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<h1 class="title">Kalman Filter From The Ground Up</h1>
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    <div class="quarto-title-meta-heading">Author</div>
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             <p>Sergey Dolin </p>
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<p>This is a R-Lab to reproduce the equations and plots from “Kalman Filter From The Ground Up” by Phil Kim. <a href="https://www.kalmanfilter.net">https://www.kalmanfilter.net</a></p>
<section id="essential-background" class="level2">
<h2 class="anchored" data-anchor-id="essential-background">ESSENTIAL BACKGROUND</h2>
<section id="mean-and-expected-value" class="level3">
<h3 class="anchored" data-anchor-id="mean-and-expected-value">MEAN AND EXPECTED VALUE</h3>
<p>Suppose we want to compare the heights of two high school basketball teams. The following table provides the players’ heights and the mean height of each team.</p>
<div class="cell">
<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>teamA <span class="ot">&lt;-</span> <span class="fu">c</span>(<span class="fl">1.89</span>, <span class="fl">2.1</span>, <span class="fl">1.75</span>, <span class="fl">1.98</span>, <span class="fl">1.85</span>)</span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a>meanTeamA <span class="ot">&lt;-</span> <span class="fu">mean</span>(teamA)</span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a>meanTeamA</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>[1] 1.914</code></pre>
</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>teamB <span class="ot">&lt;-</span> <span class="fu">c</span>(<span class="fl">1.94</span>, <span class="fl">1.9</span>, <span class="fl">1.97</span>, <span class="fl">1.89</span>, <span class="fl">1.87</span>)</span>
<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a>meanTeamB <span class="ot">&lt;-</span> <span class="fu">mean</span>(teamB)</span>
<span id="cb3-3"><a href="#cb3-3" aria-hidden="true" tabindex="-1"></a>meanTeamB</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>[1] 1.914</code></pre>
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</div>
<p>The distance from the mean for each variable would be</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb5"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a>distanceFromMeanTeamA <span class="ot">&lt;-</span> teamA <span class="sc">-</span> meanTeamA</span>
<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a>distanceFromMeanTeamB <span class="ot">&lt;-</span> teamB <span class="sc">-</span> meanTeamB</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>let’s avoid negative numbers by squaring the distances</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb6"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a>squaredDistanceFromMeanTeamA <span class="ot">&lt;-</span> distanceFromMeanTeamA<span class="sc">^</span><span class="dv">2</span></span>
<span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a>squaredDistanceFromMeanTeamB <span class="ot">&lt;-</span> distanceFromMeanTeamB<span class="sc">^</span><span class="dv">2</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<p>Sampled variance is the sum of squared distances from the mean divided by the number of observations minus one.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb7"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a>sampledVarianceTeamA <span class="ot">&lt;-</span> <span class="fu">sum</span>(squaredDistanceFromMeanTeamA) <span class="sc">/</span> (<span class="fu">length</span>(teamA) <span class="sc">-</span> <span class="dv">1</span>)</span>
<span id="cb7-2"><a href="#cb7-2" aria-hidden="true" tabindex="-1"></a>sampledVarianceTeamA</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>[1] 0.01763</code></pre>
</div>
<div class="sourceCode cell-code" id="cb9"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a>sampledVarianceTeamB <span class="ot">&lt;-</span> <span class="fu">sum</span>(squaredDistanceFromMeanTeamB) <span class="sc">/</span> (<span class="fu">length</span>(teamB) <span class="sc">-</span> <span class="dv">1</span>)</span>
<span id="cb9-2"><a href="#cb9-2" aria-hidden="true" tabindex="-1"></a>sampledVarianceTeamB</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>[1] 0.00163</code></pre>
</div>
</div>
<p>The variance is the average of the squared distances from the mean</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb11"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a>varianceTeamA <span class="ot">&lt;-</span> <span class="fu">mean</span>(squaredDistanceFromMeanTeamA)</span>
<span id="cb11-2"><a href="#cb11-2" aria-hidden="true" tabindex="-1"></a>varianceTeamA</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>[1] 0.014104</code></pre>
</div>
<div class="sourceCode cell-code" id="cb13"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a>varianceTeamB <span class="ot">&lt;-</span> <span class="fu">mean</span>(squaredDistanceFromMeanTeamB)</span>
<span id="cb13-2"><a href="#cb13-2" aria-hidden="true" tabindex="-1"></a>varianceTeamB</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>[1] 0.001304</code></pre>
</div>
</div>
<p>The standard deviation is the square root of the variance</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb15"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb15-1"><a href="#cb15-1" aria-hidden="true" tabindex="-1"></a>standardDeviationTeamA <span class="ot">&lt;-</span> <span class="fu">sqrt</span>(varianceTeamA)</span>
<span id="cb15-2"><a href="#cb15-2" aria-hidden="true" tabindex="-1"></a>standardDeviationTeamA</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>[1] 0.1187603</code></pre>
</div>
<div class="sourceCode cell-code" id="cb17"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb17-1"><a href="#cb17-1" aria-hidden="true" tabindex="-1"></a>standardDeviationTeamB <span class="ot">&lt;-</span> <span class="fu">sqrt</span>(varianceTeamB)</span>
<span id="cb17-2"><a href="#cb17-2" aria-hidden="true" tabindex="-1"></a>standardDeviationTeamB</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>[1] 0.03611094</code></pre>
</div>
</div>
<p>Final plot</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb19"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb19-1"><a href="#cb19-1" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>() <span class="sc">+</span></span>
<span id="cb19-2"><a href="#cb19-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">labs</span>(<span class="at">x =</span> <span class="st">"measurements"</span>, <span class="at">y=</span> <span class="st">"height,m"</span>) <span class="sc">+</span></span>
<span id="cb19-3"><a href="#cb19-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">data =</span> <span class="fu">data.frame</span>(teamA), </span>
