Exactly self-similar process

• ${\displaystyle X}$ is called an exactly self-similar process if there exists a self-similar parameter ${\displaystyle H}$ such that ${\displaystyle X_{k}^{(m)}}$ has the same distribution as ${\displaystyle X}$. An example of exactly self-similar process with ${\displaystyle H}$ is Fractional Gaussian Noise (FGN) with ${\displaystyle {\frac {1}{2}}<H<1}$.

Definition:Fractional Gaussian Noise (FGN)

${\displaystyle X(t)=B_{H}(t+1)-B_{H}(t),~\forall t\geq 1}$ is called the Fractional Gaussian Noise, where ${\displaystyle B_{H}(\cdot )}$ is a Fractional Brownian motion.^[1]

exactly second order self-similar process

• ${\displaystyle X}$ is called an exactly second order self-similar process if there exists a self-similar parameter ${\displaystyle H}$ such that ${\displaystyle X_{k}^{(m)}}$ has the same variance and autocorrelation as ${\displaystyle X}$.

asymptotic second order self-similar process

• ${\displaystyle X}$ is called an asymptotic second order self-similar process with self-similar parameter ${\displaystyle H}$ if ${\displaystyle \gamma ^{(m)}(t)\to {\frac {1}{2}}[(t+1)^{2H}-2t^{2H}+(t-1)^{2H}]}$ as ${\displaystyle m\to \infty }$, ${\displaystyle ~\forall t=1,2,3,\ldots }$

Long-Range-Dependence(LRD)

Suppose ${\displaystyle X(t)}$ be a weakly stationary (2nd-order stationary) process with mean ${\displaystyle \mu }$ and variance ${\displaystyle \sigma ^{2}}$. The Autocorrelation Function (ACF) of lag ${\displaystyle t}$ is given by ${\displaystyle \gamma (t)={\mathrm {cov} (X(h),X(h+t)) \over \sigma ^{2}}={E[(X(h)-\mu )(X(h+t)-\mu )] \over \sigma ^{2}}}$

Definition:

A weakly stationary process is said to be "Long-Range-Dependence" if ${\displaystyle \sum _{t=0}^{\infty }|\gamma (t)|=\infty }$

A process which satisfies ${\displaystyle \gamma (t)\rightarrow t^{-\beta }L(t)}$ as ${\displaystyle t\to \infty }$ is said to have long-range dependence. The spectral density function of long-range dependence follows a power law near the origin. Equivalently to ${\displaystyle \gamma (t)\rightarrow t^{-\beta }L(t)}$, ${\displaystyle X}$ has long-range dependence if the spectral density function of autocorrelation function, ${\displaystyle f_{t}(w)=\sum _{t=0}^{\infty }\gamma (t)e^{iwt}}$, has the form of ${\displaystyle w^{-\gamma }L(w)}$ as ${\displaystyle w\to 0}$ where ${\displaystyle 0<\gamma <1}$, ${\displaystyle L}$ is slowly varying at 0.

also see

Slowly decaying variances

${\displaystyle X^{(m)}={\frac {1}{m}}(X_{1}+\cdot \cdot \cdot +X_{m})}$
When an autocorrelation function of a self-similar process satisfies ${\displaystyle \gamma (t)\rightarrow t^{-\beta }L(t)}$ as ${\displaystyle t\to \infty }$, that means it also satisfies ${\displaystyle Var(X^{(m)})\to am^{-\beta }}$ as ${\displaystyle m\to \infty }$, where ${\displaystyle a}$ is a finite positive constant independent of m, and 0<β<1.

R/S analysis

Assume that the underlying process ${\displaystyle X}$ is Fractional Gaussian Noise. Consider the series ${\displaystyle X_{(1)},\ldots ,X_{(n)}}$, and let ${\displaystyle Y_{(n)}=\sum _{i=1}^{n}X_{(i)}}$.

The sample variance of ${\displaystyle X_{(i)}}$ is ${\displaystyle S^{2}(n)={\frac {1}{n}}\sum _{i=1}^{n}X_{(i)}^{2}-({\frac {1}{n}})^{2}Y_{n}^{2}}$

Definition:R/S statistic

${\displaystyle {\frac {R}{S}}(n)={\frac {1}{S(n)}}[\max _{0\leq t\leq n}(Y_{t}-{\frac {t}{n}}Y_{n})-\min _{0\leq t\leq n}(Y_{t}-{\frac {t}{n}}Y_{n})]}$

If ${\displaystyle X_{(i)}}$ is FGN, then ${\displaystyle E({\frac {R}{S}}(n))\to C_{H}\times n^{H}}$
Consider fitting a regression model : ${\displaystyle \log {\frac {R}{S}}(n)=\log(C_{H})+H\log(n)+\epsilon _{n}}$, where ${\displaystyle \epsilon _{n}\thicksim N(0,\sigma ^{2})}$
In particular for a time series of length ${\displaystyle N}$ divide the time series data into ${\displaystyle k}$ groups each of size ${\displaystyle {\frac {N}{k}}}$, compute ${\displaystyle {\frac {R}{S}}(n)}$ for each group.
Thus for each n we have ${\displaystyle k}$ pairs of data (${\displaystyle \log(n),\log {\frac {R}{S}}(n)}$).There are ${\displaystyle k}$ points for each ${\displaystyle n}$, so we can fit a regression model to estimate ${\displaystyle H}$ more accurately. If the slope of the regression line is between 0.5~1, it is a self-similar process.

