From 3b34bae5174bc4718351a4d173e07d35f940d836 Mon Sep 17 00:00:00 2001 From: Kaushal Modi Date: Tue, 28 Nov 2017 10:49:24 -0500 Subject: [PATCH] Add a real-world math-heavy example post Thanks @lsaravia --- test/site/content-org/all-posts.org | 116 ++++++++++++++ .../multifractals-in-ecology-using-r.md | 146 ++++++++++++++++++ .../MultifractalsInR/C3_BoxCounting.png | Bin 0 -> 39550 bytes .../images/MultifractalsInR/C3_Cladoceran.png | Bin 0 -> 114612 bytes .../images/MultifractalsInR/C3_Clouds.png | Bin 0 -> 115394 bytes .../images/MultifractalsInR/fractal-ice.jpg | Bin 0 -> 72401 bytes 6 files changed, 262 insertions(+) create mode 100644 test/site/content/real-examples/multifractals-in-ecology-using-r.md create mode 100644 test/site/static/images/MultifractalsInR/C3_BoxCounting.png create mode 100644 test/site/static/images/MultifractalsInR/C3_Cladoceran.png create mode 100644 test/site/static/images/MultifractalsInR/C3_Clouds.png create mode 100644 test/site/static/images/MultifractalsInR/fractal-ice.jpg diff --git a/test/site/content-org/all-posts.org b/test/site/content-org/all-posts.org index bfc0e62..1568ee0 100644 --- a/test/site/content-org/all-posts.org +++ b/test/site/content-org/all-posts.org @@ -2594,6 +2594,122 @@ auto-prefixed with the section for the current post. :EXPORT_FILE_NAME: alias-multiple-with-section :EXPORT_HUGO_ALIASES: /alias-test/alias-g /alias-test/alias-h :END: +* Real Examples :real_examples: +:PROPERTIES: +:EXPORT_HUGO_SECTION: real-examples +:END: +** DONE Multifractals in ecology using R :math:equations: +CLOSED: [2017-11-28 Tue 10:48] +:PROPERTIES: +:EXPORT_FILE_NAME: multifractals-in-ecology-using-r +:EXPORT_HUGO_CUSTOM_FRONT_MATTER: :author Leonardo A. Saravia +:EXPORT_HUGO_CUSTOM_FRONT_MATTER+: :source https://github.com/lsaravia/MultifractalsInR/blob/master/Curso3.md +:END: +- Disclaimer :: This post is from the [[https://github.com/lsaravia/MultifractalsInR/blob/master/Curso3.md][link]] posted by GitHub user + [[https://github.com/lsaravia][*lsaravia*]] in [[https://github.com/gohugoio/hugo/issues/234#issuecomment-347532166][this comment]]. All credits for this post + go to the original author. +----- +[[file:/images/MultifractalsInR/fractal-ice.jpg]] +*** Multifractals +- Many natural systems cannot be characterized by a single number such + as the fractal dimension. Instead an infinite spectrum of dimensions + must be introduced. + [[file:/images/MultifractalsInR/C3_Clouds.png]] +*** Multifractal definition +- Consider a given object $\Omega$, its multifractal nature is + practically determined by covering the system with a set of boxes + $\{B_i(r)\}$ with $(i=1,..., N(r))$ of side lenght $r$ +- These boxes are nonoverlaping and such that + + $$\Omega = \bigcup_{i=1}^{N(r)} B_i(r)$$ + + This is the box-counting method but now a measure $\mu(B_n)$ for each + box is computed. This measure corresponds to the total population or + biomass contained in $B_n$, in general will scale as: + + $$\mu(B_n) \propto r^\alpha$$ +*** Box counting +[[file:/images/MultifractalsInR/C3_BoxCounting.png]] +*** The generalized dimensions +- The fractal dimension $D$ already defined is actually one of an + infinite spectrum of so-called correlation dimension of order $q$ or + also called Renyi entropies. + + $$D_q = \lim_{r \to 0} \frac{1}{q-1}\frac{log \left[ \sum_{i=1}^{N(r)}p_i^q \right]}{\log r}$$ + + where $p_i=\mu(B_i)$ and a normalization is assumed: + + $$\sum_{i=1}^{N(r)}p_i=1$$ + +- For $q=0$ we have the familiar definition of fractal dimension. To see + this we replace $q=0$ + + $$D_0 = -\lim_{r \to 0}\frac{N(r)}{\log r}$$ +*** Generalized dimensions 1 +- It can be shown that the inequality $D_q' \leq D_q$ holds for + $q' \geq q$ +- The sum + + $$M_q(r) = \sum_{i=1}^{N(r)}[\mu(B_i(r))]^q = \sum_{i=1}^{N(r)}p_i^q$$ + + is the so-called moment or partition function of order $q$. +- Varying q allows to measure the