<span id="cb19-4"><a href="#cb19-4" aria-hidden="true" tabindex="-1"></a>             <span class="fu">aes</span>(<span class="at">x =</span> <span class="dv">1</span><span class="sc">:</span><span class="fu">length</span>(teamA), <span class="at">y =</span> teamA, <span class="at">color =</span> <span class="st">"Team A"</span>)) <span class="sc">+</span></span>
<span id="cb19-5"><a href="#cb19-5" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">data =</span> <span class="fu">data.frame</span>(teamB), </span>
<span id="cb19-6"><a href="#cb19-6" aria-hidden="true" tabindex="-1"></a>             <span class="fu">aes</span>(<span class="at">x =</span> <span class="dv">1</span><span class="sc">:</span><span class="fu">length</span>(teamB), <span class="at">y =</span> teamB, <span class="at">color =</span> <span class="st">"Team B"</span>)) <span class="sc">+</span></span>
<span id="cb19-7"><a href="#cb19-7" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_hline</span>(<span class="at">yintercept =</span> meanTeamA, </span>
<span id="cb19-8"><a href="#cb19-8" aria-hidden="true" tabindex="-1"></a>             <span class="at">color =</span> <span class="st">"red"</span>, <span class="at">linetype =</span><span class="st">"dashed"</span>) <span class="sc">+</span></span>
<span id="cb19-9"><a href="#cb19-9" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_hline</span>(<span class="at">yintercept =</span> meanTeamB, </span>
<span id="cb19-10"><a href="#cb19-10" aria-hidden="true" tabindex="-1"></a>             <span class="at">color =</span> <span class="st">"blue"</span>, <span class="at">linetype =</span><span class="st">"longdash"</span>) <span class="sc">+</span></span>
<span id="cb19-11"><a href="#cb19-11" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_hline</span>(<span class="at">yintercept =</span> meanTeamA <span class="sc">+</span> standardDeviationTeamA, </span>
<span id="cb19-12"><a href="#cb19-12" aria-hidden="true" tabindex="-1"></a>             <span class="at">color =</span> <span class="st">"red"</span>, <span class="at">linetype =</span> <span class="st">"dashed"</span>) <span class="sc">+</span></span>
<span id="cb19-13"><a href="#cb19-13" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_hline</span>(<span class="at">yintercept =</span> meanTeamA <span class="sc">-</span> standardDeviationTeamA, </span>
<span id="cb19-14"><a href="#cb19-14" aria-hidden="true" tabindex="-1"></a>             <span class="at">color =</span> <span class="st">"red"</span>, <span class="at">linetype =</span> <span class="st">"dashed"</span>) <span class="sc">+</span></span>
<span id="cb19-15"><a href="#cb19-15" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_hline</span>(<span class="at">yintercept =</span> meanTeamB <span class="sc">+</span> standardDeviationTeamB, </span>
<span id="cb19-16"><a href="#cb19-16" aria-hidden="true" tabindex="-1"></a>             <span class="at">color =</span> <span class="st">"blue"</span>, <span class="at">linetype =</span> <span class="st">"dashed"</span>) <span class="sc">+</span></span>
<span id="cb19-17"><a href="#cb19-17" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_hline</span>(<span class="at">yintercept =</span> meanTeamB <span class="sc">-</span> standardDeviationTeamB, </span>
<span id="cb19-18"><a href="#cb19-18" aria-hidden="true" tabindex="-1"></a>             <span class="at">color =</span> <span class="st">"blue"</span>, <span class="at">linetype =</span> <span class="st">"dashed"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="KalmanFilterFromTheGroundUp_files/figure-html/unnamed-chunk-7-1.png" class="img-fluid" width="672"></p>
</div>
</div>
</section>
<section id="normal-distribution" class="level3">
<h3 class="anchored" data-anchor-id="normal-distribution">NORMAL DISTRIBUTION</h3>
<p>Generate a normal distribution with mean 0 and standard deviation 1</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb20"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb20-1"><a href="#cb20-1" aria-hidden="true" tabindex="-1"></a>data <span class="ot">&lt;-</span> <span class="fu">rnorm</span>(<span class="dv">10000</span>, <span class="at">mean =</span> <span class="dv">0</span>, <span class="at">sd =</span> <span class="dv">1</span>)</span>
<span id="cb20-2"><a href="#cb20-2" aria-hidden="true" tabindex="-1"></a><span class="co">#, μ = 0, σ = 1 </span></span>
<span id="cb20-3"><a href="#cb20-3" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>() <span class="sc">+</span></span>
<span id="cb20-4"><a href="#cb20-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">labs</span>(<span class="at">x =</span> <span class="st">"random attempts"</span>, <span class="at">y=</span> <span class="fu">expression</span>(<span class="st">"randoms"</span> <span class="sc">~</span> mu <span class="sc">~</span><span class="st">"= 0,"</span> <span class="sc">~</span> sigma <span class="sc">~</span><span class="st">"= 1"</span>)) <span class="sc">+</span></span>
<span id="cb20-5"><a href="#cb20-5" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">data =</span> <span class="fu">data.frame</span>(data), </span>
<span id="cb20-6"><a href="#cb20-6" aria-hidden="true" tabindex="-1"></a>             <span class="fu">aes</span>(<span class="at">x =</span> <span class="dv">1</span><span class="sc">:</span><span class="fu">length</span>(data), <span class="at">y =</span> data)) <span class="sc">+</span></span>
<span id="cb20-7"><a href="#cb20-7" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_hline</span>(<span class="at">yintercept =</span> <span class="fu">mean</span>(data), </span>
<span id="cb20-8"><a href="#cb20-8" aria-hidden="true" tabindex="-1"></a>             <span class="at">color =</span> <span class="st">"red"</span>, <span class="at">linetype =</span><span class="st">"dashed"</span>) <span class="sc">+</span></span>
<span id="cb20-9"><a href="#cb20-9" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_hline</span>(<span class="at">yintercept =</span> <span class="fu">mean</span>(data) <span class="sc">+</span> <span class="fu">sd</span>(data), </span>
<span id="cb20-10"><a href="#cb20-10" aria-hidden="true" tabindex="-1"></a>             <span class="at">color =</span> <span class="st">"red"</span>, <span class="at">linetype =</span> <span class="st">"dashed"</span>) <span class="sc">+</span></span>
<span id="cb20-11"><a href="#cb20-11" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_hline</span>(<span class="at">yintercept =</span> <span class="fu">mean</span>(data) <span class="sc">-</span> <span class="fu">sd</span>(data), </span>
<span id="cb20-12"><a href="#cb20-12" aria-hidden="true" tabindex="-1"></a>             <span class="at">color =</span> <span class="st">"red"</span>, <span class="at">linetype =</span> <span class="st">"dashed"</span>) <span class="sc">+</span></span>
<span id="cb20-13"><a href="#cb20-13" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_hline</span>(<span class="at">yintercept =</span> <span class="fu">mean</span>(data) <span class="sc">+</span> <span class="fu">var</span>(data), </span>
