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<!DOCTYPE html>
<html lang="" xml:lang="">
<head>
<title>psaw_2019_overview.utf8</title>
<meta charset="utf-8" />
<link rel="stylesheet" href="my-theme.css" type="text/css" />
</head>
<body>
<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
# Developing Forecasting Models for Fisheries Time Series with R
## PSAW 2019 --- 11 Feb 2019
.futnote[Eli Holmes, UW SAFS]
.citation[[email protected]]
---
## Today's objective
* Learn the Box-Jenkins method for developing a ARIMA forecasting model in R
* Learn how to develop an exponential smoothing (ETS) model for forecasting in R
* Main R package you will learn is **forecast** by Rob Hyndman, Monash University, Australia. You will also use the base time series functions in R but we will focus on **forecast**.
* You will be working with univariate data with no covariates.
---
## Non-seasonal Data
No season. All years. If data are missing, then they are entered as NA.
```r
load("landings.RData")
class(landings)
```
```
## [1] "data.frame"
```
```r
head( subset(landings, Species=="Anchovy") )
```
<table class="huxtable" style="border-collapse: collapse; margin-bottom: 2em; margin-top: 2em; width: 36.6666666666667%; margin-left: 0%; margin-right: auto; ">
<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;">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;">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;">log.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);">1964</td>
<td style="vertical-align: top; text-align: left; white-space: nowrap; padding: 4pt 4pt 4pt 4pt; background-color: rgb(242, 242, 242);">Anchovy</td>
<td style="vertical-align: top; text-align: right; white-space: nowrap; padding: 4pt 4pt 4pt 4pt; background-color: rgb(242, 242, 242);">5.45e+03</td>
<td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0.4pt 0pt 0pt; padding: 4pt 4pt 4pt 4pt; background-color: rgb(242, 242, 242);">8.6 </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;">1965</td>
<td style="vertical-align: top; text-align: left; white-space: nowrap; padding: 4pt 4pt 4pt 4pt;">Anchovy</td>
<td style="vertical-align: top; text-align: right; white-space: nowrap; padding: 4pt 4pt 4pt 4pt;">4.26e+03</td>
<td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0.4pt 0pt 0pt; padding: 4pt 4pt 4pt 4pt;">8.36</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);">1966</td>
<td style="vertical-align: top; text-align: left; white-space: nowrap; padding: 4pt 4pt 4pt 4pt; background-color: rgb(242, 242, 242);">Anchovy</td>
<td style="vertical-align: top; text-align: right; white-space: nowrap; padding: 4pt 4pt 4pt 4pt; background-color: rgb(242, 242, 242);">5.15e+03</td>
<td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0.4pt 0pt 0pt; padding: 4pt 4pt 4pt 4pt; background-color: rgb(242, 242, 242);">8.55</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;">1967</td>
<td style="vertical-align: top; text-align: left; white-space: nowrap; padding: 4pt 4pt 4pt 4pt;">Anchovy</td>
<td style="vertical-align: top; text-align: right; white-space: nowrap; padding: 4pt 4pt 4pt 4pt;">7.27e+03</td>
<td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0.4pt 0pt 0pt; padding: 4pt 4pt 4pt 4pt;">8.89</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);">1968</td>
<td style="vertical-align: top; text-align: left; white-space: nowrap; padding: 4pt 4pt 4pt 4pt; background-color: rgb(242, 242, 242);">Anchovy</td>
<td style="vertical-align: top; text-align: right; white-space: nowrap; padding: 4pt 4pt 4pt 4pt; background-color: rgb(242, 242, 242);">6.35e+03</td>
<td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0.4pt 0pt 0pt; padding: 4pt 4pt 4pt 4pt; background-color: rgb(242, 242, 242);">8.76</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;">1969</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;">Anchovy</td>
<td style="vertical-align: top; text-align: right; white-space: nowrap; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt; padding: 4pt 4pt 4pt 4pt;">7.04e+03</td>
<td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0.4pt 0.4pt 0pt; padding: 4pt 4pt 4pt 4pt;">8.86</td>
</tr>
</table>
---
## Seasonal Data
No season. All years and seasons (months, quarters, hours). If data are missing, then they are entered as NA. Only one season. So not daily plus monthly cycle.
