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        <title>James Lawton</title>
        <link>https://JamesL813.github.io/</link>
        <description>Recent content on James Lawton</description>
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        <lastBuildDate>Tue, 25 Apr 2023 00:00:00 +0000</lastBuildDate><atom:link href="https://JamesL813.github.io/index.xml" rel="self" type="application/rss+xml" /><item>
        <title>KD Trees and Ball Trees</title>
        <link>https://JamesL813.github.io/p/kd-trees-and-ball-trees/</link>
        <pubDate>Tue, 25 Apr 2023 00:00:00 +0000</pubDate>
        
        <guid>https://JamesL813.github.io/p/kd-trees-and-ball-trees/</guid>
        <description>&lt;img src="https://JamesL813.github.io/img/ball_tree_diagram.jpg" alt="Featured image of post KD Trees and Ball Trees" /&gt;&lt;h2 id=&#34;motivation&#34;&gt;Motivation&lt;/h2&gt;
&lt;p&gt;Imagine we are interested in determining the CMDA 4654 classmate that lives nearest to us. The “brute force” approach would be to ask each classmate where they live (i.e. make the search region , compute the distance between their house and ours, and then select the minimum distance. This is quite computationally expensive. In fact, it has a time complexity of O(2n), assuming 2-dimensional distance computations. &lt;br&gt;
&lt;br&gt;
A better approach might be to partition the class into different groups based on the broader attributes of their location. For instance, we can ask everyone who lives in Blacksburg to stand on one side of the classroom, while those that don’t (i.e. live in Christiansburg) stand on the other. Then, if we live in Blacksburg, we are able to reduce the size of the group that we compute our distance to. &lt;br&gt;
&lt;br&gt;
We can continue to partition the group based on other factors (i.e. on-campus vs off-campus, north vs south, etc) to further reduce the size of the search group. Then, the time complexity of our algorithm is going to be O(n+log(n)), which is much better than vanilla kNN.&lt;br&gt;
&lt;br&gt;
However, we might find someone who lives on the edge of campus and whose nearest neighbor is off-campus, which will never be found due to the clustering nature of this algorithm.&lt;br&gt;
&lt;br&gt;
This idea of partitioning the data into subsets that will later be used to search for nearest neighbor within that subset (and hope that it’s the global nearest neighbor) is the idea behind clustering using k-d trees and ball trees.&lt;/p&gt;
&lt;h2 id=&#34;intro-to-decision-trees&#34;&gt;Intro to Decision Trees&lt;/h2&gt;
&lt;p&gt;A tree is a common data structure composed of a hierarchical structure with branch nodes, and leaf nodes. They can be useful for providing a visual representation for a data set or model. &lt;br&gt;
&lt;br&gt;
A binary tree is one of the simplest implementations of the structure. It’s simply a way of storing a set of ordered data. The rules that make up a binary tree are as follows:&lt;br&gt;
\&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Each node must have at most 2 children&lt;/li&gt;
&lt;li&gt;A node’s left child must come before it&lt;/li&gt;
&lt;li&gt;A node’s right child must come after it.&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;Using these simple rules, we can create a data structure that is computationally efficient to retrieve data from.\&lt;/p&gt;
&lt;p&gt;Notice that each node in this example tree not only stores a data point, but it also represents a decision that each incoming data point has to make as it goes through the tree. Whether the result of this decision is true or false determines which direction the point will move next.&lt;/p&gt;
&lt;p&gt;&lt;img src=&#34;https://JamesL813.github.io/p/kd-trees-and-ball-trees/img/intro_to_decision_tree_graph.png&#34;
	width=&#34;505&#34;
	height=&#34;442&#34;
	srcset=&#34;https://JamesL813.github.io/p/kd-trees-and-ball-trees/img/intro_to_decision_tree_graph_hu04e33ba3a4d7e94c30cb7e311964b447_21767_480x0_resize_box_3.png 480w, https://JamesL813.github.io/p/kd-trees-and-ball-trees/img/intro_to_decision_tree_graph_hu04e33ba3a4d7e94c30cb7e311964b447_21767_1024x0_resize_box_3.png 1024w&#34;
	loading=&#34;lazy&#34;
	
		alt=&#34;Binary Search Tree&#34;
	
	
		class=&#34;gallery-image&#34; 
		data-flex-grow=&#34;114&#34;
		data-flex-basis=&#34;274px&#34;
	
&gt;&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&#34;kd-trees&#34;&gt;KD Trees&lt;/h2&gt;
&lt;p&gt;A K-D tree (or a K-dimensional tree) is a special type of decision tree that can organize data for efficient spatial search or nearest-neighbor search within K-dimensional space.&lt;br&gt;
&lt;br&gt;
In data modeling, each one of K dimensions normally represents a specific feature of the data set.&lt;br&gt;
&lt;br&gt;
The basic idea of k-d trees is to partition our feature space along the axes and prune subtrees that are too far away from our nearest neighbor(s) (essentially depth first search with pruning).&lt;/p&gt;
&lt;p&gt;&lt;img src=&#34;https://JamesL813.github.io/p/kd-trees-and-ball-trees/img/kdtrees_pic.png&#34;
	width=&#34;572&#34;
	height=&#34;242&#34;
	srcset=&#34;https://JamesL813.github.io/p/kd-trees-and-ball-trees/img/kdtrees_pic_huaeb8696dc0c495cc4337f29b70c7fd6b_53351_480x0_resize_box_3.png 480w, https://JamesL813.github.io/p/kd-trees-and-ball-trees/img/kdtrees_pic_huaeb8696dc0c495cc4337f29b70c7fd6b_53351_1024x0_resize_box_3.png 1024w&#34;
	loading=&#34;lazy&#34;
	
		alt=&#34;KD Trees [Source: Carnegie Mellon]&#34;
	
	
		class=&#34;gallery-image&#34; 
		data-flex-grow=&#34;236&#34;
		data-flex-basis=&#34;567px&#34;
	
&gt;&lt;/p&gt;
&lt;h2 id=&#34;process&#34;&gt;Process&lt;/h2&gt;
&lt;p&gt;The basic idea of k-d trees is to partition our feature space along the axes and prune subtrees that are too far away from our nearest neighbor(s).&lt;br&gt;
\&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;We construct the tree by partitioning our data into halves along one feature&lt;/li&gt;
&lt;li&gt;We then rotate through our features based on which of them have the largest maximum variance and continue partitioning our data.&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;&lt;br&gt;
When considering a test point and its nearest neighbors, we calculate the distance between them, call it $x_d$
\&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Compare $x_d$ to the distance between the point and the partition, $x_p$&lt;/li&gt;
&lt;li&gt;If $x_p &amp;gt; x_d$, we can prune that subtree&lt;/li&gt;
&lt;li&gt;If $x_p &amp;lt; x_d$, recalculate nearest neighbor&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;&lt;br&gt;
Continue until all leaf nodes have been visited.&lt;/p&gt;
&lt;h2 id=&#34;limitations&#34;&gt;Limitations&lt;/h2&gt;
&lt;p&gt;K-d trees are really only effective in low dimensional spaces and is not very accurate in high dimensional spaces.&lt;br&gt;
\&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;In higher dimensions, the points end up being close to every partition (i.e. a million partition lines)&lt;/li&gt;
&lt;li&gt;Partitions are along axes and span the entire feature space, which proves to be problematic as the number of features grows&lt;/li&gt;
&lt;li&gt;Nearest neighbors begins to lose meaning since the distance between the closest and farthest point are very similar.&lt;/li&gt;
&lt;/ul&gt;
&lt;h2 id=&#34;example-the-iris-dataset&#34;&gt;Example: The IRIS Dataset&lt;/h2&gt;
&lt;p&gt;We will use the Iris data to demonstrate KD trees using&lt;/p&gt;
&lt;p&gt;&lt;img src=&#34;https://JamesL813.github.io/p/kd-trees-and-ball-trees/img/kd_box1.png&#34;
	width=&#34;394&#34;
	height=&#34;320&#34;
	srcset=&#34;https://JamesL813.github.io/p/kd-trees-and-ball-trees/img/kd_box1_hu1c908732a4bdfe0f8c92c99bc0e57377_27367_480x0_resize_box_3.png 480w, https://JamesL813.github.io/p/kd-trees-and-ball-trees/img/kd_box1_hu1c908732a4bdfe0f8c92c99bc0e57377_27367_1024x0_resize_box_3.png 1024w&#34;
	loading=&#34;lazy&#34;
	
