Questions tagged [fractal]

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How to show dfferent Hurst Parameters on the same random process

I would like to show on a graph the effect of different Hurst Parameters on the same random process. I can show this using the package here: https://pypi.org/project/fbm/ By running the below (3 times ...
3
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0answers
149 views

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, ...
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0answers
20 views

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} - |...
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0answers
68 views

Compare long range dependence among non-stationary multivariate time series'

I have 5 non-stationary multivariate time series' and I need to compare the "strength" of long range dependence among them. I have found many papers on long range dependence estimation (parametric, ...
2
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0answers
279 views

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 $[\...
4
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0answers
88 views

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 ...
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0answers
544 views

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?
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1answer
306 views

Test for fractional Brownian motion in Matlab

Is there a test in Matlab if a time series satisfies properties of being a fractional Brownian motion?
1
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1answer
558 views

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 $q \in \mathbb{N}$, ...
13
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2answers
1k views

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, ...
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0answers
34 views

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 ...
0
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1answer
370 views

Quantification of the extent of periodicity in a time series using fractal analyses

I need metrics to quantify the extent of periodicity between of a time series (for comparison with other time series), considering the time series is almost periodic. By almost periodic I mean: if I ...
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0answers
369 views

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 $P(f) = Cf^{-\alpha}$,...
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0answers
91 views

Finding parameters in power spectral density function

I have a process where $z$ varies with $x$. $z$ is well described by a fractal function given by $z(x)= K \Big( \frac{H}{K} \Big)^{D-1} \sum_{n=0}^{\infty} cos\Big(\frac{2 \pi \beta^n x}{K}\Big) \...
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0answers
286 views

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 ...
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0answers
39 views

Determining the roughness of a multidimensional optimization surface [duplicate]

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...
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2answers
3k views

Hurst exponent calculation methodology

I am looking for the Hurst exponent calculation methodology. Please suggest online materials / methodology papers.
7
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2answers
610 views

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 ...
12
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1answer
728 views

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 (...