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<!DOCTYPE html>
<html lang="" xml:lang="">
<head>
<title>Forecasting-3-7---ARMA-Seasonal-Models.utf8</title>
<meta charset="utf-8" />
<link rel="stylesheet" href="my-theme.css" type="text/css" />
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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
# Seasonal ARIMA Models
.futnote[Eli Holmes, UW SAFS]
.citation[[email protected]]
---
## Seasonality
Load the chinook salmon data set
```r
load("chinook.RData")
head(chinook)
```
<table class="huxtable" style="border-collapse: collapse; margin-bottom: 2em; margin-top: 2em; width: 38.8888888888889%; margin-left: 0%; margin-right: auto; ">
<col><col><col><col><col><tr>
<td style="vertical-align: top; text-align: right; white-space: nowrap; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0.4pt; padding: 4pt 4pt 4pt 4pt; font-weight: bold;">Year</td>
<td style="vertical-align: top; text-align: left; white-space: nowrap; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0pt; padding: 4pt 4pt 4pt 4pt; font-weight: bold;">Month</td>
<td style="vertical-align: top; text-align: left; white-space: nowrap; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0pt; padding: 4pt 4pt 4pt 4pt; font-weight: bold;">Species</td>
<td style="vertical-align: top; text-align: right; white-space: nowrap; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0pt; padding: 4pt 4pt 4pt 4pt; font-weight: bold;">log.metric.tons</td>
<td style="vertical-align: top; text-align: right; white-space: nowrap; border-style: solid solid solid solid; border-width: 0.4pt 0.4pt 0.4pt 0pt; padding: 4pt 4pt 4pt 4pt; font-weight: bold;">metric.tons</td>
</tr>
<tr>
<td style="vertical-align: top; text-align: right; white-space: nowrap; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0.4pt; padding: 4pt 4pt 4pt 4pt; background-color: rgb(242, 242, 242);">1990</td>
<td style="vertical-align: top; text-align: left; white-space: nowrap; padding: 4pt 4pt 4pt 4pt; background-color: rgb(242, 242, 242);">Jan</td>
<td style="vertical-align: top; text-align: left; white-space: nowrap; padding: 4pt 4pt 4pt 4pt; background-color: rgb(242, 242, 242);">Chinook</td>
<td style="vertical-align: top; text-align: right; white-space: normal; padding: 4pt 4pt 4pt 4pt; background-color: rgb(242, 242, 242);">3.4 </td>
<td style="vertical-align: top; text-align: right; white-space: nowrap; border-style: solid solid solid solid; border-width: 0pt 0.4pt 0pt 0pt; padding: 4pt 4pt 4pt 4pt; background-color: rgb(242, 242, 242);">29.9</td>
</tr>
<tr>
<td style="vertical-align: top; text-align: right; white-space: nowrap; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0.4pt; padding: 4pt 4pt 4pt 4pt;">1990</td>
<td style="vertical-align: top; text-align: left; white-space: nowrap; padding: 4pt 4pt 4pt 4pt;">Feb</td>
<td style="vertical-align: top; text-align: left; white-space: nowrap; padding: 4pt 4pt 4pt 4pt;">Chinook</td>
<td style="vertical-align: top; text-align: right; white-space: normal; padding: 4pt 4pt 4pt 4pt;">3.81</td>
<td style="vertical-align: top; text-align: right; white-space: nowrap; border-style: solid solid solid solid; border-width: 0pt 0.4pt 0pt 0pt; padding: 4pt 4pt 4pt 4pt;">45.1</td>
</tr>
<tr>
<td style="vertical-align: top; text-align: right; white-space: nowrap; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0.4pt; padding: 4pt 4pt 4pt 4pt; background-color: rgb(242, 242, 242);">1990</td>
<td style="vertical-align: top; text-align: left; white-space: nowrap; padding: 4pt 4pt 4pt 4pt; background-color: rgb(242, 242, 242);">Mar</td>
<td style="vertical-align: top; text-align: left; white-space: nowrap; padding: 4pt 4pt 4pt 4pt; background-color: rgb(242, 242, 242);">Chinook</td>