		alt=&#34;KD Tree 1&#34;
	
	
		class=&#34;gallery-image&#34; 
		data-flex-grow=&#34;123&#34;
		data-flex-basis=&#34;295px&#34;
	
&gt;&lt;/p&gt;
&lt;hr&gt;
&lt;p&gt;&lt;img src=&#34;https://JamesL813.github.io/p/kd-trees-and-ball-trees/img/kd_box2.png&#34;
	width=&#34;394&#34;
	height=&#34;321&#34;
	srcset=&#34;https://JamesL813.github.io/p/kd-trees-and-ball-trees/img/kd_box2_huafa4ea0738b4e8dbee7e39d7197906e6_27905_480x0_resize_box_3.png 480w, https://JamesL813.github.io/p/kd-trees-and-ball-trees/img/kd_box2_huafa4ea0738b4e8dbee7e39d7197906e6_27905_1024x0_resize_box_3.png 1024w&#34;
	loading=&#34;lazy&#34;
	
		alt=&#34;KD Tree 2&#34;
	
	
		class=&#34;gallery-image&#34; 
		data-flex-grow=&#34;122&#34;
		data-flex-basis=&#34;294px&#34;
	
&gt;&lt;/p&gt;
&lt;hr&gt;
&lt;p&gt;&lt;img src=&#34;https://JamesL813.github.io/p/kd-trees-and-ball-trees/img/kd_box3.png&#34;
	width=&#34;979&#34;
	height=&#34;792&#34;
	srcset=&#34;https://JamesL813.github.io/p/kd-trees-and-ball-trees/img/kd_box3_hueb654af92ec2427c686e6d81f28d5ea9_119668_480x0_resize_box_3.png 480w, https://JamesL813.github.io/p/kd-trees-and-ball-trees/img/kd_box3_hueb654af92ec2427c686e6d81f28d5ea9_119668_1024x0_resize_box_3.png 1024w&#34;
	loading=&#34;lazy&#34;
	
		alt=&#34;KD Tree 3&#34;
	
	
		class=&#34;gallery-image&#34; 
		data-flex-grow=&#34;123&#34;
		data-flex-basis=&#34;296px&#34;
	
&gt;&lt;/p&gt;
&lt;hr&gt;
&lt;p&gt;Code to draw these rectangles in R:&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;div class=&#34;chroma&#34;&gt;
&lt;table class=&#34;lntable&#34;&gt;&lt;tr&gt;&lt;td class=&#34;lntd&#34;&gt;
&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code&gt;&lt;span class=&#34;lnt&#34;&gt; 1
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 2
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 3
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 4
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 5
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 6
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 7
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 8
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 9
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;10
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;11
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;12
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;13
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;14
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;15
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;16
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;17
&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/td&gt;
&lt;td class=&#34;lntd&#34;&gt;
&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-fallback&#34; data-lang=&#34;fallback&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;library(ggplot2)
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;ggplot(iris, aes(x=Petal.Length, y=Petal.Width, col=Species)) + geom_point() + 
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  labs(x=&amp;#34;Petal Length&amp;#34;, y=&amp;#34;Petal Width&amp;#34;, title = &amp;#34;Petal Length vs. Petal Width&amp;#34;) + theme_bw() + 
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  #geom_vline(xintercept = median(iris$Petal.Length)) + 
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  geom_rect(aes(xmin=4.35, xmax=6.9, ymin=1.8, ymax=2.5), fill=NA, color=&amp;#34;blue&amp;#34;, alpha=0.25) + 
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  geom_rect(aes(xmin=4.35, xmax=6.9, ymin=0, ymax=1.8), fill=NA, color=&amp;#34;blue&amp;#34;, alpha=0.25) + 
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  geom_rect(aes(xmin=0, xmax=4.35, ymin=0.3, ymax=2.5), fill=NA, color=&amp;#34;blue&amp;#34;, alpha=0.25) + 
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  geom_rect(aes(xmin=0, xmax=4.35, ymin=0, ymax=0.3), fill=NA, color=&amp;#34;blue&amp;#34;, alpha=0.25) + 
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  geom_rect(aes(xmin=3.85, xmax=4.35, ymin=0.3, ymax=2.5), fill=NA, color=&amp;#34;green&amp;#34;, alpha=0.25) +
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  geom_rect(aes(xmin=0, xmax=3.85, ymin=0.3, ymax=2.5), fill=NA, color=&amp;#34;green&amp;#34;, alpha=0.25) + 
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  geom_rect(aes(xmin=0, xmax=1.4, ymin=0, ymax=0.3), fill=NA, color=&amp;#34;green&amp;#34;, alpha=0.25) + 
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  geom_rect(aes(xmin=1.4, xmax=4.35, ymin=0, ymax=0.3), fill=NA, color=&amp;#34;green&amp;#34;, alpha=0.25) + 
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  geom_rect(aes(xmin=4.35, xmax=4.6, ymin=0, ymax=1.8), fill=NA, color=&amp;#34;green&amp;#34;, alpha=0.25) + 
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  geom_rect(aes(xmin=4.6, xmax=6.9, ymin=0, ymax=1.8), fill=NA, color=&amp;#34;green&amp;#34;, alpha=0.25) + 
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  geom_rect(aes(xmin=4.35, xmax=5.6, ymin=1.8, ymax=2.5), fill=NA, color=&amp;#34;green&amp;#34;, alpha=0.25) +
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  geom_rect(aes(xmin=5.6, xmax=6.9, ymin=1.8, ymax=2.5), fill=NA, color=&amp;#34;green&amp;#34;, alpha=0.25)
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;
&lt;/div&gt;
&lt;/div&gt;&lt;hr&gt;
&lt;p&gt;&lt;img src=&#34;https://JamesL813.github.io/p/kd-trees-and-ball-trees/img/kd_box4.png&#34;
	width=&#34;965&#34;
	height=&#34;789&#34;
	srcset=&#34;https://JamesL813.github.io/p/kd-trees-and-ball-trees/img/kd_box4_hub6e0809dc7a252b9d414e61530065603_85177_480x0_resize_box_3.png 480w, https://JamesL813.github.io/p/kd-trees-and-ball-trees/img/kd_box4_hub6e0809dc7a252b9d414e61530065603_85177_1024x0_resize_box_3.png 1024w&#34;
	loading=&#34;lazy&#34;
	
		alt=&#34;KD Tree 4&#34;
	
	
		class=&#34;gallery-image&#34; 
		data-flex-grow=&#34;122&#34;
		data-flex-basis=&#34;293px&#34;
	
