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<!DOCTYPE html>
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<title>Exercise Solutions and Notes for R for Data Science</title>
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<li><strong><a href="./">R for Data Science</a></strong></li>
<li class="divider"></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html"><i class="fa fa-check"></i>Welcome</a></li>
<li class="part"><span><b>I Explore</b></span></li>
<li class="chapter" data-level="1" data-path="explore-intro.html"><a href="explore-intro.html"><i class="fa fa-check"></i><b>1</b> Introduction</a></li>
<li class="chapter" data-level="2" data-path="visualize.html"><a href="visualize.html"><i class="fa fa-check"></i><b>2</b> Visualize</a><ul>
<li class="chapter" data-level="2.1" data-path="visualize.html"><a href="visualize.html#introduction"><i class="fa fa-check"></i><b>2.1</b> Introduction</a><ul>
<li class="chapter" data-level="2.1.1" data-path="visualize.html"><a href="visualize.html#prerequisites"><i class="fa fa-check"></i><b>2.1.1</b> Prerequisites</a></li>
<li class="chapter" data-level="2.1.2" data-path="visualize.html"><a href="visualize.html#first-steps"><i class="fa fa-check"></i><b>2.1.2</b> First Steps</a></li>
<li class="chapter" data-level="2.1.3" data-path="visualize.html"><a href="visualize.html#aesthetic-mappings"><i class="fa fa-check"></i><b>2.1.3</b> Aesthetic mappings</a></li>
<li class="chapter" data-level="2.1.4" data-path="visualize.html"><a href="visualize.html#facets"><i class="fa fa-check"></i><b>2.1.4</b> Facets</a></li>
<li class="chapter" data-level="2.1.5" data-path="visualize.html"><a href="visualize.html#geometric-objects"><i class="fa fa-check"></i><b>2.1.5</b> Geometric Objects</a></li>
<li class="chapter" data-level="2.1.6" data-path="visualize.html"><a href="visualize.html#statistical-transformations"><i class="fa fa-check"></i><b>2.1.6</b> Statistical Transformations</a></li>
</ul></li>
<li class="chapter" data-level="2.2" data-path="visualize.html"><a href="visualize.html#position-adjustments"><i class="fa fa-check"></i><b>2.2</b> Position Adjustments</a></li>
<li class="chapter" data-level="2.3" data-path="visualize.html"><a href="visualize.html#coordinate-systems"><i class="fa fa-check"></i><b>2.3</b> Coordinate Systems</a><ul>
<li class="chapter" data-level="2.3.1" data-path="visualize.html"><a href="visualize.html#exercises-3"><i class="fa fa-check"></i><b>2.3.1</b> Exercises</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="3" data-path="workflow-basics.html"><a href="workflow-basics.html"><i class="fa fa-check"></i><b>3</b> Workflow Basics</a><ul>
<li class="chapter" data-level="3.1" data-path="workflow-basics.html"><a href="workflow-basics.html#practice"><i class="fa fa-check"></i><b>3.1</b> Practice</a><ul>
<li class="chapter" data-level="3.1.1" data-path="workflow-basics.html"><a href="workflow-basics.html#exercises-4"><i class="fa fa-check"></i><b>3.1.1</b> Exercises</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="4" data-path="data-transformation.html"><a href="data-transformation.html"><i class="fa fa-check"></i><b>4</b> Data Transformation</a><ul>
<li class="chapter" data-level="4.1" data-path="data-transformation.html"><a href="data-transformation.html#prerequisites-1"><i class="fa fa-check"></i><b>4.1</b> Prerequisites</a></li>
<li class="chapter" data-level="4.2" data-path="data-transformation.html"><a href="data-transformation.html#filter"><i class="fa fa-check"></i><b>4.2</b> Filter</a></li>
<li class="chapter" data-level="4.3" data-path="data-transformation.html"><a href="data-transformation.html#exercises-5"><i class="fa fa-check"></i><b>4.3</b> Exercises</a></li>
<li class="chapter" data-level="4.4" data-path="data-transformation.html"><a href="data-transformation.html#arrange"><i class="fa fa-check"></i><b>4.4</b> Arrange</a><ul>
<li class="chapter" data-level="4.4.1" data-path="data-transformation.html"><a href="data-transformation.html#exercises-6"><i class="fa fa-check"></i><b>4.4.1</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="4.5" data-path="data-transformation.html"><a href="data-transformation.html#mutate"><i class="fa fa-check"></i><b>4.5</b> Mutate</a><ul>
<li class="chapter" data-level="4.5.1" data-path="data-transformation.html"><a href="data-transformation.html#exercises-7"><i class="fa fa-check"></i><b>4.5.1</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="4.6" data-path="data-transformation.html"><a href="data-transformation.html#grouped-summaries-with-summarise"><i class="fa fa-check"></i><b>4.6</b> Grouped summaries with <code>summarise()</code></a><ul>
<li class="chapter" data-level="4.6.1" data-path="data-transformation.html"><a href="data-transformation.html#exercises-8"><i class="fa fa-check"></i><b>4.6.1</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="4.7" data-path="data-transformation.html"><a href="data-transformation.html#grouped-mutates-and-filters"><i class="fa fa-check"></i><b>4.7</b> Grouped mutates and filters</a><ul>
<li class="chapter" data-level="4.7.1" data-path="data-transformation.html"><a href="data-transformation.html#exercises-9"><i class="fa fa-check"></i><b>4.7.1</b> Exercises</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="5" data-path="exploratory-data-analysis.html"><a href="exploratory-data-analysis.html"><i class="fa fa-check"></i><b>5</b> Exploratory Data Analysis</a><ul>
<li class="chapter" data-level="5.1" data-path="exploratory-data-analysis.html"><a href="exploratory-data-analysis.html#introduction-1"><i class="fa fa-check"></i><b>5.1</b> Introduction</a><ul>
<li class="chapter" data-level="5.1.1" data-path="exploratory-data-analysis.html"><a href="exploratory-data-analysis.html#questions"><i class="fa fa-check"></i><b>5.1.1</b> Questions</a></li>
<li class="chapter" data-level="5.1.2" data-path="exploratory-data-analysis.html"><a href="exploratory-data-analysis.html#variation"><i class="fa fa-check"></i><b>5.1.2</b> Variation</a></li>
</ul></li>
<li class="chapter" data-level="5.2" data-path="exploratory-data-analysis.html"><a href="exploratory-data-analysis.html#missing-values"><i class="fa fa-check"></i><b>5.2</b> Missing Values</a><ul>
