---
title: "GP Theory Part 2"
output: pdf_document
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
library(tidyverse)
library(knitr)
library(mnormt)
library(plgp)
library(reshape2)
set.seed(02092021)
```


#### Correlation function: more details

Recall the variant of the exponential covariance function that we have previously seen. Where $d$ as the Euclidean distance between $x_1$ and $x_2$, such that $d = \sqrt{(x_i - x_j)^2}$

$$\rho_{i,j} = \exp \left(- d \right)$$

\vfill

Lets view the exponential correlation as a function of distance between the two points.

```{r, echo = F}
dist <- seq(0, 5, by = .1)

tibble(rho = exp(-dist), dist = dist) %>%
  ggplot(aes(y = rho, x = dist)) + geom_line() +
  theme_bw() + ylab(expression(rho))

```

\vfill


\vfill

\newpage

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


\vfill



\vfill

Modify the following code (from last lecture) to gain an intuition about these parameters.
\vfill

```{r}
phi <- 1
sigmasq <- 1
x <- seq(0, 10, by = .1)
n <- length(x)
d <- plgp::distance(x)
eps <- sqrt(.Machine$double.eps) 
H <- exp(-d/phi) + diag(eps, n) 
H[1:3,1:3]
y <- rmnorm(1, rep(0,n),sigmasq * H)
tibble(y = y, x = x) %>% ggplot(aes(y=y, x=x)) +
  geom_line() + theme_bw() + ggtitle('Random realization of a GP with phi = 1 and sigmasq = 1') +
  geom_point(size = .5)
```

\newpage

```{r}
phi <- .1
sigmasq <- 5
H <- exp(-d/phi) + diag(eps, n) 
H[1:3,1:3]
y <- rmnorm(1, rep(0,n),sigmasq * H)
tibble(y = y, x = x) %>% ggplot(aes(y=y, x=x)) +
  geom_line() + theme_bw() + ggtitle('Random realization of a GP with phi = .1 and sigmasq = 5') +
  geom_point(size = .5)
```

\newpage


\vfill

### Geostatistical Data
At last, we will look at simulated 2-d "spatial" data.


##### 1. Sampling Locations

```{r}
num.locations <- 50
coords <- data.frame(x = runif(num.locations), y = runif(num.locations))
coords %>% ggplot(aes(x=x,y=y)) + geom_point() + 
  ggtitle('Hypothetical Sampling Locations') + xlim(0,1) + 
  ylim(0,1) + theme_bw()
```

##### 2. Calculate Distances

```{r, echo = T}
dist.mat <- plgp::distance(coords)
```

##### 3. Define Covariance Function and Set Parameters

Use exponential covariance 

```{r}
sigma.sq <- 1
phi <- .1
Sigma <- sigma.sq * exp(- dist.mat/phi) + diag(eps, num.locations) 
```


##### 4. Sample realization of the process

Note this requires a distributional assumption, we will use the Gaussian distribution

```{r}
Y <- rmnorm(n=1, mean = 0, varcov = Sigma)
```



##### 5. Vizualize Spatial Process

```{r}
coords %>% mutate(Y = Y) %>% ggplot(aes(x=x,y=y)) + geom_point(aes(color=Y), size=2) + 
  ggtitle(label = 'Simulated Spatial Process', 
          subtitle = 'Exponential Covariance: sigma.sq = 1, phi = .1') +  
  xlim(0,1) + ylim(0,1) +   scale_colour_gradient2() + theme_dark()
```

\newpage

Now we can look at more sampling locations

```{r, echo = F}
dim.grid <- 10
grid.coords <- data.frame(x.grid = rep(seq(.05, .95, length.out=dim.grid), dim.grid),
  y.grid = rep(seq(.05, .95, length.out = dim.grid), each = dim.grid))

dist.grid <- distance(grid.coords)

sigma.sq <- 1
phi <- .1
Sigma <- sigma.sq * exp(- dist.grid/phi) + diag(eps, dim.grid ^ 2) 

Y <- rmnorm(n=1, mean = 0, varcov = Sigma)

grid.coords %>% mutate(Y = Y) %>% ggplot(aes(x=x.grid,y=y.grid)) + geom_point(aes(color=Y), size=3) + ggtitle('Simulated Spatial Process', subtitle = 'Exponential Covariance: sigma.sq = 1, phi = .1') + xlim(0,1) + ylim(0,1) +   scale_colour_gradient2() + theme_dark()

```


```{r, echo = F}
dim.grid <- 50
grid.coords <- data.frame(x.grid = rep(seq(.05, .95, length.out=dim.grid), dim.grid),
  y.grid = rep(seq(.05, .95, length.out = dim.grid), each = dim.grid))

dist.grid <- distance(grid.coords)

sigma.sq <- 1
phi <- .1
Sigma <- sigma.sq * exp(- dist.grid/phi) + diag(eps, dim.grid ^ 2) 

Y <- rmnorm(n=1, mean = 0, varcov = Sigma)

grid.coords %>% mutate(Y = Y) %>% ggplot(aes(x=x.grid,y=y.grid)) + geom_point(aes(color=Y), size=3) + ggtitle('Simulated Spatial Process', subtitle = 'Exponential Covariance: sigma.sq = 1, phi = .1') + xlim(0,1) + ylim(0,1) +   scale_colour_gradient2() + theme_dark()
```




How does the spatial process change with:

- another draw with same parameters?
\vfill

- a different value of $\phi$
\vfill

- a different value of $\sigma^2$
\vfill