Fractals and Stochastic Calculus

R/s Analysis to estimate the Hurst exponent

R/s analysis (or Rescaled Range analysis) was originally devised by Harold Edwin Hurst in its studies of the Nile discharge in 1951. It is a rather simple method, easily implemented in a program and that provides a direct estimation of the Hurst Exponent which is a precious indicator of the state of randomness of a time-series. It is especially interesting in revealing the existence of long-term dependence, which prevents, when it exists, the time-series to be reasonably modelised by a random walk.

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 $X_1, X_2,...,X_n$ , the R/s statistic is defined as:

$R/s(n)= \frac{1}{s}[Max_k{\sum_{i=1}^{k}(x_i-\bar{x})}-Min_k{\sum_{i=1}^{k}(x_i-\bar{x})}]$

Where $1\leq k \leq n,$
$\bar{x}$ is the arithmetic mean
And $s=\sqrt{\frac{1}{n}\sum(x_i-\bar{x})^2}$ 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:

$E[R/s(n)]=Cn^H \text{ as }n\rightarrow\infty \,\,\,\,(1)$

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 $X_j (j=1,..,N)$
We divide the time series into $K^u$ (*) non-overlapping blocks of length $d^u=\frac{N}{K^u}$
And we fix: $t_i=d^u(i-1)+1$

Next we get a new time series $W(i,k):$

$W(i,k)=\sum_{j=1}^{k}{[X_{t_i+j-1}-\frac{1}{d^u}\sum_{v=1}^{d^u}{X_{t_i+v-1}}]},\,k=1,..,d^u$

From there, we get the following rescaled range:
$R/s(i,u)=\frac{R(i,d^u)}{s(i,d^u)}$

With:

$R(i,d^u)=Max\{0,W(i,1),...,W(i,d^u)\}-Min\{0,W(i,1),...,W(i,d^u)\}$
And
$s(i,d^u)=\sqrt{\frac{1}{d^u}\sum_{j=1}^{d^u}[X_{t_i+j-1}-\frac{1}{d^u}\sum_{v=1}^{d^u}X_{t_i+v-1}]^2}$

Taking the mean over $i$ , we then get $R/s(d^u)$ :

$R/s(d^u)=\frac{1}{K^u}\sum_{i=1}^{K^u}R/s(i,d^u)$

Considering equation (1): $log(R/s(d^u))=log(C) +Hlog(d^u)$

We can plot $log(R/s(d^u))$ vs $log(d^u)$ for u varying, $H$ is then the slope of the regression line which we simply get from the linear least squares method.
Fixing:
$x_u=log(d^u)$ and $y_u=log(R/s(d^u))$

We get:

$\boxed{H=\frac{U\sum_{u}x_uy_u -(\sum_{u}x_u)(\sum_{u}y_u)}{U(\sum_{u}x_{u}^2)-(\sum_{u}x_u)^2}} \text{ with }u \text{ varying from }1 \text{ to } U$

(*): $N$ and $K^u$ are chosen adequately so that $d^u$ 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

October 14, 2009 Posted by Jean-Philippe | Uncategorized | Leave a Comment

Standard Deviation of Fractional Brownian Motion

Let $B_{H}(t)$ be the zero-mean Fractional Brownian Motion process of Hurst parameter $H (0 \le H \le 1)$ , such a process is defined as having the following covariance structure:

$E[B_{H}(t)B_{H}(s)]=\frac{1}{2}(|t|^{2H}+|s|^{2H}-|t-s|^{2H})$ (1)

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

$Var(B_{H}(t))=E[(B_{H}(t)-E(B_{H}(t)))^2]=E[B_{H}^{2}(t)]=|t|^{2H}$

Which gives the standard deviation as $|t|^H$ (and for the Wiener Brownian Motion, we indeed get a standard deviation of $\sqrt{t}$ )

May 5, 2009 Posted by Jean-Philippe | Fractional Brownian Motion | Leave a Comment

Maximum of Wiener Brownian Motion

Let $W_0$ be the standard Wiener Brownian Motion process, and $M(t)$ defined as $M(t)=\max_{0 \leq s \leq t} W_{0}(t)$ , we then have the following result: