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I'll try to make the matter as clear as possible given the circumstances. My boss asked me to look at an old report a former employee wrote around a couple of months ago. Apparently the report contains some very useful information but, unfortunately, no one is able to understand it because it is written in "mathematics".

The main node of the report is a linear regression formula for the predictions of future data for non-linear function. Consider to have the following:

enter image description here

The Y-axis report a percentage (1 equals 100%). Now imagine to only have the part of the data before the red line. My former colleague figured out that the formula to estimate the second part of the data was:

$$R(t) = 1 - d t^{1/\alpha}$$

While I did understand most of the formula I cannot understand what $d$ and $\alpha$ represent. By any chance, are they a common sign for something I am missing?

Please, ask if something is not clear.

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What formula do you use before the red line ? – Claude Leibovici Nov 11 '13 at 16:42
@ClaudeLeibovici No formula, those are real data. – Secret Parrot Nov 11 '13 at 16:45
Could you tell me how looks a plot of Log[1 - R] as a function of Log[t] for the range where data are available ? – Claude Leibovici Nov 11 '13 at 16:50
@ClaudeLeibovici I am afraid I did not understand the request. By the way, T stands for time (in this context, days). – Secret Parrot Nov 11 '13 at 16:53

$d$ and $\alpha$ are the parameters of the fit. Presumably he decided on this form of a function to model the data, maybe for some theoretical reason or by trying a number of functions and deciding this looked best. Then fed the data to a function minimizer to find the best fit $d$ and $\alpha$, finding the values that minimize the error. If you have the values for $d$ and $\alpha$ you can calculate the values of $R(t)$ They don't really have to represent anything except the parameters of the fit.

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I may have missed something but how can I minimize the error if I do not have the data I am trying to predict? Or should I find the parameters on the basis of the data I do have? One addition: on the report I found a footnote saying "parameters d and α are found with estimation techniques". Does this one add some extra useful information? – Secret Parrot Nov 11 '13 at 16:57
You use the data you do have to estimate $d$ and $\alpha$, then hope that the functional form is correct so that the values you calculate are close. "Estimation techniques" could mean that instead of using a minimizer he has just played with the parameters until the curve looked good, but I am guessing. – Ross Millikan Nov 11 '13 at 17:17

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