---
title: "Week 7: Activity"
format: pdf
editor: source
---

\renewcommand{\vec}[1]{\mathbf{#1}}

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE, fig.height = 4, fig.width = 6)
library(tidyverse) 
library(gridExtra)
library(arm)
library(rstanarm)
library(knitr)
```


### Previous Section's Recap

- Spatial data visualization
- Point pattern data: # of points and locations are random
- Parametric modeling of surface intensity

### This Week's Key Concepts

- Linear Algebra 
- Linear Model Overview
- Simulating Data in R
- Fitting Linear Models in R
- Bayesian Inference

---


### Matrices / Vectors

A matrix is an $n \times p$ object. Matrices are often denoted by a capital letter (or Greek symbol). A few common matrices will be

$$X = \begin{pmatrix}
1 & x_{11} & x_{12}\\
1 & x_{21} & x_{22}\\
\vdots & \vdots & \vdots \\
1 & x_{n1} & x_{n2}
\end{pmatrix}$$

or

$$\Sigma = \begin{pmatrix}
\sigma^2_1 & \sigma_{12} & \cdots &\sigma_{1n}\\
\sigma_{21} & \sigma^2_{2} & \cdots &\sigma_{2n}\\
\vdots & \vdots & \ddots &\vdots\\
\sigma_{n1} & \sigma_{n2} & \ddots &\sigma^2_{n}\\
\end{pmatrix}$$

\vfill

Vectors are essentially one-dimension vectors and will be denoted with an underline. We will assume vectors are $q \times 1$ dimension unless noted with a transpose.

$$\underline{y} = \begin{pmatrix}
  y_1 \\
  y_2 \\
  \vdots\\
  y_n \end{pmatrix}$$
or

$$\underline{\beta} = \begin{pmatrix}
  \beta_0 \\
  \beta_1 \\
  \vdots\\
  \beta_{p-1} \end{pmatrix}$$

\vfill

The transpose operator will be denoted by $\underline{y}^T = \begin{pmatrix} y_1 & y_2 & \cdots & y_n \end{pmatrix}$ or $\underline{y}^{'}$, both of which would result in a $1 \times n$ vector.

\newpage

##### Matrix Multiplication

The most important component in matrix multiplication is tracking dimensions.

\vfill

Consider a simple case with 

$$\hat{\underline{y}} = X \times \hat{\underline{\beta}},$$

where $X$ is a $2 \times 2$ matrix, $\begin{pmatrix} 1 & 2 \\ 1 & -1   \end{pmatrix}$ and $\hat{\underline{\beta}} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}$.

\vfill

Then $$\hat{\underline{y}}= \begin{bmatrix}
1 \times 3 + 2 \times 2 \\
1 \times 3 + (-1)\times 2 \\
\end{bmatrix} = 
\begin{bmatrix}
7 \\
-1 \\
\end{bmatrix}$$

In R, we use `%*%` for matrix multiplication. Compute $\hat{\underline{y}}$ in R.

\vfill

\newpage

### Linear Model Specification

Linear models provide the foundation for most statistical analyses: 

- _regression models(matrix notation): $y = X\underline{\beta} + \underline{\epsilon}$, where $\underline{\epsilon} \sim N(0,\Sigma)$ and $\Sigma = \sigma^2 I$_


One assumption, that is often violated in spatial statistics is that the errors are independently distributed.


### Simulating Data in R

Simulating "fake" data will be a cornerstone of fitting models in this class. Simulate data from a regression model with one covariate. Then create a scatterplot showing that data.


\newpage

### Fitting Linear Models in R

The standard method for fitting linear models in R is with `lm()`. Use `lm()` to fit a model to your simulated data. Do the results meet your expectation?



A Bayesian alternative framework for fitting regression models is to use the `rstanarm` package and the associated `stan_glm()` functionality. Fit your model and compare the results to `lm()`.





##### Motivating Dataset: Washington (DC) housing dataset

Hopefully the connections to statistics are clear, using $X$ and $\beta$, but let's consider a motivating dataset.

\vfill
This dataset contains housing information from Washington, D.C. 