<span id="cb20-14"><a href="#cb20-14" aria-hidden="true" tabindex="-1"></a>             <span class="at">color =</span> <span class="st">"blue"</span>, <span class="at">linetype =</span> <span class="st">"dashed"</span>) <span class="sc">+</span></span>
<span id="cb20-15"><a href="#cb20-15" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_hline</span>(<span class="at">yintercept =</span> <span class="fu">mean</span>(data) <span class="sc">-</span> <span class="fu">var</span>(data), </span>
<span id="cb20-16"><a href="#cb20-16" aria-hidden="true" tabindex="-1"></a>             <span class="at">color =</span> <span class="st">"blue"</span>, <span class="at">linetype =</span> <span class="st">"dashed"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="KalmanFilterFromTheGroundUp_files/figure-html/unnamed-chunk-8-1.png" class="img-fluid" width="672"></p>
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</div>
<p>Plot distribution density with library function geom_density</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb21"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb21-1"><a href="#cb21-1" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>() <span class="sc">+</span></span>
<span id="cb21-2"><a href="#cb21-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">labs</span>(<span class="at">x =</span> <span class="st">"value"</span>, <span class="at">y=</span> <span class="st">"density"</span>) <span class="sc">+</span></span>
<span id="cb21-3"><a href="#cb21-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_density</span>(<span class="at">data =</span> <span class="fu">data.frame</span>(data), <span class="fu">aes</span>(<span class="at">x =</span> data))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="KalmanFilterFromTheGroundUp_files/figure-html/unnamed-chunk-9-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<p>Calculated distribution density manually</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb22"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb22-1"><a href="#cb22-1" aria-hidden="true" tabindex="-1"></a>mean <span class="ot">&lt;-</span> <span class="fu">mean</span>(data)</span>
<span id="cb22-2"><a href="#cb22-2" aria-hidden="true" tabindex="-1"></a>variance <span class="ot">&lt;-</span> <span class="fu">var</span>(data)</span>
<span id="cb22-3"><a href="#cb22-3" aria-hidden="true" tabindex="-1"></a>sdeviation <span class="ot">&lt;-</span> <span class="fu">sd</span>(data)</span>
<span id="cb22-4"><a href="#cb22-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb22-5"><a href="#cb22-5" aria-hidden="true" tabindex="-1"></a>dencity <span class="ot">&lt;-</span> <span class="cf">function</span>(x) {</span>
<span id="cb22-6"><a href="#cb22-6" aria-hidden="true" tabindex="-1"></a>  <span class="dv">1</span> <span class="sc">/</span> <span class="fu">sqrt</span>(<span class="dv">2</span> <span class="sc">*</span> pi <span class="sc">*</span> variance) <span class="sc">*</span> <span class="fu">exp</span>(<span class="sc">-</span><span class="dv">1</span><span class="sc">/</span><span class="dv">2</span> <span class="sc">*</span> (x <span class="sc">-</span> mean)<span class="sc">^</span><span class="dv">2</span> <span class="sc">/</span> variance)</span>
<span id="cb22-7"><a href="#cb22-7" aria-hidden="true" tabindex="-1"></a>}</span>
<span id="cb22-8"><a href="#cb22-8" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb22-9"><a href="#cb22-9" aria-hidden="true" tabindex="-1"></a>dencityData <span class="ot">&lt;-</span> <span class="fu">dencity</span>(data)</span>
<span id="cb22-10"><a href="#cb22-10" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>() <span class="sc">+</span></span>
<span id="cb22-11"><a href="#cb22-11" aria-hidden="true" tabindex="-1"></a>  <span class="fu">labs</span>(<span class="at">x =</span> <span class="st">"value"</span>, <span class="at">y=</span> <span class="st">"density"</span>) <span class="sc">+</span></span>
<span id="cb22-12"><a href="#cb22-12" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">data =</span> <span class="fu">data.frame</span>(data, dencityData), </span>
<span id="cb22-13"><a href="#cb22-13" aria-hidden="true" tabindex="-1"></a>             <span class="fu">aes</span>(<span class="at">x =</span> data, <span class="at">y =</span> dencityData), <span class="at">color =</span> <span class="st">"red"</span>)<span class="sc">+</span></span>
<span id="cb22-14"><a href="#cb22-14" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_density</span>(<span class="at">data =</span> <span class="fu">data.frame</span>(data), </span>
<span id="cb22-15"><a href="#cb22-15" aria-hidden="true" tabindex="-1"></a>               <span class="fu">aes</span>(<span class="at">x =</span> data))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="KalmanFilterFromTheGroundUp_files/figure-html/unnamed-chunk-10-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<p>Map each data to count of data rounded to 2 decimal places</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb23"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb23-1"><a href="#cb23-1" aria-hidden="true" tabindex="-1"></a>dataCount <span class="ot">&lt;-</span> <span class="fu">data.frame</span>(data) <span class="sc">%&gt;%</span></span>
<span id="cb23-2"><a href="#cb23-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">data =</span> <span class="fu">round</span>(data, <span class="dv">1</span>)) <span class="sc">%&gt;%</span></span>
<span id="cb23-3"><a href="#cb23-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">group_by</span>(data) <span class="sc">%&gt;%</span></span>
<span id="cb23-4"><a href="#cb23-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">summarise</span>(<span class="at">count =</span> <span class="fu">n</span>())</span>
<span id="cb23-5"><a href="#cb23-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb23-6"><a href="#cb23-6" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>() <span class="sc">+</span></span>
<span id="cb23-7"><a href="#cb23-7" aria-hidden="true" tabindex="-1"></a>  <span class="fu">labs</span>(<span class="at">x =</span> <span class="st">"value"</span>, <span class="at">y=</span> <span class="st">"count"</span>) <span class="sc">+</span></span>
<span id="cb23-8"><a href="#cb23-8" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">data =</span> dataCount, <span class="fu">aes</span>(<span class="at">x =</span> data, <span class="at">y =</span> count), <span class="at">color =</span> <span class="st">"red"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="KalmanFilterFromTheGroundUp_files/figure-html/unnamed-chunk-11-1.png" class="img-fluid" width="672"></p>
</div>
</div>
</section>
</section>
<section id="the-alpha-beta-gamma-filter" class="level2">
