The point of my post (and graph) above was a slightly different one, though. Aimed more at Peter R.'s models, now that I think of it (since he is the one who uses 'no of tx' as a proxy for adoption in a Metcalfe based model).
What I had in mind is: if Bitcoin adoption is, against earlier optimistic assumptions, neither accurately captured by your logistic model nor by aconstant exponent exponential growth model based on a constant rate of growth, then all the current models could be missing something that the market / price discovery (perhaps) is picking up already: adoption continues, but the growth rate of adoption declines over time.
Your model would "miss" it because the growth function is hardcoded into it (the S-shape), Peter R.'s model would "miss" it, for a while at least, because the coefficient that relates price/mcap and adoption proxy is a global value, and will take time to adjust to a lower value.
Do you see the point I'm (clumsily) trying to make here?
What I had in mind is: if Bitcoin adoption is, against earlier optimistic assumptions, neither accurately captured by your logistic model nor by a
Your model would "miss" it because the growth function is hardcoded into it (the S-shape), Peter R.'s model would "miss" it, for a while at least, because the coefficient that relates price/mcap and adoption proxy is a global value, and will take time to adjust to a lower value.
Do you see the point I'm (clumsily) trying to make here?
Yes, and thank you for the helpful clarification.
Mathematically, the logistic model has the property of decreasing exponential growth, which is most obvious on a log graph as we near full adoption. It is only a falsifiable hypothesis that a logistic model can explain bitcoin prices. My reason for choosing this model was to fit the obvious constraint that exponential price growth must eventually end. Perhaps we are at that point now, but I believe not based upon the steady improvement in Bitcoin transactional infrastructure. http://www.bitcoinpulse.com/
There is at least one mathematical theory of price bubbles that could be used to elaborate this logistic model, but I hesitate to combine them for fear of unsound overfitting given the additional number of parameters whose values must be set. The bitcoin logistic model has only two parameters: the maximum price and the full adoption duration.
Alright, playing around with my idea some more...
Extrapolating from the interpretation of the graph I posted above, we get something like: the first factor 10 increase takes 12 months, the second 18, the third 24 months, etc.
In other words, to apply n times the factor g increase of the starting value, we require n-th triangle number time steps.
Which leads to the formula:
f(0.5*(t^2+t)) = s*g^t
or equivalently:
f(t)=s*g^(0.5*(sqrt(8t+1)-1))
where s is the starting value and g the growth factor.
Now, where that one[1] falls in terms of functional growth, I'm not sure. Doesn't seem to fit the definition of exponential growth anymore, but doesn't just grow linearly either... At first I thought it would be another example of bounded growth (similar to your logistic function then), but plotting it, it very much looks like it is still exponentially growing (just as a slightly compressed looking curve, compared to "regular" exponential growth) ...
[1] O(c^sqrt(n))
