STA6257_Project/index.html at main · javier-e94/STA6257_Project · GitHub

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<h1 class="title">Report Sample</h1>
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    <div class="quarto-title-meta-heading">Author</div>
    <div class="quarto-title-meta-contents">
             <p>Student name </p>
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    <div class="quarto-title-meta-heading">Published</div>
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      <p class="date">October 25, 2022</p>
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</header>

<section id="introduction" class="level2">
<h2 class="anchored" data-anchor-id="introduction">Introduction</h2>
<p>Kernel regression is a non-parametric estimator that estimates the conditional expectation of two variables which is random. The goal of a kernel regression is to discover the non-linear relationship between two random variables. To discover the non-linear relationship, kernel estimator or kernel smoothing is the main method to estimate the curve for non-parametric statistics. In kernel estimator, weight function is known as kernel function <span class="citation" data-cites="efr2008">(<a href="#ref-efr2008" role="doc-biblioref">Efromovich 2008</a>)</span>. Cite this paper <span class="citation" data-cites="bro2014principal">(<a href="#ref-bro2014principal" role="doc-biblioref">Bro and Smilde 2014</a>)</span>.</p>
</section>
<section id="methods" class="level2">
<h2 class="anchored" data-anchor-id="methods">Methods</h2>
<p>The common non-parametric regression model is <span class="math inline">\(Y_i = m(X_i) + \varepsilon_i\)</span>, where <span class="math inline">\(Y_i\)</span> can be defined as the sum of the regression function value <span class="math inline">\(m(x)\)</span> for <span class="math inline">\(X_i\)</span>. Here <span class="math inline">\(m(x)\)</span> is unknown and <span class="math inline">\(\varepsilon_i\)</span> some errors. With the help of this definition, we can create the estimation for local averaging i.e.&nbsp;<span class="math inline">\(m(x)\)</span> can be estimated with the product of <span class="math inline">\(Y_i\)</span> average and <span class="math inline">\(X_i\)</span> is near to <span class="math inline">\(x\)</span>. In other words, this means that we are discovering the line through the data points with the help of surrounding data points. The estimation formula is printed below <span class="citation" data-cites="R-base">(<a href="#ref-R-base" role="doc-biblioref">R Core Team 2019</a>)</span>:</p>
<p><span class="math display">\[
M_n(x) = \sum_{i=1}^{n} W_n (X_i) Y_i  \tag{1}
\]</span></p>
<p><span class="math inline">\(W_n(x)\)</span> is the sum of weights that belongs to all real numbers. Weights are positive numbers and small if <span class="math inline">\(X_i\)</span> is far from <span class="math inline">\(x\)</span>.</p>
</section>
<section id="analysis-and-results" class="level2">
<h2 class="anchored" data-anchor-id="analysis-and-results">Analysis and Results</h2>
<section id="data-and-vizualisation" class="level3">
<h3 class="anchored" data-anchor-id="data-and-vizualisation">Data and Vizualisation</h3>
<p>A study was conducted to determine how…</p>
<div class="cell">
<details>
<summary>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><span class="co"># loading packages </span></span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(tidyverse)</span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(knitr)</span>
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(ggthemes)</span>
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(ggrepel)</span>
<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(dslabs)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
</div>
<div class="cell">
<details>
<summary>Code</summary>
<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><span class="co"># Load Data</span></span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a><span class="fu">kable</span>(<span class="fu">head</span>(murders))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output-display">
<table class="table table-sm table-striped">
<thead>
<tr class="header">
<th style="text-align: left;">state</th>
<th style="text-align: left;">abb</th>
<th style="text-align: left;">region</th>
<th style="text-align: right;">population</th>
<th style="text-align: right;">total</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: left;">Alabama</td>
<td style="text-align: left;">AL</td>
<td style="text-align: left;">South</td>
<td style="text-align: right;">4779736</td>
<td style="text-align: right;">135</td>
</tr>
<tr class="even">
<td style="text-align: left;">Alaska</td>
<td style="text-align: left;">AK</td>
<td style="text-align: left;">West</td>
<td style="text-align: right;">710231</td>
<td style="text-align: right;">19</td>
</tr>
<tr class="odd">
<td style="text-align: left;">Arizona</td>
