Timeline for Is least squares the standard method to fit a 3 parameters Gaussian function to some x and y data?
Current License: CC BY-SA 3.0
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| Sep 25, 2013 at 14:25 | comment | added | Nick Cox | Agreed. Although Gaussian logit and your method both use three parameters, the other two parameters don't line up so easily. As your method seems closer in spirit and in substance to what the OP desires, the matter can perhaps rest here, beyond a signal to others interested in related problems that Gaussian logit is a fairly easy method of fitting bell-shapes, as it's a standard special case of generalized linear models. | |
| Sep 25, 2013 at 14:18 | comment | added | whuber♦ | I agree that the modes are similar in the two fits, but the other two parameters may (ought) to be of interest, too. Their estimates differ distinctly, especially the estimates of the amplitude. A goodness of fit test (or equivalently a comparison of AIC or BIC) will be decisive. This is an important point because it shows that even a small amount of data can discriminate among competing models. | |
| Sep 25, 2013 at 12:24 | comment | added | Nick Cox | Rounding now more sensibly to 3 d.p. @whuber's method gives $−0.033$ s as compared with $-0.036$ s from the other methods. I would guess that the differences are not worth trying to interpret. | |
| Sep 24, 2013 at 14:44 | comment | added | Nick Cox | In what's above, "value of $x$" should be "value for $x$ so calculated", i.e. the value of $x$ is the position of the maximum predicted for $y$ (in both the usual and the OP's notation). | |
| Sep 24, 2013 at 13:52 | comment | added | Nick Cox | It sounds as if the offset is the parameter of most interest. With a logit model, response the proportion, predictors $x$ and $x^2$, the position of the maximum is $-$coefficient of $x$/(2 * coefficient of $x^2)$. This is just the usual calculus argument that $a + bx + cx^2$ has zero derivative when $x = -b/2c$; we need to check that the value of $x$ really is a maximum. For your data I get $-.03574713$ from a Gaussian logit model. Your R code gives $-0.03636307$. I've given more d.p. than is sensible, but the agreement between slightly different procedures looks good. | |
| Sep 24, 2013 at 13:37 | vote | accept | danilinares | ||
| Sep 25, 2013 at 1:24 | |||||
| Sep 24, 2013 at 13:36 | comment | added | danilinares | Typically for this kind of problem the data is coarse. Many times a gaussian function is fitted and the mean is called the point of subjective simultaneity. This point often corresponds to an asynchrony in which the flash is presented before the sound and sometimes this is taken as evidence that the perception of the flash needs more time than the sound. So I guess a gaussian is used to easily get some measure of central tendency. Thanks very much for the extensive and clear answer. | |
| Sep 24, 2013 at 10:54 | history | answered | Nick Cox | CC BY-SA 3.0 |