In siliusmv/tma4250ex2:

\section{Problem 3: Clustered event spatial variables}

a)

The Neumann-Scott cluster model is a model for describing a clustered event RF. Mother-locations are distributed according to a stationary Poisson process with intensity $\lambda_M$ inside an area $\mathbf{D}$. Centered at each mother-location, a set of child-events are independently distributed with count pdf $p\left(k^c\right)$ and intensity pdf $p\left(\bm{x} | \bm{x}_j^M\right)$, where $\bm{x}_j^M$ is mother-location number $j$. For the chosen version of the model, the count pdf $p\left(k^c\right)$ is a Poisson distribution with intensity $\lambda_c$, and the intensity $p\left(\bm{x} | \bm{x}_j^M\right)$ is a normal distribution centered at $\bm{x}_j^M$ with covariance matrix $\sigma_c^2 \mathbf{I}$. All children-events that are located outside $\mathbf{D}$ must be removed. Only the children-events are counted as events in the resulting model.

This gives the pdf

\begin{equation} \label{eq:mother-child} \begin{split} \mathbf{X}\mathbf{D} &\sim p\left(\bm{x}{ji}; j = 1, 2, ..., k_\mathbf{D}^M; i = 1, 2, ..., k_{Dj}^c\right) \ &= \prod_{j = 1}^{k_\mathbf{D}^M} \left(\left[\prod_{i = 1}^{k_{Dj}^c} \frac{I\left(\bm{x}{ji} \in \mathbf{D}\right) p\left(\bm{x}{ji} | \bm{x}j^M\right)}{P\left(\mathbf{D} | \bm{x}_j^M\right)} \right] \times p\left(k{Dj}^c | k_j^c, \bm{x}j^M\right) p\left(k_j^c\right) \times \frac{1}{|\mathbf{D}|^{k\mathbf{D}^M}} p\left(k_\mathbf{D}^M\right) \right). \end{split} \end{equation}

The pdf in \eqref{eq:mother-child} is divided into three parts. The first factor explains the probability of the location of a child-event given its mother-location. It is equal to zero when a child-event is outside $\mathbf{D}$. The second factor explains the probability of $k_{Dj}^c$ child-events happening inside $\mathbf{D}$ given that the realisation from the count pdf $p(k^c)$ is $k_j^c$. The last factor is the pdf for the number and placement of the mother-locations.

The Neumann-Scott event RF model in not analytically tractable, but realisations are simple to generate. First, the number of mother-locations is generated, and the locations are distributed uniformly on $\mathbf{D}$. Then, for each mother location, the number of child-events are generated, and the child-events are distributed according to their intensity pdf's. Lastly, all child-events outside $\mathbf{D}$ are removed.

By studying the redwood tree data in figure \@ref(fig:processes), we try to fit the Neumann-Scott model using the parameters $\lambda_M = 12$, $\lambda_c = 5$, $\sigma_c^2 = 0.05^2$. 100 realisations of the model is simulated, and 0.9-confidence intervals for the L-function is displayed along with the empirical L-function for the redwood tree data. The result can be seen in figure \@ref(fig:mother-child).

(ref:mother-child) The left figure shows an example of a simulated Neumann-Scott model with parameters $\lambda_M = 12$, $\lambda_c = 5$, $\sigma_c^2 = 0.05^2$. The right figure displays the empirical L-function for the redwood data along with 0.9-confidence intervals for the L-function. These are made from simulation of 100 Neumann-Scott event RF models with the same parameters.

############
## Problem 3

mothers <- 12
children <- 5
sigma <- 0.05
area <- c(0, 1, 0, 1)

cluster_args <- list(lambda_m = mothers, lambda_c = children, sigma = sigma, area = area)


data <- clusterMC(cluster_args)

plotProcess(data = data,
            jitter = TRUE,
            title = "Redwood trees")


testModel(data = red_data,
           simulatePois = clusterMC,
           args = cluster_args,
           title = "Redwood trees")

The empirical L-function for the redwood point process fits well within the confidence intervals from the Neumann-Scott model. This indicates that the data can be explained using such a model.

siliusmv/tma4250ex2 documentation built on May 26, 2019, 4:32 a.m.