```r
load("chinook.RData")
head( subset(chinook, Species=="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>
---
## State-space time series models (Mark's workshop) vs ARIMA
Let's imagine that we can describe our data as a combination of the mean trend and error.
`$$x_t = m_t + e_t$$`
What if our main goal is to predict `\(x_{t+1}\)`?
Note we (fisheries biologists and ecologists) often want to know `\(m_t\)`. In fact, that is often our main and sole goal. But let's say our goal is to predict `\(x_{t+1}\)`.
---
## Approach #1 State-space models as applied in Mark's workshop
[Time Series Analysis, PSAW II AM](https://mdscheuerell.github.io/PSAW2/)
We estimate `\(m_t\)` from the data. Once we have an estimate of `\(m_t\)`, we have an estimate of `\(e_t\)`. We can model that error (think AR and MA) to help us predict future `\(x_t\)` from `\(x_{t-1}\)`.
We will have a model for the `\(m_t\)` and we will fit
`$$x_t = m_t + e_t$$`
directly. We will get an estimate of the model that produced `\(m_t\)`. The `\(x_t\)` and `\(m_t\)` models will have a biological interpretation.
---
So for example if our model is
`$$m_t = m_{t-1}+\mu \\ x_t = m_t + e_t, \,\, e_t \sim N(0,\sigma)$$`
Then we fit that model without taking any transforming our data.
Note the above is just a linear deterministic trend observed with error, i.e. a linear regression against time.
---
## Approach #2 Box-Jenkins
The Box-Jenkins approach is different. In this approach to predicting `\(x_{t+1}\)`, we remove `\(m_t\)` from our data using differencing. We don't have to worry about a model for `\(m_t\)` because we have removed it!
And we don't actually need `\(m_t\)` to predict `\(x_{t+1}\)` since we have `\(x_{t}\)`, `\(x_{t-1}\)`, etc.
We model `\(\Delta x_t\)` and then predict `\(\Delta x_{t+1}\)`. `\(\Delta x_t\)` is `\(x_t-x_{t-1}\)`. Once we have that we can predict `\(x_{t+1}\)` by using `\(x_t\)`, `\(x_{t-1}\)`, etc with some algebra.
*We might also use a d-th difference.
---
The error structure of `\(\Delta x_{t+1}\)` is NOT the same as `\(e_t\)`.
`$$\Delta x_{t} = \phi_1\Delta x_{t-1} + \phi_2\Delta x_{t-2} + \dots + z_t$$`
This is a model for the DIFFERENCES and `\(z_t\)` is the error of the differences. It is more difficult to make a biological interpretation of the ARMA model. But remember, the objective was to predict `\(x_{t+1}\)` not to fit a model with a biological interpretation.
---
## Examples of differencing, linear trend
`\(m_t\)` is linear and deterministic.
`$$x_t = \mu t + w_t$$`
<!-- -->
---
When we take the first difference, we get
`$$\Delta x_t = \mu (t - (t-1)) + w_t - w_{t-1} = \mu + w_t - w_{t-1}$$`
`\(w_t - w_{t-1}\)` are now the errors in the model.
---
## More resources
Rob Hyndman book on forecasting and the **forecast** package: [Forecasting Principals and Practice](https://otexts.org/fpp2/)
NWFSC Time Series Analysis: https://nwfsc-timeseries.github.io/ (PIs: Eli Holmes, Mark Scheurell, Eric Ward)
* Courses on multivariate and univariate time series analysis applied to messy fisheries and ecological data.
* R packages for fitting time series models. Bayesian and maximum-likelihood.
* online books:
- [Applied Time-Series Analysis for Fisheries and Environmental Data](https://nwfsc-timeseries.github.io/atsa-labs/)
- [Fisheries Catch Forecasting](https://fish-forecast.github.io/Fish-Forecast-Bookdown/)
</textarea>
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