<h2 class="anchored" data-anchor-id="the-alpha-beta-gamma-filter">THE <span class="math inline">\(\alpha-\beta-\gamma\)</span> FILTER</h2>
<section id="weighting-the-gold-price" class="level3">
<h3 class="anchored" data-anchor-id="weighting-the-gold-price">Weighting the Gold Price</h3>
<p>Collection of measurements</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb24"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb24-1"><a href="#cb24-1" aria-hidden="true" tabindex="-1"></a>measurements <span class="ot">&lt;-</span> <span class="fu">c</span>(<span class="dv">996</span>,  <span class="dv">994</span>,    <span class="dv">1021</span>,   <span class="dv">1000</span>,   <span class="dv">1002</span>,   <span class="dv">1010</span>,   <span class="dv">983</span>,    <span class="dv">971</span>,    <span class="dv">993</span>,    <span class="dv">1023</span>)</span>
<span id="cb24-2"><a href="#cb24-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb24-3"><a href="#cb24-3" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>() <span class="sc">+</span></span>
<span id="cb24-4"><a href="#cb24-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">labs</span>(<span class="at">x =</span> <span class="st">"measurements"</span>, <span class="at">y=</span> <span class="st">"gramms"</span>) <span class="sc">+</span></span>
<span id="cb24-5"><a href="#cb24-5" aria-hidden="true" tabindex="-1"></a>  <span class="fu">scale_x_continuous</span>( <span class="at">breaks=</span><span class="fu">pretty_breaks</span>()) <span class="sc">+</span></span>
<span id="cb24-6"><a href="#cb24-6" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">data =</span> <span class="fu">data.frame</span>(measurements), </span>
<span id="cb24-7"><a href="#cb24-7" aria-hidden="true" tabindex="-1"></a>             <span class="fu">aes</span>(<span class="at">x =</span> <span class="dv">1</span><span class="sc">:</span><span class="fu">length</span>(measurements), <span class="at">y =</span> measurements)) <span class="sc">+</span></span>
<span id="cb24-8"><a href="#cb24-8" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_hline</span>(<span class="at">yintercept =</span> <span class="fu">mean</span>(measurements), </span>
<span id="cb24-9"><a href="#cb24-9" aria-hidden="true" tabindex="-1"></a>             <span class="at">color =</span> <span class="st">"red"</span>, <span class="at">linetype =</span><span class="st">"solid"</span>) <span class="sc">+</span></span>
<span id="cb24-10"><a href="#cb24-10" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_hline</span>(<span class="at">yintercept =</span> <span class="fu">mean</span>(measurements) <span class="sc">+</span> <span class="fu">sd</span>(measurements), </span>
<span id="cb24-11"><a href="#cb24-11" aria-hidden="true" tabindex="-1"></a>             <span class="at">color =</span> <span class="st">"red"</span>, <span class="at">linetype =</span> <span class="st">"dashed"</span>) <span class="sc">+</span></span>
<span id="cb24-12"><a href="#cb24-12" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_hline</span>(<span class="at">yintercept =</span> <span class="fu">mean</span>(measurements) <span class="sc">-</span> <span class="fu">sd</span>(measurements), </span>
<span id="cb24-13"><a href="#cb24-13" aria-hidden="true" tabindex="-1"></a>             <span class="at">color =</span> <span class="st">"red"</span>, <span class="at">linetype =</span> <span class="st">"dashed"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="KalmanFilterFromTheGroundUp_files/figure-html/unnamed-chunk-12-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<p><em>Dynamic model</em> of the system</p>
<p><span class="math display">\[\begin{equation}
x_{n+1} = x_{n}
\end{equation}\]</span></p>
<div class="cell">
<div class="sourceCode cell-code" id="cb25"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb25-1"><a href="#cb25-1" aria-hidden="true" tabindex="-1"></a><span class="co">#' @description  dynamic model of the system</span></span>
<span id="cb25-2"><a href="#cb25-2" aria-hidden="true" tabindex="-1"></a><span class="co">#' </span></span>
<span id="cb25-3"><a href="#cb25-3" aria-hidden="true" tabindex="-1"></a><span class="co">#' @param x_n current state</span></span>
<span id="cb25-4"><a href="#cb25-4" aria-hidden="true" tabindex="-1"></a><span class="co">#' @retuns predicted state</span></span>
<span id="cb25-5"><a href="#cb25-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb25-6"><a href="#cb25-6" aria-hidden="true" tabindex="-1"></a>dynamicModel <span class="ot">&lt;-</span> <span class="cf">function</span>(x_n) {</span>
<span id="cb25-7"><a href="#cb25-7" aria-hidden="true" tabindex="-1"></a>  <span class="fu">return</span>(x_n)</span>
<span id="cb25-8"><a href="#cb25-8" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p><em>State Update Equation</em></p>
<p><span class="math display">\[\begin{equation}
x_{n} = x_{n-1} + \alpha(z_n - x_{n-1})
\end{equation}\]</span></p>
<p>where <span class="math inline">\(\alpha=\frac{1}{n}\)</span></p>
<div class="cell">
<div class="sourceCode cell-code" id="cb26"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb26-1"><a href="#cb26-1" aria-hidden="true" tabindex="-1"></a><span class="co">#' @description State Update Equation</span></span>
<span id="cb26-2"><a href="#cb26-2" aria-hidden="true" tabindex="-1"></a><span class="co">#' </span></span>
<span id="cb26-3"><a href="#cb26-3" aria-hidden="true" tabindex="-1"></a><span class="co">#' @param x_n_1 previous estimation</span></span>
<span id="cb26-4"><a href="#cb26-4" aria-hidden="true" tabindex="-1"></a><span class="co">#' @param z_n current measurement</span></span>
<span id="cb26-5"><a href="#cb26-5" aria-hidden="true" tabindex="-1"></a><span class="co">#' @returns current updated estimation</span></span>
<span id="cb26-6"><a href="#cb26-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb26-7"><a href="#cb26-7" aria-hidden="true" tabindex="-1"></a>stateEquation <span class="ot">&lt;-</span> <span class="cf">function</span>(n, x_n_1, z_n) {</span>
<span id="cb26-8"><a href="#cb26-8" aria-hidden="true" tabindex="-1"></a>  alpha_n <span class="ot">=</span> <span class="dv">1</span><span class="sc">/</span>n</span>
<span id="cb26-9"><a href="#cb26-9" aria-hidden="true" tabindex="-1"></a>  innovation <span class="ot">=</span> (z_n <span class="sc">-</span> x_n_1)</span>
<span id="cb26-10"><a href="#cb26-10" aria-hidden="true" tabindex="-1"></a>  <span class="fu">return</span>(x_n_1 <span class="sc">+</span> alpha_n<span class="sc">*</span>innovation)</span>
<span id="cb26-11"><a href="#cb26-11" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>The Filter (<span class="math inline">\(\alpha\)</span>, <span class="math inline">\(\beta\)</span> often are replaced with <code>g</code> and <code>h</code>)</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb27"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb27-1"><a href="#cb27-1" aria-hidden="true" tabindex="-1"></a><span class="co">#' @description The Filter</span></span>
