In antiphon/sseg: Compute various multitype-summaries for multitype point patterns

\newcommand{\R}{{\bf R}} \newcommand{\P}{{\bf P}} \newcommand{\E}{{\bf E}} \newcommand{\x}{{\bf x}} \newcommand{\simr}{\stackrel{r}{\sim}}

Multitype point pattern

Feel free to replace "species" in the place of "type" in the following.

Write $X=(X_1, ..., X_S)$ for the multi-type point pattern where each component $X_i$ is a univariate point pattern, $i=1,...,S$, and $S$ is the number of types. We assume each component (and therefore the joint) pattern is observed in an observation window $W\in \R^d$.

For $X$ with $S$ components, write $t(x)$ for the type of point $x\in X$. Then we can write equivalently to the above that $X={(x,t(x))\in W\times {1,...,S}}$.

Intensity

Write $X_i(A):=#{X_i\cap A}$ for the number of points of $X_i$ that are in some bounded set $A\subset\R^d$. We assume the each component process $X_i$ fulfills

$$\E X_i(A) := \int_A \lambda_i(x)dx$$

were $\lambda_i(x)\ge 0$ is called the intensity function. The total intensity of $X$ is $\lambda(x) := \lambda_X(x) = \sum_i \lambda_i(x)$.

The process $X_i$ is called stationary if $X_i$ and $X_i+x$ for any $x$ have the same distribution. For stationary processes the intensity is a constant, $\lambda_i(x)\equiv\lambda_i>0$ for all $x$.

The multi-type process $X$ is called stationary if all its components are stationary.

Write also $p_i(u)=\lambda_i(u)/\lambda(u)$ for the relative intensities.

Individual species-area-relationship ISAR

The original ISAR

Assuming $X$ is stationary, the original definition of ISAR by @Wiegand2007 can be written as

[I_i(r) := \E_{o,i}\sum_{j\neq i}1(X_j(b(o,r))>0) ]

where $b(o,r)$ is the origin-center disc of radius $r$, and the expectation is over the distribution of $X_i$ conditional on the event ${o\in X_i}$. Heuristically, $I_i(r)$ is the expected count of other types inside a disc of radius $r$ around a typical member of type $i$. Two other forms are handy:

[I_i(r)=\sum_{j\neq i} [1-\P_{o,i}(X_j(b(o,r))=0)] = \sum_{j\neq i} D_{ij}(r)]

where $D_{ij}(r)$ is the nearest neighbour distance distribution function from type $i$ to type $j$. The estimator for ISAR is given by the estimators of $D_{ij}'s$, implemented as the function Gcross in spatstat.

Inhomogeneous ISAR

Assume that for each $i=1,...,S$ the sub-processes $X_i$ have a locally finite intensity function $\lambda_i(u), u\in W$ which is bounded away from zero such that $\inf \lambda_i(u) = \bar\lambda_i > 0$. Assume further that $X_i$ are intensity-reweighted moment stationary point processes, that is to say all factorial moment measures are translation invariant.

We can then define the inhomogeneous ISAR (iISAR) as

[I_i(r) := \sum_{j\neq i}[1-\P_{o,i}(X_j(b(o,r))=0)]]

where we use

$$\P_{o,i}(X_j(b(o,r))=E_{o,i}\prod_{x\in X_j}\left[1-\frac{\bar\lambda 1(x \in b(o,r))}{\lambda(x)} \right]$$

The border corrected estimator is given by

$$\hat D_{ij}(r) = \frac{1}{\x_i(\cap W_r)}\sum_{x\in \x_i\cap W_r}\prod_{y\in \x_j\cap b(x,r)\cap W}\left[1-\frac{\bar\lambda_j}{\lambda_j(y)}\right]$$

where $W_r=W\ominus b(o,r)$ is the eroded window.

Note that ISAR is a special case of iISAR with constant intensities per type.

Mingling index

Original Mingling index

Define a graph $G(X,r)$ with nodes $X$ and neighbourhood relation $x\simr y$ parameterised by $r$. Two main examples are the geometric graph, where $x\simr y \Leftrightarrow ||x-y||0$, and the $r$-nearest neighbours with $r=1,2,...$.

For stationary $X$ we define the mingling index version 1

$$M^1(r) := \frac{\E_o\sum_{x\in X_{t(o)}} 1[t(o)\neq t(x)] 1[o\simr x]}{\E_o\sum_{x\in X}1[o\simr x]} = 1- \frac{\E_o\sum_{x\in X_{t(o)}} 1[o\simr x]}{\E_o\sum_{x\in X}1[o\simr x]} $$

and version 2

$$M^2(r) := \E_o\frac{\sum_{x\in X} 1[t(o)\neq t(x)]1[o\simr x]}{\sum_{x\in X}1[o\simr x]} = 1- \E_o\frac{\sum_{x\in X_{t(o)}}1[o\simr x]}{\sum_{x\in X}1[o\simr x]}$$

for $r$-nearest neighbour the version are equal since the denominator equals $r$.

Heuristically, the mingling index computes the fraction of neighbours that are of different type than the focal individual.

We define the type-wise mingling index version 1

$$M^1_i(r) := 1- \frac{\E_{o,i}\sum_{x\in X_{i}} 1[o\simr x]}{\E_{o,i}\sum_{x\in X}1[o\simr x]} $$

and version 2

$$M^2_i(r) := 1- \E_{o,i}\frac{\sum_{x\in X_i}1[o\simr x]}{\sum_{x\in X}1[o\simr x]}$$

antiphon/sseg documentation built on May 10, 2019, 12:22 p.m.