Weber–Fechner law

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The Weber–Fechner law refers to two related laws in the field of psychophysics, known as Weber's law and Fechner's law. Both laws relate to human perception; more specifically the relation between the actual change in a physical stimuli and the perceived change. This includes stimuli to all senses; vision, hearing, taste, touch and smell.

History and formulation of the laws[edit]

Both Weber's law and Fechner's law were formulated by Gustav Theodor Fechner (1801–1887). They were first published in 1860 in the work Elemente der Psychophysik (Elements of psychophysics). This publication was the first work ever in this field, and were Fechner coined the term psychophysics to describe the interdisciplinary study of how humans perceive physical magnitudes.[1]

Weber's law[edit]

Ernst Heinrich Weber (1795–1878) was one of the first people to approach the study of the human response to a physical stimulus in a quantitative fashion. Fechner was a scholar of Weber and named his firs law in honor of his mentor, since it was Weber who had conducted the experiments needed to formulate the law.[2]

Fechner formulated several versions of the law, all stating the same thing. One formulation states:

"Simple differential sensitivity is inversely proportional to the size of the components of the difference; relative differential sensitivity remains the same regardless of size."[1]

What this means is that the perceived change in stimuli is proportional to the initial stimuli.

Weber's law also incorporates the just-noticeable difference (JND). This is the smallest change in stimuli that can be perceived. As stated above, the JND is proportional to the initial stimuli. Fechner found that the JND is constant for any sense.

Weber contrast[edit]

Although Weber's law includes a statement of the proportionality of a perceived change to initial stimuli, Fechner never formulated this statement as a mathematical expression. Instead, he only refers to this as a rule of thumb regarding human perception.[1]

However, this rule of thumb has been expressed mathematically, as Weber contrast.

[3][4]

Fechner's law[edit]

Fechner noticed in his own studies that different individuals have different sensitivity to certain stimuli. For example the ability to perceive differences in light intensity could be related to how good that individuals vision is.[1] He also noted that the human sensitivity to stimuli changes depends on which sense in affected. He used this to formulate another version of Weber's law that he named after himself.[5]

Fechner's law states that the subjective sensation is proportional to the logarithm of the stimulus intensity. According to this law, human perceptions of sight and sound work as follows: Perceived loudness/brightness is proportional to logarithm of the actual intensity measured with an accurate nonhuman instrument.[5]

The relationship between stimulus and perception is logarithmic. This logarithmic relationship means that if a stimulus varies as a geometric progression (i.e., multiplied by a fixed factor), the corresponding perception is altered in an arithmetic progression (i.e., in additive constant amounts). For example, if a stimulus is tripled in strength (i.e., 3 x 1), the corresponding perception may be two times as strong as its original value (i.e., 1 + 1). If the stimulus is again tripled in strength (i.e., 3 x 3 x 1), the corresponding perception will be three times as strong as its original value (i.e., 1 + 1 + 1). Hence, for multiplications in stimulus strength, the strength of perception only adds. The mathematical derivations of the torques on a simple beam balance produce a description that is strictly compatible with Weber's law.[6][7]

Deriving Fechner's law[edit]

Fechner's law is a mathematical derivation of Weber's law.

Integrating the mathematical expression for Weber's law gives:

where is the constant of integration and ln is the natural logarithm.

To solve for , assume that the perceived stimuli becomes zero at some threshold stimuli . Using this as a constraint, set and . This gives:

Substituting in the integrated expression for Weber's law, the expression can be written as:

The constant k is sense-specific and much be determined depending on the sense and type of stimuli.[5]

Types of perception[edit]

Weber and Fechner conducted research on differences in light intensity and the perceived difference in weight.[1] Other sense modalities provide only mixed support for either Weber's law or Fechner's law.

Weight perception[edit]

Weber found that the just-noticeable difference (JND) between two weights was approximately proportional to the weights. Thus, if the weight of 105 g can (only just) be distinguished from that of 100 g, the JND (or differential threshold) is 5 g, or in the SI system, a force or weight of 0.005 kg N. If the mass is doubled, the differential threshold also doubles to 10 g, so that 210 g can be distinguished from 200 g. In this example, a weight (any weight) seems to have to increase by 5% for someone to be able to reliably detect the increase, and this minimum required fractional increase (of 5/100 of the original weight) is referred to as the "Weber fraction" for detecting changes in weight. Other discrimination tasks, such as detecting changes in brightness, or in tone height (pure tone frequency), or in the length of a line shown on a screen, may have different Weber fractions, but they all obey Weber's law in that observed values need to change by at least some small but constant proportion of the current value to ensure human observers will reliably be able to detect that change.