&gt;&lt;/p&gt;
&lt;h2 id=&#34;ball-trees&#34;&gt;Ball trees&lt;/h2&gt;
&lt;p&gt;A ball tree is a space partitioning data structure for organizing points in a multidimensional space. &lt;br&gt;
&lt;br&gt;
Each internal node partitions the data into two disjoint sets which are associated with different balls. Each point is assigned to the ball with the closest center.&lt;/p&gt;
&lt;p&gt;$$
D^B (t) =
\begin{cases}
\max( |t - B.pivot | - B.radius, D^{B.parent} ),\quad \text{ if }B \neq R \
\max( |t - B.pivot | - B.radius, D^{B.parent}, 0 ), \text{ if }B = R \
\end{cases}
$$&lt;/p&gt;
&lt;p&gt;&lt;img src=&#34;https://JamesL813.github.io/p/kd-trees-and-ball-trees/img/ball_tree_diagram.png&#34;
	width=&#34;1319&#34;
	height=&#34;739&#34;
	srcset=&#34;https://JamesL813.github.io/p/kd-trees-and-ball-trees/img/ball_tree_diagram_hud8d3d15a230b5af5a6a5d95c7875bd46_191053_480x0_resize_box_3.png 480w, https://JamesL813.github.io/p/kd-trees-and-ball-trees/img/ball_tree_diagram_hud8d3d15a230b5af5a6a5d95c7875bd46_191053_1024x0_resize_box_3.png 1024w&#34;
	loading=&#34;lazy&#34;
	
		alt=&#34;ball tree diagram&#34;
	
	
		class=&#34;gallery-image&#34; 
		data-flex-grow=&#34;178&#34;
		data-flex-basis=&#34;428px&#34;
	
&gt;&lt;/p&gt;
&lt;h2 id=&#34;motivation-1&#34;&gt;Motivation&lt;/h2&gt;
&lt;p&gt;The ball tree algorithm is similar to k-d trees in that they both partition in terms of the feature space, but unlike k-d trees, the trees are partitioned into hyperplane spheres instead of boxes. &lt;br&gt;
&lt;br&gt;
While ball tree performance may suffer in lower dimensional feature spaces,
Ball tree performance of with O(n log(n)) is still better than kNN training speed of O(nd)&lt;/p&gt;
&lt;h2 id=&#34;example-the-iris-dataset-1&#34;&gt;Example: The IRIS dataset&lt;/h2&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;div class=&#34;chroma&#34;&gt;
&lt;table class=&#34;lntable&#34;&gt;&lt;tr&gt;&lt;td class=&#34;lntd&#34;&gt;
&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code&gt;&lt;span class=&#34;lnt&#34;&gt; 1
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 2
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 3
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 4
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 5
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 6
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 7
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 8
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 9
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;10
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;11
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;12
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;13
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;14
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;15
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;16
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;17
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;18
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;19
&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/td&gt;
&lt;td class=&#34;lntd&#34;&gt;
&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-fallback&#34; data-lang=&#34;fallback&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;source(&amp;#34;ball_tree.R&amp;#34;)
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; # scales all numeric columns
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;iris_scaled &amp;lt;- iris %&amp;gt;%
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  mutate_if(is.numeric, scale)
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;#                    columns to plot
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;isc &amp;lt;- iris_scaled[, c(3:4)]
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;# builds the tree
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;bt &amp;lt;- build_ball_tree(isc, c(1:nrow(isc), min_points = 2))
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;ggplot() + geom_balltree(bt, max_depth = 4) +
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    geom_point(
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        data = iris_scaled,
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        pch = 21,
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        size = 3,
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        aes(x = Petal.Length, y = Petal.Width, fill = Species)
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    ) +
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    theme_classic() + scale_colour_stepsn(colors = c(&amp;#34;purple4&amp;#34;,&amp;#34;red&amp;#34;,&amp;#34;orange1&amp;#34;))
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;
&lt;/div&gt;
&lt;/div&gt;&lt;hr&gt;
&lt;p&gt;&lt;img src=&#34;https://JamesL813.github.io/p/kd-trees-and-ball-trees/img/iris_petal_ball.png&#34;
	width=&#34;3000&#34;
	height=&#34;2550&#34;
	srcset=&#34;https://JamesL813.github.io/p/kd-trees-and-ball-trees/img/iris_petal_ball_hue56fb58cda12bf7feaa75f33b6f278e0_584223_480x0_resize_box_3.png 480w, https://JamesL813.github.io/p/kd-trees-and-ball-trees/img/iris_petal_ball_hue56fb58cda12bf7feaa75f33b6f278e0_584223_1024x0_resize_box_3.png 1024w&#34;
	loading=&#34;lazy&#34;
	
		alt=&#34;image&#34;
	
	
		class=&#34;gallery-image&#34; 
		data-flex-grow=&#34;117&#34;
		data-flex-basis=&#34;282px&#34;
	
