Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

Questions tagged [fractal]

19 questions
13
votes
2 answers

What are the known, existing practical applications of chaos theory in data mining?

While casually reading some mass market works on chaos theory over the last few years I began to wonder how various aspects of it could be applied to data mining and related fields, like neural nets, pattern recognition, uncertainty management, etc.…
SQLServerSteve
  • 1,121
  • 1
  • 13
  • 34
12
votes
1 answer

Statistics based on fractal mathematics

I am looking for books / textbooks on statistics based on fractal mathematics. I know it is not a very well known area and it is rather difficult to find good literature. Any suggestions are welcome (books, textbooks, online materials).
Eduardas
  • 2,239
  • 4
  • 23
  • 22
7
votes
2 answers

Fractal alternative to correlation

I am looking for a fractal-based statistical measure which could be used as alternative to correlation between two variables (I know that hurst exponent can be used for auto-correlation). Is anyone aware of such measures?
Eduardas
  • 2,239
  • 4
  • 23
  • 22
5
votes
0 answers

Why is generating fractional Brownian motion (fBm) so complicated?

An fBm is characterized by a power spectrum $P(f) = Cf^{-(2H + 1)}$ with $0 < H < 1$ being the Hurst parameter. Why can't I just take the square root of the power spectrum $P(f) = Cf^{-\alpha}$, multiply with $e^{i\theta_n}$ ($\theta_n$ being $N/2$…
StrangeAttractor
  • 181
  • 3
4
votes
2 answers

Hurst exponent calculation methodology

I am looking for the Hurst exponent calculation methodology. Please suggest online materials / methodology papers.
Eduardas
  • 2,239
  • 4
  • 23
  • 22
4
votes
0 answers

Hausdorff (fractal) Dimension of a Stochastic Process

It is well known that Brownian motion (BM) has a Hausdorff dimension (ie fractal dimension) of 2, for topological dimension >= 2. In other words, BM always "behaves like" a plane surface, no matter the dimensionality of the BM process itself (so…
Simone
  • 41
  • 1
3
votes
0 answers

Interpretation of box-counting method from R

I tried to calculate the fractal dimension of a dataset using the box-counting method with R programming. I used two packages: The first one is fractaldim, specifically, the function fd.estim.boxcount. The second one is Rdimtools, particularly the…
jeza
  • 1,527
  • 2
  • 16
  • 37
2
votes
0 answers

Determining the roughness of a multidimensional optimization surface

Is there a way to determine the roughness of an n-dimensional optimization surface (n > 3)? Preferably a method that uses measures from fractal geometry/chaos theory...
classifire
  • 121
  • 1
2
votes
0 answers

How to simulate anomalous diffusion of a 1D point like particle?

I want to simulate 3 types of diffusion processes: normal diffusion $[\langle x^2(t)\rangle \propto t ]$. subdiffusion $[\langle x^2(t)\rangle \propto t^\alpha ; \alpha<1 ]$ superdiffusion $[\langle x^2(t)\rangle \propto t^\alpha ; \alpha>1]$ I…
0x90
  • 687
  • 1
  • 5
  • 17
2
votes
1 answer

Generalized Hurst Exponent - What value to specify for $\tau_{\max}$?

Consider a time series $X: S \to \mathbb{R}$, where $S := \{\nu, 2\nu, 3\nu, \ldots T\}$, and $T$ is a multiple of $\nu > 0$. For each $\tau \in (0, \tau_{\max}] \cap S$ and $q \in \mathbb{N}$, define $$K_q(\tau) := \frac{\langle \vert X(t + \tau)…
user3294195
  • 723
  • 1
  • 4
  • 16
1
vote
0 answers

An Intuitive Explanation of Multifractality in Financial Time Series

Can anyone please give an intuitive explanation of multifractality in financial time series? Most definitions I came across are either purely mathematical or not in relation to finance. As for the research papers that analyze multifractality in…
Blg Khalil
  • 608
  • 5
  • 12
1
vote
0 answers

Assumptions for Hurst exponent calculation

Are there any general assumptions for the calculation of the Hurst exponent? Does the signal need to be stationary, for example? Does it depend on the method? What about the length of the time series, is longer better? I am interested in…
sviter
  • 131
  • 1
  • 10
1
vote
0 answers

Fractional Brownian motion with additional terms

Fractional Brownian motion seems fairly a straight forward random process with a kind of auto-correlation function, $$ \mathbb{E}\left[ B^H_t B^H_s \right] = \frac{1}{2} \left( |t|^{2H} + |s|^{2H} - |t-s|^{2H} \right) $$ But this can easily be…
ignorance
  • 183
  • 6
1
vote
0 answers

Fractal dimension of time series

What does fractal dimension values of 1.02 and 1.6 indicate? How is fractional differencing related to fractal dimension?
user6460588
  • 103
  • 11
1
vote
0 answers

Intensity of fractional Gaussian noise

I try to understand the 2nd formula stated in the picture. It yields the intensity/volatility of an fGN process. It depends solely on H ?? Why is that? How is this volatility different from simply calculating standard deviation of an empirical or…
user3817704
  • 323
  • 1
  • 8
1
2