<li class="chapter" data-level="5.2.1" data-path="exploratory-data-analysis.html"><a href="exploratory-data-analysis.html#exercises-11"><i class="fa fa-check"></i><b>5.2.1</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="5.3" data-path="exploratory-data-analysis.html"><a href="exploratory-data-analysis.html#covariation"><i class="fa fa-check"></i><b>5.3</b> Covariation</a><ul>
<li class="chapter" data-level="5.3.1" data-path="exploratory-data-analysis.html"><a href="exploratory-data-analysis.html#a-categorical-and-continuous-variable"><i class="fa fa-check"></i><b>5.3.1</b> A categorical and continuous variable</a></li>
<li class="chapter" data-level="5.3.2" data-path="exploratory-data-analysis.html"><a href="exploratory-data-analysis.html#two-categorical-variables"><i class="fa fa-check"></i><b>5.3.2</b> Two categorical variables</a></li>
<li class="chapter" data-level="5.3.3" data-path="exploratory-data-analysis.html"><a href="exploratory-data-analysis.html#two-continuous-variables"><i class="fa fa-check"></i><b>5.3.3</b> Two continuous variables</a></li>
</ul></li>
</ul></li>
<li class="part"><span><b>II Wrangle</b></span></li>
<li class="chapter" data-level="6" data-path="tibbles.html"><a href="tibbles.html"><i class="fa fa-check"></i><b>6</b> Tibbles</a><ul>
<li class="chapter" data-level="6.1" data-path="tibbles.html"><a href="tibbles.html#prerquisites"><i class="fa fa-check"></i><b>6.1</b> Prerquisites</a></li>
<li class="chapter" data-level="6.2" data-path="tibbles.html"><a href="tibbles.html#creating-tibbles"><i class="fa fa-check"></i><b>6.2</b> Creating Tibbles</a></li>
<li class="chapter" data-level="6.3" data-path="tibbles.html"><a href="tibbles.html#tibbles-vs.data.frame"><i class="fa fa-check"></i><b>6.3</b> Tibbles vs. data.frame</a></li>
<li class="chapter" data-level="6.4" data-path="tibbles.html"><a href="tibbles.html#subsetting"><i class="fa fa-check"></i><b>6.4</b> Subsetting</a></li>
<li class="chapter" data-level="6.5" data-path="tibbles.html"><a href="tibbles.html#interacting-with-older-code"><i class="fa fa-check"></i><b>6.5</b> Interacting with older code</a></li>
<li class="chapter" data-level="6.6" data-path="tibbles.html"><a href="tibbles.html#exercises-12"><i class="fa fa-check"></i><b>6.6</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="7" data-path="data-import.html"><a href="data-import.html"><i class="fa fa-check"></i><b>7</b> Data Import</a><ul>
<li class="chapter" data-level="7.1" data-path="data-import.html"><a href="data-import.html#introduction-2"><i class="fa fa-check"></i><b>7.1</b> Introduction</a></li>
<li class="chapter" data-level="7.2" data-path="data-import.html"><a href="data-import.html#getting-started"><i class="fa fa-check"></i><b>7.2</b> Getting started</a><ul>
<li class="chapter" data-level="7.2.1" data-path="data-import.html"><a href="data-import.html#exercises-13"><i class="fa fa-check"></i><b>7.2.1</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="7.3" data-path="data-import.html"><a href="data-import.html#parsing-a-vector"><i class="fa fa-check"></i><b>7.3</b> Parsing a vector</a><ul>
<li class="chapter" data-level="7.3.1" data-path="data-import.html"><a href="data-import.html#exercises-14"><i class="fa fa-check"></i><b>7.3.1</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="7.4" data-path="data-import.html"><a href="data-import.html#other-types-of-data"><i class="fa fa-check"></i><b>7.4</b> Other Types of Data</a></li>
</ul></li>
<li class="chapter" data-level="8" data-path="tidy-data.html"><a href="tidy-data.html"><i class="fa fa-check"></i><b>8</b> Tidy Data</a><ul>
<li class="chapter" data-level="8.1" data-path="tidy-data.html"><a href="tidy-data.html#introduction-3"><i class="fa fa-check"></i><b>8.1</b> Introduction</a></li>
<li class="chapter" data-level="8.2" data-path="tidy-data.html"><a href="tidy-data.html#tidy-data-1"><i class="fa fa-check"></i><b>8.2</b> Tidy Data</a><ul>
<li class="chapter" data-level="8.2.1" data-path="tidy-data.html"><a href="tidy-data.html#exercises-15"><i class="fa fa-check"></i><b>8.2.1</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="8.3" data-path="tidy-data.html"><a href="tidy-data.html#spreading-and-gathering"><i class="fa fa-check"></i><b>8.3</b> Spreading and Gathering</a><ul>
<li class="chapter" data-level="8.3.1" data-path="tidy-data.html"><a href="tidy-data.html#exercises-16"><i class="fa fa-check"></i><b>8.3.1</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="8.4" data-path="tidy-data.html"><a href="tidy-data.html#separating-and-uniting"><i class="fa fa-check"></i><b>8.4</b> Separating and Uniting</a><ul>
<li class="chapter" data-level="8.4.1" data-path="tidy-data.html"><a href="tidy-data.html#exercises-17"><i class="fa fa-check"></i><b>8.4.1</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="8.5" data-path="tidy-data.html"><a href="tidy-data.html#missing-values-1"><i class="fa fa-check"></i><b>8.5</b> Missing Values</a><ul>
<li class="chapter" data-level="8.5.1" data-path="tidy-data.html"><a href="tidy-data.html#exercises-18"><i class="fa fa-check"></i><b>8.5.1</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="8.6" data-path="tidy-data.html"><a href="tidy-data.html#case-study"><i class="fa fa-check"></i><b>8.6</b> Case Study</a><ul>
<li class="chapter" data-level="8.6.1" data-path="tidy-data.html"><a href="tidy-data.html#exercises-19"><i class="fa fa-check"></i><b>8.6.1</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="8.7" data-path="tidy-data.html"><a href="tidy-data.html#non-tidy-data"><i class="fa fa-check"></i><b>8.7</b> Non-Tidy Data</a></li>
</ul></li>
<li class="chapter" data-level="9" data-path="relational-data.html"><a href="relational-data.html"><i class="fa fa-check"></i><b>9</b> Relational Data</a><ul>
<li class="chapter" data-level="9.1" data-path="relational-data.html"><a href="relational-data.html#prerequisites-2"><i class="fa fa-check"></i><b>9.1</b> Prerequisites</a></li>
<li class="chapter" data-level="9.2" data-path="relational-data.html"><a href="relational-data.html#nycflights13"><i class="fa fa-check"></i><b>9.2</b> nycflights13</a><ul>
