I wish to expose here the minimal information for understanding the method, I will also provide a few references for those who wish to deepen their understanding of the matter.

Given a time-series with n elements , the R/s statistic is defined as:

Where

is the arithmetic mean

And is the standard deviation from the mean.

With this R/s value, Hurst found a generalization of a result found by Einstein in 1905 (Investigations on the Theory of the Brownian Movement ) as equation (11) (in the cited paper) in the following formula:

Where H is the Hurst exponent.

From there, it is clear that we can obtain an estimation of the Hurst exponent pretty easily from an R/s analysis.

Several sites and articles propose a detailed methodology to implement R/s Analysis, I primarily use the approach exposed in a paper by O. Rose from February 1996:

Estimation of the Hurst Parameter of Long-Range Dependent Time Series

With a slight difference, however, I shall only plot one value of R/s for each value of d, in a manner similar to the following site: Estimating the Hurst Exponent

Anyway, I feel both articles are not very clear in their notations, and I therefore will detail the analysis I wish to implement.

**I- RESCALED RANGE ANALYSIS**

Considering the time series above

We divide the time series into (*) non-overlapping blocks of length

And we fix:

Next we get a new time series

From there, we get the following rescaled range:

With:

And

Taking the mean over , we then get :

Considering equation (1):

We can plot vs for u varying, is then the slope of the regression line which we simply get from the linear least squares method.

Fixing:

and

We get:

(*): and are chosen adequately so that is always an integer.

**II- IMPLEMENTATION**

I implemented this method within an indicator for Metatrader 4 in order to compute a Fractalised Simple Moving Average for FOREX fluctuations. The implementation is not very interesting, partly because the computation time on this platform are not very good, I was therefore constraint to use the R/s method on a very limited number of data.

Anyway, the implementation can be seen on my other blog: Rescaled Range Analysis

]]>

(1)

Then(by taking s=t in (1))

Which gives the standard deviation as (and for the Wiener Brownian Motion, we indeed get a standard deviation of )

]]>

where denotes the standard normal cumulative distribution function:

This result is simply a direct application of Theorem 2 from this paper with

]]>