Expressing relative differences (in percent) by the difference of natural logarithms
| العنوان: | Expressing relative differences (in percent) by the difference of natural logarithms |
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| المؤلفون: | Christian Graff |
| المساهمون: | Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Laboratoire de Psychologie et NeuroCognition (LPNC), Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry]) |
| المصدر: | Journal of Mathematical Psychology Journal of Mathematical Psychology, Elsevier, 2014, 60, pp.82-85. ⟨10.1016/j.jmp.2014.02.001⟩ |
| بيانات النشر: | HAL CCSD, 2014. |
| سنة النشر: | 2014 |
| مصطلحات موضوعية: | Relative difference, Logarithm, Weber–Fechner, Applied Mathematics, [SDV]Life Sciences [q-bio], Mean value, Analytical chemistry, Value (computer science), DNL %, Napierian logarithm, Additive function, Psychophysics, Timing, Arithmetic, General Psychology, Mathematics |
| الوصف: | International audience; Most psychophysical investigations measure stimuli or performance in Système International units and use relative differences between them for comparison. In this theoretical note, we propose the ratio’s natural logarithm or the difference between the Napierian logarithms, as a desirable measure of relative differences between two psychophysical quantities. It challenges the more frequently used (x2−x1)/x1(x2−x1)/x1, (x2−x1)/x2(x2−x1)/x2, as well as (x2−x1)/xM(x2−x1)/xM, where x1,x2x1,x2, and xMxM are the initial value in a change, the larger value, and a mean value between x1x1 and x2x2, respectively. As for the three aforementioned expressions, it can be conveniently expressed as a percentage. For two physical measures, x1x1 and View the MathML sourcex2(x1>0;x2>0), the difference between natural logarithms View the MathML sourceDNL=Ln(x2)−Ln(x1)=Ln(x2/x1) sits between (x2−x1)/x2(x2−x1)/x2 and (x2−x1)/x1(x2−x1)/x1; it is actually the mean value of (x2−x1)/x(x2−x1)/x for all xx values between x1x1 and x2x2. Unlike other estimates, it satisfies all three of the following properties: symmetry, i.e. Δ(x1;x2)=−Δ(x2;x1)Δ(x1;x2)=−Δ(x2;x1); agreement between inverted units, such as hertz and second, i.e. Δ(x1;x2)=−Δ(k/x1;k/x2)Δ(x1;x2)=−Δ(k/x1;k/x2) thus |Δ(x1;x2)|=|Δ(k/x1;k/x2)||Δ(x1;x2)|=|Δ(k/x1;k/x2)|; and additivity “à la Chasles”, i.e. Δ(x1;x2)+Δ(x2;x3)=Δ(x1;x3)Δ(x1;x2)+Δ(x2;x3)=Δ(x1;x3). Finally, it complies with the Weber–Fechner and Stevens laws. |
| اللغة: | English |
| تدمد: | 0022-2496 1096-0880 |
| DOI: | 10.1016/j.jmp.2014.02.001⟩ |
| URL الوصول: | https://explore.openaire.eu/search/publication?articleId=doi_dedup___::e89767a6d1e10f924b0ca15eb93b7cdc https://hal.archives-ouvertes.fr/hal-01419360 |
| Rights: | CLOSED |
| رقم الانضمام: | edsair.doi.dedup.....e89767a6d1e10f924b0ca15eb93b7cdc |
| قاعدة البيانات: | OpenAIRE |
| تدمد: | 00222496 10960880 |
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| DOI: | 10.1016/j.jmp.2014.02.001⟩ |