Fechnerian scaling: Metric from discriminability

本课题对多维技术标度（MDFS）的理论和应用进行了阐述和拓展。直观地说，MDFS是关于如何“从感知者（人类观察者、技术系统、一群人、神经生理系统）的角度”计算物体之间的距离，基于感知者区分（区分）非常相似物体的概率。判别概率赋予感知对象集合一个局部几何结构（广义Finsler几何），该局部结构可以被提取并扩展为全局度量。从本质上讲，这是G.T.费希纳在大约150年前创立科学心理学的一个想法，尽管背景非常有限。MDFS的动机是这样一种期望，即刺激之间的区分可以说是感知系统最基本的能力，而歧视概率是可区别性的普遍衡量标准，从歧视概率计算的距离可能在社会和行为测量中具有基本地位。MDFS最初是为连续参数化刺激（如颜色、声音或食品成分的混合物）的集合而开发的，并且主要基于衍生物的计算，在目前的项目中，MDFS扩展到离散的对象集合，如字母、单词或消费品。当一个离散的目标集受到MDFS的修正时，它被转换成一个具有目标间距离的网络。随后将该网络浸入欧几里得空间（通过传统的度量多维尺度），可以识别确定物体之间差异的相关特征。该部分项目的理论工作将通过收集熟悉和不熟悉的字母和示意图面等对象的识别概率的大型数据集来补充和指导。MDFS对离散对象空间的适应是一项重要的工作，只有通过最近发现和记录判别概率的两个基本属性（规则极小性和非恒定自相似性）才能实现。他们进一步的实验分析构成了该项目的第二部分。这两个属性在MDFS内部和外部都具有深远的影响。因此，已经证明，没有“随机效用”类型的模型，其中随机表示充分平滑地依赖于对象，可以解释这些属性（这里提到的模型是那些对象被映射到一些不可观察的“内部”空间中的随机实体的模型，以及关于对象是相同还是不同的决定是由这些随机实体的实现决定的）。该项目的第三部分旨在深入分析MDFS、“随机效用”类型模型和最近提出的替代这种模型的模型之间的关系，其中随机实体被感知对象的“不确定性斑点”所取代。该项目的成功完成将提高我们对歧视和差异的基本概念的理解，并将推进社会和行为科学测量的基础。该项目也有明确的应用重点。因此，离散对象空间中MDFS算法和软件的开发将为分析大规模民意调查、消费者调查和教育测试提供一个新的有用工具。

This project elaborates and expands the theory and applications of Multidimensional Fechnerian Scaling (MDFS). Intuitively, MDFS is about how to compute distances among objects "from the point of view" of a perceiver (human observer, technical system, group of people, neurophysiological system), based on the probabilities with which this perceiver discriminates (tells apart) very similar objects. Discrimination probabilities impose on the set of perceived objects a local geometric structure (generalized Finsler geometry), and this local structure can be extracted and expanded into a global metric. Essentially, this is an idea with which, albeit in a very limited context, G.T. Fechner launched scientific psychology some 150 years ago. MDFS is motivated by the expectation that, the discrimination among stimuli being arguably the most basic ability of a perceiving system, and the probability of discrimination being a universal measure of discriminability, distances computed from discrimination probabilities may have a fundamental status among social and behavioral measurements. Originally developed for sets of continuously parametrized stimuli (such as colors, sounds, or mixtures of food ingredients) and critically based on computation of derivatives, in the present project MDFS is expanded to discrete sets of objects, such as alphabets, words, or consumer products. When a discrete set of objects is subjected to the proposed modification of MDFS, it is transformed into a network with inter-object distances. A subsequent immersion of this network in a Euclidean space (by means of conventional metric multidimensional scaling) allows one to identify the relevant features that determine the dissimilarities among the objects. The theoretical work in this part of the project will be complemented and guided by the collection of large data sets on discrimination probabilities for such objects as letters of familiar and unfamiliar alphabets and schematic faces. The adaptation of MDFS to discrete object spaces is a non-trivial enterprise that is only enabled by the recent discovery and documentation of two basic properties of discrimination probabilities (regular minimality and nonconstant self-similarity). Their further experimental analysis constitutes the second part of the project. These two properties have far reaching consequences both within and without MDFS. Thus it has been shown that no "random utility" type model in which random representations depend on objects sufficiently smoothly can account for these properties (the models referred to are those in which objects are mapped into random entities in some unobservable "internal" space, and the decision as to whether the objects are the same or different is determined by the realizations of these random entities). The third part of the project is aimed at an in-depth analysis of the relationship between MDFS, the "random utility" type models, and the recently proposed alternative to such models in which random entities are replaced by "uncertainty blobs" of perceived objects.A successful completion of the project will improve our understanding of the fundamental notions of discrimination and dissimilarity, and it will advance the foundations of measurement in social and behavioral sciences. The project also has a clear applied focus. Thus, the development of algorithms and software for MDFS in discrete object spaces will provide a new useful tool for analyzing large-scale polls of public opinion, for consumer surveys, and for educational testing.