A general solution of the weighted orthonormal procrustes problem

A general solution for weighted orthonormal Procrustes problem is offered in terms of the least squares criterion. For the two-demensional case. this solution always gives the global minimum; for the general case, an algorithm is proposed that must converge, although not necessarily to the global minimum. In general, the algorithm yields a solution for the problem of how to fit one matrix to another under the condition that the dimensions of the latter matrix first are allowed to be transformed orthonormally and then weighted differentially, which is the task encountered in fitting analogues of the IDIOSCAL and INDSCAL models to a set of configurations.The authors are grateful to the Editor and the anonymous reviewers for their helpful comments on an earlier draft of this paper.     

On the definition of differential reinforcement of alternative behavior

Differential reinforcement of alternative behavior (DRA) has a long history as a behavioral treatment. The term has usually been defined in a manner that suggests one form of behavior (usually some appropriate alternative) is reinforced, while another form of behavior (usually problem behavior) is placed on extinction. In this paper, we will suggest that DRA should not require extinction, either procedurally or by definition. Ideally, problem behavior would be placed on extinction when possible; however, when problem behavior is not or cannot be placed on extinction, the procedure used is still DRA. Thus, we propose the following definition: Providing greater reinforcement, along at least one dimension, contingent on the occurrence of one form or type of behavior, while minimizing reinforcement for another form or type of behavior.     

The reconstruction of static visual forms from sparse dotted samples

This study explored the ability of observers to recognize very sparsely sampled, stereoscopic, dotted surfaces. The observers’ performance exceeded that of a simple surface-fitting algorithm that served as a first approximation model to the solution of the perceptual problem. We explored various attributes ofthe stimulus surfaces, but were unable to find any single attribute that could account for the measured performance. Therefore, we propose that the observers used several global surface attributes jointly to distinguish one form from another. Collectively, these attributes define what is meant by form in this study. We suggest that our results argue for a globally precedent, multidimensional model of form reconstruction as opposed to a local-feature, unidimensional interpretation.     

The rigid orthogonal Procrustes rotation problem

The problem of rotating a matrix orthogonally to a best least squares fit with another matrix of the same order has a closed-form solution based on a singular value decomposition. The optimal rotation matrix is not necessarily rigid, but may also involve a reflection. In some applications, only rigid rotations are permitted. Gower (1976) has proposed a method for suppressing reflections in cases where that is necessary. This paper proves that Gower’s solution does indeed give the best least squares fit over rigid rotation when the unconstrained solution is not rigid. Also, special cases that have multiple solutions are discussed. The author is obliged to Henk Kiers and Alwin Stegeman for helpful comments on a previous draft.     

A circuit to permit an electronic drinkometer to operate during footshock

A circuit to permit an electronic drinkometer to operate during footshock is described. The circuit employs a diode matrix that allows all grid bars to be connected at one point to form one electrode of the drinkometer; it is isolated at another point so as not to shortcircuit the impressed footshock.     

Procrustes matching by congruence coefficients

Matching by Procrustes methods involves the transformation of one matrix to match with another. A special least squares criterion, the congruence coefficient, has advantages as a criterion for some factor analytic interpretations. A Procrustes method maximizing the congruence coefficient is given. This solution is identical to Mosier's [1939] approximate solution, but is an exact solution for maximum congruence.     

Levels of Criticism: Handling Popperian Problems in a Popperian Way

Popper emphasised both the problem-solving nature of human knowledge, and the need to criticise a scientific theory as strongly as possible. These aims seem to contradict each other, in that the former stresses the problems that motivate scientific theories while the one ignores the character of the problems that led to the formation of the theories against which the criticism is directed. A resolution is proposed in which problems as such are taken as prime in the search for knowledge, and subject to discussion. This approach is then applied to the problem of induction. Popper set great stake to his solution of it, but others doubted its legitimacy, in ways that are clarified by changing the form of the induction problem itself. That change draws upon logic, which is the subject of another application: namely, in contrast to Popper’s adhesion to classical logic as the only welcome form (because of the maximal strength of criticism that it dispenses), can other logics be used without abandoning his philosophy of criticism?     

Orthogonal inter-battery factor analysis

It is the purpose of this paper to present a method of analysis for obtaining (i) inter-battery factors and (ii) battery specific factors for two sets of tests when the complete correlation matrix including communalities is given. In particular, the procedure amounts to constructing an orthogonal transformation such that its application to an orthogonal factor solution of the combined sets of tests results in a factor matrix of a certain desired form. The factors isolated are orthogonal but may be subjected to any suitable final rotation, provided the above classification of factors into (i) and (ii) is preserved. The general coordinate-free solution of the problem is obtained with the help of methods pertaining to the theory of linear spaces. The actual numerical analysis determined by the coordinate-free solution turns out to be a generalization of the formalism of canonical correlation analysis for two sets of variables. A numerical example is provided.This investigation has been supported by the U.S. Office of Naval Research under Contract Nonr-2752(00).     