<span id="cb27-2"><a href="#cb27-2" aria-hidden="true" tabindex="-1"></a><span class="co">#' </span></span>
<span id="cb27-3"><a href="#cb27-3" aria-hidden="true" tabindex="-1"></a><span class="co">#' @param x_0 initial state</span></span>
<span id="cb27-4"><a href="#cb27-4" aria-hidden="true" tabindex="-1"></a><span class="co">#' @param Z measurements</span></span>
<span id="cb27-5"><a href="#cb27-5" aria-hidden="true" tabindex="-1"></a><span class="co">#' @returns array of calculated states</span></span>
<span id="cb27-6"><a href="#cb27-6" aria-hidden="true" tabindex="-1"></a>filterG <span class="ot">&lt;-</span> <span class="cf">function</span>(x_0, Z) {</span>
<span id="cb27-7"><a href="#cb27-7" aria-hidden="true" tabindex="-1"></a>  <span class="co"># predict</span></span>
<span id="cb27-8"><a href="#cb27-8" aria-hidden="true" tabindex="-1"></a>  results <span class="ot">=</span> <span class="fu">c</span>()</span>
<span id="cb27-9"><a href="#cb27-9" aria-hidden="true" tabindex="-1"></a>  x_n_1 <span class="ot">=</span> <span class="fu">dynamicModel</span>(x_0)</span>
<span id="cb27-10"><a href="#cb27-10" aria-hidden="true" tabindex="-1"></a>  n <span class="ot">=</span> <span class="dv">1</span></span>
<span id="cb27-11"><a href="#cb27-11" aria-hidden="true" tabindex="-1"></a>  <span class="cf">for</span> (z_n <span class="cf">in</span> measurements) {</span>
<span id="cb27-12"><a href="#cb27-12" aria-hidden="true" tabindex="-1"></a>    <span class="co"># estimate</span></span>
<span id="cb27-13"><a href="#cb27-13" aria-hidden="true" tabindex="-1"></a>    x_n <span class="ot">=</span> <span class="fu">stateEquation</span>(n, x_n_1, z_n)</span>
<span id="cb27-14"><a href="#cb27-14" aria-hidden="true" tabindex="-1"></a>    <span class="co"># add to estimations</span></span>
<span id="cb27-15"><a href="#cb27-15" aria-hidden="true" tabindex="-1"></a>    results <span class="ot">=</span> <span class="fu">c</span>(results, x_n)</span>
<span id="cb27-16"><a href="#cb27-16" aria-hidden="true" tabindex="-1"></a>    <span class="co"># update</span></span>
<span id="cb27-17"><a href="#cb27-17" aria-hidden="true" tabindex="-1"></a>    x_n_1 <span class="ot">=</span> <span class="fu">dynamicModel</span>(x_n)</span>
<span id="cb27-18"><a href="#cb27-18" aria-hidden="true" tabindex="-1"></a>    n <span class="ot">=</span> n <span class="sc">+</span> <span class="dv">1</span></span>
<span id="cb27-19"><a href="#cb27-19" aria-hidden="true" tabindex="-1"></a>  }</span>
<span id="cb27-20"><a href="#cb27-20" aria-hidden="true" tabindex="-1"></a>  <span class="fu">return</span>(results)</span>
<span id="cb27-21"><a href="#cb27-21" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>Apply the filter with the initial state of the first measurement</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb28"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb28-1"><a href="#cb28-1" aria-hidden="true" tabindex="-1"></a>estimations <span class="ot">&lt;-</span> <span class="fu">filterG</span>(measurements[<span class="fu">c</span>(<span class="dv">1</span>)], measurements)</span>
<span id="cb28-2"><a href="#cb28-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb28-3"><a href="#cb28-3" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>() <span class="sc">+</span></span>
<span id="cb28-4"><a href="#cb28-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">labs</span>(<span class="at">x =</span> <span class="st">"measurements"</span>, <span class="at">y=</span> <span class="st">"gramms"</span>) <span class="sc">+</span></span>
<span id="cb28-5"><a href="#cb28-5" aria-hidden="true" tabindex="-1"></a>  <span class="fu">scale_x_continuous</span>( <span class="at">breaks=</span><span class="fu">pretty_breaks</span>()) <span class="sc">+</span></span>
<span id="cb28-6"><a href="#cb28-6" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_line</span>(<span class="at">data =</span> <span class="fu">data.frame</span>(measurements), </span>
<span id="cb28-7"><a href="#cb28-7" aria-hidden="true" tabindex="-1"></a>            <span class="fu">aes</span>(<span class="at">x =</span> <span class="dv">1</span><span class="sc">:</span><span class="fu">length</span>(measurements), <span class="at">y =</span> measurements)) <span class="sc">+</span></span>
<span id="cb28-8"><a href="#cb28-8" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_hline</span>(<span class="at">yintercept =</span> <span class="fu">mean</span>(measurements), </span>
<span id="cb28-9"><a href="#cb28-9" aria-hidden="true" tabindex="-1"></a>             <span class="at">linetype =</span> <span class="st">"longdash"</span>) <span class="sc">+</span></span>
<span id="cb28-10"><a href="#cb28-10" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_hline</span>(<span class="at">yintercept =</span> <span class="fu">mean</span>(measurements) <span class="sc">+</span> <span class="fu">sd</span>(measurements), </span>
<span id="cb28-11"><a href="#cb28-11" aria-hidden="true" tabindex="-1"></a>             <span class="at">linetype =</span> <span class="st">"dashed"</span>) <span class="sc">+</span></span>
<span id="cb28-12"><a href="#cb28-12" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_hline</span>(<span class="at">yintercept =</span> <span class="fu">mean</span>(measurements) <span class="sc">-</span> <span class="fu">sd</span>(measurements), </span>
<span id="cb28-13"><a href="#cb28-13" aria-hidden="true" tabindex="-1"></a>             <span class="at">linetype =</span> <span class="st">"dashed"</span>) <span class="sc">+</span></span>
<span id="cb28-14"><a href="#cb28-14" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_line</span>(<span class="at">data =</span> <span class="fu">data.frame</span>(estimations), </span>
<span id="cb28-15"><a href="#cb28-15" aria-hidden="true" tabindex="-1"></a>            <span class="fu">aes</span>(<span class="at">x =</span> <span class="dv">1</span><span class="sc">:</span><span class="fu">length</span>(estimations), <span class="at">y =</span> estimations), <span class="at">color =</span> <span class="st">"red"</span>)<span class="sc">+</span></span>
<span id="cb28-16"><a href="#cb28-16" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_hline</span>(<span class="at">yintercept =</span> <span class="fu">mean</span>(estimations), </span>
<span id="cb28-17"><a href="#cb28-17" aria-hidden="true" tabindex="-1"></a>             <span class="at">color =</span> <span class="st">"red"</span>, <span class="at">linetype =</span> <span class="st">"longdash"</span>) <span class="sc">+</span></span>
<span id="cb28-18"><a href="#cb28-18" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_hline</span>(<span class="at">yintercept =</span> <span class="fu">mean</span>(estimations) <span class="sc">+</span> <span class="fu">sd</span>(estimations), </span>
