Least squares metric,unidimensional scaling of multivariate linear models

The squared error loss function for the unidimensional metric scaling problem has a special geometry. It is possible to efficiently find the global minimum for every coordinate conditioned on every other coordinate being held fixed. This approach is generalized to the case in which the coordinates are polynomial functions of exogenous variables. The algorithms shown in the paper are linear in the number of parameters. They always descend and, at convergence, every coefficient of every polynomial is at its global minimum conditioned on every other parameter being held fixed. Convergence is very rapid and Monte Carlo tests show the basic procedure almost always converges to the overall global minimum.The author thanks Ivo Molenaar, three anonymous referees, and Howard Rosenthal for their many helpful comments.     

An efficient alternating least-squares algorithm to perform multidimensional unfolding

We consider the problem of least-squares fitting of squared distances in unfolding. An alternating procedure is proposed which fixes the row or column configuration in turn and finds the global optimum of the objective criterion with respect to the free parameters, iterating in this fashion until convergence is reached. A considerable simplification in the algorithm results, namely that this conditional global optimum is identified by performing a single unidimensional search for each point, irrespective of the dimensionality of the unfolding solution.This work originally formed part of a doctoral thesis (Greenacre, 1978) presented at the University of Paris VI. The authors acknowledge the helpful comments of John Gower during the first author's sabbatical at Rothamsted Experimental Station. The authors are also indebted to Alexander Shapiro, who came up with the proof of the key result which the authors had long suspected, but had not proved, namely that the smallest root of function (13) provides the global minimum of function (7). The constructive comments of the referees of this paper are acknowledged with thanks. This research was supported in part by the South African Council for Scientific and Industrial Research.     

A Class of Probabilistic Unfolding Models for Polytomous Responses

By revisiting the approaches used to present the Rasch model for polytomous response, this paper uses the principle of the rating formulation (Andrich, 1978) to construct a class of unfolding models for polytomous responses in terms of a set of latent dichotomous unfolding variables. By anchoring the dichotomous unfolding variables involved at the same location, this paper presents a formulation of a very general class of unfolding models for ordered polytomous responses, of which the unfolding models for ordered polytomous responses proposed hitherto are special cases. Within this class, the analytic and measurement properties of the probabilistic functions are well interpreted in terms of the latitudes of acceptance parameters of the dichotomous unfolding models. Based on the general form of this class of unfolding models, some new models are readily specified. Copyright 2001 Academic Press.     

可数图灵理想的恰对和一致上界

一个（图灵）理想,是满足两个封闭条件的图灵度集合：向下封闭;任意,中一对图灵度的上确界也在,中。可数理想不仅在图灵度整体性质的研究中有着重要意义,而且在对哥德尔可构成集合L精细结构的早期研究中也发挥过重要作用。研究可数理想的两个重要概念是：恰对和一致上界。借助这两个概念,我们可以将可数理想简化为一个（一致上界）或者一对（恰对）图灵度。通过前人的研究,我们可以发现这两个概念是紧密相连的,同时我们也可以对它们的关系提出进一步的问题。在本文中,我们证明以下定理：任给一个可数理想I,都存在两个I的一致上界a0和a1,同时a0和a1构成,的一个恰对。此定理从正面回答了Lerman提出的关于算术图灵度构成的理想的一个问题。此定理的证明实际上是经过小心修改的、典型的恰对构造。我们在典型恰对构造的过程中,加入一些微妙的限制,使得形成恰对的两个图灵度a0和a1可以各自独立地在一定程度上用逼近的办法还原整个构造,从而分别给出可数理想I的一致枚举。在a0和a1分别的逼近中,我们引入了有穷损坏方法。本文的最后指出a0和a1的图灵跃迁的一些性质。     

A Discrimination Model For Multivariate Q Representations

The proposed discrimination model for Q representation is a generalization of the method of unfolding in an algebraic rather than a geometric field. Placements of Q statements relative to one another in joint space so as to form a quasi-normal distribution of the q stimulus items are assigned Q values which form a bounded set of symbols with specified order in arithmetic progression; for example, a finite set from "one" to "nine" wherein "five" is the midpoint of an ordered metric within persons. An algebraic theory of numbers permits mathematical operations to be carried out with sets of sets of such symbols. Using illustrative data, a mathematical model is developed for the analysis of Q values, each of which represents the intersect of two or more dimensions of behavior structured into a Q set, and for testing its stability, consistency, and reliability. The Q check procedures permit comparisons among individuals and subsamples of population samples.     