The detailed method of optimal regions

The detailed method of optimal regions is an extended form of the method of optimal regions which has been found effective in solving the personnel classification problem when the number of job categories is small. The automatic determination of the successive values of thev _i , made possible by the more exact techniques of the detailed method, provide easier solutions for the more complex problems and provide solutions, which, for the most part, can be mechanized. In a sense the detailed method of optimal regions is more than a detailed form of the method of optimal regions. It is essentially a method of transformations by which the original matrix is reduced to a matrix from which the solution is easily obtained.Much of the basic research covered in this paper was carried out while the author was working on the problem of personnel classification in his capacity as Consultant, Personnel Research Branch, Department of the Army. The author wishes to express his appreciation to the Department of the Army for permission to use these materials in this paper. The opinions expressed are those of the author and are not to be construed as official or as those of the Department of the Army.     

A method for finding the inverse of a matrix

The problem of solving simultaneous linear equations is one that frequently confronts the scientist. Whenever it is desired to state the parameters as expressions of the constant terms of these equations, the problem resolves itself into the discovery of the inverse of the matrix of coefficients. The method here described gives a routine for the solution of this problem.     

Least-squares approximation of an improper correlation matrix by a proper one

An algorithm is presented for the best least-squares fitting correlation matrix approximating a given missing value or improper correlation matrix. The proposed algorithm is based upon a solution for Mosier's oblique Procrustes rotation problem offered by ten Berge and Nevels. A necessary and sufficient condition is given for a solution to yield the unique global minimum of the least-squares function. Empirical verification of the condition indicates that the occurrence of non-optimal solutions with the proposed algorithm is very unlikely. A possible drawback of the optimal solution is that it is a singular matrix of necessity. In cases where singularity is undesirable, one may impose the additional nonsingularity constraint that the smallest eigenvalue of the solution be , where is an arbitrary small positive constant. Finally, it may be desirable to weight the squared errors of estimation differentially. A generalized solution is derived which satisfies the additional nonsingularity constraint and also allows for weighting. The generalized solution can readily be obtained from the standard unweighted singular solution by transforming the observed improper correlation matrix in a suitable way.     

Confucianism as canonic culture

The present article is inspired by Liu's ( 2017 ) idea on how Asian philosophy can, in general, and Confucianism, in particular, contribute to psychological science and practice. We first clarify potential misunderstandings of Liu's ( 2017 ) paper as a theoretical argument for indigenous psychology or as a debate on philosophy or the philosophy of science. To interpret and develop Liu's idea, we then conceptualize the concept of culture as canonic (abstract, philosophical and as it appears in classical books or articles) vs. popular (concrete, experiential and as it appears in people's daily lives). Further, since Liu's main point is about canonic rather than popular culture, we focus on canonic culture and propose three principles of it, namely the principle of diversity, the principle of hermeneutics and the principle of inspiration. These principles respectively reveal the characteristics, the interpretation process and the beneficial function of a canonic culture like Confucianism.     

Causation, coincidence, and commensuration

What does it take to solve the exclusion problem? An ingenious strategy is Stephen Yablo’s idea that causes must be commensurate with their effects. Commensuration is a relation between events. Roughly, events are commensurate with one another when one contains all that is required for the occurrence of the other, and as little as possible that is not required. According to Yablo, one event is a cause of another only if they are commensurate. I raise three reasons to doubt that this account solves the exclusion problem successfully. First, it leaves a mystery about what determines a particular’s causal capacities. Second, because there are two ways of construing coincidence between particulars, a dilemma arises: either the solution to the exclusion problem is threatened, or the account of coincidence loses an attractive feature concerning ontological economy. Third, even if we assume the commensuration constraint, a plausible principle about overdetermination seems to regenerate the exclusion problem.     

The Russian Cards Problem

Suppose we have a stack of cards that is divided over some players. For certain distributions of cards it is possible to communicate your hand of cards to another player by public announcements, without yet another player learning any of your cards. A solution to this problem consists of some sequence of announcements and is called an exchange. It is called a direct exchange if it consists of (the minimum of) two announcements only. The announcements in an exchange have a special form: they are safe communications, an interesting new form of update. Certain unsafe communications turn out to be unsuccessful updates. A communication is a public announcement that is known to be true. Each communication may be about a set of alternative card deals only, and even about a set of alternatives to the communicating player's own hand only. We list the direct exchanges for a deal of seven cards where the two players holding three cards communicate their hands to each other. Our work may be applicable to the design of cryptographic protocols.     

