---
title: "STAT 436 / 536 - Lecture 7: Key"
date: September 26, 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)

```

## Stochastic Models

- Thus far we have seen two approaches for estimating a time series.
    1. The `decompose` function estimates the trend and seasonal patterns for a time series. \vfill
    2. The `HoltWinters` function uses exponentially weighted averages to estimate the mean, trend, and seasonal components. \vfill
    
- When fitting time series models, most of the deterministic features of the time series can be captured in various ways, but regardless of the approach, we still have a  
\vfill
- Sometimes the deterministic features capture the time series behavior, so that the residual error series is 
\vfill
- Otherwise, if the residual error series contains 
\vfill

#### White Noise
- If the time series model is defined for value $y$, then the residual time series can be defined as: 
\vfill
- A time series exhibits white noise if $x_t = w_t$, where $w_t$ are 
\vfill
```{r}
set.seed(09192018)
library(dplyr)
library(ggfortify)
rnorm(100) %>% as.ts() %>% autoplot() + ggtitle('White Noise')
```

\vfill
\newpage
-The second order properties for white noise are: the mean term 
\vfill
- the covariance $\gamma_k(w_t, w_{t+k}) = 0$ 
\vfill
- the correlation $\rho_k(w_t, w_{t+k}) =0$
\vfill
- Q: will the sample correlation necessarily be zero from a simulation?
\vfill

```{r, eval = F}
w <- ts(rnorm(100))
acf.obj <- acf(w)
acf.obj
```
\vfill
\vfill
- When fitting this model, what parameters would we need to estimate? 
\vfill

\newpage


#### Random Walks
- Let $\{x_t\}$ be a time series object, then this is a random walk if...
\vfill
\vfill
- Using back substitution, this series can be written as:
\vfill
\vfill
- The textbook defines **B** 
\vfill
- The second order properties of the random walk are $\mu_x = 0$ and $\gamma_k(t) = t \sigma^2$ (Note this is on HW4).
\vfill
- The autocorrelation $\rho_k(t) = \frac{1}{\sqrt{1+k/t}}$
\vfill
- Note this results in a non-stationary time series as the covariance depends on $t$.
\vfill
- A common approach with a non-stationary time series, such as a random walk, is to take the difference between consecutive time points. This is denoted as 
\vfill
- **Q:** what is the resulting time series after applying the differencing operator to a random walk time series $\{x_t\}$?  \vfill
- Sketch out pseudocode to simulate a random walk. 
\vfill
\newpage
```{r}
time.pts <- 100
random.walk <- rep(0,time.pts)
sigma.w <- 1
for (t in 2:time.pts){
  random.walk[t] <- random.walk[t-1] + rnorm(sigma.w)
}
random.walk %>% as.ts() %>% autoplot() + ggtitle('Simulated Random Walk')
```
\vfill
- ACF plots for random walk and differenced random walk.
```{r, fig.height=4, fig.width=6}
par(mfcol=c(1,2))
acf(random.walk); acf(diff(random.walk))
```
\newpage
- In some situations, a purely random walk model may not be appropriate. Consider the following figure.

```{r, echo = FALSE}
par(mfcol=c(1,1))
time.pts <- 100
random.walk.drift <- rep(0,time.pts)
const <- .5
sigma.w <- 1
for (t in 2:time.pts){
  random.walk.drift[t] <- random.walk.drift[t-1] + rnorm(sigma.w) + const
}
random.walk.drift %>% as.ts() %>% autoplot() + ggtitle('Simulated Random Walk?')
```
\vfill
- This is a random walk
\vfill
\vfill
- The
\vfill
```{r}
drift.est <- random.walk.drift %>% diff() %>% mean() %>% round(digits = 2)
```
where the estimate of $\delta$ is `r drift.est`.
\vfill
- The Holt-Winters function can be used to estimate both of these time series data sets.

**Random Walk**
```{r}
HW.rw <- HoltWinters(random.walk, gamma = FALSE, beta = FALSE)
HW.rw$alpha
random.walk[time.pts]
```
\vfill
\newpage
```{r, eval = F}
rw.pred <- predict(HW.rw, n.ahead = 5, prediction.interval = T);
rw.pred
plot(HW.rw, rw.pred)
```


**Random Walk with Drift**
```{r}
HW.rw.drift <- HoltWinters(random.walk.drift, gamma = FALSE)
HW.rw.drift$alpha; HW.rw.drift$coefficients['b']
rw.drift.pred <- predict(HW.rw.drift, n.ahead = 5, prediction.interval = T)
rw.drift.pred
plot(HW.rw.drift, rw.drift.pred)
```