-
Notifications
You must be signed in to change notification settings - Fork 5
/
Forecasting-3-5---ARMA-Forecasting.html
338 lines (246 loc) · 9.6 KB
/
Forecasting-3-5---ARMA-Forecasting.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
<!DOCTYPE html>
<html lang="" xml:lang="">
<head>
<title>Forecasting-3-5---ARMA-Forecasting.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
# ARIMA Models
## Forecasting
.futnote[Eli Holmes, UW SAFS]
.citation[[email protected]]
---
## Forecasting
The basic idea of forecasting with an ARIMA model is the same as forecasting with a time-varying regressiion model.
We estimate a model and the parameters for that model. For example, let's say we want to forecast with ARIMA(2,1,0) model:
`$$y_t = \beta_1 y_{t-1} + \beta_2 y_{t-2} + e_t$$`
where `\(y_t\)` is the first difference of our anchovy data.
---
Let's estimate the `\(\beta\)`'s:
```r
fit <- Arima(anchovy, order=c(2,1,0))
coef(fit)
```
```
## ar1 ar2
## -0.3347994 -0.1453928
```
So we will forecast with this model:
`$$y_t = -0.3348 y_{t-1} - 0.1454 y_{t-2} + e_t$$`
So to get our forecast for 1988, we do this
`$$(y_{1988}-y_{1987}) = -0.3348 (y_{1987}-y_{1986}) - 0.1454 (y_{1986}-y_{1985})$$`
`$$y_{1988} = y_{1987}-0.3348 (y_{1987}-y_{1986}) - 0.1454 (y_{1986}-y_{1985})$$`
---
`$$y_{1988} = y_{1987}-0.3348 (y_{1987}-y_{1986}) - 0.1454 (y_{1986}-y_{1985})$$`
Here is R code to do that:
```r
anchovy[24]+coef(fit)[1]*(anchovy[24]-anchovy[23])+
coef(fit)[2]*(anchovy[23]-anchovy[22])
```
```
## ar1
## 10.00938
```
Thankfully, `forecast()` automates this calculation for us.
```r
forecast(fit, h=1)
```
```
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 25 10.00938 9.727733 10.29102 9.57864 10.44011
```
---
## Forecasting
We can forecast from a fit in R using the `forecast()` function. `h` is the number of time steps forward to forecast. The upper and lower prediction intervals are shown.
```r
fit <- auto.arima(sardine, test="adf")
fr <- forecast(fit, h=5)
fr
```
```
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 25 9.178334 9.015992 9.340675 8.930053 9.426614
## 26 9.178334 8.948748 9.407919 8.827212 9.529455
## 27 9.178334 8.897150 9.459518 8.748300 9.608367
## 28 9.178334 8.853650 9.503017 8.681773 9.674894
## 29 9.178334 8.815327 9.541341 8.623162 9.733505
```
---
We can plot our forecast with prediction intervals. Here is the sardine forecast:
```r
plot(fr, xlab="Year")
```
<img src="Forecasting-3-5---ARMA-Forecasting_files/figure-html/unnamed-chunk-5-1.png" style="display: block; margin: auto;" />
---
# Repeat for anchovy
```r
fit <- auto.arima(anchovy)
fr <- forecast(fit, h=5)
plot(fr)
```
<img src="Forecasting-3-5---ARMA-Forecasting_files/figure-html/unnamed-chunk-6-1.png" style="display: block; margin: auto;" />
---
# What happens if I have missing values?
```r
anchovy.miss <- anchovy
anchovy.miss[10:14] <- NA
fit <- auto.arima(anchovy.miss)
fr <- forecast(fit, h=5)
plot(fr)
```
<img src="Forecasting-3-5---ARMA-Forecasting_files/figure-html/unnamed-chunk-7-1.png" style="display: block; margin: auto;" />
---
# Repeat for Chub Mackerel
```r
dat <- subset(landings, Species=="Chub.mackerel")$log.metric.tons
fit <- auto.arima(dat)
fr <- forecast(fit, h=5)
plot(fr, ylim=c(6,10))
```
<img src="Forecasting-3-5---ARMA-Forecasting_files/figure-html/unnamed-chunk-8-1.png" style="display: block; margin: auto;" />
---
## The "Naive" forecast
The "naive" forecast is simply the last value observed. If we want to prediction landings in 2019, the naive forecast would be the landings in 2018. This is a difficult forecast to beat! It has the advantage of having no parameters.
In forecast, we can fit this model with the `naive()` function. Note this is the same as the `rwf()` function.
---
```r
fit.naive <- naive(anchovy)
fr.naive <- forecast(fit.naive, h=5)
plot(fr.naive)
```
<img src="Forecasting-3-5---ARMA-Forecasting_files/figure-html/unnamed-chunk-9-1.png" style="display: block; margin: auto;" />
---
## The "Naive" forecast with drift
The "naive" forecast is equivalent to a random walk with no drift. So this
`$$x_t = x_{t-1} + e_t$$`
As you saw with the anchovy fit, it doesn't allow an upward trend. Let's make it a little more flexible by add `drift`. This means we estimate one term, the trend.
