---
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 


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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 $\boldsymbol{s}$.
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With CSR, $K(d) = \frac{\lambda \pi d^2}{\lambda} = \lambda d^2.$
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To estimate $K(d)$, *we use*
$$\hat{K}(d) = (\hat{\lambda})^{-1}\sum_i \sum_j 1(||\boldsymbol{s}_i - \boldsymbol{s}_j|| \leq d)/n$$
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The empirical $K$ statistic is compared with $\pi d^2$.

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\newpage

###  More `spatstat`

Consider a dataset with medieval grave site information.
```{r, echo = T, message = F}
data(grave)
summary(grave)
plot(grave)
plot(envelope(grave, Kest, verbose = F))
plot(envelope(grave, Fest, verbose = F))
plot(envelope(grave, Gest, verbose = F))
```
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### Estimating the intensity Function 

- With CSR, the intensity function is trivial 
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- __Discuss:__ given a realization of a point process, how could an intensity function be estimated?
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\newpage
Now using the `plot(density(.))` function, plot and interpret the empirical intensity for the grave dataset along with the four synthetic examples.

```{r}
plot(density(grave))
```


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