Summary characterization of distributions over time

The first set of graphs show several statistics about the age of spent outputs of BTC, BCH, LTC, and DOGE since 2015. The age units are in terms of blocks. For BTC and BCH the interval between blocks is 10 minutes. For LTC it is 2.5 minutes and for DOGE it is 1 minute. The unit of observation is the ISO week, a natural unit of economic time.

The first line graphs show the mean, median, standard deviation, skewness, and kurtosis. The skewness and kurtosis statistics may be unfamiliar. They are the third and fourth standardized moments of a distribution, respectively. The moments of a distribution is defined by a power of the expectation \mathrm{E}[X], i.e. theoretical mean, of the distribution. The kth moment is \mathrm{E}[X^{k}]. Moments extend the concept of moving from expectation to variance (which is the square of the standard deviation). The mean (expectation) of a random variable is simply the first moment:

\mu=\mathrm{E}[X^{1}]

The variance is the second central moment:

\sigma^{2}=\mathrm{E}\left[\left(X-\mathrm{E}[X]\right)^{2}\right]

The skewness is the third standardized moment (standardized by the standard deviation \sigma):

\tilde{\mu}_{3}=\mathrm{E}\left[\left(\dfrac{X-\mathrm{E}[X]}{\sigma}\right)^{3}\right]

The kurtosis is the fourth standardized moment:

\tilde{\mu}_{4}=\mathrm{E}\left[\left(\dfrac{X-\mathrm{E}[X]}{\sigma}\right)^{4}\right]

The mean is a measure of the central tendency of a distribution. The standard deviation is a measure of its dispersion (spread). The skewness and kurtosis of a distribution involve its characteristics in its tail or tails. A positive skew means that the distribution’s tail is on the right side of the distribution. All the age distributions analyzed here tend to have a positive skew. High kurtosis means that large outliers are more likely. Kurtosis of greater than 3 suggests that a distribution has a fatter tail than the normal distribution. The skewness and kurtosis become relevant when very old outputs “wake up” during periods of exchange rate volatility to participate in speculative activity, i.e. buying and selling on exchanges.

The fitting function is essentially a minimum discrepancy estimator. This means that the parameters of the parametric probability density function (PDF) are chosen to minimize the distance between the parametric PDF and the PDF formed by the data (the empirical PDF), for some specified metric of “distance”.

There are several measures of distance that could be used. For the purpose of this exploration of output age distribution forecasting, the distance metric to be minimized will be the total linear sum of the mass of the estimated parametric PDF that falls below the empirical PDF. With the “loss function” specified this way, the optimization algorithm attempts to minimize the probability that the real spends are much more likely to come from a block of a particular age compared to a potential decoy. Ideally, the decoy distribution would be identical to the real spend distribution, but parametric PDFs are not flexible enough to perfectly match an empirical PDF.

Define f_{S}(x) as the empirical spent output age distribution at block age x and f_{D}(x;\boldsymbol{\beta}) as a potential “decoy” distribution with some parameter vector \boldsymbol{\beta} (with the transparent blockchains presented here, there is no actual decoy mechanism of course). Then for each week of spent output age data the parameter vector \boldsymbol{\beta} can be chosen to minimize this quantity:

L(\boldsymbol{\beta})=\sum_{i\in\{x_{i}:f_{D}(x_{i};\boldsymbol{\beta})<f_{S}(x_{i})\}}^{N}f_{S}(x_{i})-f_{D}(x_{i};\boldsymbol{\beta})

That is, minimize the sum of the difference between the real spend age distribution for blocks x_{i} where the decoy distribution is less than the real spend age distribution. The minimization is performed by computer numerical minimization methods similar to gradient descent.

The two candidate “decoy” parametric PDFs under consideration in this exercise are the Log-gamma (lgamma) distribution and the Right-Pareto Log-normal (rpln) distribution. The lgamma distribution, with two parameters, was used in Moser et al. (2018) to suggest a decoy distribution that was later incorporated into Monero’s reference wallet software. The rpln distribution is a more flexible distribution, with three parameters. The PDFs of these two distributions are:

f_{lgamma}(x)=\dfrac{ratelog^{shapelog}}{\Gamma(shapelog)}\cdot\dfrac{(\ln x)^{shapelog-1}}{x^{ratelog+1}}

with ratelog>0 and shapelog>0 and where \Gamma is the gamma function and

f_{rpln}(x)=shape2\cdot x^{-shape2-1}e^{shape2\cdot meanlog+\frac{shape2^{2}sdlog^{2}}{2}}\Phi(\frac{\ln x-meanlog-shape2\cdot sdlog^{2}}{sdlog})

with shape2>0 and sdlog>0 and where \Phi is the cumulative distribution function of the standard Normal distribution.