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9h
comment What the minimum of infinite tetration divided by $\sqrt{x}$?
The key, why $x=\sqrt{2}$ works is not because it is a root, but an exponential of the form $x=b^{1\over b}$ at $b=2$. Thus a better generalization is for instance $ ^3 \sqrt{3}$ etc. After that try to find the number $b$ where $x=2 = b^{1 \over b}$ and look what happens when you evaluate x^b,x^x^b,x^x^x^b,... resp 2^b,2^2^b, 2^2^2^b,...
17h
comment How to do a regression which includes reciprocals?
Step 1: Yes, I've done the same regression and got the same/a similar result. Step 2: But after that I looked at the residuals and noticed, that they have a trend to become absolutely smaller. So I considered to do a further improvement to find a characteristic of that. Looked at the residuals and at their reciprocals - observing/guessing that there is another linear trend. Thus the residuals of Step 1(as of your finding) are in the denominator of my formula and I tried -as a higher order approximation- to get something useful by removing that trend. That's the reason for my complete formula.
19h
comment How to do a regression which includes reciprocals?
I think now, that a simple answer is not possible since it is a non-linear regression-problem and I've to finetune any final approximation manually. So I just want to say thank you very much for all your help!
20h
comment How to do a regression which includes reciprocals?
I added a link to a *.csv-file in my answer which contains the sample of data
20h
revised How to do a regression which includes reciprocals?
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20h
comment How to do a regression which includes reciprocals?
@Yves - well actually I used that linear interpolation as rough approximation (as well as a more sophisticated formula, but that's complete different path, see the new link in my updated answer). I just saw room for improvement and tried to make sense of the above regression-model. My problem is simply to derive the solution for the parameters for a regression, where the data-to-be-fit (resp. the intended residuals) are in the denominator of the regression-formula which is quite unusual for me. I thought there might be some (simple) canonical answer for such a question.
20h
revised How to do a regression which includes reciprocals?
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21h
comment How to do a regression which includes reciprocals?
Hmm, the rationale of the "regression" is the minimization of the error of prognosis in the sense of the least (sum-of) squares of the residuals. Now I'm missing the "squares" and the "sum" and the "minimum" in your first equation where you equal the lhs to zero. Speaking of "minimization" the derivative of a function should become zero (and the second derivative should become positive), not the function itself. Did I misread something in your ansatz?
21h
comment How to do a regression which includes reciprocals?
... was an instrument to uncover the inappropriateness of the polynomial interpolation at all. I think the same applies to any method - even with this regression-model.
21h
comment How to do a regression which includes reciprocals?
@Yves: the splitting is done to have some sanity check of the procedure. For how many current data-points I'll do any optimizaton, there will be infinitely many more for which that optimization must be sensical. The problem is that an estimation will always face the problem that I have to extrapolate to hiher indexes. So this splitting shall give some indication, whether the found optimization can be meaningful at all. Consider the polynomial interpolation. I did it with first forty points and got exact fit. But extrapolated to further points showed catastrophic error, so this splitting...
22h
revised An obvious pattern to $i\uparrow\uparrow n$ that is eluding us all?
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22h
revised An obvious pattern to $i\uparrow\uparrow n$ that is eluding us all?
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22h
revised How to do a regression which includes reciprocals?
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22h
comment How to do a regression which includes reciprocals?
@Yves : I've put the data into an answer-box to not to clutter the question too much.
22h
answered How to do a regression which includes reciprocals?
22h
revised How to do a regression which includes reciprocals?
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23h
asked How to do a regression which includes reciprocals?
1d
revised An obvious pattern to $i\uparrow\uparrow n$ that is eluding us all?
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1d
revised An obvious pattern to $i\uparrow\uparrow n$ that is eluding us all?
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1d
answered An obvious pattern to $i\uparrow\uparrow n$ that is eluding us all?