Abstract Writing ξ s ( x ) for the stimulus intensity judged greater (louder, heavier, brighter) than stimulus intensity x with criterion s , Iverson (2006b) proposed a law of similarity ξ s ( λ x ) = γ ( λ , s ) ξ η ( λ , s ) ( x ) to model the dependence of ξ s ( x ) on x . This model, which has η ( λ , s ) and γ ( λ , s ) as parameters, is quite general and may be applied in a number of situations in psychophysics. Iverson (2006b) analyzed this model assuming the representation s = u ( ξ s ( x ) ) − u ( x ) and derived the possible functional forms for the scale u . In the present work, we extend the analysis to the more general s = u ( ξ s ( x ) ) − w ( x ) and derive the forms for the scales u and w . We avoid the assumption of differentiability and replace it with an assumption either of nonconstancy or of dependence on only one input variable. We find that for some solutions, w has the same form as u , possibly with different parameters, while for other solutions, w takes a different form than u . Comparisons are made to Iverson (2006b) and to other work.

中文翻译：

---

减法表示下的心理物理学相似律中的函数形式

摘要 将 ξ s ( x ) 写成比刺激强度 x 更大（更响亮、更重、更亮）的刺激强度 x 与标准 s ，艾弗森 (2006b) 提出了相似定律 ξ s ( λ x ) = γ ( λ , s ) ξ η ( λ , s ) ( x ) 来模拟 ξ s ( x ) 对 x 的依赖性。该模型以 η ( λ , s ) 和 γ ( λ , s ) 作为参数，非常通用，可以应用于心理物理学的许多情况。Iverson (2006b) 分析了这个模型，假设表示 s = u ( ξ s ( x ) ) − u ( x ) 并推导出尺度 u 的可能函数形式。在目前的工作中，我们将分析扩展到更一般的 s = u ( ξ s ( x ) ) − w ( x ) 并推导出尺度 u 和 w 的形式。我们避免了可微性的假设，并用非恒定性或仅依赖于一个输入变量的假设来代替它。我们发现，对于某些解，w 与 u 具有相同的形式，可能具有不同的参数，而对于其他解，w 与 u 具有不同的形式。与 Iverson (2006b) 和其他工作进行了比较。