<li class="chapter" data-level="9.2.1" data-path="relational-data.html"><a href="relational-data.html#exercises-20"><i class="fa fa-check"></i><b>9.2.1</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="9.3" data-path="relational-data.html"><a href="relational-data.html#keys"><i class="fa fa-check"></i><b>9.3</b> Keys</a></li>
<li class="chapter" data-level="9.4" data-path="relational-data.html"><a href="relational-data.html#mutating-joins"><i class="fa fa-check"></i><b>9.4</b> Mutating Joins</a><ul>
<li class="chapter" data-level="9.4.1" data-path="relational-data.html"><a href="relational-data.html#exercises-21"><i class="fa fa-check"></i><b>9.4.1</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="9.5" data-path="relational-data.html"><a href="relational-data.html#filtering-joins"><i class="fa fa-check"></i><b>9.5</b> Filtering Joins</a><ul>
<li class="chapter" data-level="9.5.1" data-path="relational-data.html"><a href="relational-data.html#exercises-22"><i class="fa fa-check"></i><b>9.5.1</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="9.6" data-path="relational-data.html"><a href="relational-data.html#set-operations"><i class="fa fa-check"></i><b>9.6</b> Set operations</a></li>
</ul></li>
<li class="chapter" data-level="10" data-path="strings.html"><a href="strings.html"><i class="fa fa-check"></i><b>10</b> Strings</a><ul>
<li class="chapter" data-level="10.1" data-path="strings.html"><a href="strings.html#introduction-4"><i class="fa fa-check"></i><b>10.1</b> Introduction</a></li>
<li class="chapter" data-level="10.2" data-path="strings.html"><a href="strings.html#string-basics"><i class="fa fa-check"></i><b>10.2</b> String Basics</a><ul>
<li class="chapter" data-level="10.2.1" data-path="strings.html"><a href="strings.html#exercises-23"><i class="fa fa-check"></i><b>10.2.1</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="10.3" data-path="strings.html"><a href="strings.html#matching-patterns-and-regular-expressions"><i class="fa fa-check"></i><b>10.3</b> Matching Patterns and Regular Expressions</a><ul>
<li class="chapter" data-level="10.3.1" data-path="strings.html"><a href="strings.html#exercises-24"><i class="fa fa-check"></i><b>10.3.1</b> Exercises</a></li>
<li class="chapter" data-level="10.3.2" data-path="strings.html"><a href="strings.html#repitition"><i class="fa fa-check"></i><b>10.3.2</b> Repitition</a></li>
<li class="chapter" data-level="10.3.3" data-path="strings.html"><a href="strings.html#grouping-and-backreferences"><i class="fa fa-check"></i><b>10.3.3</b> Grouping and backreferences</a></li>
</ul></li>
<li class="chapter" data-level="10.4" data-path="strings.html"><a href="strings.html#tools"><i class="fa fa-check"></i><b>10.4</b> Tools</a><ul>
<li class="chapter" data-level="10.4.1" data-path="strings.html"><a href="strings.html#detect-matches"><i class="fa fa-check"></i><b>10.4.1</b> Detect matches</a></li>
<li class="chapter" data-level="10.4.2" data-path="strings.html"><a href="strings.html#exercises-29"><i class="fa fa-check"></i><b>10.4.2</b> Exercises</a></li>
<li class="chapter" data-level="10.4.3" data-path="strings.html"><a href="strings.html#extract-matches"><i class="fa fa-check"></i><b>10.4.3</b> Extract Matches</a></li>
<li class="chapter" data-level="10.4.4" data-path="strings.html"><a href="strings.html#grouped-matches"><i class="fa fa-check"></i><b>10.4.4</b> Grouped Matches</a></li>
<li class="chapter" data-level="10.4.5" data-path="strings.html"><a href="strings.html#splitting"><i class="fa fa-check"></i><b>10.4.5</b> Splitting</a></li>
</ul></li>
<li class="chapter" data-level="10.5" data-path="strings.html"><a href="strings.html#other-types-of-patterns"><i class="fa fa-check"></i><b>10.5</b> Other types of patterns</a><ul>
<li class="chapter" data-level="10.5.1" data-path="strings.html"><a href="strings.html#exercises-33"><i class="fa fa-check"></i><b>10.5.1</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="10.6" data-path="strings.html"><a href="strings.html#stringi"><i class="fa fa-check"></i><b>10.6</b> stringi</a><ul>
<li class="chapter" data-level="10.6.1" data-path="strings.html"><a href="strings.html#exercises-34"><i class="fa fa-check"></i><b>10.6.1</b> Exercises</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="11" data-path="factors.html"><a href="factors.html"><i class="fa fa-check"></i><b>11</b> Factors</a><ul>
<li class="chapter" data-level="11.1" data-path="factors.html"><a href="factors.html#introduction-5"><i class="fa fa-check"></i><b>11.1</b> Introduction</a></li>
<li class="chapter" data-level="11.2" data-path="factors.html"><a href="factors.html#creating-factors"><i class="fa fa-check"></i><b>11.2</b> Creating Factors</a></li>
<li class="chapter" data-level="11.3" data-path="factors.html"><a href="factors.html#general-social-survey"><i class="fa fa-check"></i><b>11.3</b> General Social Survey</a><ul>
<li class="chapter" data-level="11.3.1" data-path="factors.html"><a href="factors.html#exercises-35"><i class="fa fa-check"></i><b>11.3.1</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="11.4" data-path="factors.html"><a href="factors.html#modifying-factor-order"><i class="fa fa-check"></i><b>11.4</b> Modifying factor order</a><ul>
<li class="chapter" data-level="11.4.1" data-path="factors.html"><a href="factors.html#exercises-36"><i class="fa fa-check"></i><b>11.4.1</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="11.5" data-path="factors.html"><a href="factors.html#modifying-factor-levels"><i class="fa fa-check"></i><b>11.5</b> Modifying factor levels</a><ul>
<li class="chapter" data-level="11.5.1" data-path="factors.html"><a href="factors.html#exercises-37"><i class="fa fa-check"></i><b>11.5.1</b> Exercises</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="12" data-path="dates-and-times.html"><a href="dates-and-times.html"><i class="fa fa-check"></i><b>12</b> Dates and Times</a><ul>
<li class="chapter" data-level="12.1" data-path="dates-and-times.html"><a href="dates-and-times.html#prerequisite"><i class="fa fa-check"></i><b>12.1</b> Prerequisite</a></li>
<li class="chapter" data-level="12.2" data-path="dates-and-times.html"><a href="dates-and-times.html#creating-datetimes"><i class="fa fa-check"></i><b>12.2</b> Creating date/times</a><ul>
