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Forecasting-3-1---ARMA-Intro.html
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<!DOCTYPE html>
<html lang="" xml:lang="">
<head>
<title>Forecasting-3-1---ARMA-Intro.utf8</title>
<meta charset="utf-8" />
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<textarea id="source">
layout: true
.hheader[<a href="index.html"><svg style="height:0.8em;top:.04em;position:relative;fill:steelblue;" viewBox="0 0 576 512"><path d="M280.37 148.26L96 300.11V464a16 16 0 0 0 16 16l112.06-.29a16 16 0 0 0 15.92-16V368a16 16 0 0 1 16-16h64a16 16 0 0 1 16 16v95.64a16 16 0 0 0 16 16.05L464 480a16 16 0 0 0 16-16V300L295.67 148.26a12.19 12.19 0 0 0-15.3 0zM571.6 251.47L488 182.56V44.05a12 12 0 0 0-12-12h-56a12 12 0 0 0-12 12v72.61L318.47 43a48 48 0 0 0-61 0L4.34 251.47a12 12 0 0 0-1.6 16.9l25.5 31A12 12 0 0 0 45.15 301l235.22-193.74a12.19 12.19 0 0 1 15.3 0L530.9 301a12 12 0 0 0 16.9-1.6l25.5-31a12 12 0 0 0-1.7-16.93z"/></svg></a>]
---
class: center, middle, inverse
# Forecasting Time Series
## ARIMA Models
.futnote[Eli Holmes, UW SAFS]
.citation[[email protected]]
---
## Basic idea
Past values in the time series have information about the current state. An ARMA model models the current state as a linear function of past values:
`$$x_t = \phi_1 x_{t-1} + \phi_2 x_{t-2} + ... + \phi_p x_{t-p} + e_t$$`
.center[
]
---
## ARIMA models (Box-Jenkins models)
You will commonly see ARIMA models referred to as *Box-Jenkins* models. This model has 3 components (p, d, q):
- **AR autoregressive** `\(y_t\)` depends on past values. The AR level is maximum lag `\(p\)`.
`$$x_t = \phi_1 x_{t-1} + \phi_2 x_{t-2} + ... + \phi_p x_{t-p} + e_t$$`
- **I differencing** `\(x_t\)` may be a difference of the observed time series. The number of differences is denoted `\(d\)`. First difference is `\(d=1\)`:
`$$x_t = y_t - y_{t-1}$$`
- **MA moving average** The error `\(e_t\)` can be a sum of a time series of independent random errors. The maximum lag is denoted `\(q\)`.
`$$e_t = \eta_t + \theta_1 \eta_{t-1} + \theta_2 \eta_{t-2} + ... + \theta_q \eta_{t-q},\quad \eta_t \sim N(0, \sigma)$$`
---
## Create some data from an AR(2) Model
`$$x_t = 0.5 x_{t-1} + 0.3 x_{t-2} + e_t$$`
```r
dat = arima.sim(n=1000, model=list(ar=c(.5,.3)))
plot(dat)
abline(h=0, col="red")
```
<img src="Forecasting-3-1---ARMA-Intro_files/figure-html/arima.sim-1.png" style="display: block; margin: auto;" />
---
<img src="Forecasting-3-1---ARMA-Intro_files/figure-html/arimavsrn-1.png" style="display: block; margin: auto;" />
---
## AR(2) is auto-correlated
Plot the data at time `\(t\)` against the data at time `\(t-1\)`
<img src="Forecasting-3-1---ARMA-Intro_files/figure-html/arimavsrncorr-1.png" style="display: block; margin: auto;" />
---
## Auto-correlation function
The auto-correlation function is the correlation between the data at time `\(t\)` and `\(t+1\)`. This is one of the basic diagnostic plots for time series data.
```r
acf(dat[1:50])
```
<img src="Forecasting-3-1---ARMA-Intro_files/figure-html/acf-1.png" style="display: block; margin: auto;" />
---
`cor()` is the correlation function.
```r
cor(dat[2:TT], dat[1:(TT-1)])
```
```
## [1] 0.7401032
```
```r
cor(dat[11:TT], dat[1:(TT-10)])
```
```
## [1] 0.198833
```
<img src="Forecasting-3-1---ARMA-Intro_files/figure-html/acf2-1.png" style="display: block; margin: auto;" />
---
## ACF for independent data
<img src="Forecasting-3-1---ARMA-Intro_files/figure-html/acf.random-1.png" style="display: block; margin: auto;" />
---
## Partial ACF
`$$x_t = 0.5 x_{t-1} + 0.3 x_{t-2} + e_t$$`
```r
pacf(dat)
```
<img src="Forecasting-3-1---ARMA-Intro_files/figure-html/pacf.ar2-1.png" style="display: block; margin: auto;" />
---
## Partial ACF for AR(1)
`$$x_t = 0.5 x_{t-1} + e_t$$`
```r
dat <- arima.sim(TT, model=list(ar=c(.5)))
pacf(dat)
```
<img src="Forecasting-3-1---ARMA-Intro_files/figure-html/pacf.ar3-1.png" style="display: block; margin: auto;" />
---
## Box-Jenkins Method
This refers to a step-by-step process of selecting a forecasting model. You need to go through the steps otherwise you could end up fitting a nonsensical model or using fitting a sensible model with an algorithm that will not work on your data.
A. Model form selection
1. Evaluate stationarity and seasonality
2. Selection of the differencing level (d)
3. Selection of the AR level (p)
4. Selection of the MA level (q)
B. Parameter estimation
C. Model checking
---
.pull-left.left[
## Lectures
1. Stationarity
2. Model Selection
3. Fitting Models (parameter estimation)
4. Forecasting
5. Model Checking
]
.pull-right.left[
## Labs
1. Create and plot data from ARMA processes
2. Test the Greek catch data for stationarity
3. Fit ARMA models to the Greek catch data
4. Create forecasts
5. Test forecast performance
]
</textarea>
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