Fechnarian Scaling: Metric from Discriminability

这项研究是基于一种新的理论，即如何根据每个对象与其无限小的近邻的可区分度来计算形成连续参数空间的对象之间的主观(费氏)距离。这一理论使用现代微分几何(广义Finsler空间，以前成功地应用于物理中的各种问题，最近又应用于生物学中的各种问题)来提炼G.T.Fechner 150年前的思想，并将它们推广到任意维度的刺激空间。这项研究旨在探索由这一发展而产生的几个重要理论问题，并实证测试其对一维和二维刺激空间的适用性。研究的理论部分包括如下问题：(1)由刺激间距离确定区分概率的可能方式，以及这些距离与根据这些区分概率计算的费氏度量之间的关系；(2)不同版本的知觉可分离性和独立性假设对费氏距离施加的约束；以及(3)通过假设每个刺激由感知空间中的随机变量表示的模型来产生区分概率的可能方法，当且仅当两个刺激的随机表示相隔超过某一临界距离时，才能在给定的试验中区分两个刺激。这项研究的经验部分旨在实现以下目标：(A)重建歧视概率函数在其极小值附近的垂直和水平横截面；(B)测试根据与不同主观置信度水平相对应的歧视概率计算的费氏距离的不变性；(C)比较根据不同判断方案计算的费氏度量，例如“相同-不同”和两种选择的强迫选择。所有实验中使用的刺激空间都是通过改变小视觉目标的瞬时位移的距离和方向来形成的。这一项目的成功完成有望极大地推进心理学测量的理论基础，以及在任何其他领域，在这些领域中，被研究对象可以通过物理测量或其他定标程序被分配连续变化的坐标，并且具有相似坐标的对象可以与可确定的概率区分或混淆。因此，尽管本研究中使用和检验的理论语言是面向感官知觉的，但其适用范围包括电生理学、教育测试、态度调查、消费者选择分析、决策研究等各种范式。

This research is based on a new theory of how subjective (Fechnerian) distances among objects that form a continuously parametrized space can be computed from the degree of discriminability of each object from its infinitesimally close neighbors. This theory uses the apparatus of modern differential geometry (generalized Finsler spaces, previously successfully applied to a variety of problems in physics and, more recently, biology) to refine the 150 year-old ideas of G.T. Fechner and generalize them to stimulus spaces of arbitrary dimensionality. The research is aimed at exploring several important theoretical issues stemming from this development, and at empirically testing its applicability to one- and two-dimensional stimulus spaces. The theoretical part of the research includes such issues as (1) the possible ways in which discrimination probabilities may be determined by inter-stimulus distances, and the relationship between these distances and the Fechnerian metric computed from these discrimination probabilities; (2) the constraints imposed on Fechnerian distances by various versions of the perceptual separability and independence assumptions; and (3) the possible ways of generating discrimination probabilities by means of a model in which each stimulus is assumed to be represented by a random variable in a perceptual space, and two stimuli be discriminated in a given trial if and only if their random representations are separated by more than a certain critical distance. The empirical part of the research is aimed at such goals as (A) reconstruction of the vertical and horizontal cross-sections of discrimination probability functions in small vicinities of their minima; (B) testing the invariance of Fechnerian distances computed from discrimination probabilities corresponding to different levels of subjective confidence; and (C) the comparison of the Fechnerian metrics computed from different judgment schemes, such as "same-different" and two-alternative forced choice. The stimulus space used in all the experiments is formed by varying the distance and direction of instantaneous displacements of a small visual target. A successful completion of this project promises to significantly advance the theoretical foundations of measurement in psychology, as well as in any other domain where the objects being studied can be assigned continuously varying coordinates, either by physical measurements or by other scaling procedures, and where the objects with similar coordinates can be discriminated or confused with determinable probabilities. Thus, although the language of the theory that is used and tested in this research is geared towards sensory perception, its sphere of applicability includes a variety of paradigms in electrophysiology, educational testing, attitudinal surveys, consumer choice analysis, decision making research, etc.