---
title: "STAT 436 / 536 - Lecture 4"
date: September 10, 2018
output: pdf_document
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
knitr::opts_chunk$set(warning = FALSE)
knitr::opts_chunk$set(message = FALSE)
```

### 1.  Correlation Structure and Motivation
- We have seen how to decompose a time series model to remove a trend and seasonal components. So what remains?
    -  \vfill
    -  \vfill
    -  \vfill
    
### 2. Expectation, Variance, and Auto Correlation

- The expected value, or expectation, or a random variable is defined as:  \vfill
- In the context of annual measurements of Nile River flows, what is an interpretation of the expectation?  \vfill
- A times series model is stationary in the mean if:  \vfill
- What is $E[(x-\mu_x)(y-\mu_y)]$?
    - \vfill
    - \vfill
    
\newpage
    
##### Sample Based Moments
- Sample based calculations can be made in R mean using `mean(x)`, variance `var(x)`, covariance `cov(x,y)`, and correlation `cor(x,y)`. \vfill
- Using a dataset containing information on housing sales in King County, WA [http://math.montana.edu/ahoegh/teaching/stat408/datasets/SeattleHousing.csv](http://math.montana.edu/ahoegh/teaching/stat408/datasets/SeattleHousing.csv), compute the following quantities: 
```{r}
Seattle <- read.csv('http://math.montana.edu/ahoegh/teaching/stat408/datasets/SeattleHousing.csv')
```
\vfill
  - mean sales price 
 
  \vfill
  - standard deviation of sales price 
 
  \vfill
  - correlation between sales price and square footage (sqft_living)
 
 - Intepret the three quantities above. 
 
 \vfill

\newpage

### 3. Autocorrelation and the Correlogram
 
- In addition to mean and variance, the serial correlation (or autocorrelation) is an important element in time series modeling. 
\vfill
- Autocovariance is defined as:  \vfill
- A time series model is second-order stationary if  \vfill
- The autocorrelation function is defined as  \vfill
- Similar to variance calculations, the sample autocovariance and autocorrelation functions can be computed:
  - sample acvf: \vfill
  - sample acf:  \vfill
- Note these properties require a stationary process, hence trends and cyclical patterns need to be removed when considering correlated random noise. 
\vfill

##### Simulating Correlated Time Series Data
- As was mentioned earlier in class, we can think of time series modeling as similar to mixed models. Specifically, there is a specific correlation structure defined for each type of model.
\vfill
\newpage
- First construct a covariance matrix between all of the observations.
```{r}
set.seed(09062018)
time.pts <- 200
auto.corr <- 0.9
evolution.matrix <- diag(time.pts)
for (column in 1:time.pts){
  evolution.matrix[,column] <- auto.corr ^ abs((1:time.pts) - column)
}
library(knitr) # for kable
kable(evolution.matrix[1:5,1:5],caption = "Covariance matrix for first 5 time points")
```
\vfill
- Simulate a vector of correlated normal random variables.
```{r}
library(mnormt) # for rmnorm
Y <- as.ts(rmnorm(n=1, mean=0, varcov=evolution.matrix))
```
\vfill
- Create time series figure
```{r, fig.align='center',fig.width=6, fig.height=4}
library(ggfortify) # for autoplot
autoplot(Y) + ggtitle(expression(paste('Simulated Time Series where ', rho[1], '= 0.9')))
```
\newpage

##### Exercises:
1. Is the simulated time series stationary in the mean, why or why not? \vfill
2. What is $\gamma_k$ + \vfill
3. What is $\rho_k$ = \vfill
4. Change the `auto.corr` variable, rerun the simulation and describe how your figures are different.
    a. `auto.corr = 0`\vfill
    b. `auto.corr = .5`\vfill
    c. `auto.corr = -.9`\vfill
5. (536) Adapt the code to include a trend and seasonal cycle in addition to the serial correlated random innovations. \vfill

\newpage

- A useful tool for identifying autocorrelation structure in a time series dataset is the correlogram. The command for this in R is `acf()`.
```{r, fig.align='center',fig.height=4, fig.width=6}
acf(Y)
```
\vfill
- Correlograms have the following properties:

    - The  \vfill
    - The lag 0 autocorrelation is always 1 and is include for comparison purposes. \vfill
    - If $\rho_k$ = 0, then the sampling distribution of $r_k$ is (approximately) normal with mean -1/n and variance of 1/n.  \vfill
- It is important to have a stationary time series that does not include deterministic signals, such as a trend or cycle. \vfill

##### Airline Passenger Example Section 2.3.2

Load Data and Decompose Time Series
```{r}
data("AirPassengers")
AP.decomp <- decompose(AirPassengers, 'additive')
str(AP.decomp)
```

\newpage
ACF Plot with Air Passengers Data (Covariance)
```{r, fig.align='center',fig.height=4, fig.width=6}
par(mfcol=(c(1,2)))
acf(AirPassengers, type = 'covariance'); acf(AirPassengers)
```
\vfill

ACF Plot on Decomposed Random Component (Covariance)
```{r, fig.align='center',fig.height=4, fig.width=6}
#exclude NA's
random.AP <- AP.decomp$random[!is.na(AP.decomp$random)]
par(mfcol=(c(1,2)))
acf(random.AP, type='covariance'); acf(random.AP)
```