Questions tagged [fractal]

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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 ...
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3 votes
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190 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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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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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, ...
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308 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 $[\...
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4 votes
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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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576 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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330 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?
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2 votes
1 answer
582 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 $q \in \mathbb{N}$, ...
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13 votes
2 answers
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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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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 ...
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1 answer
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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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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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1 vote
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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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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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  • 131
2 votes
0 answers
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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4 votes
2 answers
3k views

Hurst exponent calculation methodology

I am looking for the Hurst exponent calculation methodology. Please suggest online materials / methodology papers.
7 votes
2 answers
618 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 ...
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12 votes
1 answer
732 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 (...