non-homogeneity of the pattern. The + moments with larger $q$ will be dominated by the densest boxes. For + $q<0$ will come from small $p_i$'s. +- Alternatively we can think that for $q>0$, $D_q$ reflects the scaling + of the large fluctuations and strong singularities. In contrast, for + $q<0$, $D_q$ reflects the scaling of the small fluctuations and weak + singularities. +*** Exercise +- Calculate the partition function for the center and lower images of + the figure: + [[file:/images/MultifractalsInR/C3_BoxCounting.png]] +*** Two important dimensions +- Two particular cases are $q=1$ and $q=2$. The dimension for $q=1$ is + the Shannon entropy or also called by ecologist the Shannon's index of + diversity. + + $$D_1 = -\lim_{r \to 0}\sum_{i=1}^{N(r)} p_i \log p_i$$ + + and the second is the so-called correlation dimension: + + $$D_2 = -\lim_{r \to 0} \frac{\log \left[ \sum_{i=1}^{N(r)} p_i^2 \right]}{\log r} $$ + + the numerator is the log of the Simpson index. +*** Application +- Salinity stress in the cladoceran Daphniopsis Australis. Behavioral + experiments were conducted on individual males, and their successive + displacements analyzed using the generalized dimension function $D_q$ + and the mass exponent function $\tau_q$ + [[file:/images/MultifractalsInR/C3_Cladoceran.png]] + both functions indicate that the successive displacements of male D. + australis have weaker multifractal properties. This is consistent with + and generalizes previous results showing a decrease in the complexity + of behavioral sequences under stressful conditions for a range of + organisms. +- A shift between multifractal and fractal properties or a change in + multifractal properties, in animal behavior is then suggested as a + potential diagnostic tool to assess animal stress levels and health. +*** Mass exponent and Hurst exponent +- The same information contained in the generalized dimensions can be + expressed using mass exponents: + + $$M_q(r) \propto r^{-\tau_q}$$ + + This is the scaling of the partition function. For monofractals + $\tau_q$ is linear and related to the Hurst exponent: + + $$\tau_q = q H - 1$$ + + For multifractals we have + + $$\tau_q = (q -1) D_q$$ + + Note that for $q=0$, $D_q = \tau_q$ and for $q=1$, $\tau_q=0$ +*** Paper +1. Kellner JR, Asner GP (2009) Convergent structural responses of + tropical forests to diverse disturbance regimes. Ecology Letters 12: + 887--897. doi:10.1111/j.1461-0248.2009.01345.x. * TODO Pre-Draft State :PROPERTIES: :EXPORT_FILE_NAME: draft-state-todo diff --git a/test/site/content/real-examples/multifractals-in-ecology-using-r.md b/test/site/content/real-examples/multifractals-in-ecology-using-r.md new file mode 100644 index 0000000..b1fc81d --- /dev/null +++ b/test/site/content/real-examples/multifractals-in-ecology-using-r.md @@ -0,0 +1,146 @@ ++++ +title = "Multifractals in ecology using R" +date = 2017-11-28T10:48:00-05:00 +tags = ["real-examples", "math", "equations"] +draft = false +author = "Leonardo A. Saravia" +source = "https://github.com/lsaravia/MultifractalsInR/blob/master/Curso3.md" ++++ + +- **Disclaimer:** This post is from the [link](https://github.com/lsaravia/MultifractalsInR/blob/master/Curso3.md) posted by GitHub user + [**lsaravia**](https://github.com/lsaravia) in [this comment](https://github.com/gohugoio/hugo/issues/234#issuecomment-347532166). All credits for this post + go to the original author. + +--- + +{{
}} + + +## Multifractals {#multifractals} + +- Many natural systems cannot be characterized by a single number such + as the fractal dimension. Instead an infinite spectrum of dimensions + must be introduced. + {{