```{r}
DC <- read_csv('https://math.montana.edu/ahoegh/teaching/stat532/data/DC.csv')
```


There are many factors in this dataset that can are useful to predict housing prices.

\vfill
\begin{equation}
y_i = \beta_0 + \beta_1 * x_{SQFT,i} + \beta_2 x_{BEDRM,i} + \epsilon_i,
\end{equation}
where $y_i$ is the sales price of the $i^{th}$ house, $x_{SQFT,i}$ is the living square footage of the $i^{th}$ house, and $x_{BEDRM,i}$ is the number of bedrooms for the $i^{th}$ house. Note this implies that we are treating bedrooms as continuous variables as opposed to categorical.

\vfill

we usually write $\epsilon_i \sim N(0,\sigma^2)$. More on that soon.

\vfill

In R we often write something like: `price ~  LANDUSE + BEDRM`. Use `lm()` or `stan_glm()` to fit this model.



\vfill

This model written in matrix notation:

\begin{equation}
\underline{y} = X \underline{\beta} + \underline{\epsilon},
\end{equation}

where $\underline{y} = \begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{bmatrix}$, $X = \begin{bmatrix} 1 & x_{SQFT,1} & x_{BEDRM,1}\\
1 & x_{SQFT,2} & x_{BEDRM,2} \\ 
\vdots & \vdots & \vdots \\
1 & x_{SQFT,n} & x_{BEDRM,n}
\end{bmatrix}$ ,$\underline{\beta} = \begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \end{bmatrix}$, and $\underline{\epsilon} = \begin{bmatrix} \epsilon_1 \\ \epsilon_2 \\ \vdots \\ \epsilon_n \end{bmatrix}$

\vfill

Now what are the implications of:

$\epsilon_i \sim N(0,\sigma^2)$ or $\underline{\epsilon} \sim N(\underline{0}, \Sigma)$, where $$\Sigma = \sigma \times \begin{bmatrix} 1 & 0 & \cdots & 0\\
0 & 1 & \cdots & 0\\
\vdots & \vdots &\ddots  & \vdots\\
0 & 0 & \cdots & 1\\
\end{bmatrix}$$

\vfill

These are equivalent statements and both imply that $y_i$ and $y_j$ are conditionally independent given X. In other words, after controlling for predictors (ward, square footage), then the price of house $i$ gives us no additional information about price of house $j$.  


\newpage

###### Diagonal Matrices 
The matrix we previously specified, is referred to as a diagonal matrix. This is often denoted with 
$$I_n = \begin{bmatrix} 1 & 0 & \cdots & 0\\
0 & 1 & \cdots & 0\\
\vdots & \vdots &\ddots  & \vdots\\
0 & 0 & \cdots & 1\\
\end{bmatrix},$$ where $n$ is the dimension. Note this is also just shortened to $I$.

\vfill


##### Correlation Matrices

It turns out that $I$ is the special case of what is referred to as a correlation matrix.

\vfill

A correlation matrix is:

- is symmetric

\vfill

- contains ones on the diagonal

\vfill

- contains correlation terms on the off diagonal

\vfill

- is positive definite (more later)

\vfill

Similarly $\Sigma$ is often referred to as a variance - covariance matrix (or just a covariance matrix). A covariance matrix:

- is symmetric

\vfill

- contains variance terms on the diagonal

\vfill

- contains covariance terms on the off diagonal

\vfill

- is positive definite (more later)

\vfill

$$\Sigma = \begin{pmatrix}
\sigma^2_1 & \sigma_{12} & \cdots &\sigma_{1n}\\
\sigma_{21} & \sigma^2_{2} & \cdots &\sigma_{2n}\\
\vdots & \vdots & \ddots &\vdots\\
\sigma_{n1} & \sigma_{n2} & \ddots &\sigma^2_{n}\\
\end{pmatrix}$$

\newpage

##### Multivariate Normal Distribution

Formally, our matrix notation has used a multivariate normal distribution.

\begin{equation}
\underline{y} = X \underline{\beta} + \underline{\epsilon},
\end{equation}

where $\underline{\epsilon} \sim N(\underline{0}, \Sigma),$ which also implies $\underline{y} \sim N(X \underline{\beta}, \Sigma)$.

\vfill

###### Partitioned Matrices

Now consider splitting the sampling units into two partitions such that $\underline{y} = \begin{bmatrix} \underline{y}_1 \\ \underline{y}_2 \end{bmatrix}$. Then,

\vfill

$$\begin{bmatrix}\underline{y}_1 \\ \underline{y}_2 \end{bmatrix} \sim N \left( \begin{bmatrix}X_1 \\ X_2 \end{bmatrix} \underline{\beta}  ,
\begin{bmatrix} \Sigma_{11} & \Sigma_{12} \\ \Sigma_{12} & \Sigma_{22} \end{bmatrix}\right)$$

\vfill

Fundamentally, there is no change to the model, we have just created "groups" by partitioning the model. Do note that $\Sigma_{11}$ is an $n_1 \times n_1$ covariance matrix.

$$\Sigma_{11} = \begin{bmatrix}
\sigma^2_1 & \sigma_{12} & \cdots &\sigma_{1n_1} \\
\sigma_{22} & \sigma^2_2 &   \cdots &\sigma_{2n_1} \\
\sigma_{31} & \sigma_{32} &   \ddots &\sigma_{3n_1} \\
\vdots & \vdots &   \ddots &\vdots \\
\sigma_{n_1 1} & \sigma_{n_1 2} &   \ddots &\sigma^2_{n_1} 
\end{bmatrix}$$

\vfill

However, while $\Sigma_{12} = \Sigma_{21}^T$, neither of these are necessarily symmetric matrices. They also do not have any variance components, but rather just covariance terms. $\Sigma_{12}$ will be an $n_1 \times n_2$ matrix.

$$\Sigma_{11} = \begin{bmatrix}
\sigma_{1,n_1 +1} & \sigma_{1,n_1 + 2} & \cdots &\sigma_{1,n_1 + n_2} \\
\sigma_{2, n_1 + 1} & \sigma_{2, n1 + 2} &   \cdots &\sigma_{2,n_1 + n_2} \\
\vdots & \vdots &   \ddots &\vdots \\
\sigma_{n_1, n_1 + 1} & \sigma_{n_1, n_1 + 2} &   \ddots &\sigma_{n_1, n_1 + n_2} 
\end{bmatrix}$$

\newpage

### Conditional Multivariate Normal

Here is where the magic happens with correlated data. Let $\underline{y_1}|\underline{Y_2}=\underline{y_2}$ be a conditional distribution for $\underline{y_1}$ given that $\underline{y_2}$ is known. Then

\vfill

$$\underline{y_1}|\underline{y_2} \sim N \left( X_1\beta + \Sigma_{12} \Sigma_{22}^{-1}\left(\underline{y_2} - X_2\beta \right), \Sigma_{11} - \Sigma_{12} \Sigma_{22}^{-1} \Sigma_{21} \right)$$
\vfill

Now let's consider a few special cases (in the context of the DC housing dataset.)

\vfill

1. Let $\Sigma = \sigma^2 I$, then the batch of houses in group 1 are conditionally dependent from the houses in group 2 and 

$$\underline{y_1}|\underline{y_2}  \sim N \left( X_1\beta, \Sigma_{11} \right)$$
\vfill

2. Otherwise, let $\Sigma = \sigma^2 H$ and we'll assume $\Sigma_{12}$ has some non-zero elements. Then we have a more precise estimate of $\underline{y_1}$ as $\Sigma_{11} - \Sigma_{12} \Sigma_{22}^{-1} \Sigma_{21}$ will be "less than" $\Sigma_{11}$ (that positive definite thing). Furthermore, the mean will shift such that highly correlated observations such as houses in close proximity (local model structure) will tend to differ from the global mean in the same fashion.

\vfill

First a quick interlude about matrix inversion. The inverse of a symmetric matrix is defined such that $E \times E^{-1} = I$. We can calculate the inverse of a matrix for a $1 \times 1$ matrix, perhaps as $2 \times 2$, matrix and maybe even a $3 \times 3$ matrix. However, beyond that it is quite challenging and time consuming. Furthermore, it is also (relatively) time intensive for your computer.

\vfill

\newpage

3. Let $n_1 = 1$ and $n_2 = 1$, then

$$\begin{bmatrix}y_1 \\ y_2 \end{bmatrix} \sim N \left( \begin{bmatrix}\mu_1 \\ \mu_2 \end{bmatrix}   ,
\begin{bmatrix} \sigma^2_{1} & \sigma_{12} \\ \sigma_{12} & \sigma_{2}^2 \end{bmatrix}\right)$$

and

$$y_1|y_2 \sim N \left( \mu_1 + \sigma_{12} (\sigma_{2}^{2})^{-1}\left(y_2 - \mu_2 \right), \sigma_{1}^2 - \sigma_{12} (\sigma_{2}^{2})^{-1} \sigma_{21} \right)$$
\vfill

Now consider an illustration for a couple simple scenarios. Let $\mu_1 = \mu_2 = 0$ and $\sigma^2_1 = \sigma^2_2 = 1$. Now assume $y_2 = -2$ and we compare the conditional distribution for a few values of $\sigma_{12}$.
\vfill