<li class="chapter" data-level="12.2.1" data-path="dates-and-times.html"><a href="dates-and-times.html#exercises-38"><i class="fa fa-check"></i><b>12.2.1</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="12.3" data-path="dates-and-times.html"><a href="dates-and-times.html#date-time-components"><i class="fa fa-check"></i><b>12.3</b> Date-Time Components</a><ul>
<li class="chapter" data-level="12.3.1" data-path="dates-and-times.html"><a href="dates-and-times.html#exercises-39"><i class="fa fa-check"></i><b>12.3.1</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="12.4" data-path="dates-and-times.html"><a href="dates-and-times.html#time-spans"><i class="fa fa-check"></i><b>12.4</b> Time Spans</a><ul>
<li class="chapter" data-level="12.4.1" data-path="dates-and-times.html"><a href="dates-and-times.html#durations"><i class="fa fa-check"></i><b>12.4.1</b> Durations</a></li>
<li class="chapter" data-level="12.4.2" data-path="dates-and-times.html"><a href="dates-and-times.html#periods"><i class="fa fa-check"></i><b>12.4.2</b> Periods</a></li>
<li class="chapter" data-level="12.4.3" data-path="dates-and-times.html"><a href="dates-and-times.html#intervals"><i class="fa fa-check"></i><b>12.4.3</b> Intervals</a></li>
<li class="chapter" data-level="12.4.4" data-path="dates-and-times.html"><a href="dates-and-times.html#exercises-40"><i class="fa fa-check"></i><b>12.4.4</b> Exercises</a></li>
<li class="chapter" data-level="12.4.5" data-path="dates-and-times.html"><a href="dates-and-times.html#time-zones"><i class="fa fa-check"></i><b>12.4.5</b> Time Zones</a></li>
</ul></li>
</ul></li>
<li class="part"><span><b>III Program</b></span></li>
<li class="chapter" data-level="13" data-path="program-intro.html"><a href="program-intro.html"><i class="fa fa-check"></i><b>13</b> Introduction</a></li>
<li class="chapter" data-level="14" data-path="pipes.html"><a href="pipes.html"><i class="fa fa-check"></i><b>14</b> Pipes</a></li>
<li class="chapter" data-level="15" data-path="vectors.html"><a href="vectors.html"><i class="fa fa-check"></i><b>15</b> Vectors</a><ul>
<li class="chapter" data-level="15.1" data-path="vectors.html"><a href="vectors.html#introduction-6"><i class="fa fa-check"></i><b>15.1</b> Introduction</a></li>
<li class="chapter" data-level="15.2" data-path="vectors.html"><a href="vectors.html#important-types-of-atomic-vector"><i class="fa fa-check"></i><b>15.2</b> Important types of Atomic Vector</a><ul>
<li class="chapter" data-level="15.2.1" data-path="vectors.html"><a href="vectors.html#exercises-41"><i class="fa fa-check"></i><b>15.2.1</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="15.3" data-path="vectors.html"><a href="vectors.html#using-atomic-vectors"><i class="fa fa-check"></i><b>15.3</b> Using atomic vectors</a></li>
<li class="chapter" data-level="15.4" data-path="vectors.html"><a href="vectors.html#recursive-vectors-lists"><i class="fa fa-check"></i><b>15.4</b> Recursive Vectors (lists)</a><ul>
<li class="chapter" data-level="15.4.1" data-path="vectors.html"><a href="vectors.html#exercises-42"><i class="fa fa-check"></i><b>15.4.1</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="15.5" data-path="vectors.html"><a href="vectors.html#augmented-vectors"><i class="fa fa-check"></i><b>15.5</b> Augmented Vectors</a><ul>
<li class="chapter" data-level="15.5.1" data-path="vectors.html"><a href="vectors.html#exercises-43"><i class="fa fa-check"></i><b>15.5.1</b> Exercises</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="16" data-path="iteration.html"><a href="iteration.html"><i class="fa fa-check"></i><b>16</b> Iteration</a><ul>
<li class="chapter" data-level="16.1" data-path="iteration.html"><a href="iteration.html#introduction-7"><i class="fa fa-check"></i><b>16.1</b> Introduction</a></li>
<li class="chapter" data-level="16.2" data-path="iteration.html"><a href="iteration.html#for-loops"><i class="fa fa-check"></i><b>16.2</b> For Loops</a><ul>
<li class="chapter" data-level="16.2.1" data-path="iteration.html"><a href="iteration.html#exercises-44"><i class="fa fa-check"></i><b>16.2.1</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="16.3" data-path="iteration.html"><a href="iteration.html#for-loop-variations"><i class="fa fa-check"></i><b>16.3</b> For loop variations</a><ul>
<li class="chapter" data-level="16.3.1" data-path="iteration.html"><a href="iteration.html#section"><i class="fa fa-check"></i><b>16.3.1</b> </a></li>
</ul></li>
<li class="chapter" data-level="16.4" data-path="iteration.html"><a href="iteration.html#for-loops-vs.functionals"><i class="fa fa-check"></i><b>16.4</b> For loops vs. functionals</a><ul>
<li class="chapter" data-level="16.4.1" data-path="iteration.html"><a href="iteration.html#exercises-45"><i class="fa fa-check"></i><b>16.4.1</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="16.5" data-path="iteration.html"><a href="iteration.html#the-map-functions"><i class="fa fa-check"></i><b>16.5</b> The map functions</a><ul>
<li class="chapter" data-level="16.5.1" data-path="iteration.html"><a href="iteration.html#shortcuts"><i class="fa fa-check"></i><b>16.5.1</b> Shortcuts</a></li>
<li class="chapter" data-level="16.5.2" data-path="iteration.html"><a href="iteration.html#exercises-46"><i class="fa fa-check"></i><b>16.5.2</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="16.6" data-path="iteration.html"><a href="iteration.html#dealing-with-failure"><i class="fa fa-check"></i><b>16.6</b> Dealing with Failure</a></li>
<li class="chapter" data-level="16.7" data-path="iteration.html"><a href="iteration.html#mapping-over-multiple-arguments"><i class="fa fa-check"></i><b>16.7</b> Mapping over multiple arguments</a></li>
<li class="chapter" data-level="16.8" data-path="iteration.html"><a href="iteration.html#walk"><i class="fa fa-check"></i><b>16.8</b> Walk</a></li>
<li class="chapter" data-level="16.9" data-path="iteration.html"><a href="iteration.html#other-patterns-of-for-loops"><i class="fa fa-check"></i><b>16.9</b> Other patterns of for loops</a><ul>
<li class="chapter" data-level="16.9.1" data-path="iteration.html"><a href="iteration.html#exercises-47"><i class="fa fa-check"></i><b>16.9.1</b> Exercises</a></li>
</ul></li>
</ul></li>
<li class="part"><span><b>IV Model</b></span></li>
