---
title: "K function"
output: pdf_document
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = FALSE)
knitr::opts_chunk$set(warning = FALSE)
knitr::opts_chunk$set(message = FALSE)
knitr::opts_chunk$set(fig.align= 'center')
knitr::opts_chunk$set(fig.height= 4)
knitr::opts_chunk$set(fig.width = 6)
library(tidyverse)
library(gridExtra)
library(smacpod)
library(spatstat)
set.seed(04132021)
```

### K function

We previously looked at the $F(d)$ and $G(d)$ functions, which corresponded to *CDFs, as a function of distance, for open space and distance between points.*

\vfill

Another interesting feature of a point process is the number of points in a specified area. Consider $E(Num(\boldsymbol{s},d,\boldsymbol{S}))$, the expected number of points in $\delta_d\boldsymbol{s}$, a circle of radius $d$ centered at $\boldsymbol{s}$.
\vfill

*Ripley's K or just the K function, considers the expected number of points within a distance $d$ of an arbitrary point. Formally this is defined for CSR as*
$$K(d) = \frac{E(\text{number of points within d})}{\lambda}$$.
*In other words, this is scaled by $\lambda$*
\vfill

With CSR, $K(d) = \frac{\lambda \pi d^2}{\lambda} = \lambda d^2.$
\vfill

To estimate $K(d)$, *we use*
$$\hat{K}(d) = (\hat{\lambda})^{-1}\sum_i \sum_j 1(||\boldsymbol{s}_i - \boldsymbol{s}_j|| \leq d)/n$$
*where $\hat{\lambda} = n / |\mathcal{D}|$, note the typo in the book.*
\vfill

The empirical $K$ statistic is compared with $\pi d^2$. *For $K > \pi d^2$, the series exhibits clustering, for $K < \pi d^2$ the process exhibits inhibition.*
\vfill

\newpage


Recall the  dataset with medieval grave site information.
```{r, echo = T, message = F}
data(grave)
plot(envelope(grave, Kest, verbose = F))
```
\vfill