&gt;&lt;/p&gt;
&lt;h2 id=&#34;housing-data&#34;&gt;Housing Data&lt;/h2&gt;
&lt;p&gt;The housing data we chose to use analyzed 12 different building shapes. Each building differs with respect to the glazing area, the glazing area distribution, and the orientation, amongst other parameters. With the data being found &lt;a class=&#34;link&#34; href=&#34;https://drive.google.com/drive/u/1/folders/1W_Qc1XoOJUjNWN14925pfaIVZKvrBahP&#34;  target=&#34;_blank&#34; rel=&#34;noopener&#34;
    &gt;here&lt;/a&gt;.&lt;/p&gt;
&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;Name&lt;/th&gt;
&lt;th&gt;Definition&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;X1&lt;/td&gt;
&lt;td&gt;Relative Compactness&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;X2&lt;/td&gt;
&lt;td&gt;Surface Area&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;X3&lt;/td&gt;
&lt;td&gt;Wall Area&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;X4&lt;/td&gt;
&lt;td&gt;Roof Area&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;X5&lt;/td&gt;
&lt;td&gt;Overall Height&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;X6&lt;/td&gt;
&lt;td&gt;Orientation&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;X7&lt;/td&gt;
&lt;td&gt;Glazing Area&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;X8&lt;/td&gt;
&lt;td&gt;Glazing Area Distribution&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;y1&lt;/td&gt;
&lt;td&gt;Heating Load&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;y2&lt;/td&gt;
&lt;td&gt;Cooling Load&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;h2 id=&#34;kd-tree-approach&#34;&gt;KD tree approach&lt;/h2&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;div class=&#34;chroma&#34;&gt;
&lt;table class=&#34;lntable&#34;&gt;&lt;tr&gt;&lt;td class=&#34;lntd&#34;&gt;
&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code&gt;&lt;span class=&#34;lnt&#34;&gt; 1
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 2
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 3
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 4
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 5
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 6
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 7
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 8
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 9
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;10
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;11
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;12
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;13
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;14
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;15
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;16
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;17
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;18
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;19
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;20
&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/td&gt;
&lt;td class=&#34;lntd&#34;&gt;
&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-fallback&#34; data-lang=&#34;fallback&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;library(FNN)
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;library(caret)
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;house = read.csv(&amp;#34;data/ENB2012_data.csv&amp;#34;)
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;house1 = house[c(1:768),-c(6,8)]
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;index &amp;lt;- sample(1:nrow(house1), round(nrow(house1) * 0.7))
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;training_df &amp;lt;- house1[index, ]
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;testing_df &amp;lt;- house1[-index, ]
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;# Store the training/testing data features
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;train_features &amp;lt;- training_df[,-3]
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;test_features &amp;lt;- testing_df[, -3]
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;# Store the actual labels
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;train_classes &amp;lt;- training_df$X3
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;test_classes &amp;lt;- testing_df$X3
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;house_kd = knn(train = train_features, test = test_features, cl = train_classes, k = 5, algorithm=c(&amp;#34;kd_tree&amp;#34;))
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;confusionMatrix(data = house_kd, reference = as.factor(test_classes))
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;
&lt;/div&gt;
&lt;/div&gt;&lt;h2 id=&#34;confusion-matrix&#34;&gt;Confusion Matrix&lt;/h2&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;div class=&#34;chroma&#34;&gt;
&lt;table class=&#34;lntable&#34;&gt;&lt;tr&gt;&lt;td class=&#34;lntd&#34;&gt;
&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code&gt;&lt;span class=&#34;lnt&#34;&gt; 1
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 2
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 3
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 4
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 5
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 6
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 7
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 8
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 9
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;10
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;11
&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/td&gt;
&lt;td class=&#34;lntd&#34;&gt;
&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-fallback&#34; data-lang=&#34;fallback&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;confusion.table = table(&amp;#34;predicted&amp;#34; = house_kd, &amp;#34;actual&amp;#34; = as.factor(test_classes))
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;plt &amp;lt;- as.data.frame(confusion.table)
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;plt$predicted &amp;lt;- factor(plt$predicted, levels=rev(levels(plt$predicted)))
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;ggplot(plt, aes(actual,predicted, fill= Freq)) +
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        geom_tile() + geom_text(aes(label=Freq)) +
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        scale_fill_gradient(low=&amp;#34;white&amp;#34;, high=&amp;#34;#CF4420&amp;#34;) +
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        labs(x = &amp;#34;Reference&amp;#34;,y = &amp;#34;Prediction&amp;#34;) +
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        scale_x_discrete(labels=c(&amp;#34;245&amp;#34;,&amp;#34;269.5&amp;#34;,&amp;#34;294&amp;#34;,&amp;#34;318.5&amp;#34;,&amp;#34;343&amp;#34;,&amp;#34;367.5&amp;#34;,&amp;#34;416.5&amp;#34;),  position = &amp;#34;top&amp;#34;) +
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        scale_y_discrete(labels=c(&amp;#34;416.5&amp;#34;,&amp;#34;367.5&amp;#34;,&amp;#34;343&amp;#34;,&amp;#34;318.5&amp;#34;,&amp;#34;294&amp;#34;,&amp;#34;269.5&amp;#34;,&amp;#34;245&amp;#34;))
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;
&lt;/div&gt;
&lt;/div&gt;&lt;hr&gt;
&lt;p&gt;&lt;img src=&#34;https://JamesL813.github.io/p/kd-trees-and-ball-trees/img/kd_house_cm.png&#34;
	width=&#34;2689&#34;
	height=&#34;1823&#34;
	srcset=&#34;https://JamesL813.github.io/p/kd-trees-and-ball-trees/img/kd_house_cm_hu18608a10de6c6646d36f37e532f86e11_209258_480x0_resize_box_3.png 480w, https://JamesL813.github.io/p/kd-trees-and-ball-trees/img/kd_house_cm_hu18608a10de6c6646d36f37e532f86e11_209258_1024x0_resize_box_3.png 1024w&#34;
	loading=&#34;lazy&#34;
	
		alt=&#34;Confusion Matrix&#34;
	
	
		class=&#34;gallery-image&#34; 
		data-flex-grow=&#34;147&#34;
		data-flex-basis=&#34;354px&#34;
	
&gt;&lt;/p&gt;
&lt;h2 id=&#34;ball-tree-approach&#34;&gt;Ball Tree Approach&lt;/h2&gt;
&lt;h2 id=&#34;knn-approach&#34;&gt;kNN Approach&lt;/h2&gt;
&lt;p&gt;For reference, we will show the kNN approach of classifying our data set&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;div class=&#34;chroma&#34;&gt;
&lt;table class=&#34;lntable&#34;&gt;&lt;tr&gt;&lt;td class=&#34;lntd&#34;&gt;
&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code&gt;&lt;span class=&#34;lnt&#34;&gt;1
&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/td&gt;
&lt;td class=&#34;lntd&#34;&gt;
&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-fallback&#34; data-lang=&#34;fallback&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;knn(train = train_features, test = test_features, cl = train_classes, k = 5, algorithm=c(&amp;#34;brute&amp;#34;))
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;
&lt;/div&gt;
&lt;/div&gt;</description>
        </item>
        <item>
        <title>Raindrop Air Resistance Simulation</title>
        <link>https://JamesL813.github.io/p/raindrop-air-resistance-simulation/</link>
        <pubDate>Thu, 29 Sep 2022 00:00:00 +0000</pubDate>
        
        <guid>https://JamesL813.github.io/p/raindrop-air-resistance-simulation/</guid>
        <description>&lt;img src="https://JamesL813.github.io/p/raindrop-air-resistance-simulation/cover.jpg" alt="Featured image of post Raindrop Air Resistance Simulation" /&gt;&lt;p&gt;A simple python simulation&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Simulates a falling raindrop with air resistance and mass&lt;/li&gt;
&lt;li&gt;Used to write a paper analyzing how air resistance affects terminal velocity and distance traveled&lt;/li&gt;
&lt;/ul&gt;
</description>
        </item>
        <item>
        <title>Schrodinger Equation Simulation</title>
        <link>https://JamesL813.github.io/p/schrodinger-equation-simulation/</link>
        <pubDate>Thu, 29 Sep 2022 00:00:00 +0000</pubDate>
        
        <guid>https://JamesL813.github.io/p/schrodinger-equation-simulation/</guid>
        <description>&lt;img src="https://JamesL813.github.io/p/schrodinger-equation-simulation/cover.jpg" alt="Featured image of post Schrodinger Equation Simulation" /&gt;&lt;p&gt;A simple python simulation&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Simulates a falling raindrop with air resistance and mass&lt;/li&gt;
&lt;li&gt;Used to write a paper analyzing how air resistance affects terminal velocity and distance traveled&lt;/li&gt;
&lt;/ul&gt;
</description>
        </item>
        <item>
        <title>Dijkstra’s Path-finding Algorithm Simulation</title>
        <link>https://JamesL813.github.io/p/dijkstras-path-finding-algorithm-simulation/</link>
        <pubDate>Wed, 29 Jun 2022 00:00:00 +0000</pubDate>
        
        <guid>https://JamesL813.github.io/p/dijkstras-path-finding-algorithm-simulation/</guid>
        <description>&lt;img src="https://JamesL813.github.io/p/dijkstras-path-finding-algorithm-simulation/cover.jpg" alt="Featured image of post Dijkstra’s Path-finding Algorithm Simulation" /&gt;&lt;p&gt;A simple python simulation&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Utilizes Dijkstra’s Path-finding Algorithm&lt;/li&gt;
&lt;li&gt;Finds the shortest paths to each point from a starting point on a directed graph.&lt;/li&gt;
&lt;/ul&gt;
</description>
        </item>
        <item>
        <title>Geographic Information System</title>
        <link>https://JamesL813.github.io/p/geographic-information-system/</link>
        <pubDate>Sun, 29 May 2022 00:00:00 +0000</pubDate>
        
