Hmm - do you say that the line you are watching is not 'USD/BTC = exp(-2.869800 + 0.003012 * D), D being the number of days' anymore? Have you changed the coefficients or have you changed it altogether so some other function?
It is always, every day, the line (or other construct) that gives the highest R^2 fit with the USD/BTC price data between 2009-1-3 and present_day. For all the time it has been an exponential function, which is linear when plotted in logarithmic space as I do.
Your comment about only one best fitting trendline only makes sense if you constrain your search space - for example by choosing only exponential functions.
A side note - if you for example allow for trendlines to be polynomials of unrestricted degree - then you'd be able to fit the trendline to the price chart exactly (with no divergencies at all).
1. Not really. Others just don't come close. 2. That's quite theoretical, since I cannot convince myself that a model with more than 2nd degree term is anything but noise with no predictive power, and Excel allows construction to 6th degree, with no improvement in R^2.
What IS important is if the growth trend is slowing or not. I currently hold the opinion that the trend is pretty much intact and price is about to increase 10x in a year. AnonyMint thinks it has slowed.
How about trigonometric functions? Have you tried them? Or polynomials with trigonometric functions? I am sure Excel have many many functions and you can combine them in many many ways - I am sure you have not tried them all. So my question is how do you chose your functions - why are you sure that exp is good and cos is not?
I am sure that you are trolling, yet want to explain the scientific method to others:
- The model needs to have a good fit to the data (best fit is usually best)
- The model needs to be in unison with the observed mechanisms that produce the data
- In absence of exact mechanisms (which is usually the case), the model should rely on general events (time passing) more than special (number of sunspots), if they give the fit that is equally good to explain the historical data.
- The model is used to predict the future. Therefore its future predictions have to be reasonable. In most other contexts, predicting a market cap of $17,000 billion dollars in 4 years is not credible. Here disregarding it as an impossibility may be a grave mistake, as it was to refuse to invest a few grand into Bitcoin 4 years back when it was available to all with little effort at $0.08. If Bitcoin has done something that other haven't, ever, it has a nonzero possibility of repeating the behaviour, and the model is better knowing it.
OK - so let's go through your list:
1. Fit to the data - here you have not really explained what did you try. I take your word that for all functions that you did exp was the best fitting - but I have the feeling that your imagination about the possibilities is limited.
2. Relation to the observed mechanisms that produce the data - You don't mention any mechanism. You write that relation to time passing is better then relation to sun spots - cos(t) would also be related to time passing. A dumped oscillator function would actually have some relation to the observed mechanisms that produce the data - if we agree that people tend to predict that the price will go in the direction it goes now and that people predictions do influence the price.
3. Predicting future -
perl -e 'print 10**(-2.869800 + 0.003012 * 6000)'
1.59294213512763e+15
And that is a 1000 times more than all money supply on Earth (M0 is around 1.2e+12).
Evidently exp function cannot predict the future too far away and eventually it will have to be replaced by some S curve or maybe a dumped oscillator or something else. Maybe even something with a trigonometric function in it (the simplest oscillators)?