An n-dimensional Weber Law and the Corresponding Fechner Law

ISSN： 0022-2496

Source： Journal of Mathematical Psychology, Vol.44, Iss.2, 2000-06, pp. : 330-335

Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.

Previous Menu Next

Abstract

Weber's law of 1834, ΔS/S=c for the just noticeable difference (jnd), can be written as S+ΔS=kS, k=1+c. It follows that the stimulus decrement required to elicit one jnd of sensation is S-ΔS*=k^-1S. If generalized for two stimulus dimensions and two corresponding response dimensions, Weber's law would have to state such equations for all directions of change in the plane. A two-dimensional Weber law with exactly these properties is realized by [S_x+ΔS_x(θ), S_y+ΔS_y(θ)]=[k^sin(θ)S_x, k^cos(θ)S_y] which determines the stimulus coordinates for all stimuli just noticeably different from the stimulus (S_x, S_y) in all directions 0θ2π. Fechner's problem now is understood as finding a transformation of the plane which maps the set of stimuli one jnd apart from the standard stimulus onto a unit circle around the standard stimulus' image. This transformation (R²₊→R²) is [x, y]↦[log_k(x), log_k(y)]. The solution is generalized to arbitrarily many dimensions by substituting the sin and cos in the generalized Weber law by the standard coordinates of a unit vector. Copyright 2000 Academic Press.