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@ -28,10 +28,10 @@ source = "https://github.com/lsaravia/MultifractalsInR/blob/master/Curso3.md" |
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- Consider a given object \\(\Omega\\), its multifractal nature is |
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practically determined by covering the system with a set of boxes |
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\\(\{B\_i(r)\}\\) with \\((i=1,..., N(r))\\) of side lenght \\(r\\) |
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\\(\{B\_i( r)\}\\) with \\((i=1,..., N( r))\\) of side length \\(r\\) |
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- These boxes are nonoverlaping and such that |
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\\[\Omega = \bigcup\_{i=1}^{N(r)} B\_i(r)\\] |
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\\[\Omega = \bigcup\_{i=1}^{N( r)} B\_i( r)\\] |
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This is the box-counting method but now a measure \\(\mu(B\_n)\\) for each |
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box is computed. This measure corresponds to the total population or |
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@ -51,16 +51,16 @@ source = "https://github.com/lsaravia/MultifractalsInR/blob/master/Curso3.md" |
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infinite spectrum of so-called correlation dimension of order \\(q\\) or |
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also called Renyi entropies. |
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\\[D\_q = \lim\_{r \to 0} \frac{1}{q-1}\frac{log \left[ \sum\_{i=1}^{N(r)}p\_i^q \right]}{\log r}\\] |
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\\[D\_q = \lim\_{r \to 0} \frac{1}{q-1}\frac{log \left[ \sum\_{i=1}^{N( r)}p\_i^q \right]}{\log r}\\] |
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where \\(p\_i=\mu(B\_i)\\) and a normalization is assumed: |
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\\[\sum\_{i=1}^{N(r)}p\_i=1\\] |
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\\[\sum\_{i=1}^{N( r)}p\_i=1\\] |
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- For \\(q=0\\) we have the familiar definition of fractal dimension. To see |
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this we replace \\(q=0\\) |
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\\[D\_0 = -\lim\_{r \to 0}\frac{N(r)}{\log r}\\] |
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\\[D\_0 = -\lim\_{r \to 0}\frac{N( r)}{\log r}\\] |
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## Generalized dimensions 1 {#generalized-dimensions-1} |
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@ -69,7 +69,7 @@ source = "https://github.com/lsaravia/MultifractalsInR/blob/master/Curso3.md" |
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\\(q' \geq q\\) |
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- The sum |
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\\[M\_q(r) = \sum\_{i=1}^{N(r)}[\mu(B\_i(r))]^q = \sum\_{i=1}^{N(r)}p\_i^q\\] |
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\\[M\_q( r) = \sum\_{i=1}^{N( r)}[\mu(B\_i( r))]^q = \sum\_{i=1}^{N( r)}p\_i^q\\] |
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is the so-called moment or partition function of order \\(q\\). |
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- Varying q allows to measure the non-homogeneity of the pattern. The |
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@ -94,11 +94,11 @@ source = "https://github.com/lsaravia/MultifractalsInR/blob/master/Curso3.md" |
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the Shannon entropy or also called by ecologist the Shannon's index of |
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diversity. |
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\\[D\_1 = -\lim\_{r \to 0}\sum\_{i=1}^{N(r)} p\_i \log p\_i\\] |
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\\[D\_1 = -\lim\_{r \to 0}\sum\_{i=1}^{N( r)} p\_i \log p\_i\\] |
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and the second is the so-called correlation dimension: |
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\\[D\_2 = -\lim\_{r \to 0} \frac{\log \left[ \sum\_{i=1}^{N(r)} p\_i^2 \right]}{\log r} \\] |
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\\[D\_2 = -\lim\_{r \to 0} \frac{\log \left[ \sum\_{i=1}^{N( r)} p\_i^2 \right]}{\log r} \\] |
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the numerator is the log of the Simpson index. |
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@ -125,7 +125,7 @@ source = "https://github.com/lsaravia/MultifractalsInR/blob/master/Curso3.md" |
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- The same information contained in the generalized dimensions can be |
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expressed using mass exponents: |
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\\[M\_q(r) \propto r^{-\tau\_q}\\] |
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\\[M\_q( r) \propto r^{-\tau\_q}\\] |
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This is the scaling of the partition function. For monofractals |
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\\(\tau\_q\\) is linear and related to the Hurst exponent: |
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