<td style="vertical-align: top; text-align: right; white-space: normal; padding: 4pt 4pt 4pt 4pt; background-color: rgb(242, 242, 242);">3.51</td>
<td style="vertical-align: top; text-align: right; white-space: nowrap; border-style: solid solid solid solid; border-width: 0pt 0.4pt 0pt 0pt; padding: 4pt 4pt 4pt 4pt; background-color: rgb(242, 242, 242);">33.5</td>
</tr>
<tr>
<td style="vertical-align: top; text-align: right; white-space: nowrap; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0.4pt; padding: 4pt 4pt 4pt 4pt;">1990</td>
<td style="vertical-align: top; text-align: left; white-space: nowrap; padding: 4pt 4pt 4pt 4pt;">Apr</td>
<td style="vertical-align: top; text-align: left; white-space: nowrap; padding: 4pt 4pt 4pt 4pt;">Chinook</td>
<td style="vertical-align: top; text-align: right; white-space: normal; padding: 4pt 4pt 4pt 4pt;">4.25</td>
<td style="vertical-align: top; text-align: right; white-space: nowrap; border-style: solid solid solid solid; border-width: 0pt 0.4pt 0pt 0pt; padding: 4pt 4pt 4pt 4pt;">70 </td>
</tr>
<tr>
<td style="vertical-align: top; text-align: right; white-space: nowrap; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0.4pt; padding: 4pt 4pt 4pt 4pt; background-color: rgb(242, 242, 242);">1990</td>
<td style="vertical-align: top; text-align: left; white-space: nowrap; padding: 4pt 4pt 4pt 4pt; background-color: rgb(242, 242, 242);">May</td>
<td style="vertical-align: top; text-align: left; white-space: nowrap; padding: 4pt 4pt 4pt 4pt; background-color: rgb(242, 242, 242);">Chinook</td>
<td style="vertical-align: top; text-align: right; white-space: normal; padding: 4pt 4pt 4pt 4pt; background-color: rgb(242, 242, 242);">5.2 </td>
<td style="vertical-align: top; text-align: right; white-space: nowrap; border-style: solid solid solid solid; border-width: 0pt 0.4pt 0pt 0pt; padding: 4pt 4pt 4pt 4pt; background-color: rgb(242, 242, 242);">181 </td>
</tr>
<tr>
<td style="vertical-align: top; text-align: right; white-space: nowrap; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0.4pt; padding: 4pt 4pt 4pt 4pt;">1990</td>
<td style="vertical-align: top; text-align: left; white-space: nowrap; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt; padding: 4pt 4pt 4pt 4pt;">Jun</td>
<td style="vertical-align: top; text-align: left; white-space: nowrap; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt; padding: 4pt 4pt 4pt 4pt;">Chinook</td>
<td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt; padding: 4pt 4pt 4pt 4pt;">4.37</td>
<td style="vertical-align: top; text-align: right; white-space: nowrap; border-style: solid solid solid solid; border-width: 0pt 0.4pt 0.4pt 0pt; padding: 4pt 4pt 4pt 4pt;">79.2</td>
</tr>
</table>
---
The data are monthly and start in January 1990. To make this into a ts object do
```r
chinookts <- ts(chinook$log.metric.tons, start=c(1990,1),
frequency=12)
```
`start` is the year and month and frequency is the number of months in the year.
Use `?ts` to see more examples of how to set up ts objects.
---
## Plot seasonal data
```r
plot(chinookts)
```
<img src="Forecasting-3-7---ARMA-Seasonal-Models_files/figure-html/unnamed-chunk-3-1.png" style="display: block; margin: auto;" />
---
## Seasonal ARIMA model
Seasonally differenced data:
`$$z_t = x_t - x_{t+s} - m$$`
Basic structure of a seasonal AR model
`\(z_t\)` = AR(p) + AR(season) + AR(p+season)
Example AR(1) non-seasonal part + AR(1) seasonal part
`$$z_t = \phi_1 z_{t-1} + \Phi_1 z_{t-12} - \phi_1\Phi_1 z_{t-13}$$`
---
## Notation
ARIMA (p,d,q)(ps,ds,qs)S
ARIMA (1,0,0)(1,1,0)[12]
Notice we are modeling `\(x\)` this year in Jan (say) as a function of `\(x\)` in Jan last year.
