Fechnerian Scaling: Metric from Discriminability

该项目将研究支配知觉歧视的基本规律，并将进一步发展广义费氏标度理论。歧视一词指的是决定两个物体是否相同或不同的能力(在所有方面，在特定方面，或在属于同一类别的意义上)。这是生物有机体的基本认知能力之一，也是人工智能系统的基本要求之一。正则极小原理被提出作为同异判断的一个基本性质。从本质上讲，刺激Y与刺激X最难区分当且仅当X与Y最小可区分时。这一直观上可信且得到经验证实的原理是广义费氏标度理论的基石，在广义费氏标度理论中，刺激之间的主观距离(即，从观察者的角度出发的距离，无论是人类观察者、一群人、一个技术系统，还是“纸和笔”的计算过程)是根据区分概率计算的。然而，直到最近，这种计算的逻辑被认为对于连续刺激空间，如颜色空间，以及离散空间，如颜色名称空间，是不同的。该项目的一个目标是通过开发一种适用于所有可能的刺激空间的计算逻辑来纠正这种差异(“普适费克纳标度”)。这一目标是通过一个新的数学概念-相异函数来实现的，该函数的公理理论推广了度量空间的公理理论。除正则极小值外，区分概率通常还表现出另一个特性，称为非常数自相异度。如果X与Y是最小可区分性的，A与B是最小可区分性的，则X与Y的可区分性与A与B的可区分性一般不同。正则极小值与非常数自相异项的结合对于理解异同比较具有重要的结果。因此，这种结合与X和Y之间的主观距离与X与Y被区分的概率单调相关的假设是不相容的(该假设尤其是在应用于区分概率时广泛使用的多维尺度技术的基础)。规则极小值和非常数自相异也被证明与广泛使用的良好的瑟斯顿型模型不相容，根据该模型，X和Y是不同的还是相同的是基于X和Y的随机图像，每个刺激影响其图像的分布“足够平稳”。然而，瑟斯顿型模型可以近似正则极小值，这使得确定正则极小值在经验数据中所具有的精度极限至关重要。现有数据不能回答这一问题，本项目旨在通过专门设计的调整/匹配程序来调查这一问题。该项目的一个相关目标是通过分析和计算机模拟来研究在实验误差范围内能够逼近规则极小值的瑟斯顿型模型的传统变体是否能够产生现实的判别概率函数。广义费氏标度和同异判断理论在教育评估和专业培训、应用统计学、人工认知系统和知觉辅助工具的构建以及大规模民意调查和消费者调查的分析中具有应用价值。该项目本质上是跨学科的，连接了行为科学、社会科学、数学科学和计算机科学。该项目也是国际性的。研究人员来自不同的国家，其他合作将通过在国际会议上组织的一系列研讨会和研讨会得到支持。普渡大学和奥尔登堡大学的研究人员讲授的课程将使用与该项目有关的主题。将努力吸引本科生参与该项目，特别强调妇女和少数民族的参与。

This project will investigate the fundamental laws governing perceptual discrimination and will further develop the theory of Generalized Fechnerian Scaling. The term discrimination refers to the ability to decide whether two objects are the same or different (in all respects, in a specified respect, or in the sense of belonging to one and the same category). This is one of the basic cognitive abilities of living organisms and one of the basic requirements of artificial intelligent systems. The principle of Regular Minimality has been proposed as a fundamental property of same-different judgments. It says, in essence, that stimulus Y is least discriminable from stimulus X if and only if X is least discriminable from Y. This intuitively plausible and empirically corroborated principle is the cornerstone of the theory of Generalized Fechnerian Scaling, in which subjective distances among stimuli (i.e., distances "from the point of view" of a perceiver, be it a human observer, group of people, a technical system, or a "paper-and-pencil" computational procedure) are computed from discrimination probabilities. Until recently, however, the logic of this computation was posited to be different for continuous stimulus spaces, such as a space of colors, and for discrete spaces, such as a space of color names. One aim of the project is to remedy this discrepancy by developing a computational logic which would apply to all possible stimulus spaces ("Universal Fechnerian Scaling"). This aim is achieved through a new mathematical concept, Dissimilarity Function, whose axiomatic theory generalizes that of metric spaces. In addition to Regular Minimality, discrimination probabilities typically exhibit another property, called Nonconstant Self-Dissimilarity. If X is least discriminable from Y, and so is A from B, the discriminability of X from Y is not generally the same as that of A from B. The conjunction of Regular Minimality with Nonconstant Self-Dissimilarity has important consequences for understanding same-different comparisons. Thus, this conjunction is incompatible with the hypothesis that the subjective distance between X and Y is monotonically related to the probability with which X is discriminated from Y (the hypothesis underlying, among other things, the widely used techniques of Multidimensional Scaling, when applied to discrimination probabilities). Regular Minimality and Nonconstant Self-Dissimilarity are also shown to be incompatible with the widely used well-behaved Thurstonian-type models, according to which the decision whether X and Y are different or the same is based on random images of X and Y, with each stimulus affecting its image's distribution "sufficiently smoothly." The Thurstonian-type models can, however, approximate Regular Minimality, which makes it critical to determine the limits of precision with which Regular Minimality holds in empirical data. Available data do not answer this question, and this project aims at investigating it by means of specially designed adjustment/matching procedures. A related aim of the project is to investigate, analytically and by means of computer simulations, whether the conventional variants of Thurstonian-type models which can approximate Regular Minimality within limits of experimental error can generate realistic discrimination probability functions.Generalized Fechnerian Scaling and theory of same-different judgments have applications in educational assessment and professional training, in applied statistics, in the construction of artificial cognitive systems and perceptual aids, and in the analysis of large-scale polls of public opinion and consumer surveys. The project is interdisciplinary in its nature, bridging behavioral, social, mathematical, and computer sciences. The project is also international. The investigators are from different countries, and other collaborations will be bolstered by a series of symposia and workshops organized at international conferences. Topics related to this project will be used in courses taught by the investigators at Purdue and Oldenburg Universities. An effort will be made to attract undergraduate students to participate in the project, with a special emphasis on the involvement of women and minorities.