}} + + +## Multifractal definition {#multifractal-definition} + +- Consider a given object \\(\Omega\\), its multifractal nature is + practically determined by covering the system with a set of boxes + \\(\{B\_i(r)\}\\) with \\((i=1,..., N(r))\\) of side lenght \\(r\\) +- These boxes are nonoverlaping and such that + + \\[\Omega = \bigcup\_{i=1}^{N(r)} B\_i(r)\\] + + This is the box-counting method but now a measure \\(\mu(B\_n)\\) for each + box is computed. This measure corresponds to the total population or + biomass contained in \\(B\_n\\), in general will scale as: + + \\[\mu(B\_n) \propto r^\alpha\\] + + +## Box counting {#box-counting} + +{{
}} + + +## The generalized dimensions {#the-generalized-dimensions} + +- The fractal dimension \\(D\\) already defined is actually one of an + infinite spectrum of so-called correlation dimension of order \\(q\\) or + also called Renyi entropies. + + \\[D\_q = \lim\_{r \to 0} \frac{1}{q-1}\frac{log \left[ \sum\_{i=1}^{N(r)}p\_i^q \right]}{\log r}\\] + + where \\(p\_i=\mu(B\_i)\\) and a normalization is assumed: + + \\[\sum\_{i=1}^{N(r)}p\_i=1\\] + +- For \\(q=0\\) we have the familiar definition of fractal dimension. To see + this we replace \\(q=0\\) + + \\[D\_0 = -\lim\_{r \to 0}\frac{N(r)}{\log r}\\] + + +## Generalized dimensions 1 {#generalized-dimensions-1} + +- It can be shown that the inequality \\(D\_q' \leq D\_q\\) holds for + \\(q' \geq q\\) +- The sum + + \\[M\_q(r) = \sum\_{i=1}^{N(r)}[\mu(B\_i(r))]^q = \sum\_{i=1}^{N(r)}p\_i^q\\] + + is the so-called moment or partition function of order \\(q\\). +- Varying q allows to measure the non-homogeneity of the pattern. The + moments with larger \\(q\\) will be dominated by the densest boxes. For + \\(q<0\\) will come from small \\(p\_i\\)'s. +- Alternatively we can think that for \\(q>0\\), \\(D\_q\\) reflects the scaling + of the large fluctuations and strong singularities. In contrast, for + \\(q<0\\), \\(D\_q\\) reflects the scaling of the small fluctuations and weak + singularities. + + +## Exercise {#exercise} + +- Calculate the partition function for the center and lower images of + the figure: + {{
}} + + +## Two important dimensions {#two-important-dimensions} + +- Two particular cases are \\(q=1\\) and \\(q=2\\). The dimension for \\(q=1\\) is + the Shannon entropy or also called by ecologist the Shannon's index of + diversity. + + \\[D\_1 = -\lim\_{r \to 0}\sum\_{i=1}^{N(r)} p\_i \log p\_i\\] + + and the second is the so-called correlation dimension: + + \\[D\_2 = -\lim\_{r \to 0} \frac{\log \left[ \sum\_{i=1}^{N(r)} p\_i^2 \right]}{\log r} \\] + + the numerator is the log of the Simpson index. + + +## Application {#application} + +- Salinity stress in the cladoceran Daphniopsis Australis. Behavioral + experiments were conducted on individual males, and their successive + displacements analyzed using the generalized dimension function \\(D\_q\\) + and the mass exponent function \\(\tau\_q\\) + {{
}} + both functions indicate that the successive displacements of male D. + australis have weaker multifractal properties. This is consistent with + and generalizes previous results showing a decrease in the complexity + of behavioral sequences under stressful conditions for a range of + organisms. +- A shift between multifractal and fractal properties or a change in + multifractal properties, in animal behavior is then suggested as a + potential diagnostic tool to assess animal stress levels and health. + + +## Mass exponent and Hurst exponent {#mass-exponent-and-hurst-exponent} + +- The same information contained in the generalized dimensions can be + expressed using mass exponents: + + \\[M\_q(r) \propto r^{-\tau\_q}\\] + + This is the scaling of the partition function. For monofractals + \\(\tau\_q\\) is linear and related to the Hurst exponent: + + \\[\tau\_q = q H - 1\\] + + For multifractals we have + + \\[\tau\_q = (q -1) D\_q\\] + + Note that for \\(q=0\\), \\(D\_q = \tau\_q\\) and for \\(q=1\\), \\(\tau\_q=0\\) + + +## Paper {#paper} + +1. Kellner JR, Asner GP (2009) Convergent structural responses of + tropical forests to diverse disturbance regimes. 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