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@eladar saidy = Int[ e^(-x^2) ]
y=e^(-x^2)
That is an interesting curve.
Its a non-elementary function. I've only ever recall using its tabulated values to solve Non Steady State Diffusion problems in Engineering Material Science coursework. That is the extent of my familiarity!
@eladar saidYes, but the function does not have an elementary definite integral. The function they are trying to fit is only defined by its integral form. So our curve would have to be y = Int[ e^(-x^2) dx ] . Starting with a non-closed form function I think is going to be very different than starting with a closed form function like we did for the logistical regression.
@joe-shmo
That would just be the area under that curve.
@joe-shmo saidexp[-x²] is an elementary function. Elementary functions are polynomials, quotients, trigonometric functions, exponentials (including sinh, cosh and tanh), their inverses and any function that can be produced by combining them. However, the error function, giving the area under the curve of exp[-x²] up to some limit y, cannot be expressed in closed form i.e. as an elementary function. It's important in all sorts of statistical fields.
y = Int[ e^(-x^2) ]
Its a non-elementary function. I've only ever recall using its tabulated values to solve Non Steady State Diffusion problems in Engineering Material Science coursework. That is the extent of my familiarity!
It is possible to get the integral over the entire real line, you just work out the square of the integral by noticing that you can do it as a two dimensional integral and take the limit. Happily Wikipedia has a nice demonstration of how to do it:
https://en.wikipedia.org/wiki/Gaussian_integral
@joe-shmo saidBased on an interview on Channel 4 news they included early data points which will have been due to infections really early on, before the lock down. The problems with this are two fold, first the policy changed, and secondly it's a statistically small sample. Their method seems too sensitive to the early part of the curve.
For a purely academic experience I would like to see what the methodology might be for setting up that regression based on the error function ( just to understand what they were grappling with). Surely it has to be a more complicated procedure?
On a side note: either/or have been very effective at making accurate predictions for any sizable future date. The IMHE model ...[text shortened]... uture, but I'm sure there will be some controversy over it ( and rightfully so ) when this subsides.
The catch is we had a huge jump in new cases today. Unless it's due to more testing this will translate into about 800 extra deaths. A problem is that alteration in testing policy messes up predictions based on them.
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@deepthought saidYeah, I was pointing out to Eladar that we are not using y = e^(-x^2) as the function in the way we did the logistical function.
exp[-x²] is an elementary function. Elementary functions are polynomials, quotients, trigonometric functions, exponentials (including sinh, cosh and tanh), their inverses and any function that can be produced by combining them. However, the error function, giving the area under the curve of exp[-x²] up to some limit y, cannot be expressed in closed form i.e. as an eleme ...[text shortened]... kipedia has a nice demonstration of how to do it:
https://en.wikipedia.org/wiki/Gaussian_integral
I was saying they are using "Int[y dx}" as the starting point.
"However, the error function, giving the area under the curve of exp[-x²] up to some limit y, cannot be expressed in closed form i.e. as an elementary function. It's important in all sorts of statistical fields."
I stated this in my following reply to Eladar, where he said "That would just be the area under that curve."
I thought though his statement is true, it presents the matter of fitting Y = int [ y dx ] as trivial. I cant imagine that to be the case, but we all have different levels of triviality I suppose.
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@eladar saidThat's not really helpful, because that is not the difficult part. The difficulty arises in trying to determine the parameters, what are they, how can they be formed, etc...?
The graphing calculator can easily give you a decimal approximation give an interval.
So even if you cannot know how to express the answer you can know any specific answer to the millionth place easy enough.
If we say that our Cumulative Deaths is f(x), then we have to solve
f(x) = Int ( e^(-x^2) ) for some group of parameters that fully manipulates e^(-x^2)
The result of that should be some non-linear system of equations, for which we try to minimize the error by traditional methods. Or find some ingenious way of distilling it ( which perhaps Deepthought has already considered).
Its not just a matter of plugging some "x" into the integral and evaluating. The parameters will effect the value of that integral.
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@ponderable said@ponderable
Taking the data from
https://www.worldometers.info/coronavirus/country/us/
which report 256 dead as of yesterday (and who give sources) and their daily number starting with one dead at 29th of February I can model the curve death over day using the exponential curve
1.61*exp(0.24*d) with a r-value of 0.97 which is reasonable.
If I calculate to day 42 (which is three we ...[text shortened]... Worldometers.info just changed to 276 death today it should be around 316 if the curve was correct.
Today is the big day.
38,416 or bust...
To be fair though, at the time if you would have said 20k, I still would have been skeptical.