        <guid>https://JamesL813.github.io/p/geographic-information-system/</guid>
        <description>&lt;img src="https://JamesL813.github.io/p/geographic-information-system/cover.jpg" alt="Featured image of post Geographic Information System" /&gt;&lt;ul&gt;
&lt;li&gt;Tool to parse through Geographic Data&lt;/li&gt;
&lt;li&gt;Uses a custom built PR Quad-Tree and Hash Table to store locations and names.&lt;/li&gt;
&lt;li&gt;Can easily query by location range and name range.&lt;/li&gt;
&lt;/ul&gt;
</description>
        </item>
        <item>
        <title>Hash Table</title>
        <link>https://JamesL813.github.io/p/hash-table/</link>
        <pubDate>Fri, 29 Apr 2022 00:00:00 +0000</pubDate>
        
        <guid>https://JamesL813.github.io/p/hash-table/</guid>
        <description>&lt;img src="https://JamesL813.github.io/p/hash-table/cover.jpg" alt="Featured image of post Hash Table" /&gt;&lt;ul&gt;
&lt;li&gt;Data structure made to be versatile with a variable bucket size.&lt;/li&gt;
&lt;li&gt;Less space efficiency but more speed when querying&lt;/li&gt;
&lt;/ul&gt;
</description>
        </item>
        <item>
        <title>PR Quad Tree</title>
        <link>https://JamesL813.github.io/p/pr-quad-tree/</link>
        <pubDate>Tue, 29 Mar 2022 00:00:00 +0000</pubDate>
        
        <guid>https://JamesL813.github.io/p/pr-quad-tree/</guid>
        <description>&lt;img src="https://JamesL813.github.io/p/pr-quad-tree/cover.jpg" alt="Featured image of post PR Quad Tree" /&gt;&lt;ul&gt;
&lt;li&gt;Data structure similar to a binary tree.&lt;/li&gt;
&lt;li&gt;Each node contains four, a Northwest, Northeast, Southwest, and Southeast&lt;/li&gt;
&lt;li&gt;Efficient when storing and querying data that can be identified by a location coordinate.&lt;/li&gt;
&lt;/ul&gt;
</description>
        </item>
        <item>
        <title>Archives</title>
        <link>https://JamesL813.github.io/archives/</link>
        <pubDate>Tue, 28 May 2019 00:00:00 +0000</pubDate>
        
        <guid>https://JamesL813.github.io/archives/</guid>
        <description></description>
        </item>
        <item>
        <title>Markdown Syntax Guide</title>
        <link>https://JamesL813.github.io/p/markdown-syntax-guide/</link>
        <pubDate>Mon, 11 Mar 2019 00:00:00 +0000</pubDate>
        