---
## Seasonal models
Let's imagine that we can describe our data as a combination of the mean trend, a seasonal term, and error.
`$$x_t = \mu t+ s_t + w_t$$`
Let's imagine that the seasonal term is just a constant based on month and doesn't change with time.
`$$s_t = f(month)$$`
---
We want to remove the `\(s_t\)` with differencing so that we can model `\(e_t\)`. We can solve for `\(x_{t+1}\)` by using `\(x_{t-s}\)` where `\(s\)` is the seasonal length (e.g. 12 if season is yearly).
When we take the first seasonal difference, we get
`$$\Delta_s x_t = \mu(t-(t-s)) + s_t - s_{t-s} + w_t - w_{t-s} = \mu s + w_t - w_{t-s}$$`
The `\(s_t-s_{t-s}\)` disappears because `\(s_t = s_{t-s}\)` when the seasonal effect is just a function of the month. Depending on what `\(m_t\)` is, we might be done or we might have to do a first difference. Notice that the error term is a moving average in the seasonal part.
---
<img src="Forecasting-3-7---ARMA-Seasonal-Models_files/figure-html/unnamed-chunk-4-1.png" style="display: block; margin: auto;" />
```r
plot(diff(xt,lag=12))
```
<img src="Forecasting-3-7---ARMA-Seasonal-Models_files/figure-html/unnamed-chunk-5-1.png" style="display: block; margin: auto;" />
---
We can recover the model parameters with `Arima()`. Note the drift term is returned at `\(\mu\)` not `\(\mu s\)`.
```r
forecast::Arima(xt, seasonal=c(0,1,1), include.drift=TRUE)
```
```
## Series: xt
## ARIMA(0,0,0)(0,1,1)[12] with drift
##
## Coefficients:
## sma1 drift
## -1.0000 0.0496
## s.e. 0.1845 0.0008
##
## sigma^2 estimated as 0.08162: log likelihood=-30.74
## AIC=67.49 AICc=67.72 BIC=75.53
```
---
`auto.arima()` identifies a ARIMA(0,0,0)(0,1,2)[12].
```r
forecast::auto.arima(xt, stepwise=FALSE)
```
```
## Series: xt
## ARIMA(0,0,0)(0,1,2)[12] with drift
##
## Coefficients:
## sma1 sma2 drift
## -1.0488 0.1765 0.0496
## s.e. 0.1606 0.1148 0.0007
##
## sigma^2 estimated as 0.08705: log likelihood=-29.56
## AIC=67.12 AICc=67.51 BIC=77.85
```
---
## Seasonal model with changing season
Let's imagine that our seasonality is increasing over time.
`$$s_t = \beta \times year \times f(month)\times$$`
When we take the first seasonal difference, we get
`$$\Delta_s x_t = \mu(t-(t-s)) + \beta f(month)\times (year - (year-1)) + w_t - w_{t-s} \\ = \mu s + \beta f(month) + w_t - w_{t-s}$$`
---
We need to take another seasonal difference to get rid of the `\(f(month)\)` which is not a constant; it is different for different months as it is our seasonality.
`$$\Delta^2_{s} x_t = w_t - w_{t-s} - w_{t-s} + w_{t-2s}=w_t - 2w_{t-s} + w_{t-2s}$$`
So our ARIMA model should be ARIMA(0,0,0)(0,2,2)
---
<img src="Forecasting-3-7---ARMA-Seasonal-Models_files/figure-html/unnamed-chunk-8-1.png" style="display: block; margin: auto;" />
---
But we can recover the model with `Arima()`. Note the drift term is returned at `\(\mu\)` not `\(\mu s\)`.
