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<p>Recall the time series representing the monthly relative number of Google searches for the term “chocolate.” Here is a time plot of the data from 2021 to 2023. Month 1 is January 2021, month 13 is January 2022, and month 25 is January 2023.</p>
<p>The folded chunk of code below fits the model to the data and computes the estimated parameter values.</p>
<div class="cell">
<details>
<summary>Show the code</summary>
<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a>chocolate_month <span class="ot"><-</span> rio<span class="sc">::</span><span class="fu">import</span>(<span class="st">"https://byuistats.github.io/timeseries/data/chocolate.csv"</span>) <span class="sc">|></span></span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">mutate</span>(</span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a> <span class="at">dates =</span> <span class="fu">yearmonth</span>(<span class="fu">ym</span>(Month)),</span>
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a> <span class="at">month =</span> <span class="fu">month</span>(dates),</span>
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a> <span class="at">year =</span> <span class="fu">year</span>(dates),</span>
<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a> <span class="at">stats_time =</span> year <span class="sc">+</span> (month <span class="sc">-</span> <span class="dv">1</span>) <span class="sc">/</span> <span class="dv">12</span>,</span>
<span id="cb1-7"><a href="#cb1-7" aria-hidden="true" tabindex="-1"></a> <span class="at">month_seq =</span> <span class="dv">1</span><span class="sc">:</span><span class="fu">n</span>()</span>
<span id="cb1-8"><a href="#cb1-8" aria-hidden="true" tabindex="-1"></a> ) <span class="sc">|></span></span>
<span id="cb1-9"><a href="#cb1-9" aria-hidden="true" tabindex="-1"></a> <span class="fu">mutate</span>(<span class="at">month =</span> <span class="fu">factor</span>(month)) <span class="sc">|></span></span>
<span id="cb1-10"><a href="#cb1-10" aria-hidden="true" tabindex="-1"></a> <span class="fu">as_tsibble</span>(<span class="at">index =</span> dates)</span>
<span id="cb1-11"><a href="#cb1-11" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-12"><a href="#cb1-12" aria-hidden="true" tabindex="-1"></a><span class="co"># Fit regression model</span></span>
<span id="cb1-13"><a href="#cb1-13" aria-hidden="true" tabindex="-1"></a>chocolate_lm <span class="ot"><-</span> chocolate_month <span class="sc">|></span></span>
<span id="cb1-14"><a href="#cb1-14" aria-hidden="true" tabindex="-1"></a> <span class="fu">model</span>(<span class="fu">TSLM</span>(chocolate <span class="sc">~</span> <span class="dv">0</span> <span class="sc">+</span> stats_time <span class="sc">+</span> month))</span>
<span id="cb1-15"><a href="#cb1-15" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-16"><a href="#cb1-16" aria-hidden="true" tabindex="-1"></a><span class="co"># Estimated parameter values</span></span>
<span id="cb1-17"><a href="#cb1-17" aria-hidden="true" tabindex="-1"></a>param_est <span class="ot"><-</span> chocolate_lm <span class="sc">|></span></span>
<span id="cb1-18"><a href="#cb1-18" aria-hidden="true" tabindex="-1"></a> <span class="fu">tidy</span>() <span class="sc">|></span></span>
<span id="cb1-19"><a href="#cb1-19" aria-hidden="true" tabindex="-1"></a> <span class="fu">pull</span>(estimate)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
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<div class="cell">
<div class="sourceCode cell-code" id="cb2"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a>chocolate_lm <span class="sc">|></span></span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">residuals</span>() <span class="sc">|></span></span>
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a> <span class="fu">acf</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<p><img src="_ch6_lesson2_chocolate_files/figure-html/unnamed-chunk-5-1.png" class="img-fluid" width="672"></p>
</div>
<div class="sourceCode cell-code" id="cb3"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a>chocolate_lm <span class="sc">|></span></span>
<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">residuals</span>() <span class="sc">|></span></span>
<span id="cb3-3"><a href="#cb3-3" aria-hidden="true" tabindex="-1"></a> <span class="fu">pacf</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<p><img src="_ch6_lesson2_chocolate_files/figure-html/unnamed-chunk-5-2.png" class="img-fluid" width="672"></p>
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<div class="cell">
<div class="sourceCode cell-code" id="cb4"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a>model_resid <span class="ot"><-</span> chocolate_lm <span class="sc">|></span></span>
<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">residuals</span>() <span class="sc">|></span></span>
<span id="cb4-3"><a href="#cb4-3" aria-hidden="true" tabindex="-1"></a> <span class="fu">select</span>(<span class="sc">-</span>.model) <span class="sc">|></span></span>
<span id="cb4-4"><a href="#cb4-4" aria-hidden="true" tabindex="-1"></a> <span class="fu">model</span>(</span>