        <guid>https://JamesL813.github.io/p/markdown-syntax-guide/</guid>
        <description>&lt;img src="https://JamesL813.github.io/p/markdown-syntax-guide/pawel-czerwinski-8uZPynIu-rQ-unsplash.jpg" alt="Featured image of post Markdown Syntax Guide" /&gt;&lt;p&gt;This article offers a sample of basic Markdown syntax that can be used in Hugo content files, also it shows whether basic HTML elements are decorated with CSS in a Hugo theme.&lt;/p&gt;
&lt;h2 id=&#34;headings&#34;&gt;Headings&lt;/h2&gt;
&lt;p&gt;The following HTML &lt;code&gt;&amp;lt;h1&amp;gt;&lt;/code&gt;—&lt;code&gt;&amp;lt;h6&amp;gt;&lt;/code&gt; elements represent six levels of section headings. &lt;code&gt;&amp;lt;h1&amp;gt;&lt;/code&gt; is the highest section level while &lt;code&gt;&amp;lt;h6&amp;gt;&lt;/code&gt; is the lowest.&lt;/p&gt;
&lt;h1 id=&#34;h1&#34;&gt;H1&lt;/h1&gt;
&lt;h2 id=&#34;h2&#34;&gt;H2&lt;/h2&gt;
&lt;h3 id=&#34;h3&#34;&gt;H3&lt;/h3&gt;
&lt;h4 id=&#34;h4&#34;&gt;H4&lt;/h4&gt;
&lt;h5 id=&#34;h5&#34;&gt;H5&lt;/h5&gt;
&lt;h6 id=&#34;h6&#34;&gt;H6&lt;/h6&gt;
&lt;h2 id=&#34;paragraph&#34;&gt;Paragraph&lt;/h2&gt;
&lt;p&gt;Xerum, quo qui aut unt expliquam qui dolut labo. Aque venitatiusda cum, voluptionse latur sitiae dolessi aut parist aut dollo enim qui voluptate ma dolestendit peritin re plis aut quas inctum laceat est volestemque commosa as cus endigna tectur, offic to cor sequas etum rerum idem sintibus eiur? Quianimin porecus evelectur, cum que nis nust voloribus ratem aut omnimi, sitatur? Quiatem. Nam, omnis sum am facea corem alique molestrunt et eos evelece arcillit ut aut eos eos nus, sin conecerem erum fuga. Ri oditatquam, ad quibus unda veliamenimin cusam et facea ipsamus es exerum sitate dolores editium rerore eost, temped molorro ratiae volorro te reribus dolorer sperchicium faceata tiustia prat.&lt;/p&gt;
&lt;p&gt;Itatur? Quiatae cullecum rem ent aut odis in re eossequodi nonsequ idebis ne sapicia is sinveli squiatum, core et que aut hariosam ex eat.&lt;/p&gt;
&lt;p&gt;😄&lt;/p&gt;
&lt;h2 id=&#34;blockquotes&#34;&gt;Blockquotes&lt;/h2&gt;
&lt;p&gt;The blockquote element represents content that is quoted from another source, optionally with a citation which must be within a &lt;code&gt;footer&lt;/code&gt; or &lt;code&gt;cite&lt;/code&gt; element, and optionally with in-line changes such as annotations and abbreviations.&lt;/p&gt;
&lt;h4 id=&#34;blockquote-without-attribution&#34;&gt;Blockquote without attribution&lt;/h4&gt;
&lt;blockquote&gt;
&lt;p&gt;Tiam, ad mint andaepu dandae nostion secatur sequo quae.
&lt;strong&gt;Note&lt;/strong&gt; that you can use &lt;em&gt;Markdown syntax&lt;/em&gt; within a blockquote.&lt;/p&gt;
&lt;/blockquote&gt;
&lt;h4 id=&#34;blockquote-with-attribution&#34;&gt;Blockquote with attribution&lt;/h4&gt;
&lt;blockquote&gt;
&lt;p&gt;Don&amp;rsquo;t communicate by sharing memory, share memory by communicating.&lt;!-- raw HTML omitted --&gt;
— &lt;!-- raw HTML omitted --&gt;Rob Pike&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;&lt;!-- raw HTML omitted --&gt;&lt;/p&gt;
&lt;/blockquote&gt;
&lt;h2 id=&#34;tables&#34;&gt;Tables&lt;/h2&gt;
&lt;p&gt;Tables aren&amp;rsquo;t part of the core Markdown spec, but Hugo supports supports them out-of-the-box.&lt;/p&gt;
&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;Name&lt;/th&gt;
&lt;th&gt;Age&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;Bob&lt;/td&gt;
&lt;td&gt;27&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Alice&lt;/td&gt;
&lt;td&gt;23&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;h4 id=&#34;inline-markdown-within-tables&#34;&gt;Inline Markdown within tables&lt;/h4&gt;
&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;Italics&lt;/th&gt;
&lt;th&gt;Bold&lt;/th&gt;
&lt;th&gt;Code&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;&lt;em&gt;italics&lt;/em&gt;&lt;/td&gt;
&lt;td&gt;&lt;strong&gt;bold&lt;/strong&gt;&lt;/td&gt;
&lt;td&gt;&lt;code&gt;code&lt;/code&gt;&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;A&lt;/th&gt;
&lt;th&gt;B&lt;/th&gt;
&lt;th&gt;C&lt;/th&gt;
&lt;th&gt;D&lt;/th&gt;
&lt;th&gt;E&lt;/th&gt;
&lt;th&gt;F&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;Lorem ipsum dolor sit amet, consectetur adipiscing elit.&lt;/td&gt;
&lt;td&gt;Phasellus ultricies, sapien non euismod aliquam, dui ligula tincidunt odio, at accumsan nulla sapien eget ex.&lt;/td&gt;
&lt;td&gt;Proin eleifend dictum ipsum, non euismod ipsum pulvinar et. Vivamus sollicitudin, quam in pulvinar aliquam, metus elit pretium purus&lt;/td&gt;
&lt;td&gt;Proin sit amet velit nec enim imperdiet vehicula.&lt;/td&gt;
&lt;td&gt;Ut bibendum vestibulum quam, eu egestas turpis gravida nec&lt;/td&gt;
&lt;td&gt;Sed scelerisque nec turpis vel viverra. Vivamus vitae pretium sapien&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;h2 id=&#34;code-blocks&#34;&gt;Code Blocks&lt;/h2&gt;
&lt;h4 id=&#34;code-block-with-backticks&#34;&gt;Code block with backticks&lt;/h4&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;div class=&#34;chroma&#34;&gt;
&lt;table class=&#34;lntable&#34;&gt;&lt;tr&gt;&lt;td class=&#34;lntd&#34;&gt;
&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code&gt;&lt;span class=&#34;lnt&#34;&gt; 1
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 2
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 3
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 4
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 5
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 6
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 7
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 8
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 9
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;10
&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/td&gt;
&lt;td class=&#34;lntd&#34;&gt;
&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-html&#34; data-lang=&#34;html&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;cp&#34;&gt;&amp;lt;!doctype html&amp;gt;&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&#34;nt&#34;&gt;html&lt;/span&gt; &lt;span class=&#34;na&#34;&gt;lang&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;en&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&#34;nt&#34;&gt;head&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&#34;nt&#34;&gt;meta&lt;/span&gt; &lt;span class=&#34;na&#34;&gt;charset&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;utf-8&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&#34;nt&#34;&gt;title&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;Example HTML5 Document&lt;span class=&#34;p&#34;&gt;&amp;lt;/&lt;/span&gt;&lt;span class=&#34;nt&#34;&gt;title&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;&amp;lt;/&lt;/span&gt;&lt;span class=&#34;nt&#34;&gt;head&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&#34;nt&#34;&gt;body&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&#34;nt&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;Test&lt;span class=&#34;p&#34;&gt;&amp;lt;/&lt;/span&gt;&lt;span class=&#34;nt&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;&amp;lt;/&lt;/span&gt;&lt;span class=&#34;nt&#34;&gt;body&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;&amp;lt;/&lt;/span&gt;&lt;span class=&#34;nt&#34;&gt;html&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;
&lt;/div&gt;
&lt;/div&gt;&lt;h4 id=&#34;code-block-indented-with-four-spaces&#34;&gt;Code block indented with four spaces&lt;/h4&gt;
&lt;pre&gt;&lt;code&gt;&amp;lt;!doctype html&amp;gt;
&amp;lt;html lang=&amp;quot;en&amp;quot;&amp;gt;
&amp;lt;head&amp;gt;
  &amp;lt;meta charset=&amp;quot;utf-8&amp;quot;&amp;gt;
  &amp;lt;title&amp;gt;Example HTML5 Document&amp;lt;/title&amp;gt;
&amp;lt;/head&amp;gt;
&amp;lt;body&amp;gt;
  &amp;lt;p&amp;gt;Test&amp;lt;/p&amp;gt;
&amp;lt;/body&amp;gt;
&amp;lt;/html&amp;gt;
&lt;/code&gt;&lt;/pre&gt;
&lt;h4 id=&#34;code-block-with-hugos-internal-highlight-shortcode&#34;&gt;Code block with Hugo&amp;rsquo;s internal highlight shortcode&lt;/h4&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;div class=&#34;chroma&#34;&gt;
&lt;table class=&#34;lntable&#34;&gt;&lt;tr&gt;&lt;td class=&#34;lntd&#34;&gt;
&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code&gt;&lt;span class=&#34;lnt&#34;&gt; 1
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 2
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 3
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 4
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 5
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 6
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 7
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 8
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt; 9
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;10
&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/td&gt;
&lt;td class=&#34;lntd&#34;&gt;
&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-html&#34; data-lang=&#34;html&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;cp&#34;&gt;&amp;lt;!doctype html&amp;gt;&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&#34;nt&#34;&gt;html&lt;/span&gt; &lt;span class=&#34;na&#34;&gt;lang&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;en&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&#34;nt&#34;&gt;head&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&#34;nt&#34;&gt;meta&lt;/span&gt; &lt;span class=&#34;na&#34;&gt;charset&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;utf-8&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&#34;nt&#34;&gt;title&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;Example HTML5 Document&lt;span class=&#34;p&#34;&gt;&amp;lt;/&lt;/span&gt;&lt;span class=&#34;nt&#34;&gt;title&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;&amp;lt;/&lt;/span&gt;&lt;span class=&#34;nt&#34;&gt;head&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&#34;nt&#34;&gt;body&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&#34;nt&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;Test&lt;span class=&#34;p&#34;&gt;&amp;lt;/&lt;/span&gt;&lt;span class=&#34;nt&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;&amp;lt;/&lt;/span&gt;&lt;span class=&#34;nt&#34;&gt;body&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;&amp;lt;/&lt;/span&gt;&lt;span class=&#34;nt&#34;&gt;html&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;h4 id=&#34;diff-code-block&#34;&gt;Diff code block&lt;/h4&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;div class=&#34;chroma&#34;&gt;
&lt;table class=&#34;lntable&#34;&gt;&lt;tr&gt;&lt;td class=&#34;lntd&#34;&gt;
&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code&gt;&lt;span class=&#34;lnt&#34;&gt;1
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;2
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;3
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;4
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;5
&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/td&gt;
&lt;td class=&#34;lntd&#34;&gt;
&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-diff&#34; data-lang=&#34;diff&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;[dependencies.bevy]
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;git = &amp;#34;https://github.com/bevyengine/bevy&amp;#34;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;rev = &amp;#34;11f52b8c72fc3a568e8bb4a4cd1f3eb025ac2e13&amp;#34;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;gd&#34;&gt;- features = [&amp;#34;dynamic&amp;#34;]
&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;gd&#34;&gt;&lt;/span&gt;&lt;span class=&#34;gi&#34;&gt;+ features = [&amp;#34;jpeg&amp;#34;, &amp;#34;dynamic&amp;#34;]
&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;
&lt;/div&gt;
&lt;/div&gt;&lt;h2 id=&#34;list-types&#34;&gt;List Types&lt;/h2&gt;
&lt;h4 id=&#34;ordered-list&#34;&gt;Ordered List&lt;/h4&gt;
&lt;ol&gt;
&lt;li&gt;First item&lt;/li&gt;
&lt;li&gt;Second item&lt;/li&gt;
&lt;li&gt;Third item&lt;/li&gt;
&lt;/ol&gt;
&lt;h4 id=&#34;unordered-list&#34;&gt;Unordered List&lt;/h4&gt;
&lt;ul&gt;
&lt;li&gt;List item&lt;/li&gt;
&lt;li&gt;Another item&lt;/li&gt;
&lt;li&gt;And another item&lt;/li&gt;
&lt;/ul&gt;
&lt;h4 id=&#34;nested-list&#34;&gt;Nested list&lt;/h4&gt;
&lt;ul&gt;
&lt;li&gt;Fruit
&lt;ul&gt;
&lt;li&gt;Apple&lt;/li&gt;
&lt;li&gt;Orange&lt;/li&gt;
&lt;li&gt;Banana&lt;/li&gt;
&lt;/ul&gt;
&lt;/li&gt;
&lt;li&gt;Dairy
&lt;ul&gt;
&lt;li&gt;Milk&lt;/li&gt;
&lt;li&gt;Cheese&lt;/li&gt;
&lt;/ul&gt;
&lt;/li&gt;
&lt;/ul&gt;
&lt;h2 id=&#34;other-elements--abbr-sub-sup-kbd-mark&#34;&gt;Other Elements — abbr, sub, sup, kbd, mark&lt;/h2&gt;
&lt;p&gt;&lt;!-- raw HTML omitted --&gt;GIF&lt;!-- raw HTML omitted --&gt; is a bitmap image format.&lt;/p&gt;
&lt;p&gt;H&lt;!-- raw HTML omitted --&gt;2&lt;!-- raw HTML omitted --&gt;O&lt;/p&gt;
&lt;p&gt;X&lt;!-- raw HTML omitted --&gt;n&lt;!-- raw HTML omitted --&gt; + Y&lt;!-- raw HTML omitted --&gt;n&lt;!-- raw HTML omitted --&gt; = Z&lt;!-- raw HTML omitted --&gt;n&lt;!-- raw HTML omitted --&gt;&lt;/p&gt;
&lt;p&gt;Press &lt;!-- raw HTML omitted --&gt;&lt;!-- raw HTML omitted --&gt;CTRL&lt;!-- raw HTML omitted --&gt;+&lt;!-- raw HTML omitted --&gt;ALT&lt;!-- raw HTML omitted --&gt;+&lt;!-- raw HTML omitted --&gt;Delete&lt;!-- raw HTML omitted --&gt;&lt;!-- raw HTML omitted --&gt; to end the session.&lt;/p&gt;
&lt;p&gt;Most &lt;!-- raw HTML omitted --&gt;salamanders&lt;!-- raw HTML omitted --&gt; are nocturnal, and hunt for insects, worms, and other small creatures.&lt;/p&gt;
&lt;h2 id=&#34;hyperlinked-image&#34;&gt;Hyperlinked image&lt;/h2&gt;
&lt;p&gt;&lt;a class=&#34;link&#34; href=&#34;https://google.com&#34;  target=&#34;_blank&#34; rel=&#34;noopener&#34;
    &gt;&lt;img src=&#34;https://www.google.com/images/branding/googlelogo/1x/googlelogo_light_color_272x92dp.png&#34;
	
