Benford's Law?

[phatsphere]

Has anyone so far looked into Benford's Law for the highest mantissa digits of all prices? It's kind of a rule, that for arbitrary numbers and prices, the highest digit tends to be the digit 1 or 2, but not 7, 8 or 9.
What the script below does: daily average since 2011 from blockchain.info, do some transformations with pandas, and then plot it. It almost fits, but a bit too much for 1 and not enough for 2 and 3.

Based on that, I speculate that we'll either see a long period of $ 200-299  or $ 2000-2999.

Ideas welcome, improvements are certainly possible ...

Code: (python)

#!/usr/bin/env python
import numpy as np
import pandas as pd

pricedata = "http://blockchain.info/charts/market-price?showDataPoints=false&timespan=all&show_header=true&daysAverageString=1&scale=0&format=csv"
data = pd.read_csv(pricedata, names = ["day", "price"], parse_dates=[0], index_col=0)

#start at 2011-01-01
data = data.ix['2011-01-01':]
print data.head()

price100 = data["price"] * 100
pcolstr = price100.map(str)
highest = pcolstr.map(lambda _ : _[0]).map(int)

print highest.describe()
val, _ = np.histogram(highest.as_matrix(), bins=9)

digits = np.arange(1,10)
benford = np.log10(1 + 1. / digits)

vals = val.astype(np.float) / highest.size

print(np.c_[benford, vals])

import matplotlib.pyplot as plt
plt.plot(digits, benford)
plt.plot(digits, vals, color="green")
plt.show()

[jamesc760]

I've been speculating that the end of 2014 price will be around $2500, after a rise to $5000.

[frienemy]

In case of a "bubble" this year, the subsequent correction to a 2000ish level and a slow advance from there would pretty much fit into this picture. One of the explanations for Benford's Law is exponential growth anyway, isn't it?

[OnkelPaul]

Benford's law basically describes the distribution of first digits in a data set with exponential variation. Since the price curve of bitcoin approximately follows an exponential curve, the law applies to bitcoin prices relatively well.
However, as with all real-world data for which an approximate mathematical model exists, future datapoints will not be skewed to "fill" differences between historical datapoints and the model.
So while it is very likely that future prices will more often have 1, 2 or 3 as their first digit than 7, 8 or 9, there is no reason to believe that 2 will be more common in future since it was a bit less common in the past.
This is similar to dice throws: Just because you did not roll a 6 for a number of times does not make it more probable that your next roll will be a 6.

On the contrary: It might be possible that a mechanism is at work that somehow makes 2 less probable, so after going through 1000..1999 the time for 2000..2999 might be much shorter due to psychological effects.

Onkel Paul

[pinksheep]

Has anyone so far looked into Benford's Law for the highest mantissa digits of all prices? It's kind of a rule, that for arbitrary numbers and prices, the highest digit tends to be the digit 1 or 2, but not 7, 8 or 9.
What the script below does: daily average since 2011 from blockchain.info, do some transformations with pandas, and then plot it. It almost fits, but a bit too much for 1 and not enough for 2 and 3.

Based on that, I speculate that we'll either see a long period of $ 200-299  or $ 2000-2999.

Ideas welcome, improvements are certainly possible ...

Code: (python)

#!/usr/bin/env python
import numpy as np
import pandas as pd

pricedata = "http://blockchain.info/charts/market-price?showDataPoints=false&timespan=all&show_header=true&daysAverageString=1&scale=0&format=csv"
data = pd.read_csv(pricedata, names = ["day", "price"], parse_dates=[0], index_col=0)

#start at 2011-01-01
data = data.ix['2011-01-01':]
print data.head()

price100 = data["price"] * 100
pcolstr = price100.map(str)
highest = pcolstr.map(lambda _ : _[0]).map(int)

print highest.describe()
val, _ = np.histogram(highest.as_matrix(), bins=9)

digits = np.arange(1,10)
benford = np.log10(1 + 1. / digits)

vals = val.astype(np.float) / highest.size

print(np.c_[benford, vals])

import matplotlib.pyplot as plt
plt.plot(digits, benford)
plt.plot(digits, vals, color="green")
plt.show()

So do you think after that we'll have a long period with the price around $20,000-$29,999?

[phatsphere]

This is similar to dice throws: Just because you did not roll a 6 for a number of times does not make it more probable that your next roll will be a 6.

Don't be confused with "gambler's fallacy". What I've posted doesn't concern a single event, but the overall summary.

[phatsphere]

So do you think after that we'll have a long period with the price around $20,000-$29,999?

Well, if the "2" and "3" are filled up, miraculously, in the near future ... there is absolutely no implication for the far future. Hence no.

[Foxpup]

Classic gambler's fallacy. "I'm more likely to roll a two, because I pre-rolled the ones out."

So do you think after that we'll have a long period with the price around $20,000-$29,999?

Of course. It'll certainly last longer than the $30,000-$39,999 range... because $20k-$30k is a 50% increase, but $30k-$40k is only a 33% increase. (How do you think Benford's law works in the first place?)

[Alonzo Ewing]

This is awesome.  Thank you for posting this and teaching me something.

I'm wondering if you add an analysis: the number of consecutive days in various groupings:

Group 1: 1 and 2 first digit
Group 2: 2, 3, 5 first digit
Group 3: 6-9 first digit

When I look back at the long term chart, there's really only been two periods when there were numerous consecutive days (weeks actually) with the first digit between 6 and 9: back in 2011 when the price ranged from 0.5 through 0.9, and the last 5-6 months.

Edit: Actually, you don't necessarily need to break it down into groups--using the same single number histogram is fine.  What I'm curious about is the consecutiveness of the days (or weeks).

[glub0x]

Makes sens to me. It is really not a obvious one.
But i m not sure you can actually make profit with it, can you?

Thanks for sharing

[btc-owl]

The issue with these kind of projections is that the curve can change dramatically when major events occur (as it happens on stock markets when a Tsunami/war/terrorist attacks hits any major hub).

BTC is on its early days, and major announcements are due to come. Either good or bad ones, from frauds to successful startups, volatility will still be around for a while.

[DolanDuck]

We'll either see a long period of $ 200-299  or $ 2000-2999, but what are the odds of the low and high price, 50/50?