Quote from: Peter R on September 03, 2014, 10:47:22 PM

Quote from: zby on September 03, 2014, 08:39:34 AM

Quote from: rpietila on September 03, 2014, 08:36:04 AM

Quote from: zby on September 03, 2014, 08:20:02 AM

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.

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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?

Exponential growth is not some "arbitrary function." It is the solution to a very simple--and very meaningful--differential equation. It occurs whenever the growth rate of something is proportional to the size of the thing that's growing: e.g., the population of bunny rabbits in a park, bacteria in a petri dish, or users of a social networks.

Here's a simple model for bitcoin adoption:

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Let N by the number of bitcoin users. Assume that on average each user converts k non-users every year. Each year (Δt) the change in the number of users (ΔN) is then clearly k N. This allows use to write the differential equation¹:

  ΔN/Δt = k N

The solution to this equation is:

   N(t) = N₀ e^{k t}

Where N₀ is the initial number of users and N(t) is the number of users at a later time t (which clearly grows exponentially with time).
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Note that this is exactly the same rationale that we'd use to explain the growth of an intially-small population of bunny rabbits introduced into a park with abundant food:

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Let N by the number of ~~bitcoin users~~ bunnies. Assume that on average each ~~user converts~~ bunny creates k ~~non-users~~ new bunnies every year. Each year (Δt) the change in the number of ~~users~~ bunnies (ΔN) is then clearly k N. This allows use to write the differential equation:

  ΔN/Δt = k N

The solution to this equation is:

   N(t) = N₀ e^{k t}

Where N₀ is the initial number of ~~users~~ bunnies and N(t) is the number of ~~users~~ bunnies at a later time t (which clearly grows exponentially with time).

The exponential function comes from a very simple and very reasonable underlying dynamical model. You can't just say "maybe bitcoin growth is a trig function" or "maybe it's a Bessel function"--you need to refine the original differential equation with new reasonable dynamics, and then solve it to determine what the "function might be." For example, the "logistic function" arises by noting that things do not grow exponentially forever. Instead, the growth rate often slows down and then approaches zero when N reaches some saturation value N_sat. The simplest way to model this is by adding the following term to original differential equation:

ΔN/Δt = k N (1 - N / N_sat)

The solution to this is the logisitic function that SlipperySlope is using.

For bunny rabbits, the saturation level is the equilibrium population of bunnies that the park in question can support. For bitcoinwell time will tell.

TL/DR: The exponential growth model is the best/simplest model that explains bitcoin adoption to date.

¹I should use an appropriate limiting procedure here.

while I agree that number of users is increasing exponentially (so far), the number of transactions is NOT, which was posted earlier.
Possible explanation-the absence of accurate statistics, but then the whole argument is mute.