Skip to main content Accessibility help
×
Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-21T09:25:51.787Z Has data issue: false hasContentIssue false

3 - Fechnerian Scaling: Dissimilarity Cumulation Theory

Published online by Cambridge University Press:  20 April 2023

F. Gregory Ashby
Affiliation:
University of California, Santa Barbara
Hans Colonius
Affiliation:
Carl V. Ossietzky Universität Oldenburg, Germany
Ehtibar N. Dzhafarov
Affiliation:
Purdue University, Indiana
Get access

Summary

This chapter presents a systematic theory of generalized (or universal) Fechnerian scaling, based on the intuition underlying Fechner’s original theory. The intuition is that subjective distances among stimuli are computed by cumulating small discriminability values between “neighboring” stimuli. A stimulus space is supposed to be endowed by a dissimilarity function, computed from a discrimination probability function for any pair of stimuli chosen in two distinct observation areas. On the most abstract level, one considers all possible chains of stimuli leading from stimulus a to stimulus b and back to a, and takes the infimum of the sums of the dissimilarities along these chains as the subjective distance between a and b. In arc-connected spaces, the cumulation of dissimilarity values along all possible chains reduces to their cumulation along continuous paths, leading to a fully fledged metric geometry. In topologically Euclidean spaces, the cumulation along paths further reduces to integration along smooth paths, and the geometry in question acquires the form of a generalized Finsler geometry. The chapter also discusses Fechner’s original derivation of his logarithmic law, observation sorites paradox, a generalized Floyd--Warshall algorithm for computing metric distances from dissimilarities, and an ultra-metric version and data-analytic application of Fechnerian scaling.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2023

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×