied in their predictive importance depending on the behavior's situational context. Following this field demonstration, two issues related to properties affecting the model's predictive effectiveness were investigated. First, it was found that the predictability of alcohol-drinking intentions (BI) declined when attitudes (A_act) and normative beliefs (NB) did not correspond with BI in terms of alcohol type (beer, liquor, or wine) and drinking situation (home, party, or pub). A second experiment tested the hypothesis that exogenous variables (those other than A_act and NB) would have a direct influence on BI. A comprehensive test was provided within the context of Jessor's multivariate predictive system for adolescent alcohol use. Present data indicate only a marginal increase in explained variance occurred when applying this system, thereby generally indicating the sufficiency of the Fishbein model's two components.     

Pseudocontingencies derived from categorically organized memory representations

Pseudocontingencies (PCs) allow for inferences about the contingency between two variables X and Y when the conditions for genuine contingency assessment are not met. Even when joint observations X _i and Y _i about the same reference objects i are not available or are detached in time or space, the correlation r(X _i ,Y _i ) is readily inferred from base rates. Inferred correlations are positive (negative) if X and Y base rates are skewed in the same (different) directions. Such PC inferences afford useful proxies for actually existing contingencies. While previous studies have focused on PCs due to environmental base rates, the present research highlights memory organization as a natural source of PC effects. When information about two attributes X and Y is represented in a hierarchically organized categorical memory code, as category-wise base rates p(X) and p(Y), the reconstruction of item-level information from category base rates will naturally produce PC effects. Three experiments support this contention. When the yes base rates of two respondents in four questionnaire subscales (categories) were correlated, recalled and predicted item-level responses were correlated in the same direction, even when the original responses to specific items within categories were correlated in the opposite direction.     

Continuous Orthogonal Complement Functions and Distribution-Free Goodness of Fit Tests in Moment Structure Analysis

It is shown that for any full column rank matrix X ₀ with more rows than columns there is a neighborhood $\mathcal{N}$ of X ₀ and a continuous function f on $\mathcal{N}$ such that f(X) is an orthogonal complement of X for all X in $\mathcal{N}$ . This is used to derive a distribution free goodness of fit test for covariance structure analysis. This test was proposed some time ago and is extensively used. Unfortunately, there is an error in the proof that the proposed test statistic has an asymptotic χ ² distribution. This is a potentially serious problem, without a proof the test statistic may not, in fact, be asymptoticly χ ². The proof, however, is easily fixed using a continuous orthogonal complement function. Similar problems arise in other applications where orthogonal complements are used. These can also be resolved by using continuous orthogonal complement functions.     

Generalized procrustes analysis

SupposeP _i ⁽ⁱ⁾ (i = 1, 2, ...,m, j = 1, 2, ...,n) give the locations ofmn points inp-dimensional space. Collectively these may be regarded asm configurations, or scalings, each ofn points inp-dimensions. The problem is investigated of translating, rotating, reflecting and scaling them configurations to minimize the goodness-of-fit criterion Σ _i=1 ^m Σ _i=1 ⁿ Δ²(P _j ⁽ⁱ⁾ G _i ), whereG _i is the centroid of them pointsP _i ⁽ⁱ⁾ (i = 1, 2, ...,m). The rotated positions of each configuration may be regarded as individual analyses with the centroid configuration representing a consensus, and this relationship with individual scaling analysis is discussed. A computational technique is given, the results of which can be summarized in analysis of variance form. The special casem = 2 corresponds to Classical Procrustes analysis but the choice of criterion that fits each configuration to the common centroid configuration avoids difficulties that arise when one set is fitted to the other, regarded as fixed.     

A generalization of Kristof's theorem on the trace of certain matrix products

Kristof has derived a theorem on the maximum and minimum of the trace of matrix products of the form \(X_1 \hat \Gamma _1 X_2 \hat \Gamma _2 \cdots X_n \hat \Gamma _n\) where the matrices \(\hat \Gamma _i\) are diagonal and fixed and theX _i vary unrestrictedly and independently over the set of orthonormal matrices. The theorem is a useful tool in deriving maxima and minima of matrix trace functions subject to orthogonality constraints. The present paper contains a generalization of Kristof's theorem to the case where theX _i are merely required to be submatrices of orthonormal matrices and to have a specified maximum rank. The generalized theorem contains the Schwarz inequality as a special case. Various examples from the psychometric literature, illustrating the practical use of the generalized theorem, are discussed.