<span id="cb28-19"><a href="#cb28-19" aria-hidden="true" tabindex="-1"></a>             <span class="at">color =</span> <span class="st">"red"</span>, <span class="at">linetype =</span> <span class="st">"dashed"</span>) <span class="sc">+</span></span>
<span id="cb28-20"><a href="#cb28-20" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_hline</span>(<span class="at">yintercept =</span> <span class="fu">mean</span>(estimations) <span class="sc">-</span> <span class="fu">sd</span>(estimations),  </span>
<span id="cb28-21"><a href="#cb28-21" aria-hidden="true" tabindex="-1"></a>             <span class="at">color =</span> <span class="st">"red"</span>, <span class="at">linetype =</span> <span class="st">"dashed"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="KalmanFilterFromTheGroundUp_files/figure-html/unnamed-chunk-16-1.png" class="img-fluid" width="672"></p>
</div>
</div>
</section>
<section id="constant-velocity" class="level3">
<h3 class="anchored" data-anchor-id="constant-velocity">Constant Velocity</h3>
<p>We assume an aircraft that is moving radially away from the radar (or towards the radar).</p>
<p>Collection of measurements</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb29"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb29-1"><a href="#cb29-1" aria-hidden="true" tabindex="-1"></a>measurements <span class="ot">&lt;-</span> <span class="fu">c</span>(<span class="dv">30171</span>,    <span class="dv">30353</span>,  <span class="dv">30756</span>,  <span class="dv">30799</span>,  <span class="dv">31018</span>,  <span class="dv">31278</span>,  <span class="dv">31276</span>,  <span class="dv">31379</span>,</span>
<span id="cb29-2"><a href="#cb29-2" aria-hidden="true" tabindex="-1"></a>                  <span class="dv">31748</span>,    <span class="dv">32175</span>)</span>
<span id="cb29-3"><a href="#cb29-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb29-4"><a href="#cb29-4" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>() <span class="sc">+</span></span>
<span id="cb29-5"><a href="#cb29-5" aria-hidden="true" tabindex="-1"></a>  <span class="fu">labs</span>(<span class="at">x =</span> <span class="st">"measurements"</span>, <span class="at">y=</span> <span class="st">"feets"</span>) <span class="sc">+</span></span>
<span id="cb29-6"><a href="#cb29-6" aria-hidden="true" tabindex="-1"></a>  <span class="fu">scale_x_continuous</span>( <span class="at">breaks=</span><span class="fu">pretty_breaks</span>()) <span class="sc">+</span></span>
<span id="cb29-7"><a href="#cb29-7" aria-hidden="true" tabindex="-1"></a>  <span class="fu">scale_x_continuous</span>(<span class="at">breaks =</span> <span class="fu">pretty_breaks</span>()) <span class="sc">+</span></span>
<span id="cb29-8"><a href="#cb29-8" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">data =</span> <span class="fu">data.frame</span>(measurements), </span>
<span id="cb29-9"><a href="#cb29-9" aria-hidden="true" tabindex="-1"></a>             <span class="fu">aes</span>(<span class="at">x =</span> <span class="dv">1</span><span class="sc">:</span><span class="fu">length</span>(measurements), <span class="at">y =</span> measurements)) <span class="sc">+</span></span>
<span id="cb29-10"><a href="#cb29-10" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_hline</span>(<span class="at">yintercept =</span> <span class="fu">mean</span>(measurements), </span>
<span id="cb29-11"><a href="#cb29-11" aria-hidden="true" tabindex="-1"></a>             <span class="at">color =</span> <span class="st">"red"</span>, <span class="at">linetype =</span><span class="st">"solid"</span>) <span class="sc">+</span></span>
<span id="cb29-12"><a href="#cb29-12" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_hline</span>(<span class="at">yintercept =</span> <span class="fu">mean</span>(measurements) <span class="sc">+</span> <span class="fu">sd</span>(measurements), </span>
<span id="cb29-13"><a href="#cb29-13" aria-hidden="true" tabindex="-1"></a>             <span class="at">color =</span> <span class="st">"red"</span>, <span class="at">linetype =</span> <span class="st">"dashed"</span>) <span class="sc">+</span></span>
<span id="cb29-14"><a href="#cb29-14" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_hline</span>(<span class="at">yintercept =</span> <span class="fu">mean</span>(measurements) <span class="sc">-</span> <span class="fu">sd</span>(measurements), </span>
<span id="cb29-15"><a href="#cb29-15" aria-hidden="true" tabindex="-1"></a>             <span class="at">color =</span> <span class="st">"red"</span>, <span class="at">linetype =</span> <span class="st">"dashed"</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-stderr">
<pre><code>Scale for x is already present.
Adding another scale for x, which will replace the existing scale.</code></pre>
</div>
<div class="cell-output-display">
<p><img src="KalmanFilterFromTheGroundUp_files/figure-html/unnamed-chunk-17-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<p>If <span class="math inline">\(x\)</span> - is range,then velocity <span class="math inline">\(\dot{x}\)</span> can be a derivative of the range :</p>
<p><span class="math display">\[\begin{equation}
\dot{x}=v=\frac{dx}{dt} \\end{equation}
\end{equation}\]</span></p>
<p><em>Dynamic model</em> (also called <em>State Extrapolation Equation</em> or <em>Transition Equation</em> or a <em>Prediction Equation</em>):</p>
<p><span class="math display">\[\begin{equation}
\begin{cases}
x_{n+1} = x_n + \Delta t \dot{x}_n\\
\dot{x}_{n+1} = \dot{x}_n\\
\end{cases}
\end{equation}\]</span></p>
<p>In the matrix form:</p>
<p><span class="math display">\[\begin{equation}
\begin{pmatrix}
x\\
\dot{x}
\end{pmatrix}=
\begin{pmatrix}
1 &amp; \Delta t\\
0 &amp; 1
\end{pmatrix}
\begin{pmatrix}
x_n \\
\dot{x}_n
\end{pmatrix}
\end{equation}\]</span></p>
<div class="cell">
<div class="sourceCode cell-code" id="cb31"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb31-1"><a href="#cb31-1" aria-hidden="true" tabindex="-1"></a><span class="co">#' @description  dynamic model of the system</span></span>
<span id="cb31-2"><a href="#cb31-2" aria-hidden="true" tabindex="-1"></a><span class="co">#' </span></span>
<span id="cb31-3"><a href="#cb31-3" aria-hidden="true" tabindex="-1"></a><span class="co">#' @param dt time interval</span></span>
<span id="cb31-4"><a href="#cb31-4" aria-hidden="true" tabindex="-1"></a><span class="co">#' @param x_n vector of previous state</span></span>
<span id="cb31-5"><a href="#cb31-5" aria-hidden="true" tabindex="-1"></a><span class="co">#' @retuns current state</span></span>
<span id="cb31-6"><a href="#cb31-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb31-7"><a href="#cb31-7" aria-hidden="true" tabindex="-1"></a>dynamicModel <span class="ot">&lt;-</span> <span class="cf">function</span>(dt, x_n) {</span>