```{r, echo = F}
mu1 <- 0
mu2 <- 0
sigmasq1 <- sigmasq2 <- 1

dat_seq <- seq(-4,4, by = .01)
n_seq <- length(dat_seq)
tibble(group = rep(c('y2 = -2; sigma12 = 0',
                     'y2 = -2; sigma12 = .2',
                     'y2 = -2; sigma12 = .8'), each = n_seq), 
dens = c(dnorm(dat_seq, mu1 + 0*(1/sigmasq2)*(-2 - mu2),
               sqrt(sigmasq1 - 0 * (1/sigmasq2)*0 )), 
         dnorm(dat_seq, mu1 + .2*(1/sigmasq2)*(-2 - mu2),
               sqrt(sigmasq1 - .2 * (1/sigmasq2)*2 )), 
         dnorm(dat_seq, mu1 + .8*(1/sigmasq2)*(-2 - mu2),
               sqrt(sigmasq1 - .8 * (1/sigmasq2)*.8 ))), 
y = rep(dat_seq, 3)) %>% 
  ggplot(aes(x=y, y = dens, group = group, color = group)) +
  geom_line() + theme_bw()
```


\vfill

One last note, the marginal distributions for any partition $\underline{y_1}$ are quite simple.

$$\underline{y_1} \sim N \left( X_1\beta, \Sigma_{11} \right)$$
or just

$$y_1 \sim N \left( X_1\beta, \sigma^2_{1} \right)$$
if $y_1$ is scalar.
