The trend is based on a model and rpietila's model has one major shortcoming. I came to this conclusion by analysing the early 2010 and 2009 data. The fact that there is something wrong here becomes clear when one looks at the graph and focuses on 2009 and early 2010. My first thought was that the data from this period was way off; however rpietila provided enough anecdotal evidence to convince me that his estimate was at least in the right ballpark. This led me to question the model instead. My conclusion is that using price for the log fit is incorrect and that market capitalization should be used instead. The impact of this is most profound in 2009. My preliminary results predict a trend-line right now of around 4000 per BTC rather than around 900 USD per BTC. It also predicted from the 2010 and later data a constant price around 0.005 USD per BTC for a significant part of 2009, vindicating rpietila's guess that I had severely questioned.
This is a very interesting idea, modelling bitcoin market capitalization rather than price. I maintain a logistic model of price that I can adapt for market cap. Did you estimate the historical data series for the number of circulating bitcoins, or is this available for download somewhere?
I am currently away from home and will not return until Monday, at which point I will continue buying fractional coin from my local bitcoin ATM.
I used to total number of BTC created using the Blockchain data. One issue I am still trying to grapple with is accounting for irrevocably lost coins. I followed the model of accounting for stock splits when pricing a stock; however in the case of Bitcoin the "stock split" occurs approximately every 10 min. For example 12,629,850 BTC is "split" to 12,629,875 BTC when the next block is mined.