<span id="cb31-8"><a href="#cb31-8" aria-hidden="true" tabindex="-1"></a>  A <span class="ot">=</span> <span class="fu">matrix</span>(<span class="fu">c</span>(<span class="dv">1</span>, dt, <span class="dv">0</span>, <span class="dv">1</span>), <span class="dv">2</span>,<span class="dv">2</span>, <span class="cn">TRUE</span>)</span>
<span id="cb31-9"><a href="#cb31-9" aria-hidden="true" tabindex="-1"></a>  <span class="fu">return</span>(A <span class="sc">%*%</span> x_n)</span>
<span id="cb31-10"><a href="#cb31-10" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>Let’s <span class="math inline">\(z_n\)</span> be a measurement at the <span class="math inline">\(\Delta t\)</span> time interval. Then the state update equation for velocity can be written as:</p>
<p><span class="math display">\[\begin{equation}
\hat{\dot{x}}=\hat{\dot{x}}_{n-1} - \beta(\frac{z_n - \hat{x}_{n-1}}{\Delta t})
\end{equation}\]</span></p>
<p>where <span class="math inline">\(\hat{\dot{x}}\)</span> is the estimated state and <span class="math inline">\(\beta\)</span> is the weighting factor to scale the impact of the new measurement.</p>
<p>And the <em>state update equation</em> for position can be written as:</p>
<p><span class="math display">\[\begin{equation}
\hat{x} = \hat{x}_{n-1} + \alpha(z_n - \hat{x}_{n-1} )
\end{equation}\]</span></p>
<p>where <span class="math inline">\(\alpha\)</span> is the weighting factor to scale the impact of the new measurement.</p>
<p>where <span class="math inline">\(\alpha\)</span> is the weighting factor to scale the impact of the new measurement.</p>
<p>Unlike the “Weighting the Gold Price” example <span class="math inline">\(\alpha\)</span> is constant and depends onthe measurement precision.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb32"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb32-1"><a href="#cb32-1" aria-hidden="true" tabindex="-1"></a><span class="co">#' @description State Update Equation</span></span>
<span id="cb32-2"><a href="#cb32-2" aria-hidden="true" tabindex="-1"></a><span class="co">#' </span></span>
<span id="cb32-3"><a href="#cb32-3" aria-hidden="true" tabindex="-1"></a><span class="co">#' @param dt time interval needed to derive velocity</span></span>
<span id="cb32-4"><a href="#cb32-4" aria-hidden="true" tabindex="-1"></a><span class="co">#' @param x_n_1 previous estimation vector</span></span>
<span id="cb32-5"><a href="#cb32-5" aria-hidden="true" tabindex="-1"></a><span class="co">#' @param z_n current measurement</span></span>
<span id="cb32-6"><a href="#cb32-6" aria-hidden="true" tabindex="-1"></a><span class="co">#' @returns current updated estimation</span></span>
<span id="cb32-7"><a href="#cb32-7" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb32-8"><a href="#cb32-8" aria-hidden="true" tabindex="-1"></a>stateEquation <span class="ot">&lt;-</span> <span class="cf">function</span>(alpha, beta, dt, x_n_1, z_n) {</span>
<span id="cb32-9"><a href="#cb32-9" aria-hidden="true" tabindex="-1"></a>  xHat_n_1 <span class="ot">=</span> x_n_1[<span class="dv">1</span>,<span class="dv">1</span>]</span>
<span id="cb32-10"><a href="#cb32-10" aria-hidden="true" tabindex="-1"></a>  innovationX <span class="ot">=</span> (z_n <span class="sc">-</span> xHat_n_1)</span>
<span id="cb32-11"><a href="#cb32-11" aria-hidden="true" tabindex="-1"></a>  xHat_n <span class="ot">=</span> xHat_n_1 <span class="sc">+</span> alpha<span class="sc">*</span>innovationX</span>
<span id="cb32-12"><a href="#cb32-12" aria-hidden="true" tabindex="-1"></a>  </span>
<span id="cb32-13"><a href="#cb32-13" aria-hidden="true" tabindex="-1"></a>  vHat_n_1 <span class="ot">=</span> x_n_1[<span class="dv">2</span>,<span class="dv">1</span>]</span>
<span id="cb32-14"><a href="#cb32-14" aria-hidden="true" tabindex="-1"></a>  innovationV <span class="ot">=</span> (z_n <span class="sc">-</span> xHat_n_1)<span class="sc">/</span>dt</span>
<span id="cb32-15"><a href="#cb32-15" aria-hidden="true" tabindex="-1"></a>  vHat <span class="ot">=</span> (vHat_n_1 <span class="sc">+</span> beta<span class="sc">*</span>innovationV)</span>
<span id="cb32-16"><a href="#cb32-16" aria-hidden="true" tabindex="-1"></a>  </span>
<span id="cb32-17"><a href="#cb32-17" aria-hidden="true" tabindex="-1"></a>  <span class="fu">return</span>(<span class="fu">matrix</span>(<span class="fu">c</span>(xHat_n, vHat), <span class="at">nrow =</span> <span class="dv">2</span>, <span class="at">ncol =</span> <span class="dv">1</span>))</span>
<span id="cb32-18"><a href="#cb32-18" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>The filter</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb33"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb33-1"><a href="#cb33-1" aria-hidden="true" tabindex="-1"></a><span class="co">#' @description The Filter</span></span>
<span id="cb33-2"><a href="#cb33-2" aria-hidden="true" tabindex="-1"></a><span class="co">#' </span></span>
<span id="cb33-3"><a href="#cb33-3" aria-hidden="true" tabindex="-1"></a><span class="co">#' @param x_0 initial state</span></span>
<span id="cb33-4"><a href="#cb33-4" aria-hidden="true" tabindex="-1"></a><span class="co">#' @param Z measurements</span></span>
<span id="cb33-5"><a href="#cb33-5" aria-hidden="true" tabindex="-1"></a><span class="co">#' @returns array of calculated states</span></span>
<span id="cb33-6"><a href="#cb33-6" aria-hidden="true" tabindex="-1"></a>filterGF <span class="ot">&lt;-</span> <span class="cf">function</span>(alpha, beta, x_0, Z,dt) {</span>
<span id="cb33-7"><a href="#cb33-7" aria-hidden="true" tabindex="-1"></a>  <span class="co"># setup time interval between measurements 5 seconds</span></span>
<span id="cb33-8"><a href="#cb33-8" aria-hidden="true" tabindex="-1"></a>  <span class="co"># predict</span></span>
<span id="cb33-9"><a href="#cb33-9" aria-hidden="true" tabindex="-1"></a>  results <span class="ot">=</span> <span class="fu">list</span>()</span>
<span id="cb33-10"><a href="#cb33-10" aria-hidden="true" tabindex="-1"></a>  x_n_1 <span class="ot">=</span> <span class="fu">dynamicModel</span>(dt, x_0)</span>
<span id="cb33-11"><a href="#cb33-11" aria-hidden="true" tabindex="-1"></a>  <span class="cf">for</span> (z_n <span class="cf">in</span> Z) {</span>
<span id="cb33-12"><a href="#cb33-12" aria-hidden="true" tabindex="-1"></a>    <span class="co"># estimate</span></span>
<span id="cb33-13"><a href="#cb33-13" aria-hidden="true" tabindex="-1"></a>    x_n <span class="ot">=</span> <span class="fu">stateEquation</span>(alpha, beta, dt, x_n_1, z_n)</span>
<span id="cb33-14"><a href="#cb33-14" aria-hidden="true" tabindex="-1"></a>    <span class="co"># add to estimations</span></span>
<span id="cb33-15"><a href="#cb33-15" aria-hidden="true" tabindex="-1"></a>    results <span class="ot">=</span> <span class="fu">c</span>(results, <span class="fu">list</span>(x_n))</span>
<span id="cb33-16"><a href="#cb33-16" aria-hidden="true" tabindex="-1"></a>    <span class="co"># update</span></span>
