---
title: "STAT 436 / 536 - Lecture 8"
date: September 28, 2018
output: pdf_document
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
knitr::opts_chunk$set(warning = FALSE)
knitr::opts_chunk$set(message = FALSE)
knitr::opts_chunk$set(fig.align= 'center')
knitr::opts_chunk$set(fig.height= 3)
knitr::opts_chunk$set(fig.width = 5)
```

## Autoregressive Models
- The random walk model can be written more generally as
$$x_t = \alpha x_{t-1} + w_t,$$
where $\alpha = 1$. 
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- If a time series can be written as
$$x_t = \alpha_1 x_{t-1} + \alpha_2 x_{t-2} + \dots + \alpha_p x_{t-p}$$

\vfill
- The AR model can also be written in terms of the backward shift operator **B**.
$$\theta_p(\boldsymbol{B})x_t = (1 - \alpha_1\boldsymbol{B} - \alpha_2\boldsymbol{B}^2 - \dots - \alpha_p\boldsymbol{B}^p)x_t = w_t$$
\vfill
- We have seen that the random walk is a special case of an AR(1) model.
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- The name autoregressive comes from the fact 
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- The prediction (of a point estimate) at time $t$ is given by plugging in point estimates for the $\alpha$ values.
$$\hat{x_t} =  \alpha_1 x_{t-1} + \alpha_2 x_{t-2} + \dots + \alpha_p x_{t-p}$$
\vfill
\newpage
- Stationarity of the AR process can be determined using the $\theta_p(\boldsymbol{B})x_t$ representation of the series, where $\boldsymbol{B}$ is treated as a number. This equation is known as the characteristic equation.
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- The roots of the characteristic equation determine the stationarity of the series. The absolute value of all of the roots must be greater than one for stationarity.
    \vfill
- Consider the AR(1) model, $x_t = \frac{1}{2}x_{t-1} + w_t$
  \vfill
    \vfill
- Consider the AR(2) model, $x_t = x_{t-1} + \frac{1}{4}x_{t-2} + w_t$
   \vfill
   \vfill
- Consider the random walk model $x_t = x_{t-1} + w_t$
    \vfill
    \vfill
- For an AR(1) process, $x_t = \alpha x_{t-1} + w_t$, the second order properties are: 
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- The autocorrelation function for an AR(1) process is 
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\newpage
- Write a function to simulate an AR(1) process
```{r, eval = F}
simAR <- function(alpha, sigma, time.pts){
  # function to simulate and AR process
  # inputs: alpha - the alpha coefficient
  #       : sigma - standard deviation of noise
  #       : time.pts - number of time points
  # outputs: the time series vector as a ts object

  
  
  
  
  
  
  
}
```

```{r, echo = F}
simAR <- function(alpha, sigma, time.pts){
  # function to simulate and AR process
  # inputs: alpha - the alpha coefficient
  #       : sigma - standard deviation of noise
  #       : time.pts - number of time points
  # outputs: the time series vector as a ts object
  x <- rep(0, time.pts)
  for (t in 2:time.pts){
    x[t] <- alpha * x[t-1] + rnorm(1,0,sigma)
  }
  return(ts(x))
}
ar <- simAR(alpha=.8, sigma=1, time.pts = 50)
library(ggfortify)
library(dplyr)
ar %>% autoplot
```
\vfill
- Now let's examine the correlogram
```{r}
set.seed(09192018)
ar.series <- simAR(alpha=.8, sigma=1, time.pts = 500)
acf.ar <- ar.series %>% acf
acf.ar
```
this is fairly close to the empirical correlation term.
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- The autocorrelation will be non-zero for all lags, even though the model for time $t$ only depends on the value from time $t-1$. 
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\vfill
- The
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```{r}
set.seed(09192018)
ar.series <- simAR(alpha=.8, sigma=1, time.pts = 500)
pacf.ar <- ar.series %>% pacf
pacf.ar
```
\vfill
- The
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- The `ar()` function in R can be used to fit AR models and has several useful properties \vfill
- *the* \vfill
- *the* \vfill
- *the* \vfill
    
```{r}
ar.vals <- ar(ar.series, order.max = 2)
ar.vals
predict(ar.vals, n.ahead = 5)
```
\vfill