---
title: "HW 6"
format: pdf
editor: source
editor_options: 
  chunk_output_type: console
---

```{r}
#| message: false
#| echo: false
library(tidyverse)
library(spatstat)
```

## Question 1 

This question continues the exercise at the end of activity 8 / start of activity 9. Consider adding another term to our working exponential covariance function such that

$$\sigma_{i,j} = \sigma^2 \exp \left(- d_{ij} /\phi \right) + \tau^2$$

### 1. (2 points)

Create a plot that simulates a surface with $\sigma^2 = 1$ and $\tau^2 = 0$. Use a high resolution grid to visualize this surface.

### 2. (2 points)

Create a plot that simulates a surface with $\sigma^2 = 0$ and $\tau^2 = 1$. Use a high resolution grid to visualize this surface.

### 3. (2 points)

Create a plot that simulates a surface with $\phi = 0$ and $\tau^2 = 1$. Use a high resolution grid to visualize this surface.


### 4. (3 points)

Discuss the differences in the plots in parts 1.1, 1.2, and 1.3. How do the parameters contribute to those differences?

### 5. (3 points)

Broadly interpret how $\sigma^2$, $\tau^2$, and $\phi$ impact the covariance structure in the spatial surface.


Now we have introduced two new parameters into this function. What do you suppose that they do?

- $\sigma^2$: controls the magnitude of the covariance.

\vfill

- $\phi$: controls the range of the spatial correlation


## Question 2 


Building on the model framework from the previous question, simulate data from the following model

$$y \sim N(\mu, \Sigma)$$
$$\sigma_{i,j} = \sigma^2 \exp \left(- d_{ij} /\phi \right) + \tau^2$$
where $\sigma^2 = 5$, $\tau^2 = 1$, you can choose $\phi$.

### 1. (4 points)

Visualize a draw from this spatial random surface with $n = 50$ points.

### 2. (4 points)

Use STAN code to estimate the model parameters. How do your results match your expectations?