<span id="cb4-5"><a href="#cb4-5" aria-hidden="true" tabindex="-1"></a> <span class="at">auto =</span> <span class="fu">ARIMA</span>(.resid <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> <span class="fu">pdq</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">3</span>,<span class="dv">0</span>,<span class="dv">0</span><span class="sc">:</span><span class="dv">3</span>) <span class="sc">+</span> <span class="fu">PDQ</span>(<span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>)),</span>
<span id="cb4-6"><a href="#cb4-6" aria-hidden="true" tabindex="-1"></a> <span class="at">a000 =</span> <span class="fu">ARIMA</span>(.resid <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> <span class="fu">pdq</span>(<span class="dv">0</span>,<span class="dv">0</span>,<span class="dv">0</span>) <span class="sc">+</span> <span class="fu">PDQ</span>(<span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>)),</span>
<span id="cb4-7"><a href="#cb4-7" aria-hidden="true" tabindex="-1"></a> <span class="at">a001 =</span> <span class="fu">ARIMA</span>(.resid <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> <span class="fu">pdq</span>(<span class="dv">0</span>,<span class="dv">0</span>,<span class="dv">1</span>) <span class="sc">+</span> <span class="fu">PDQ</span>(<span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>)),</span>
<span id="cb4-8"><a href="#cb4-8" aria-hidden="true" tabindex="-1"></a> <span class="at">a002 =</span> <span class="fu">ARIMA</span>(.resid <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> <span class="fu">pdq</span>(<span class="dv">0</span>,<span class="dv">0</span>,<span class="dv">2</span>) <span class="sc">+</span> <span class="fu">PDQ</span>(<span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>)),</span>
<span id="cb4-9"><a href="#cb4-9" aria-hidden="true" tabindex="-1"></a> <span class="at">a100 =</span> <span class="fu">ARIMA</span>(.resid <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> <span class="fu">pdq</span>(<span class="dv">1</span>,<span class="dv">0</span>,<span class="dv">0</span>) <span class="sc">+</span> <span class="fu">PDQ</span>(<span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>)),</span>
<span id="cb4-10"><a href="#cb4-10" aria-hidden="true" tabindex="-1"></a> <span class="at">a101 =</span> <span class="fu">ARIMA</span>(.resid <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> <span class="fu">pdq</span>(<span class="dv">1</span>,<span class="dv">0</span>,<span class="dv">1</span>) <span class="sc">+</span> <span class="fu">PDQ</span>(<span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>)),</span>
<span id="cb4-11"><a href="#cb4-11" aria-hidden="true" tabindex="-1"></a> <span class="at">a102 =</span> <span class="fu">ARIMA</span>(.resid <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> <span class="fu">pdq</span>(<span class="dv">1</span>,<span class="dv">0</span>,<span class="dv">2</span>) <span class="sc">+</span> <span class="fu">PDQ</span>(<span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>)),</span>
<span id="cb4-12"><a href="#cb4-12" aria-hidden="true" tabindex="-1"></a> <span class="at">a200 =</span> <span class="fu">ARIMA</span>(.resid <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> <span class="fu">pdq</span>(<span class="dv">2</span>,<span class="dv">0</span>,<span class="dv">0</span>) <span class="sc">+</span> <span class="fu">PDQ</span>(<span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>)),</span>
<span id="cb4-13"><a href="#cb4-13" aria-hidden="true" tabindex="-1"></a> <span class="at">a201 =</span> <span class="fu">ARIMA</span>(.resid <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> <span class="fu">pdq</span>(<span class="dv">2</span>,<span class="dv">0</span>,<span class="dv">1</span>) <span class="sc">+</span> <span class="fu">PDQ</span>(<span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>)),</span>
<span id="cb4-14"><a href="#cb4-14" aria-hidden="true" tabindex="-1"></a> <span class="at">a202 =</span> <span class="fu">ARIMA</span>(.resid <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> <span class="fu">pdq</span>(<span class="dv">2</span>,<span class="dv">0</span>,<span class="dv">2</span>) <span class="sc">+</span> <span class="fu">PDQ</span>(<span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>))) </span>
<span id="cb4-15"><a href="#cb4-15" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb4-16"><a href="#cb4-16" aria-hidden="true" tabindex="-1"></a><span class="co"># best order is 2, 0, 0</span></span>
<span id="cb4-17"><a href="#cb4-17" aria-hidden="true" tabindex="-1"></a><span class="co"># auto would pick it without running each </span></span>
<span id="cb4-18"><a href="#cb4-18" aria-hidden="true" tabindex="-1"></a>model_best <span class="ot"><-</span> <span class="fu">select</span>(model_resid, <span class="st">"a200"</span>)</span>
<span id="cb4-19"><a href="#cb4-19" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb4-20"><a href="#cb4-20" aria-hidden="true" tabindex="-1"></a>model_resid <span class="sc">|></span></span>
<span id="cb4-21"><a href="#cb4-21" aria-hidden="true" tabindex="-1"></a> <span class="fu">glance</span>() <span class="sc">|></span></span>