<li class="chapter" data-level="17" data-path="model-intro.html"><a href="model-intro.html"><i class="fa fa-check"></i><b>17</b> Introduction</a></li>
<li class="chapter" data-level="18" data-path="model-basics.html"><a href="model-basics.html"><i class="fa fa-check"></i><b>18</b> Model Basics</a><ul>
<li class="chapter" data-level="18.1" data-path="model-basics.html"><a href="model-basics.html#prerequisites-3"><i class="fa fa-check"></i><b>18.1</b> Prerequisites</a></li>
<li class="chapter" data-level="18.2" data-path="model-basics.html"><a href="model-basics.html#a-simple-model"><i class="fa fa-check"></i><b>18.2</b> A simple model</a><ul>
<li class="chapter" data-level="18.2.1" data-path="model-basics.html"><a href="model-basics.html#exercises-48"><i class="fa fa-check"></i><b>18.2.1</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="18.3" data-path="model-basics.html"><a href="model-basics.html#visualizing-models"><i class="fa fa-check"></i><b>18.3</b> Visualizing Models</a><ul>
<li class="chapter" data-level="18.3.1" data-path="model-basics.html"><a href="model-basics.html#exercises-49"><i class="fa fa-check"></i><b>18.3.1</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="18.4" data-path="model-basics.html"><a href="model-basics.html#formulas-and-model-families"><i class="fa fa-check"></i><b>18.4</b> Formulas and Model Families</a><ul>
<li class="chapter" data-level="18.4.1" data-path="model-basics.html"><a href="model-basics.html#categorical-variables"><i class="fa fa-check"></i><b>18.4.1</b> Categorical Variables</a></li>
<li class="chapter" data-level="18.4.2" data-path="model-basics.html"><a href="model-basics.html#exercises-50"><i class="fa fa-check"></i><b>18.4.2</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="18.5" data-path="model-basics.html"><a href="model-basics.html#missing-values-2"><i class="fa fa-check"></i><b>18.5</b> Missing values</a></li>
<li class="chapter" data-level="18.6" data-path="model-basics.html"><a href="model-basics.html#other-model-families"><i class="fa fa-check"></i><b>18.6</b> Other model families</a></li>
</ul></li>
<li class="part"><span><b>V Communicate</b></span></li>
<li class="chapter" data-level="19" data-path="communicate-intro.html"><a href="communicate-intro.html"><i class="fa fa-check"></i><b>19</b> Introduction</a></li>
<li class="chapter" data-level="20" data-path="r-markdown.html"><a href="r-markdown.html"><i class="fa fa-check"></i><b>20</b> R Markdown</a><ul>
<li class="chapter" data-level="20.1" data-path="r-markdown.html"><a href="r-markdown.html#r-markdown-basics"><i class="fa fa-check"></i><b>20.1</b> R Markdown Basics</a><ul>
<li class="chapter" data-level="20.1.1" data-path="r-markdown.html"><a href="r-markdown.html#exercise"><i class="fa fa-check"></i><b>20.1.1</b> Exercise</a></li>
</ul></li>
<li class="chapter" data-level="20.2" data-path="r-markdown.html"><a href="r-markdown.html#text-formatting-with-r-markdown"><i class="fa fa-check"></i><b>20.2</b> Text formatting with R Markdown</a></li>
</ul></li>
<li class="chapter" data-level="21" data-path="r-markdown-formats.html"><a href="r-markdown-formats.html"><i class="fa fa-check"></i><b>21</b> R Markdown Formats</a></li>
<li class="chapter" data-level="22" data-path="r-markdown-workflow.html"><a href="r-markdown-workflow.html"><i class="fa fa-check"></i><b>22</b> R Markdown Workflow</a></li>
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<i class="fa fa-circle-o-notch fa-spin"></i><a href="./">Exercise Solutions and Notes for “R for Data Science”</a>
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<div id="model-basics" class="section level1">
<h1><span class="header-section-number">18</span> Model Basics</h1>
<p>Distinction between <em>family of models</em> and <em>fitted model</em> is a useful way to think about models. Especially as we can abstract some families of models to be themselves a fitted model of a more flexible family of models. For example, linear regression is a special case of GLM or Gaussian Processes etc.</p>
<div id="prerequisites-3" class="section level2">
<h2><span class="header-section-number">18.1</span> Prerequisites</h2>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">library</span>(tidyverse)
<span class="kw">library</span>(modelr)
<span class="kw">options</span>(<span class="dt">na.action =</span> na.warn)</code></pre></div>
<p>The option <code>na.action</code> determines how missing values are handled. It is a function. <code>na.warn</code> sets it so that there is a warning if there are any missing values (by default, R will just silently drop them).</p>
</div>
<div id="a-simple-model" class="section level2">
<h2><span class="header-section-number">18.2</span> A simple model</h2>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">ggplot</span>(sim1, <span class="kw">aes</span>(x, y)) +
<span class="st"> </span><span class="kw">geom_point</span>()</code></pre></div>
<p><img src="model-basics_files/figure-html/unnamed-chunk-3-1.png" width="70%" style="display: block; margin: auto;" /></p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">models <-<span class="st"> </span><span class="kw">tibble</span>(
<span class="dt">a1 =</span> <span class="kw">runif</span>(<span class="dv">250</span>, -<span class="dv">20</span>, <span class="dv">40</span>),
<span class="dt">a2 =</span> <span class="kw">runif</span>(<span class="dv">250</span>, -<span class="dv">5</span>, <span class="dv">5</span>)
)
<span class="kw">ggplot</span>(sim1, <span class="kw">aes</span>(x, y)) +
<span class="st"> </span><span class="kw">geom_abline</span>(<span class="kw">aes</span>(<span class="dt">intercept =</span> a1, <span class="dt">slope =</span> a2), <span class="dt">data =</span> models, <span class="dt">alpha =</span> <span class="dv">1</span>/<span class="dv">4</span>) +
<span class="st"> </span><span class="kw">geom_point</span>()</code></pre></div>
<p><img src="model-basics_files/figure-html/unnamed-chunk-4-1.png" width="70%" style="display: block; margin: auto;" /></p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">model1 <-<span class="st"> </span>function(a, data) {
a[<span class="dv">1</span>] +<span class="st"> </span>data$x *<span class="st"> </span>a[<span class="dv">2</span>]
