# 9 September 15

## 9.1 Announcements

• Go over assignment #2
• Oppportunity to redo the assignment #2
• Why you may want to come to office hours
• Common issues
• New resource

## 9.2 Extreme precipitation in Kansas

• What we will need to learn
• How to use R as a geographic information system
• New general tools from statistics
• Gaussian process
• Metropolis and Metropolis–Hastings algorithms
• Gibbs sampler
• How to use the hierarchical modeling framework to describe Kriging
• Hierarchical Bayesian model vs. “empirical” hierarchical model
• Specialized language used in spatial statistics (e.g., range, nugget, variogram)

## 9.3 Gaussian process

• See bottom of pg. 139 in Wikle et al. (2019)
• A Gaussian process is a probability distribution over functions
• If the function is observed at a finite number of points or “locations,” then the vector of values follows a multivariate normal distribution.

### 9.3.1 Multivariate normal distribution

• Multivariate normal distribution
• $$\boldsymbol{\eta}\sim\text{N}(\mathbf{0},\sigma^{2}\mathbf{R})$$
• Definitions
• Correlation matrix – A positive semi-definite matrix whose elements are the correlation between observations
• Correlation function – A function that describes the correlation between observations
• Example correlation matrices
• Compound symmetry $\mathbf{R}=\left[\begin{array}{cccccc} 1 & 1 & 0 & 0 & 0 & 0\\ 1 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 1 & 0 & 0\\ 0 & 0 & 1 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 1\\ 0 & 0 & 0 & 0 & 1 & 1 \end{array}\right]$
• AR(1) $\mathbf{R(\phi})=\left[\begin{array}{ccccc} 1 & \phi^{1} & \phi^{2} & \cdots & \phi^{n-1}\\ \phi^{1} & 1 & \phi^{1} & \cdots & \phi^{n-2}\\ \phi^{2} & \phi^{1} & 1 & \vdots & \phi^{n-3}\\ \vdots & \vdots & \vdots & \ddots & \ddots\\ \phi^{n-1} & \phi^{n-2} & \phi^{n-3} & \ddots & 1 \end{array}\right]\:$
• Example simulating from $$\boldsymbol{\eta}\sim\text{N}(\mathbf{0},\sigma^{2}\mathbf{R})$$ in R

n <- 200
x <- 1:n
I <- diag(1, n)
sigma2 <- 1

library(MASS)
set.seed(2034)
eta <- mvrnorm(1, rep(0, n), sigma2 * I)
cor(eta[1:(n - 1)], eta[2:n])
## [1] -0.06408623
acf(eta)

par(mfrow = c(2, 1))
par(mar = c(0, 2, 0, 0), oma = c(5.1, 3.1, 2.1, 2.1))
plot(x, eta, typ = "l", xlab = "", xaxt = "n")
abline(a = 0, b = 0)

n <- 200
x <- 1:n  # Must be equally spaced
phi <- 0.7
R <- phi^abs(outer(x, x, "-"))
sigma2 <- 1
set.seed(1330)
eta <- mvrnorm(1, rep(0, n), sigma2 * R)
cor(eta[1:(n - 1)], eta[2:n])
## [1] 0.7062142
plot(x, eta, typ = "l")
abline(a = 0, b = 0)

acf(eta)
• Example correlation functions
• Gaussian correlation function $r_{ij}(\phi)=e^{-\frac{d_{ij}^{2}}{\phi}}$ where $$d_{ij}$$ is the “distance” between locations i and j (note that $$d_{ij}=0$$ for $$i=j$$) and $$r_{ij}(\phi)$$ is the element in the $$i^{\textrm{th}}$$ row and $$j^{\textrm{th}}$$ column of $$\mathbf{R}(\phi)$$.
library(fields)
n <- 200
x <- 1:n
phi <- 40
d <- rdist(x)
R <- exp(-d^2/phi)
sigma2 <- 1

set.seed(4673)
eta <- mvrnorm(1, rep(0, n), sigma2 * R)
plot(x, eta, typ = "l")
abline(a = 0, b = 0)

cor(eta[1:(n - 1)], eta[2:n])
## [1] 0.9717695
• Linear correlation function $r_{ij}(\phi)=\begin{cases} 1-\frac{d_{ij}}{\phi} &\text{if}\ d_{ij}<0\\ 0 &\text{if}\ d_{ij}>0 \end{cases}$
library(fields)
n <- 200
x <- 1:n
phi <- 40
d <- rdist(x)
R <- ifelse(d < phi, 1 - d/phi, 0)
sigma2 <- 1

set.seed(4803)
eta <- mvrnorm(1, rep(0, n), sigma2 * R)
plot(x, eta, typ = "l")
abline(a = 0, b = 0)

cor(eta[1:(n - 1)], eta[2:n])
## [1] 0.9779363