A Set Theory with Support for Partial Functions

Partial functions can be easily represented in set theory as certain sets of ordered pairs. However, classical set theory provides no special machinery for reasoning about partial functions. For instance, there is no direct way of handling the application of a function to an argument outside its domain as in partial logic. There is also no utilization of lambda-notation and sorts or types as in type theory. This paper introduces a version of von-Neumann-Bernays-Gödel set theory for reasoning about sets, proper classes, and partial functions represented as classes of ordered pairs. The underlying logic of the system is a partial first-order logic, so class-valued terms may be nondenoting. Functions can be specified using lambda-notation, and reasoning about the application of functions to arguments is facilitated using sorts similar to those employed in the logic of the IMPS Interactive Mathematical Proof System. The set theory is intended to serve as a foundation for mechanized mathematics systems.     

The tunneling method for global optimization in multidimensional scaling

This paper focuses on the problem of local minima of the STRESS function. It turns out that unidimensional scaling is particularly prone to local minima, whereas full dimensional scaling with Euclidean distances has a local minimum that is global. For intermediate dimensionality with Euclidean distances it depends on the dissimilarities how severe the local minimum problem is. For city-block distances in any dimensionality many different local minima are found. A simulation experiment is presented that indicates under what conditions local minima can be expected. We introduce the tunneling method for global minimization, and adjust it for multidimensional scaling with general Minkowski distances. The tunneling method alternates a local search step, in which a local minimum is sought, with a tunneling step in which a different configuration is sought with the same STRESS as the previous local minimum. In this manner successively better local minima are obtained, and experimentation so far shows that the last one is often a global minimum.This paper is based on the 1994 Psychometric Society's outstanding thesis award of the first author. The authros would like to thank Robert Tijssen of the CWTS Leiden for kindly making available the co-citation data of the Psychometric literature. This paper is an extended version of the paper presented at the Annual Meeting of the Psychometric Society at Champaign-Urbana, Illin., June 1994.     

Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part II: Category Theoretic Approaches

For classical sets one has with the cumulative hierarchy of sets, with axiomatizations like the system ZF, and with the category SET of all sets and mappings standard approaches toward global universes of all sets.We discuss here the corresponding situation for fuzzy set theory. Our emphasis will be on various approaches toward (more or less naively formed) universes of fuzzy sets as well as on axiomatizations, and on categories of fuzzy sets.What we give is a (critical) survey of quite a lot of such approaches which have been offered in the last approximately 35 years.Part I was devoted to model based and to axiomatic approaches; the present Part II is devoted to category theoretic approaches.This paper is a version of the invited talk given by the author at the conference Trends in Logic III, dedicated to the memory of A. MOSTOWSKI, H. RASIOWA and C. RAUSZER, and held in Warsaw and Ruciane-Nida from 23rd to 25th September 2005.     

A stochastic multidimensional unfolding approach for representing phased decision outcomes

This paper presents a stochastic multidimensional unfolding (MDU) procedure to spatially represent individual differences in phased or sequential decision processes. The specific application or scenario to be discussed involves the area of consumer psychology where consumers form judgments sequentially in their awareness, consideration, and choice set compositions in a phased or sequential manner as more information about the alternative brands in a designated product/service class are collected. A brief review of the consumer psychology literature on these nested congnitive sets as stages in phased decision making is provided. The technical details of the proposed model, maximum likelihood estimation framework, and algorithm are then discussed. A small scale Monte Carlo analysis is presented to demonstrate estimation proficiency and the appropriateness of the proposed model selection heuristic. An application of the methodology to capture awareness, consideration, and choice sets in graduate school applicants is presented. Finally, directions for future research and other potential applications are given.     

Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part I: Model-Based and Axiomatic Approaches

For classical sets one has with the cumulative hierarchy of sets, with axiomatizations like the system ZF, and with the category SET of all sets and mappings standard approaches toward global universes of all sets. We discuss here the corresponding situation for fuzzy set theory.Our emphasis will be on various approaches toward (more or less naively formed)universes of fuzzy sets as well as on axiomatizations, and on categories of fuzzy sets. What we give is a (critical)survey of quite a lot of such approaches which have been offered in the last approximately 35 years. The present Part I is devoted to model based and to axiomatic approaches; the forth-coming Part II will be devoted to category theoretic approaches. This paper is a version of the invited talk given by the author at the conference Trends in Logic III, dedicated to the memory of A. MOSTOWSKI, H. RASIOWA and C. RAUSZER, and held in Warsaw and Ruciane-Nida from 23rd to 25th September 2005. Presented by Jacek Malinowski     

A marginal maximum likelihood method for the vector threshold model to analyze dichotomous choice data

Abstract: At least two types of models, the vector model and the unfolding model can be used for the analysis of dichotomous choice data taken from, for example, the pick any/ n method. The previous vector threshold models have a difficulty with estimation of the nuisance parameters such as the individual vectors and thresholds. This paper proposes a new probabilistic vector threshold model, where, unlike the former vector models, the angle that defines an individual vector is a random variable, and where the marginal maximum likelihood estimation method using the expectation-maximization algorithm is adopted to avoid incidental parameters. The paper also attempts to discuss which of the two models is more appropriate to account for dichotomous choice data. Two sets of dichotomous choice data are analyzed by the model.     

Goodness-of-Fit Assessment in Multidimensional Scaling and Unfolding

Judging goodness of fit in multidimensional scaling requires a comprehensive set of diagnostic tools instead of relying on stress rules of thumb. This article elaborates on corresponding strategies and gives practical guidelines for researchers to obtain a clear picture of the goodness of fit of a solution. Special emphasis will be placed on the use of permutation tests. The second part of the article focuses on goodness-of-fit assessment of an important variant of multidimensional scaling called unfolding, which can be applied to a broad range of psychological data settings. Two real-life data sets are presented in order to walk the reader through the entire set of diagnostic measures, tests, and plots. R code is provided as supplementary information that makes the whole goodness-of-fit assessment workflow, as presented in this article, fully reproducible.     

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.     

A linear conservative extension of Zermelo-Fraenkel set theory

In this paper, we develop the system LZF of set theory with the unrestricted comprehension in full linear logic and show that LZF is a conservative extension of ZF^– i.e., the Zermelo-Fraenkel set theory without the axiom of regularity. We formulate LZF as a sequent calculus with abstraction terms and prove the partial cut-elimination theorem for it. The cut-elimination result ensures the subterm property for those formulas which contain only terms corresponding to sets in ZF^–. This implies that LZF is a conservative extension of ZF^– and therefore the former is consistent relative to the latter. Hiroakira Ono     

Perceptual identification thresholds for 150 fragmented pictures from the Snodgrass and Vanderwart picture set

This paper reports perceptual identification thresholds for 150 pictures from the 1980 Snodgrass and Vanderwart picture set. These pictures were fragmented and presented on the Apple Macintosh microcomputer in a picture-fragment completion task in which identification thresholds were obtained at three phases of learning: Train (initial presentation), New (initial presentation after training on a different set), and Old (repeated presentation of the Train set). Pictures were divided into five sets of two subsets of 15 pictures each, which served alternately as the Train and New sets. A total of 100 subjects participated in the task, with 10 subjects assigned to each subset. Individual thresholds for each picture at each phase of learning are presented, along with the fragmented pictures identified by 35% of the subjects across the Train and New learning phases. This set of fragmented pictures is provided for use in experiments in which a single level of fragmented image is presented for identification after a priming phase. Correlations between the Snodgrass and Vanderwart norms and identification thresholds at the three phases of learning are also reported.     