Variance-time plot

Variance of the sample mean is given by ${\displaystyle Var({\bar {X}}_{n})\to cn^{2H-2},~\forall c>0}$.
For estimating H, calculate sample means ${\displaystyle {\bar {X}}_{1},{\bar {X}}_{2},\cdots ,{\bar {X}}_{m_{k}}}$ for ${\displaystyle m_{k}}$ sub-series of length ${\displaystyle k}$.
Overall mean can be given by ${\displaystyle {\bar {X}}(k)={\frac {1}{m_{k}}}\sum _{i=1}^{m_{k}}{\bar {X}}_{i}(k)}$, sample variance ${\displaystyle S^{2}(k)={\frac {1}{m_{k}-1}}\sum _{i=1}^{m_{k}}({\bar {X}}_{i}(k)-{\bar {X}}(k))^{2}}$.
The variance-time plots are obtained by plotting ${\displaystyle \log S^{2}(k)}$ against ${\displaystyle \log k}$ and we can fit a simple least square line through the resulting points in the plane ignoring the small values of k.

For large values of ${\displaystyle k}$, the points in the plot are expected to be scattered around a straight line with a negative slope ${\displaystyle 2H-2}$.For short-range dependence or independence among the observations, the slope of the straight line is equal to -1.
Self-similarity can be inferred from the values of the estimated slope which is asymptotically between –1 and 0, and an estimate for the degree of self-similarity is given by ${\displaystyle {\hat {H}}=1+{\frac {1}{2}}(slope).}$

Periodogram-based analysis

Whittle's approximate maximum likelihood estimator (MLE) is applied to solve the Hurst's parameter via the spectral density of ${\displaystyle X}$. It is not only a tool for visualizing the Hurst's parameter, but also a method to do some statistical inference about the parameters via the asymptotic properties of the MLE. In particular, ${\displaystyle X}$ follows a Gaussian process. Let the spectral density of ${\displaystyle X}$, ${\displaystyle f_{x}(w;\theta )=\sigma _{\epsilon }^{2}f_{x}(w;(1,\eta ))}$, where ${\displaystyle \theta =(\sigma _{\epsilon }^{2},\eta )=(\sigma _{\epsilon }^{2},H,\theta _{3},\ldots ,\theta _{k}),H={\frac {\gamma +1}{2}}}$, and ${\displaystyle \theta _{3},\ldots ,\theta _{k}}$ construct a short-range time series autoregression (AR) model, that is ${\displaystyle X_{j}=\sum _{i=1}^{k}\alpha _{i}X_{j-i}+\epsilon _{j}}$, with ${\displaystyle Var(\epsilon _{j})=\sigma _{\epsilon }^{2}}$.

Thus, the Whittle's estimator ${\displaystyle {\hat {\eta }}}$ of ${\displaystyle \eta }$ minimizes the function ${\displaystyle Q(\eta )=\int _{-\pi }^{\pi }{\frac {I(w)}{f(w;(1,\eta ))}}\,dw}$ , where ${\displaystyle I(w)}$ denotes the periodogram of X as ${\displaystyle (2\pi n)^{-1}|\sum _{j=1}^{n}X_{j}e^{iwj}|^{2}}$ and ${\displaystyle {\hat {\sigma }}^{2}=\int _{-\pi }^{\pi }{\frac {I(w)}{f(w;(1,{\hat {\eta }}))}}\,dw}$. These integrations can be assessed by Riemann sum.

Then ${\displaystyle n^{1/2}({\hat {\theta }}-\theta )}$ asymptotically follows a normal distribution if ${\displaystyle X_{j}}$ can be expressed as a form of an infinite moving average model.

To estimate ${\displaystyle H}$, first, one has to calculate this periodogram. Since ${\displaystyle I_{n}(w)}$ is an estimator of the spectral density, a series with long-range dependence should have a periodogram, which is proportional to ${\displaystyle |\lambda |^{1-2H}}$ close to the origin. The periodogram plot is obtained by plotting ${\displaystyle \log(I_{n}(w))}$ against ${\displaystyle \log(w)}$.
Then fitting a regression model of the ${\displaystyle \log(I_{n}(w))}$ on the ${\displaystyle \log(w)}$ should give a slope of ${\displaystyle {\hat {\beta }}}$. The slope of the fitted straight line is also the estimation of ${\displaystyle 1-2H}$. Thus, the estimation ${\displaystyle {\hat {H}}}$ is obtained.

Note:
There are two common problems when we apply the periodogram method. First, if the data does not follow a Gaussian distribution, transformation of the data can solve this kind of problems. Second, the sample spectrum which deviates from the assumed spectral density is another one. An aggregation method is suggested to solve this problem. If ${\displaystyle X}$ is a Gaussian process and the spectral density function of ${\displaystyle X}$ satisfies ${\displaystyle w^{-\gamma }L(w)}$ as ${\displaystyle w\to \infty }$, the function, ${\displaystyle m^{-H}L^{-{\frac {1}{2}}}(m)\sum _{i=(j-1)m+1}^{m}k(X_{i}-E(|X_{i}|)),~j=1,2,\ldots ,[{\tfrac {n}{m}}]}$, converges in distribution to FGN as ${\displaystyle m\to \infty }$.