Fechner did not conduct any experiments on how perceived heaviness increased with the mass of the stimulus. Instead, he assumed that all kinds are subjectively equal, and argued mathematically that this would produce a logarithmic relation between the stimulus intensity and the sensation. These assumptions have both been questioned.[8][9] Most researchers nowadays accept that a power law is a more realistic relationship, or that a logarithmic function is just one of a family of possible functions.[10]

Sound[edit]

Weber's law does not quite hold for loudness. It is a fair approximation for higher intensities, but not for lower amplitudes.[11]

"Near miss" of Weber's law in the auditory system[edit]

Weber's law does not hold at perception of higher intensities. Intensity discrimination improves at higher intensities. The first demonstration of the phenomena were presented by Riesz in 1928, in Physical Review. This deviation of the Weber's law is known as the "near miss" of the Weber's law. This term was coined by McGill and Goldberg in their paper of 1968 in Perception & Psychophysics. Their study consisted of intensity discrimination in pure tones. Further studies have shown that the near miss is observed in noise stimuli as well. Jesteadt et al. (1977)[12] demonstrated that the near miss holds across all the frequencies, and that the intensity discrimination is not a function of frequency, and that the change in discrimination with level can be represented by a single function across all frequencies.

Vision[edit]

The eye senses brightness approximately logarithmically over a moderate range (but more like a power law over a wider range), and stellar magnitude is measured on a logarithmic scale.[13] This magnitude scale was invented by the ancient Greek astronomer Hipparchus in about 150 B.C. He ranked the stars he could see in terms of their brightness, with 1 representing the brightest down to 6 representing the faintest, though now the scale has been extended beyond these limits; an increase in 5 magnitudes corresponds to a decrease in brightness by a factor of 100.[13] Modern researchers have attempted to incorporate such perceptual effects into mathematical models of vision.[14][15]

"Near miss" of Weber's law in visual regularity perception[edit]

Perception of glass patterns and mirror symmetries in the presence of noise follows Weber's law in the middle range of regularity-to-noise ratios (S), but in both outer ranges, sensitivity to variations is disproportionally lower. As Maloney, Mitchison, & Barlow (1987)[16] showed for Glass patterns, and as van der Helm (2010)[17] showed for mirror symmetries, perception of these visual regularities in the whole range of regularity-to-noise ratios follows the law p = g/(2+1/S) with parameter g to be estimated using experimental data.

Numerical cognition[edit]

Psychological studies show that it becomes increasingly difficult to discriminate among two numbers as the difference between them decreases. This is called the distance effect.[18][19] This is important in areas of magnitude estimation, such as dealing with large scales and estimating distances. It may also play a role in explaining why consumers neglect to shop around to save a small percentage on a large purchase, but will shop around to save a large percentage on a small purchase which represents a much smaller absolute dollar amount.[20]

Pharmacology[edit]

It has been hypothesized that dose–response relationships can follow Weber's law[21] which suggests this law – which is often applied at the sensory level – originates from underlying chemoreceptor responses to cellular signaling dose relationships within the body. Dose response can be related to the Hill equation, which is closer to a power law.

Public finance[edit]

There is a new branch of the literature on public finance hypothesizing that the Weber–Fechner law can explain the increasing levels of public expenditures in mature democracies. Election after election, voters demand more public goods to be effectively impressed; therefore, politicians try to increase the magnitude of this "signal" of competence – the size and composition of public expenditures – in order to collect more votes.[22]

See also[edit]

References[edit]