}
<span class="kw">model1</span>(<span class="kw">c</span>(<span class="dv">7</span>, <span class="fl">1.5</span>), sim1)
<span class="co">#> [1] 8.5 8.5 8.5 10.0 10.0 10.0 11.5 11.5 11.5 13.0 13.0 13.0 14.5 14.5</span>
<span class="co">#> [15] 14.5 16.0 16.0 16.0 17.5 17.5 17.5 19.0 19.0 19.0 20.5 20.5 20.5 22.0</span>
<span class="co">#> [29] 22.0 22.0</span></code></pre></div>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">measure_distance <-<span class="st"> </span>function(mod, data) {
diff <-<span class="st"> </span>data$y -<span class="st"> </span><span class="kw">model1</span>(mod, data)
<span class="kw">sqrt</span>(<span class="kw">mean</span>(diff ^<span class="st"> </span><span class="dv">2</span>))
}
<span class="kw">measure_distance</span>(<span class="kw">c</span>(<span class="dv">7</span>, <span class="fl">1.5</span>), sim1)
<span class="co">#> [1] 2.67</span></code></pre></div>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">sim1_dist <-<span class="st"> </span>function(a1, a2) {
<span class="kw">measure_distance</span>(<span class="kw">c</span>(a1, a2), sim1)
}
models <-<span class="st"> </span>models %>%<span class="st"> </span>
<span class="st"> </span><span class="kw">mutate</span>(<span class="dt">dist =</span> purrr::<span class="kw">map2_dbl</span>(a1, a2, sim1_dist))
models
<span class="co">#> # A tibble: 250 × 3</span>
<span class="co">#> a1 a2 dist</span>
<span class="co">#> <dbl> <dbl> <dbl></span>
<span class="co">#> 1 -15.15 0.0889 30.8</span>
<span class="co">#> 2 30.06 -0.8274 13.2</span>
<span class="co">#> 3 16.05 2.2695 13.2</span>
<span class="co">#> 4 -10.57 1.3769 18.7</span>
<span class="co">#> 5 -19.56 -1.0359 41.8</span>
<span class="co">#> 6 7.98 4.5948 19.3</span>
<span class="co">#> # ... with 244 more rows</span></code></pre></div>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">ggplot</span>(sim1, <span class="kw">aes</span>(x, y)) +<span class="st"> </span>
<span class="st"> </span><span class="kw">geom_point</span>(<span class="dt">size =</span> <span class="dv">2</span>, <span class="dt">colour =</span> <span class="st">"grey30"</span>) +<span class="st"> </span>
<span class="st"> </span><span class="kw">geom_abline</span>(
<span class="kw">aes</span>(<span class="dt">intercept =</span> a1, <span class="dt">slope =</span> a2, <span class="dt">colour =</span> -dist),
<span class="dt">data =</span> <span class="kw">filter</span>(models, <span class="kw">rank</span>(dist) <=<span class="st"> </span><span class="dv">10</span>)
)</code></pre></div>
<p><img src="model-basics_files/figure-html/unnamed-chunk-8-1.png" width="70%" style="display: block; margin: auto;" /></p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">grid <-<span class="st"> </span><span class="kw">expand.grid</span>(
<span class="dt">a1 =</span> <span class="kw">seq</span>(-<span class="dv">5</span>, <span class="dv">20</span>, <span class="dt">length =</span> <span class="dv">25</span>),
<span class="dt">a2 =</span> <span class="kw">seq</span>(<span class="dv">1</span>, <span class="dv">3</span>, <span class="dt">length =</span> <span class="dv">25</span>)
) %>%<span class="st"> </span>
<span class="st"> </span><span class="kw">mutate</span>(<span class="dt">dist =</span> purrr::<span class="kw">map2_dbl</span>(a1, a2, sim1_dist))
grid %>%<span class="st"> </span>
<span class="st"> </span><span class="kw">ggplot</span>(<span class="kw">aes</span>(a1, a2)) +
<span class="st"> </span><span class="kw">geom_point</span>(<span class="dt">data =</span> <span class="kw">filter</span>(grid, <span class="kw">rank</span>(dist) <=<span class="st"> </span><span class="dv">10</span>), <span class="dt">size =</span> <span class="dv">4</span>, <span class="dt">colour =</span> <span class="st">"red"</span>) +
<span class="st"> </span><span class="kw">geom_point</span>(<span class="kw">aes</span>(<span class="dt">colour =</span> -dist)) </code></pre></div>
<p><img src="model-basics_files/figure-html/unnamed-chunk-9-1.png" width="70%" style="display: block; margin: auto;" /></p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">ggplot</span>(sim1, <span class="kw">aes</span>(x, y)) +<span class="st"> </span>
<span class="st"> </span><span class="kw">geom_point</span>(<span class="dt">size =</span> <span class="dv">2</span>, <span class="dt">colour =</span> <span class="st">"grey30"</span>) +<span class="st"> </span>
<span class="st"> </span><span class="kw">geom_abline</span>(
<span class="kw">aes</span>(<span class="dt">intercept =</span> a1, <span class="dt">slope =</span> a2, <span class="dt">colour =</span> -dist),
<span class="dt">data =</span> <span class="kw">filter</span>(grid, <span class="kw">rank</span>(dist) <=<span class="st"> </span><span class="dv">10</span>)
)</code></pre></div>
<p><img src="model-basics_files/figure-html/unnamed-chunk-10-1.png" width="70%" style="display: block; margin: auto;" /></p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">best <-<span class="st"> </span><span class="kw">optim</span>(<span class="kw">c</span>(<span class="dv">0</span>, <span class="dv">0</span>), measure_distance, <span class="dt">data =</span> sim1)
best$par
<span class="co">#> [1] 4.22 2.05</span>
<span class="kw">ggplot</span>(sim1, <span class="kw">aes</span>(x, y)) +<span class="st"> </span>
<span class="st"> </span><span class="kw">geom_point</span>(<span class="dt">size =</span> <span class="dv">2</span>, <span class="dt">colour =</span> <span class="st">"grey30"</span>) +<span class="st"> </span>
<span class="st"> </span><span class="kw">geom_abline</span>(<span class="dt">intercept =</span> best$par[<span class="dv">1</span>], <span class="dt">slope =</span> best$par[<span class="dv">2</span>])</code></pre></div>
<p><img src="model-basics_files/figure-html/unnamed-chunk-11-1.png" width="70%" style="display: block; margin: auto;" /></p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">sim1_mod <-<span class="st"> </span><span class="kw">lm</span>(y ~<span class="st"> </span>x, <span class="dt">data =</span> sim1)
<span class="kw">coef</span>(sim1_mod)
<span class="co">#> (Intercept) x </span>
<span class="co">#> 4.22 2.05</span></code></pre></div>
