Fechnarian Scaling: Metric from Discriminability

这项研究基于一种新的理论，即形成连续参数化空间的物体之间的主观（Fechnerian）距离可以通过每个物体与其无限小近邻的可分辨程度来计算。这个理论使用现代微分几何（广义的芬斯勒空间，以前成功地应用于物理学中的各种问题，最近应用于生物学）的仪器来完善150年前的G.T.费希纳的思想，并将其推广到任意维度的刺激空间。本研究的目的是探索源于这一发展的几个重要理论问题，并在经验上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以下问题：(1)识别概率可能由刺激间距离决定的方式，以及这些距离与由这些识别概率计算的费希内度量之间的关系；(2)不同版本的感知可分离性和独立性假设对技术距离的约束；(3)利用一个模型产生判别概率的可能方法，该模型假设每个刺激都由感知空间中的一个随机变量表示，并且当且仅当两个刺激的随机表示间隔超过某一临界距离时，在给定的试验中可以判别两个刺激。实证部分的研究目标是：(A)在判别概率函数的极小值的小附近重建判别概率函数的垂直和水平横截面；(B)检验由不同主观置信度水平对应的判别概率计算的法希内距离的不变性；(C)不同判断方案（如“相同-不同”和两种选择强制选择）计算出的技术指标的比较。所有实验中使用的刺激空间都是通过改变视觉小目标瞬时位移的距离和方向来形成的。这个项目的成功完成有望显著推进心理学测量的理论基础，以及在任何其他领域，被研究的对象可以通过物理测量或其他缩放程序分配连续变化的坐标，并且具有相似坐标的对象可以被区分或与可确定的概率混淆。因此，尽管在本研究中使用和测试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、决策研究等领域的各种范式。

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.