Add a real-world math-heavy example post

Thanks @lsaravia

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

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.
+
+---
+
+{{<figure src="/images/MultifractalsInR/fractal-ice.jpg">}}
+
+
+## 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.
+    {{<figure src="/images/MultifractalsInR/C3_Clouds.png">}}
+
+
+## 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}
+
+{{<figure src="/images/MultifractalsInR/C3_BoxCounting.png">}}
+
+
+## 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:
+    {{<figure src="/images/MultifractalsInR/C3_BoxCounting.png">}}
+
+
+## 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\\)
+    {{<figure src="/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 {#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. Ecology Letters 12:
+    887--897. <10.1111/j.1461-0248.2009.01345.x>.