<div id="exercises-48" class="section level3">
<h3><span class="header-section-number">18.2.1</span> Exercises</h3>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">sim1a <-<span class="st"> </span><span class="kw">tibble</span>(
<span class="dt">x =</span> <span class="kw">rep</span>(<span class="dv">1</span>:<span class="dv">10</span>, <span class="dt">each =</span> <span class="dv">3</span>),
<span class="dt">y =</span> x *<span class="st"> </span><span class="fl">1.5</span> +<span class="st"> </span><span class="dv">6</span> +<span class="st"> </span><span class="kw">rt</span>(<span class="kw">length</span>(x), <span class="dt">df =</span> <span class="dv">2</span>)
)</code></pre></div>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">lm</span>(y ~<span class="st"> </span>x, <span class="dt">data =</span> sim1a)
<span class="co">#> </span>
<span class="co">#> Call:</span>
<span class="co">#> lm(formula = y ~ x, data = sim1a)</span>
<span class="co">#> </span>
<span class="co">#> Coefficients:</span>
<span class="co">#> (Intercept) x </span>
<span class="co">#> 6.05 1.53</span></code></pre></div>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">ggplot</span>(sim1a, <span class="kw">aes</span>(<span class="dt">x =</span> x, <span class="dt">y =</span> y)) +
<span class="st"> </span><span class="kw">geom_point</span>() +
<span class="st"> </span><span class="kw">geom_smooth</span>(<span class="dt">method =</span> <span class="st">"lm"</span>, <span class="dt">se =</span> <span class="ot">FALSE</span>)</code></pre></div>
<p><img src="model-basics_files/figure-html/unnamed-chunk-15-1.png" width="70%" style="display: block; margin: auto;" /></p>
<p>To re-run this a few times using <code>purrr</code></p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">simt <-<span class="st"> </span>function(i) {
<span class="kw">tibble</span>(
<span class="dt">x =</span> <span class="kw">rep</span>(<span class="dv">1</span>:<span class="dv">10</span>, <span class="dt">each =</span> <span class="dv">3</span>),
<span class="dt">y =</span> x *<span class="st"> </span><span class="fl">1.5</span> +<span class="st"> </span><span class="dv">6</span> +<span class="st"> </span><span class="kw">rt</span>(<span class="kw">length</span>(x), <span class="dt">df =</span> <span class="dv">2</span>),
<span class="dt">.id =</span> i
)
}
lm_df <-<span class="st"> </span>function(.data) {
mod <-<span class="st"> </span><span class="kw">lm</span>(y ~<span class="st"> </span>x, <span class="dt">data =</span> .data)
beta <-<span class="st"> </span><span class="kw">coef</span>(mod)
<span class="kw">tibble</span>(<span class="dt">intercept =</span> beta[<span class="dv">1</span>], <span class="dt">slope =</span> beta[<span class="dv">2</span>])
}
sims <-<span class="st"> </span><span class="kw">map</span>(<span class="dv">1</span>:<span class="dv">100</span>, simt) %>%
<span class="st"> </span><span class="kw">map_df</span>(lm_df)
<span class="kw">ggplot</span>(sims, <span class="kw">aes</span>(<span class="dt">x =</span> intercept, <span class="dt">y =</span> slope)) +
<span class="st"> </span><span class="kw">geom_point</span>()</code></pre></div>
<p><img src="model-basics_files/figure-html/unnamed-chunk-16-1.png" width="70%" style="display: block; margin: auto;" /></p>
<p><strong>NOTE</strong> It’s not entirely clear what is meant by “visualize the results”.</p>
<p>The data are generated from a low-degrees of freedmo t-distribution, so there will be outliers.r4ds Linear regression is</p>
<ol start="2" style="list-style-type: decimal">
<li>One way to make linear models more robust is to use a different distance measure. For example, instead of root-mean-squared distance, you could use mean-absolute distance:</li>
</ol>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">measure_distance <-<span class="st"> </span>function(mod, data) {
diff <-<span class="st"> </span>data$y -<span class="st"> </span><span class="kw">make_prediction</span>(mod, data)
<span class="kw">mean</span>(<span class="kw">abs</span>(diff))
}</code></pre></div>
<p>To re-run this a few times use purrr</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">simt <-<span class="st"> </span>function(i) {
<span class="kw">tibble</span>(
<span class="dt">x =</span> <span class="kw">rep</span>(<span class="dv">1</span>:<span class="dv">10</span>, <span class="dt">each =</span> <span class="dv">3</span>),
<span class="dt">y =</span> x *<span class="st"> </span><span class="fl">1.5</span> +<span class="st"> </span><span class="dv">6</span> +<span class="st"> </span><span class="kw">rt</span>(<span class="kw">length</span>(x), <span class="dt">df =</span> <span class="dv">2</span>),
<span class="dt">.id =</span> i
)
}
lm_df <-<span class="st"> </span>function(.data) {
mod <-<span class="st"> </span><span class="kw">lm</span>(y ~<span class="st"> </span>x, <span class="dt">data =</span> .data)
beta <-<span class="st"> </span><span class="kw">coef</span>(mod)
<span class="kw">tibble</span>(<span class="dt">intercept =</span> beta[<span class="dv">1</span>], <span class="dt">slope =</span> beta[<span class="dv">2</span>])
}
sims <-<span class="st"> </span><span class="kw">map</span>(<span class="dv">1</span>:<span class="dv">100</span>, simt) %>%
<span class="st"> </span><span class="kw">map_df</span>(lm_df)
<span class="kw">ggplot</span>(sims, <span class="kw">aes</span>(<span class="dt">x =</span> intercept, <span class="dt">y =</span> slope)) +
<span class="st"> </span><span class="kw">geom_point</span>()</code></pre></div>
<p><img src="model-basics_files/figure-html/unnamed-chunk-18-1.png" width="70%" style="display: block; margin: auto;" /></p>
<ol start="3" style="list-style-type: decimal">
<li>One challenge with performing numerical optimisation is that it’s only guaranteed to find one local optima. What’s the problem with optimising a three parameter model like this?</li>
</ol>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">model1 <-<span class="st"> </span>function(a, data) {
a[<span class="dv">1</span>] +<span class="st"> </span>data$x *<span class="st"> </span>a[<span class="dv">2</span>] +<span class="st"> </span>a[<span class="dv">3</span>]
}</code></pre></div>