	
	
	loading=&#34;lazy&#34;
	
		alt=&#34;Google&#34;
	
	
&gt;&lt;/a&gt;&lt;/p&gt;
&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
&lt;hr&gt;
&lt;ol&gt;
&lt;li id=&#34;fn:1&#34;&gt;
&lt;p&gt;The above quote is excerpted from Rob Pike&amp;rsquo;s &lt;a class=&#34;link&#34; href=&#34;https://www.youtube.com/watch?v=PAAkCSZUG1c&#34;  target=&#34;_blank&#34; rel=&#34;noopener&#34;
    &gt;talk&lt;/a&gt; during Gopherfest, November 18, 2015.&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
&lt;/li&gt;
&lt;/ol&gt;
&lt;/div&gt;</description>
        </item>
        <item>
        <title>Math Typesetting</title>
        <link>https://JamesL813.github.io/p/math-typesetting/</link>
        <pubDate>Fri, 08 Mar 2019 00:00:00 +0000</pubDate>
        
        <guid>https://JamesL813.github.io/p/math-typesetting/</guid>
        <description>&lt;p&gt;Mathematical notation in a Hugo project can be enabled by using third party JavaScript libraries.&lt;/p&gt;
&lt;p&gt;In this example we will be using &lt;a class=&#34;link&#34; href=&#34;https://katex.org/&#34;  target=&#34;_blank&#34; rel=&#34;noopener&#34;
    &gt;KaTeX&lt;/a&gt;&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Create a partial under &lt;code&gt;/layouts/partials/math.html&lt;/code&gt;&lt;/li&gt;
&lt;li&gt;Within this partial reference the &lt;a class=&#34;link&#34; href=&#34;https://katex.org/docs/autorender.html&#34;  target=&#34;_blank&#34; rel=&#34;noopener&#34;
    &gt;Auto-render Extension&lt;/a&gt; or host these scripts locally.&lt;/li&gt;
&lt;li&gt;Include the partial in your templates like so:&lt;/li&gt;
&lt;/ul&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;div class=&#34;chroma&#34;&gt;
&lt;table class=&#34;lntable&#34;&gt;&lt;tr&gt;&lt;td class=&#34;lntd&#34;&gt;
&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code&gt;&lt;span class=&#34;lnt&#34;&gt;1
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;2
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;3
&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/td&gt;
&lt;td class=&#34;lntd&#34;&gt;
&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-bash&#34; data-lang=&#34;bash&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;o&#34;&gt;{{&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; or .Params.math .Site.Params.math &lt;span class=&#34;o&#34;&gt;}}&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;o&#34;&gt;{{&lt;/span&gt; partial &lt;span class=&#34;s2&#34;&gt;&amp;#34;math.html&amp;#34;&lt;/span&gt; . &lt;span class=&#34;o&#34;&gt;}}&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;o&#34;&gt;{{&lt;/span&gt; end &lt;span class=&#34;o&#34;&gt;}}&lt;/span&gt;
&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;
&lt;/div&gt;
&lt;/div&gt;&lt;ul&gt;
&lt;li&gt;To enable KaTex globally set the parameter &lt;code&gt;math&lt;/code&gt; to &lt;code&gt;true&lt;/code&gt; in a project&amp;rsquo;s configuration&lt;/li&gt;
&lt;li&gt;To enable KaTex on a per page basis include the parameter &lt;code&gt;math: true&lt;/code&gt; in content files&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; Use the online reference of &lt;a class=&#34;link&#34; href=&#34;https://katex.org/docs/supported.html&#34;  target=&#34;_blank&#34; rel=&#34;noopener&#34;
    &gt;Supported TeX Functions&lt;/a&gt;&lt;/p&gt;

&lt;h3 id=&#34;examples&#34;&gt;Examples&lt;/h3&gt;