On Adaptation in Analogy: Tests of Pragmatic Importance and Adaptability in Analogical Problem Solving

When people use analogies to solve problems, they form an analogical mapping between two domains of knowledge. This mapping may support inferences by analogy that suggest a novel solution to a problem. Several factors have been proposed to be important in selecting this mapping from among several alternative mappings: structural factors (systematicity and structural consistency) and pragmatic factors (the exploitation of higher-order planning categories). We suggest another set of factors plays a role in selecting mappings: adaptability. Specifically, if a mapped solution can be adapted easily to a problem, then it will be preferred over an alternative mapping that is less adaptable. Two experiments are reported which test the effects of pragmatic and adaptation factors, using a novel technique in which the story analogue has two alternative plans, either of which can be used to solve an insight problem. In Experiment 1, these plans were varied in terms of their pragmatic importance (success or failure) and their adaptability. In Experiment 2, the relative adaptability of plans was manipulated. The results suggest that there is little evidence for these specific pragmatic factors, but that adaptability plays a definite role in selecting an analogous plan. The findings suggest that most models need to be extended to include adaptation constraints.     

Proportionality,Contrast and Explanation

If counterfactual dependence is sufficient for causation and if omissions can be causes, then all events have many more causes than common sense tends to recognize. This problem is standardly addressed by appeal to pragmatics. However, Carolina Sartorio [2010] has recently raised what I shall argue is a more interesting problem concerning omissions for counterfactual theories of causation—more interesting because it demands a more subtle pragmatic solution. I discuss the relationship between the idea that causes are proportional to their effects, the idea that causation is contrastive, and the question of the dimensions along which causal explanations should be evaluated with respect to one another.     

Problem-oriented training promotes spontaneous analogical transfer: Memory-oriented training promotes memory for training

Spontaneous analogical transfer is the use of information from one problem to solve another problem, without an explicit hint to use the previous information. The results of five experiments were that if subjects tried to solve a training problem before hearing its solution, or tried to explain a training story's solution before hearing the correct explanation, spontaneous transfer was more likely than it was if subjects had studied the same training passage for memory before hearing its solution or explanation. The advantage of problem-oriented processing over memory-oriented processing occurred even though solution attempts nearly always failed, and the advantage was not reduced if the target problem was tested 15 min later rather than immediately after training. We propose that problem-oriented processes performed at study are appropriate processes to use at test. Further support for the account comes from subjects' memory for the training passages; the advantage for problem-oriented processing on solution tasks was mirrored by an equally substantial advantage for the memory-oriented subjects on a recall task.     

A Tale of Two Thinkers,One Meeting,and Three Degrees of Infinity: Leibniz and Spinoza (1675–8)

The article presents Leibniz's preoccupation (in 1675–6) with the difference between the notion of infinite number, which he regards as impossible, and that of the infinite being, which he regards as possible. I call this issue ‘Leibniz's Problem’ and examine Spinoza's solution to a similar problem that arises in the context of his philosophy. ‘Spinoza's solution’ is expounded in his letter on the infinite (Ep.12), which Leibniz read and annotated in April 1676. The gist of Spinoza's solution is to distinguish between three kinds of infinity and, in particular, between one that applies to substance, and one that applies to numbers, seen as auxiliaries of the imagination. The rest of the paper examines the extent to which Spinoza's solution solves Leibniz's problem. The main thesis I advance is that, when Spinoza and Leibniz say that the divine substance is infinite, in most contexts it is to be understood in non-numerical and non-quantitative terms. Instead, for Spinoza and Leibniz, a substance is said to be infinite in a qualitative sense stressing that it is complete, perfect and indivisible. I argue that this approach solves one strand of Leibniz's problem and leaves another unsolved.     

Development of relational learning: Effects of instruction and delay of transfer

Preschool children were trained on a color relevant oddity problem by one of three methods: increased salience of the oddity relationship, instruction on the solution rule, or a combination of increased salience and rule instruction. The youngest Ss solved the training problem when the solution rule was provided but not under the salience procedure. The two older groups solved in all conditions and errors decreased with age. The generality of the solution was determined by performance on a form relevant transfer task. The youngest Ss made fewer errors on the transfer problem following rule instruction training than following training in the combined procedure. The intermediate age group made fewer errors when the transfer task was administered one week after training than one minute after training. The transfer problem was easily solved by the oldest Ss and performance was independent of training procedure and amount of delay. These results were interpreted as reflecting developmental differences in relational concept learning.     

A variable neighborhood search method for generalized blockmodeling of two-mode binary matrices

The clustering of two-mode proximity matrices is a challenging combinatorial optimization problem that has important applications in the quantitative social sciences. We focus on one particular type of problem related to the clustering of a two-mode binary matrix, which is relevant to the establishment of generalized blockmodels for social networks. In this context, clusters for the rows of the two-mode matrix intersect with clusters of the columns to form blocks, which should ideally be either complete (all 1s) or null (all 0s). A new procedure based on variable neighborhood search is presented and compared to an existing two-mode K-means clustering algorithm. The new procedure generally provided slightly greater explained variation; however, both methods yielded exceptional recovery of cluster structure.