<span id="cb4-22"><a href="#cb4-22" aria-hidden="true" tabindex="-1"></a> <span class="fu">filter</span>(AIC <span class="sc">==</span> <span class="fu">min</span>(AIC) <span class="sc">|</span> AICc <span class="sc">==</span> <span class="fu">min</span>(AICc) <span class="sc">|</span> BIC <span class="sc">==</span> <span class="fu">min</span>(BIC) <span class="sc">|</span> .model <span class="sc">==</span> <span class="st">"auto"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code># A tibble: 2 × 8
.model sigma2 log_lik AIC AICc BIC ar_roots ma_roots
<chr> <dbl> <dbl> <dbl> <dbl> <dbl> <list> <list>
1 auto 11.5 -632. 1273. 1273. 1291. <cpl [1]> <cpl [2]>
2 a200 11.5 -632. 1272. 1272. 1286. <cpl [2]> <cpl [0]></code></pre>
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appendTo: function(el) {
return el.parentElement;
},
interactive: true,
interactiveBorder: 10,
theme: 'quarto',
placement: 'bottom-start'
};
window.tippy(el, config);
}
const noterefs = window.document.querySelectorAll('a[role="doc-noteref"]');
for (var i=0; i<noterefs.length; i++) {
const ref = noterefs[i];
tippyHover(ref, function() {
// use id or data attribute instead here
let href = ref.getAttribute('data-footnote-href') || ref.getAttribute('href');
try { href = new URL(href).hash; } catch {}
const id = href.replace(/^#\/?/, "");
const note = window.document.getElementById(id);
return note.innerHTML;
});
}
let selectedAnnoteEl;
const selectorForAnnotation = ( cell, annotation) => {
let cellAttr = 'data-code-cell="' + cell + '"';
let lineAttr = 'data-code-annotation="' + annotation + '"';
const selector = 'span[' + cellAttr + '][' + lineAttr + ']';
return selector;
}
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const doc = window.document;
const targetCell = annoteEl.getAttribute("data-target-cell");
const targetAnnotation = annoteEl.getAttribute("data-target-annotation");
const annoteSpan = window.document.querySelector(selectorForAnnotation(targetCell, targetAnnotation));
const lines = annoteSpan.getAttribute("data-code-lines").split(",");
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return targetCell + "-" + line;
})
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let height = null;
let parent = null;
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top = el.offsetTop;
height = el.offsetHeight;
parent = el.parentElement.parentElement;
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const lastEl = window.document.getElementById(lineIds[lineIds.length - 1]);
const bottom = lastEl.offsetTop + lastEl.offsetHeight;
height = bottom - top;
}
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// cook up a div (if necessary) and position it
let div = window.document.getElementById("code-annotation-line-highlight");
if (div === null) {
div = window.document.createElement("div");
div.setAttribute("id", "code-annotation-line-highlight");
div.style.position = 'absolute';
parent.appendChild(div);
}
div.style.top = top - 2 + "px";
div.style.height = height + 4 + "px";
let gutterDiv = window.document.getElementById("code-annotation-line-highlight-gutter");
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gutterDiv.setAttribute("id", "code-annotation-line-highlight-gutter");
gutterDiv.style.position = 'absolute';
const codeCell = window.document.getElementById(targetCell);
const gutter = codeCell.querySelector('.code-annotation-gutter');
gutter.appendChild(gutterDiv);
}
gutterDiv.style.top = top - 2 + "px";
gutterDiv.style.height = height + 4 + "px";
}
selectedAnnoteEl = annoteEl;
}
};
const unselectCodeLines = () => {
const elementsIds = ["code-annotation-line-highlight", "code-annotation-line-highlight-gutter"];
elementsIds.forEach((elId) => {
const div = window.document.getElementById(elId);
if (div) {
div.remove();
}
});
selectedAnnoteEl = undefined;
};
// Attach click handler to the DT
const annoteDls = window.document.querySelectorAll('dt[data-target-cell]');
for (const annoteDlNode of annoteDls) {
annoteDlNode.addEventListener('click', (event) => {
const clickedEl = event.target;
if (clickedEl !== selectedAnnoteEl) {
unselectCodeLines();
const activeEl = window.document.querySelector('dt[data-target-cell].code-annotation-active');
if (activeEl) {
activeEl.classList.remove('code-annotation-active');
}
selectCodeLines(clickedEl);
clickedEl.classList.add('code-annotation-active');
} else {
// Unselect the line
unselectCodeLines();
clickedEl.classList.remove('code-annotation-active');
}
});
}
const findCites = (el) => {
const parentEl = el.parentElement;
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const cites = parentEl.dataset.cites;
if (cites) {
return {
el,
cites: cites.split(' ')
};
} else {
return findCites(el.parentElement)
}
} else {
return undefined;
}
};
var bibliorefs = window.document.querySelectorAll('a[role="doc-biblioref"]');
for (var i=0; i<bibliorefs.length; i++) {
const ref = bibliorefs[i];
const citeInfo = findCites(ref);
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tippyHover(citeInfo.el, function() {
var popup = window.document.createElement('div');
citeInfo.cites.forEach(function(cite) {
var citeDiv = window.document.createElement('div');
citeDiv.classList.add('hanging-indent');
citeDiv.classList.add('csl-entry');
var biblioDiv = window.document.getElementById('ref-' + cite);
if (biblioDiv) {
citeDiv.innerHTML = biblioDiv.innerHTML;
}
popup.appendChild(citeDiv);
});
return popup.innerHTML;
});
}
}
});
</script>
</div> <!-- /content -->
</body></html>