&lt;p&gt;Block math:
$$
\varphi = 1+\frac{1} {1+\frac{1} {1+\frac{1} {1+\cdots} } }
$$&lt;/p&gt;</description>
        </item>
        <item>
        <title>About</title>
        <link>https://JamesL813.github.io/about/</link>
        <pubDate>Thu, 28 Feb 2019 00:00:00 +0000</pubDate>
        
        <guid>https://JamesL813.github.io/about/</guid>
        <description>&lt;p&gt;Written in Go, Hugo is an open source static site generator available under the &lt;a class=&#34;link&#34; href=&#34;https://github.com/gohugoio/hugo/blob/master/LICENSE&#34;  target=&#34;_blank&#34; rel=&#34;noopener&#34;
    &gt;Apache Licence 2.0.&lt;/a&gt; Hugo supports TOML, YAML and JSON data file types, Markdown and HTML content files and uses shortcodes to add rich content. Other notable features are taxonomies, multilingual mode, image processing, custom output formats, HTML/CSS/JS minification and support for Sass SCSS workflows.&lt;/p&gt;
&lt;p&gt;Hugo makes use of a variety of open source projects including:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;a class=&#34;link&#34; href=&#34;https://github.com/yuin/goldmark&#34;  target=&#34;_blank&#34; rel=&#34;noopener&#34;
    &gt;https://github.com/yuin/goldmark&lt;/a&gt;&lt;/li&gt;
&lt;li&gt;&lt;a class=&#34;link&#34; href=&#34;https://github.com/alecthomas/chroma&#34;  target=&#34;_blank&#34; rel=&#34;noopener&#34;
    &gt;https://github.com/alecthomas/chroma&lt;/a&gt;&lt;/li&gt;
&lt;li&gt;&lt;a class=&#34;link&#34; href=&#34;https://github.com/muesli/smartcrop&#34;  target=&#34;_blank&#34; rel=&#34;noopener&#34;
    &gt;https://github.com/muesli/smartcrop&lt;/a&gt;&lt;/li&gt;
&lt;li&gt;&lt;a class=&#34;link&#34; href=&#34;https://github.com/spf13/cobra&#34;  target=&#34;_blank&#34; rel=&#34;noopener&#34;
    &gt;https://github.com/spf13/cobra&lt;/a&gt;&lt;/li&gt;
&lt;li&gt;&lt;a class=&#34;link&#34; href=&#34;https://github.com/spf13/viper&#34;  target=&#34;_blank&#34; rel=&#34;noopener&#34;
    &gt;https://github.com/spf13/viper&lt;/a&gt;&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;Hugo is ideal for blogs, corporate websites, creative portfolios, online magazines, single page applications or even a website with thousands of pages.&lt;/p&gt;
&lt;p&gt;Hugo is for people who want to hand code their own website without worrying about setting up complicated runtimes, dependencies and databases.&lt;/p&gt;
&lt;p&gt;Websites built with Hugo are extremelly fast, secure and can be deployed anywhere including, AWS, GitHub Pages, Heroku, Netlify and any other hosting provider.&lt;/p&gt;
&lt;p&gt;Learn more and contribute on &lt;a class=&#34;link&#34; href=&#34;https://github.com/gohugoio&#34;  target=&#34;_blank&#34; rel=&#34;noopener&#34;
    &gt;GitHub&lt;/a&gt;.&lt;/p&gt;
</description>
        </item>
        <item>
        <title>Links</title>
        <link>https://JamesL813.github.io/links/</link>
        <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
        
        <guid>https://JamesL813.github.io/links/</guid>
        <description>&lt;p&gt;To use this feature, add &lt;code&gt;links&lt;/code&gt; section to frontmatter.&lt;/p&gt;
&lt;p&gt;This page&amp;rsquo;s frontmatter:&lt;/p&gt;
&lt;div class=&#34;highlight&#34;&gt;&lt;div class=&#34;chroma&#34;&gt;
&lt;table class=&#34;lntable&#34;&gt;&lt;tr&gt;&lt;td class=&#34;lntd&#34;&gt;
&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code&gt;&lt;span class=&#34;lnt&#34;&gt;1
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;2
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;3
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;4
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;5
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;6
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;7
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;8
&lt;/span&gt;&lt;span class=&#34;lnt&#34;&gt;9
&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/td&gt;
&lt;td class=&#34;lntd&#34;&gt;
&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-yaml&#34; data-lang=&#34;yaml&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nt&#34;&gt;links&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;w&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;w&#34;&gt;  &lt;/span&gt;- &lt;span class=&#34;nt&#34;&gt;title&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;w&#34;&gt; &lt;/span&gt;&lt;span class=&#34;l&#34;&gt;GitHub&lt;/span&gt;&lt;span class=&#34;w&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;w&#34;&gt;    &lt;/span&gt;&lt;span class=&#34;nt&#34;&gt;description&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;w&#34;&gt; &lt;/span&gt;&lt;span class=&#34;l&#34;&gt;GitHub is the world&amp;#39;s largest software development platform.&lt;/span&gt;&lt;span class=&#34;w&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;w&#34;&gt;    &lt;/span&gt;&lt;span class=&#34;nt&#34;&gt;website&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;w&#34;&gt; &lt;/span&gt;&lt;span class=&#34;l&#34;&gt;https://github.com&lt;/span&gt;&lt;span class=&#34;w&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;w&#34;&gt;    &lt;/span&gt;&lt;span class=&#34;nt&#34;&gt;image&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;w&#34;&gt; &lt;/span&gt;&lt;span class=&#34;l&#34;&gt;https://github.githubassets.com/images/modules/logos_page/GitHub-Mark.png&lt;/span&gt;&lt;span class=&#34;w&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;w&#34;&gt;  &lt;/span&gt;- &lt;span class=&#34;nt&#34;&gt;title&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;w&#34;&gt; &lt;/span&gt;&lt;span class=&#34;l&#34;&gt;TypeScript&lt;/span&gt;&lt;span class=&#34;w&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;w&#34;&gt;    &lt;/span&gt;&lt;span class=&#34;nt&#34;&gt;description&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;w&#34;&gt; &lt;/span&gt;&lt;span class=&#34;l&#34;&gt;TypeScript is a typed superset of JavaScript that compiles to plain JavaScript.&lt;/span&gt;&lt;span class=&#34;w&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;w&#34;&gt;    &lt;/span&gt;&lt;span class=&#34;nt&#34;&gt;website&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;w&#34;&gt; &lt;/span&gt;&lt;span class=&#34;l&#34;&gt;https://www.typescriptlang.org&lt;/span&gt;&lt;span class=&#34;w&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;w&#34;&gt;    &lt;/span&gt;&lt;span class=&#34;nt&#34;&gt;image&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;w&#34;&gt; &lt;/span&gt;&lt;span class=&#34;l&#34;&gt;ts-logo-128.jpg&lt;/span&gt;&lt;span class=&#34;w&#34;&gt;
&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;
&lt;/div&gt;
&lt;/div&gt;&lt;p&gt;&lt;code&gt;image&lt;/code&gt; field accepts both local and external images.&lt;/p&gt;
</description>
        </item>
        <item>
        <title>Search</title>
        <link>https://JamesL813.github.io/search/</link>
        <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
        
        <guid>https://JamesL813.github.io/search/</guid>
        <description></description>
        </item>
        
    </channel>
</rss>