`$$x_t = \mu + x_{t-1} + e_t$$`
---
```r
fit.rwf <- rwf(anchovy, drift=TRUE)
fr.rwf <- forecast(fit.rwf, h=5)
plot(fr.rwf)
```
<img src="Forecasting-3-5---ARMA-Forecasting_files/figure-html/unnamed-chunk-10-1.png" style="display: block; margin: auto;" />
---
## The "mean" forecast
The "mean" forecast is simply the mean of the data. If we want to prediction landings in 2019, the mean forecast would be the average of all our data. This is a poor forecast typically. It uses no information about the most recent values.
In forecast, we can fit this model with the `Arima()` function and `order=c(0,0,0)`. This will fit this model:
`$$x_t = e_t$$`
where `\(e_t \sim N(\mu, \sigma)\)`.
---
```r
fit.mean <- Arima(anchovy, order=c(0,0,0))
fr.mean <- forecast(fit.mean, h=5)
plot(fr.mean)
```
<img src="Forecasting-3-5---ARMA-Forecasting_files/figure-html/unnamed-chunk-11-1.png" style="display: block; margin: auto;" />
</textarea>
<style data-target="print-only">@media screen {.remark-slide-container{display:block;}.remark-slide-scaler{box-shadow:none;}}</style>
<script src="https://remarkjs.com/downloads/remark-latest.min.js"></script>
<script>var slideshow = remark.create({
"highlightStyle": "github",
"highlightLines": true
});
if (window.HTMLWidgets) slideshow.on('afterShowSlide', function (slide) {
window.dispatchEvent(new Event('resize'));
});
(function(d) {
var s = d.createElement("style"), r = d.querySelector(".remark-slide-scaler");
if (!r) return;
s.type = "text/css"; s.innerHTML = "@page {size: " + r.style.width + " " + r.style.height +"; }";
d.head.appendChild(s);
})(document);
(function(d) {
var el = d.getElementsByClassName("remark-slides-area");
if (!el) return;
var slide, slides = slideshow.getSlides(), els = el[0].children;
for (var i = 1; i < slides.length; i++) {
slide = slides[i];
if (slide.properties.continued === "true" || slide.properties.count === "false") {
els[i - 1].className += ' has-continuation';
}
}
var s = d.createElement("style");
s.type = "text/css"; s.innerHTML = "@media print { .has-continuation { display: none; } }";
d.head.appendChild(s);
})(document);
// delete the temporary CSS (for displaying all slides initially) when the user
// starts to view slides
(function() {
var deleted = false;
slideshow.on('beforeShowSlide', function(slide) {
if (deleted) return;
var sheets = document.styleSheets, node;
for (var i = 0; i < sheets.length; i++) {
node = sheets[i].ownerNode;
if (node.dataset["target"] !== "print-only") continue;
node.parentNode.removeChild(node);
}
deleted = true;
});
})();
// adds .remark-code-has-line-highlighted class to <pre> parent elements
// of code chunks containing highlighted lines with class .remark-code-line-highlighted
(function(d) {
const hlines = d.querySelectorAll('.remark-code-line-highlighted');
const preParents = [];
const findPreParent = function(line, p = 0) {
if (p > 1) return null; // traverse up no further than grandparent
const el = line.parentElement;
return el.tagName === "PRE" ? el : findPreParent(el, ++p);
};
for (let line of hlines) {
let pre = findPreParent(line);
if (pre && !preParents.includes(pre)) preParents.push(pre);
}
preParents.forEach(p => p.classList.add("remark-code-has-line-highlighted"));
})(document);</script>
<script>
(function() {
var links = document.getElementsByTagName('a');
for (var i = 0; i < links.length; i++) {
if (/^(https?:)?\/\//.test(links[i].getAttribute('href'))) {
links[i].target = '_blank';
}
}
})();
</script>
<script>
slideshow._releaseMath = function(el) {
var i, text, code, codes = el.getElementsByTagName('code');
for (i = 0; i < codes.length;) {
code = codes[i];
if (code.parentNode.tagName !== 'PRE' && code.childElementCount === 0) {
text = code.textContent;
if (/^\\\((.|\s)+\\\)$/.test(text) || /^\\\[(.|\s)+\\\]$/.test(text) ||
/^\$\$(.|\s)+\$\$$/.test(text) ||
/^\\begin\{([^}]+)\}(.|\s)+\\end\{[^}]+\}$/.test(text)) {
code.outerHTML = code.innerHTML; // remove <code></code>
continue;
}
}
i++;
}
};
slideshow._releaseMath(document);
</script>
<!-- dynamically load mathjax for compatibility with self-contained -->
<script>
(function () {
var script = document.createElement('script');
script.type = 'text/javascript';
script.src = 'https://mathjax.rstudio.com/latest/MathJax.js?config=TeX-MML-AM_CHTML';
if (location.protocol !== 'file:' && /^https?:/.test(script.src))
script.src = script.src.replace(/^https?:/, '');
document.getElementsByTagName('head')[0].appendChild(script);
})();
</script>
</body>
</html>