论模态逻辑的集合论语义

引入非良基集合可以为模态逻辑提供一种新的语义学。这种语义是在集合上解释模态语言,使用集合中作为元素的集合之间的属于关系解释模态词,并在集合中采用命题变元作为本元,从而解释原子命题的真假。在这种新的语义下,从模型构造的角度看可以引入几种非标准的集合运算：不交并、生成子集合、p-态射、树展开等等,证明模态公式在这些运算下的保持或不变结果。利用这些结果还可以证明一些集合类不是模态可定义的。     

Picture of all Solutions of Successive 2-Block Maxbet Problems

The Maxbet method is a generalized principal components analysis of a data set, where the group structure of the variables is taken into account. Similarly, 3-block[12,13] partial Maxdiff method is a generalization of covariance analysis, where only the covariances between blocks (1, 2) and (1, 3) are taken into account. The aim of this paper is to give the global maximum for the 2-block Maxbet and 3-block[12,13] partial Maxdiff problems by picking the best solution from the complete solution set for the multivariate eigenvalue problem involved. To do this, we generalize the characteristic polynomial of a matrix to a system of two characteristic polynomials, and provide the complete solution set of the latter via Sylvester resultants. Examples are provided.     

Are There Enough Injective Sets?

The axiom of choice ensures precisely that, in ZFC, every set is projective: that is, a projective object in the category of sets. In constructive ZF (CZF) the existence of enough projective sets has been discussed as an additional axiom taken from the interpretation of CZF in Martin-Löf’s intuitionistic type theory. On the other hand, every non-empty set is injective in classical ZF, which argument fails to work in CZF. The aim of this paper is to shed some light on the problem whether there are (enough) injective sets in CZF. We show that no two element set is injective unless the law of excluded middle is admitted for negated formulas, and that the axiom of power set is required for proving that “there are strongly enough injective sets”. The latter notion is abstracted from the singleton embedding into the power set, which ensures enough injectives both in every topos and in IZF. We further show that it is consistent with CZF to assume that the only injective sets are the singletons. In particular, assuming the consistency of CZF one cannot prove in CZF that there are enough injective sets. As a complement we revisit the duality between injective and projective sets from the point of view of intuitionistic type theory.     

Statistical Evaluation of Algebraic Constraints for Social Networks

A multirelational social network on a set of individuals may be represented as a collection of binary relations. Compound relations constructed from this collection represent various labeled paths linking individuals in the network. Since many models of interest for social networks can be formulated in terms of orderings among these labeled paths, we consider the problem of evaluating an hypothesized set of orderings, termed algebraic constraints. Each constraint takes the form of an hypothesized inclusion relation for a pair of labeled paths. In this paper, we establish conditions under which sets of such constraints may be regarded as partial algebras. We describe the structure of constraint sets and show that each corresponds to a subset of consistent relation bundles between pairs of individuals. We thereby construct measures of fit for a given constraint set. Then, we show how, in combination with the assumption of various conditional uniform multigraph distributions, these measures lead to a flexible approach to the evaluation of fit of an hypothesized constraint set. Several applications are presented and some possible extensions of the approach are briefly discussed. Copyright 2000 Academic Press.     

Visual recognition under conditions of high and low discrimination redundancy

The purpose of this investigation was to measure the minimum interval for recognition under conditions of high and low discrimination redundancy. The minimum interval for recognition is the briefest interval between two stimuli at which the first stimulus can be processed. Two sets of four digits were used. Digits were selected so that one set exhibited high discrimination redundancy and the other low discrimination redundancy. Ten Ss viewed digits with high discrimination redundancy, and 10 Ss viewed digits with low discrimination redundancy. The minimum interval for recognition for Ss tested with high discrimination redundancy was nearly twice as large as the minimum interval for recognition of the digits with low discrimination redundancy.