Affine representations in psychophysics and the near-miss to Weber's law

作者：Hsu Yung Fong^*; Iverson Geoffrey J; Doble Christopher W

来源：Journal of Mathematical Psychology, 2010, 54(2): 256-265.

DOI：10.1016/j.jmp.2010.01.003

• Affine representation
• Falmagne's power law
• Fechnerian representation
• Functional equation
• Integration of jnds
• Near-miss to Weber's law
• Subtractive representation

A theoretical account for the near-miss to Weber's law in the form of a power function, with a special emphasis on the interpretation of the exponent, was proposed by Falmagne [Falmagne, J.-C. (1985). Elements of psychophysical theory. New York: Oxford University Press] within the framework of a subtractive representation, P(x, y) = F(u(x) - g(y)). In this paper, we examine a more general affine representation, P(x, y) = F(u(x)h(y) + g(y)). We first obtain a uniqueness theorem for the affine representation. We then study the conditions that force an affine representation to degenerate to a subtractive one. Part of that study involves the case for which two different affine representations coexist for the same data. We also show that the balance condition P(x, y) + P(y, x) = 1 constrains an affine representation to be a special kind of subtractive representation, a Fechnerian one. We further show that Falmagne's power law takes on a special form for a so-called weakly balanced system of probabilities, in which case the affine representation is Fechnerian. Finally, following Iverson [Iverson, G.J. (2006a). Analytical methods in the theory of psychophysical discrimination I: Inequalities, convexity and integration of just noticeable differences. Journal of Mathematical Psychology, 50, 271-282], we generalize the Fechner method to construct the sensory scales in a weakly balanced affine representation by integrating (derivatives of) just noticeable differences.

• 出版日期2010-4