<span id="cb33-17"><a href="#cb33-17" aria-hidden="true" tabindex="-1"></a>    x_n_1 <span class="ot">=</span> <span class="fu">dynamicModel</span>(dt, x_n)</span>
<span id="cb33-18"><a href="#cb33-18" aria-hidden="true" tabindex="-1"></a>  }</span>
<span id="cb33-19"><a href="#cb33-19" aria-hidden="true" tabindex="-1"></a>  <span class="fu">return</span>(results)</span>
<span id="cb33-20"><a href="#cb33-20" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>Apply the filter with the initial state of the first measurement with <span class="math inline">\(\alpha=0.2\)</span> and <span class="math inline">\(\beta=0.1\)</span></p>
<div class="cell">
<div class="sourceCode cell-code" id="cb34"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb34-1"><a href="#cb34-1" aria-hidden="true" tabindex="-1"></a>estimations <span class="ot">&lt;-</span> <span class="fu">filterGF</span>(<span class="fl">0.2</span>, <span class="fl">0.1</span>, <span class="fu">matrix</span>(<span class="fu">c</span>(<span class="dv">30000</span>, <span class="dv">40</span>),<span class="dv">2</span>), measurements,<span class="dv">5</span>)</span>
<span id="cb34-2"><a href="#cb34-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb34-3"><a href="#cb34-3" aria-hidden="true" tabindex="-1"></a>estimationsX <span class="ot">&lt;-</span> <span class="fu">unlist</span>(<span class="fu">sapply</span>(estimations, <span class="cf">function</span>(x) x[<span class="dv">1</span>]))</span>
<span id="cb34-4"><a href="#cb34-4" aria-hidden="true" tabindex="-1"></a>estimationsV <span class="ot">&lt;-</span> <span class="fu">unlist</span>(<span class="fu">sapply</span>(estimations, <span class="cf">function</span>(x) x[<span class="dv">2</span>]))</span>
<span id="cb34-5"><a href="#cb34-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb34-6"><a href="#cb34-6" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>() <span class="sc">+</span></span>
<span id="cb34-7"><a href="#cb34-7" aria-hidden="true" tabindex="-1"></a>  <span class="fu">labs</span>(<span class="at">x =</span> <span class="st">"measurements"</span>, <span class="at">y=</span> <span class="st">"feets"</span>) <span class="sc">+</span></span>
<span id="cb34-8"><a href="#cb34-8" aria-hidden="true" tabindex="-1"></a>  <span class="fu">scale_x_continuous</span>( <span class="at">breaks=</span><span class="fu">pretty_breaks</span>()) <span class="sc">+</span></span>
<span id="cb34-9"><a href="#cb34-9" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_line</span>(<span class="at">data =</span> <span class="fu">data.frame</span>(measurements), </span>
<span id="cb34-10"><a href="#cb34-10" aria-hidden="true" tabindex="-1"></a>            <span class="fu">aes</span>(<span class="at">x =</span> <span class="dv">1</span><span class="sc">:</span><span class="fu">length</span>(measurements), <span class="at">y =</span> measurements)) <span class="sc">+</span></span>
<span id="cb34-11"><a href="#cb34-11" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_line</span>(<span class="at">data =</span> <span class="fu">data.frame</span>(estimationsX), </span>
<span id="cb34-12"><a href="#cb34-12" aria-hidden="true" tabindex="-1"></a>            <span class="fu">aes</span>(<span class="at">x =</span> <span class="dv">1</span><span class="sc">:</span><span class="fu">length</span>(estimationsX), <span class="at">y =</span> estimationsX), <span class="at">color =</span> <span class="st">"red"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="KalmanFilterFromTheGroundUp_files/figure-html/unnamed-chunk-21-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<p>Apply the filter with the initial state of the first measurement with <span class="math inline">\(\alpha=0.8\)</span> and <span class="math inline">\(\beta=0.5\)</span></p>
<div class="cell">
<div class="sourceCode cell-code" id="cb35"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb35-1"><a href="#cb35-1" aria-hidden="true" tabindex="-1"></a>estimations <span class="ot">&lt;-</span> <span class="fu">filterGF</span>(<span class="fl">0.8</span>, <span class="fl">0.5</span>, <span class="fu">matrix</span>(<span class="fu">c</span>(<span class="dv">30000</span>, <span class="dv">40</span>),<span class="dv">2</span>), measurements,<span class="dv">5</span>)</span>
<span id="cb35-2"><a href="#cb35-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb35-3"><a href="#cb35-3" aria-hidden="true" tabindex="-1"></a>estimationsX <span class="ot">&lt;-</span> <span class="fu">unlist</span>(<span class="fu">sapply</span>(estimations, <span class="cf">function</span>(x) x[<span class="dv">1</span>]))</span>
<span id="cb35-4"><a href="#cb35-4" aria-hidden="true" tabindex="-1"></a>estimationsV <span class="ot">&lt;-</span> <span class="fu">unlist</span>(<span class="fu">sapply</span>(estimations, <span class="cf">function</span>(x) x[<span class="dv">2</span>]))</span>
<span id="cb35-5"><a href="#cb35-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb35-6"><a href="#cb35-6" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>() <span class="sc">+</span></span>
<span id="cb35-7"><a href="#cb35-7" aria-hidden="true" tabindex="-1"></a>  <span class="fu">labs</span>(<span class="at">x =</span> <span class="st">"measurements"</span>, <span class="at">y=</span> <span class="st">"feets"</span>) <span class="sc">+</span></span>
<span id="cb35-8"><a href="#cb35-8" aria-hidden="true" tabindex="-1"></a>  <span class="fu">scale_x_continuous</span>( <span class="at">breaks=</span><span class="fu">pretty_breaks</span>()) <span class="sc">+</span></span>
<span id="cb35-9"><a href="#cb35-9" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_line</span>(<span class="at">data =</span> <span class="fu">data.frame</span>(measurements), </span>
<span id="cb35-10"><a href="#cb35-10" aria-hidden="true" tabindex="-1"></a>            <span class="fu">aes</span>(<span class="at">x =</span> <span class="dv">1</span><span class="sc">:</span><span class="fu">length</span>(measurements), <span class="at">y =</span> measurements)) <span class="sc">+</span></span>
<span id="cb35-11"><a href="#cb35-11" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_line</span>(<span class="at">data =</span> <span class="fu">data.frame</span>(estimationsX), </span>
<span id="cb35-12"><a href="#cb35-12" aria-hidden="true" tabindex="-1"></a>            <span class="fu">aes</span>(<span class="at">x =</span> <span class="dv">1</span><span class="sc">:</span><span class="fu">length</span>(estimationsX), <span class="at">y =</span> estimationsX), <span class="at">color =</span> <span class="st">"red"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="KalmanFilterFromTheGroundUp_files/figure-html/unnamed-chunk-22-1.png" class="img-fluid" width="672"></p>
</div>
</div>
</section>
</section>

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});
</script>
</div> <!-- /content -->


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