<td style="text-align: left;">AZ</td>
<td style="text-align: left;">West</td>
<td style="text-align: right;">6392017</td>
<td style="text-align: right;">232</td>
</tr>
<tr class="even">
<td style="text-align: left;">Arkansas</td>
<td style="text-align: left;">AR</td>
<td style="text-align: left;">South</td>
<td style="text-align: right;">2915918</td>
<td style="text-align: right;">93</td>
</tr>
<tr class="odd">
<td style="text-align: left;">California</td>
<td style="text-align: left;">CA</td>
<td style="text-align: left;">West</td>
<td style="text-align: right;">37253956</td>
<td style="text-align: right;">1257</td>
</tr>
<tr class="even">
<td style="text-align: left;">Colorado</td>
<td style="text-align: left;">CO</td>
<td style="text-align: left;">West</td>
<td style="text-align: right;">5029196</td>
<td style="text-align: right;">65</td>
</tr>
</tbody>
</table>
</div>
<details>
<summary>Code</summary>
<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>ggplot1 <span class="ot">=</span> murders <span class="sc">%&gt;%</span> <span class="fu">ggplot</span>(<span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x=</span>population<span class="sc">/</span><span class="dv">10</span><span class="sc">^</span><span class="dv">6</span>, <span class="at">y=</span>total)) </span>
<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb3-3"><a href="#cb3-3" aria-hidden="true" tabindex="-1"></a>  ggplot1 <span class="sc">+</span> <span class="fu">geom_point</span>(<span class="fu">aes</span>(<span class="at">col=</span>region), <span class="at">size =</span> <span class="dv">4</span>) <span class="sc">+</span></span>
<span id="cb3-4"><a href="#cb3-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_text_repel</span>(<span class="fu">aes</span>(<span class="at">label=</span>abb)) <span class="sc">+</span></span>
<span id="cb3-5"><a href="#cb3-5" aria-hidden="true" tabindex="-1"></a>  <span class="fu">scale_x_log10</span>() <span class="sc">+</span></span>
<span id="cb3-6"><a href="#cb3-6" aria-hidden="true" tabindex="-1"></a>  <span class="fu">scale_y_log10</span>() <span class="sc">+</span></span>
<span id="cb3-7"><a href="#cb3-7" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_smooth</span>(<span class="at">formula =</span> <span class="st">"y~x"</span>, <span class="at">method=</span>lm,<span class="at">se =</span> F)<span class="sc">+</span></span>
<span id="cb3-8"><a href="#cb3-8" aria-hidden="true" tabindex="-1"></a>  <span class="fu">xlab</span>(<span class="st">"Populations in millions (log10 scale)"</span>) <span class="sc">+</span> </span>
<span id="cb3-9"><a href="#cb3-9" aria-hidden="true" tabindex="-1"></a>  <span class="fu">ylab</span>(<span class="st">"Total number of murders (log10 scale)"</span>) <span class="sc">+</span></span>
<span id="cb3-10"><a href="#cb3-10" aria-hidden="true" tabindex="-1"></a>  <span class="fu">ggtitle</span>(<span class="st">"US Gun Murders in 2010"</span>) <span class="sc">+</span></span>
<span id="cb3-11"><a href="#cb3-11" aria-hidden="true" tabindex="-1"></a>  <span class="fu">scale_color_discrete</span>(<span class="at">name =</span> <span class="st">"Region"</span>)<span class="sc">+</span></span>
<span id="cb3-12"><a href="#cb3-12" aria-hidden="true" tabindex="-1"></a>      <span class="fu">theme_wsj</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output-display">
<p><img src="index_files/figure-html/unnamed-chunk-2-1.png" class="img-fluid" width="672"></p>
</div>
</div>
</section>
<section id="statistical-modeling" class="level3">
<h3 class="anchored" data-anchor-id="statistical-modeling">Statistical Modeling</h3>
</section>
<section id="conlusion" class="level3">
<h3 class="anchored" data-anchor-id="conlusion">Conlusion</h3>
</section>
</section>
<section id="references" class="level2 unnumbered">


</section>

<div id="quarto-appendix" class="default"><section class="quarto-appendix-contents" role="doc-bibliography"><h2 class="anchored quarto-appendix-heading">References</h2><div id="refs" class="references csl-bib-body hanging-indent" role="doc-bibliography">
<div id="ref-bro2014principal" class="csl-entry" role="doc-biblioentry">
Bro, Rasmus, and Age K Smilde. 2014. <span>“Principal Component Analysis.”</span> <em>Analytical Methods</em> 6 (9): 2812–31.
</div>
<div id="ref-efr2008" class="csl-entry" role="doc-biblioentry">
Efromovich, S. 2008. <em>Nonparametric Curve Estimation: Methods, Theory, and Applications</em>. Springer Series in Statistics. Springer New York. <a href="https://books.google.com/books?id=mdoLBwAAQBAJ">https://books.google.com/books?id=mdoLBwAAQBAJ</a>.
</div>
<div id="ref-R-base" class="csl-entry" role="doc-biblioentry">
R Core Team. 2019. <em>R: A Language and Environment for Statistical Computing</em>. Vienna, Austria: R Foundation for Statistical Computing. <a href="https://www.R-project.org">https://www.R-project.org</a>.
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