```r
forecast::Arima(xt, seasonal=c(0,2,2))
```
```
## Series: xt
## ARIMA(0,0,0)(0,2,2)[12]
##
## Coefficients:
## sma1 sma2
## -1.7745 0.9994
## s.e. 0.4178 0.4624
##
## sigma^2 estimated as 0.1045: log likelihood=-55.1
## AIC=116.2 AICc=116.46 BIC=123.9
```
---
`auto.arima()` again has problems and returns many Infs; turn on `trace=TRUE` to see the problem.
```r
forecast::auto.arima(xt, stepwise=FALSE)
```
```
## Series: xt
## ARIMA(4,0,0)(1,1,0)[12] with drift
##
## Coefficients:
## ar1 ar2 ar3 ar4 sar1 drift
## 0.2548 -0.0245 0.2166 -0.2512 -0.5393 0.0492
## s.e. 0.0945 0.0947 0.0943 0.0966 0.0868 0.0025
##
## sigma^2 estimated as 0.1452: log likelihood=-48.2
## AIC=110.41 AICc=111.53 BIC=129.18
```
---
## Seasonal model with changing season #2
Let's imagine that our seasonality increases and then decreases.
`$$s_t = (a y^2-b y+h) f(month)$$`
<img src="Forecasting-3-7---ARMA-Seasonal-Models_files/figure-html/unnamed-chunk-11-1.png" style="display: block; margin: auto;" />
---
Then we need to take 3 seasonal differences to get rid of the seasonality. The first will get rid of the `\(h f(month)\)`, the next will get rid of `\(by\)` (year) terms and `\(y^2\)` terms, the third will get rid of extra `\(y\)` terms introduced by the 2nd difference. The seasonal differences will get rid of the linear trend also.
`$$\Delta^3_{s} x_t = w_t - 2w_{t-s} + w_{t-2s}-w_{t-s}+2w_{t-2s}-w_{t-3s}=w_t - 3w_{t-s} + 3w_{t-2s}-w_{t-3s}$$`
So our ARIMA model should be ARIMA(0,0,0)(0,3,3).
---
## `auto.arima()` for seasonal ts
`auto.arima()` will recognize that our data has season and fit a seasonal ARIMA model to our data by default. We will define the training data up to 1998 and use 1999 as the test data.
```r
traindat <- window(chinookts, c(1990,10), c(1998,12))
testdat <- window(chinookts, c(1999,1), c(1999,12))
fit <- forecast::auto.arima(traindat)
fit
```
```
## Series: traindat
## ARIMA(1,0,0)(0,1,0)[12] with drift
##
## Coefficients:
## ar1 drift
## 0.3676 -0.0320
## s.e. 0.1335 0.0127
##
## sigma^2 estimated as 0.8053: log likelihood=-107.37
## AIC=220.73 AICc=221.02 BIC=228.13
```
---
## Forecast using seasonal model
```r
fr <- forecast::forecast(fit, h=12)
plot(fr)
points(testdat)
```
<img src="Forecasting-3-7---ARMA-Seasonal-Models_files/figure-html/unnamed-chunk-13-1.png" style="display: block; margin: auto;" />
---
## Missing values
Missing values are ok when fitting a seasonal ARIMA model
<img src="Forecasting-3-7---ARMA-Seasonal-Models_files/figure-html/unnamed-chunk-14-1.png" style="display: block; margin: auto;" />
---
## Summary
Basic steps for identifying a seasonal model. **forecast** automates most of this.
* Check that you have specified your season correctly in your ts object.
* Plot your data. Look for trend, seasonality and random walks.
---
## Summary
* Use differencing to remove season and trend.
* Season and no trend. Take a difference of lag = season
* No seasonality but a trend. Try a first difference
* Both. Do both types of differences
* Neither. No differencing.
* Random walk. First difference.
* Parametric looking curve. Tranform.
---
## Summary
* Examine the ACF and PACF of the differenced data.
* Look for patterns (spikes) at seasonal lags
* Estimate likely models and compare with model selection criteria (or cross-validation). Use `TRACE=TRUE`
* Do residual checks.
</textarea>
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