  1. ^ a b c d e Fechner, Gustav Theodor. Howes D H, Boring E G, editors. (1966) Adler H E, translator. Elements of psychophysics, volume 1. United States of America: Holt, Rinehart and Winston. Original title: (1860) Elemente der Psychophysik. Leipzig: Breitkopf und Härtel;
  2. ^ Ross, H.E. and Murray, D. J.(Ed. and Transl.) (1996)E.H.Weber on the tactile senses. 2nd ed. Hove: Erlbaum (UK) Taylor & Francis;
  3. ^ Li, Wu-bin; Lu, Chang-hou; Zhang, Jian-chuan (February 2013). A lower envelope Weber contrast detection algorithm for steel bar surface pit defects (Thesis). Optics & Laser Technology. pp. Pages 654–659. 
  4. ^ Drew, SA; Chubb, CF; Sperling, G (2010). Precise attention filters for Weber contrast derived from centroid estimations (Article). JOURNAL OF VISION. pp. 16p. ISSN 1534-7362. 
  5. ^ a b c Fechner, Gustav Theodor. (1869) Elemente der Psychophysik, Band 2. Leipzig: Breitkopf und Härtel
  6. ^ Lanzara, Richard G. "Weber's Law Modeled by the Mathematical Description of a Beam Balance". cogprints.org. CogPrints. Retrieved 5 December 2015. 
  7. ^ "Bio Balance - Reference Library". bio-balance.com. Retrieved 5 December 2015. 
  8. ^ Heidelberger, M. (2004)Nature from within: Gustav Theodor Fechner and his psychophysical worldview. Transl. C. Klohr. Pittsburgh, USA: University of Pittsburgh Press.
  9. ^ Masin, S.C.; Zudini, V.; Antonelli, M. (2009). "Early alternative derivations of Fechner's law" (PDF). J. History of the Behavioral Sciences. 45: 56–65. doi:10.1002/jhbs.20349. 
  10. ^ Murray, D.J. (1993). "A perspective for viewing the history of psychophysics". Behavioral and Brain Sciences. 16: 115–186. doi:10.1017/s0140525x00029277. 
  11. ^ Yost, William A. (2000). Fundamentals of hearing : an introduction (4. ed.). San Diego [u.a.]: Academic Press. p. 158. ISBN 0-12-775695-7. 
  12. ^ Jesteadt, Walt, Craig C. Wier, and David M. Green. "Intensity discrimination as a function of frequency and sensation level." The Journal of the acoustical society of America 61 (1977): 169.
  13. ^ a b V. B. Bhatia (2001). Astronomy and astrophysics with elements of cosmology. CRC Press. p. 20. ISBN 978-0-8493-1013-3. 
  14. ^ Jianhong (Jackie) Shen; Yoon-Mo Jung (2006). "Weberized Mumford–Shah model with Bose–Einstein photon noise". Appl. Math. Optim. 53 (3): 331–358. doi:10.1007/s00245-005-0850-1. 
  15. ^ Jianhong (Jackie) Shen (2003). "On the foundations of vision modeling I. Weber's law and Weberized TV (total variation) restoration". Physica D: Nonlinear Phenomena. 175 (3/4): 241–251. doi:10.1016/S0167-2789(02)00734-0. 
  16. ^ Maloney, R. K., Mitchison, G. J., & Barlow, H. B. (1987). Limit to the detection of Glass patterns in the presence of noise. Journal of the Optical Society of America A, 4, 2336-2341.
  17. ^ van der Helm, P. A. (2010). Weber-Fechner behaviour in symmetry perception? Attention, Perception, & Psychophysics, 72, 1854-1864.
  18. ^ Moyer R.S., Landauer T.K. (September 1967). "Time required for judgements of numerical inequality". Nature. 215 (5109): 1519–20. doi:10.1038/2151519a0. PMID 6052760. 
  19. ^ Longo M.R., Lourenco S.F. (2007). "Spatial attention and the mental number line: evidence for characteristic biases and compression". Neuropsychologia. 45 (7): 1400–6. doi:10.1016/j.neuropsychologia.2006.11.002. 
  20. ^ http://www.npr.org/2015/01/13/376836545/consumer-agency-launches-tool-to-help-you-find-a-cheaper-mortgage
  21. ^ D. Murray Lyon (1923). "Does the reaction to adrenalin obey Weber's law?". Journal of Pharmacology. 21 (4): 229–235. 
  22. ^ Mourao, P. (2012). "The Weber-Fechner Law and Public Expenditures Impact to the Win-Margins at Parliamentary Elections". 21 (3). Prague Economic Papers: 290–308. doi:10.18267/j.pep.425. 

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