<p>The problem is that you for any values <code>a[1] = a1</code> and <code>a[3] = a3</code>, any other values of <code>a[1]</code> and <code>a[3]</code> where <code>a[1] + a[3] == (a1 + a3)</code> will have the same fit.</p>
</div>
</div>
<div id="visualizing-models" class="section level2">
<h2><span class="header-section-number">18.3</span> Visualizing Models</h2>
<p>More complicated models can be visualized with</p>
<ol style="list-style-type: decimal">
<li>predictions</li>
<li>residuals</li>
</ol>
<p>Notes</p>
<ul>
<li>look at <code>tidyr::complete</code>, <code>tidyr::expand</code>, and <code>modelr::data_grid</code> functions</li>
<li><code>modelr::add_residuals</code> and <code>modelr::add_predictions</code> functions add residuals or predictions to the original data</li>
<li><code>geom_ref_line</code></li>
</ul>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">grid <-<span class="st"> </span>sim1 %>%<span class="st"> </span><span class="kw">data_grid</span>(x)
grid
<span class="co">#> # A tibble: 10 × 1</span>
<span class="co">#> x</span>
<span class="co">#> <int></span>
<span class="co">#> 1 1</span>
<span class="co">#> 2 2</span>
<span class="co">#> 3 3</span>
<span class="co">#> 4 4</span>
<span class="co">#> 5 5</span>
<span class="co">#> 6 6</span>
<span class="co">#> # ... with 4 more rows</span></code></pre></div>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">grid <-<span class="st"> </span>grid %>%
<span class="st"> </span><span class="kw">add_predictions</span>(sim1_mod)
grid
<span class="co">#> # A tibble: 10 × 2</span>
<span class="co">#> x pred</span>
<span class="co">#> <int> <dbl></span>
<span class="co">#> 1 1 6.27</span>
<span class="co">#> 2 2 8.32</span>
<span class="co">#> 3 3 10.38</span>
<span class="co">#> 4 4 12.43</span>
<span class="co">#> 5 5 14.48</span>
<span class="co">#> 6 6 16.53</span>
<span class="co">#> # ... with 4 more rows</span></code></pre></div>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">ggplot</span>(sim1, <span class="kw">aes</span>(x)) +
<span class="st"> </span><span class="kw">geom_point</span>(<span class="kw">aes</span>(<span class="dt">y =</span> y)) +
<span class="st"> </span><span class="kw">geom_line</span>(<span class="kw">aes</span>(<span class="dt">y =</span> pred), <span class="dt">data =</span> grid, <span class="dt">colour =</span> <span class="st">"red"</span>, <span class="dt">size =</span> <span class="dv">1</span>)</code></pre></div>
<p><img src="model-basics_files/figure-html/unnamed-chunk-22-1.png" width="70%" style="display: block; margin: auto;" /></p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">sim1 <-<span class="st"> </span>sim1 %>%
<span class="st"> </span><span class="kw">add_residuals</span>(sim1_mod)
sim1
<span class="co">#> # A tibble: 30 × 3</span>
<span class="co">#> x y resid</span>
<span class="co">#> <int> <dbl> <dbl></span>
<span class="co">#> 1 1 4.20 -2.072</span>
<span class="co">#> 2 1 7.51 1.238</span>
<span class="co">#> 3 1 2.13 -4.147</span>
<span class="co">#> 4 2 8.99 0.665</span>
<span class="co">#> 5 2 10.24 1.919</span>
<span class="co">#> 6 2 11.30 2.973</span>
<span class="co">#> # ... with 24 more rows</span>
<span class="kw">ggplot</span>(sim1, <span class="kw">aes</span>(resid)) +
<span class="st"> </span><span class="kw">geom_freqpoly</span>(<span class="dt">binwidth =</span> <span class="fl">0.5</span>)</code></pre></div>
<p><img src="model-basics_files/figure-html/unnamed-chunk-23-1.png" width="70%" style="display: block; margin: auto;" /></p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">ggplot</span>(sim1, <span class="kw">aes</span>(x, resid)) +<span class="st"> </span>
<span class="st"> </span><span class="kw">geom_ref_line</span>(<span class="dt">h =</span> <span class="dv">0</span>) +
<span class="st"> </span><span class="kw">geom_point</span>()</code></pre></div>
<p><img src="model-basics_files/figure-html/unnamed-chunk-24-1.png" width="70%" style="display: block; margin: auto;" /></p>
<div id="exercises-49" class="section level3">
<h3><span class="header-section-number">18.3.1</span> Exercises</h3>
<ol style="list-style-type: decimal">
<li>nstead of using <code>lm()</code> to fit a straight line, you can use <code>loess()</code> to fit a smooth curve. Repeat the process of model fitting, grid generation, predictions, and visualisation on sim1 using <code>loess()</code> instead of <code>lm()</code>. How does the result compare to <code>geom_smooth()</code>?</li>
</ol>
<p>I’ll use <code>add_predictions</code> and <code>add_residuals</code> to add the predictions and residuals from a loess regression to the <code>sim1</code> data.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">sim1_loess <-<span class="st"> </span><span class="kw">loess</span>(y ~<span class="st"> </span>x, <span class="dt">data =</span> sim1)
grid_loess <-<span class="st"> </span>sim1 %>%
<span class="st"> </span><span class="kw">add_predictions</span>(sim1_loess)
sim1 <-<span class="st"> </span>sim1 %>%
<span class="st"> </span><span class="kw">add_residuals</span>(sim1_loess, <span class="dt">var =</span> <span class="st">"resid_loess"</span>) %>%
<span class="st"> </span><span class="kw">add_predictions</span>(sim1_loess, <span class="dt">var =</span> <span class="st">"pred_loess"</span>)
</code></pre></div>
<p>This plots the loess predictions. The loess produces a nonlinear, but smooth line through the data.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">plot_sim1_loess <-<span class="st"> </span>
<span class="st"> </span><span class="kw">ggplot</span>(sim1, <span class="kw">aes</span>(<span class="dt">x =</span> x, <span class="dt">y =</span> y)) +
<span class="st"> </span><span class="kw">geom_point</span>() +
<span class="st"> </span><span class="kw">geom_line</span>(<span class="kw">aes</span>(<span class="dt">x =</span> x, <span class="dt">y =</span> pred), <span class="dt">data =</span> grid_loess, <span class="dt">colour =</span> <span class="st">"red"</span>)
plot_sim1_loess</code></pre></div>
<p><img src="model-basics_files/figure-html/unnamed-chunk-26-1.png" width="70%" style="display: block; margin: auto;" /></p>
<p>The predictions of loess are the same as the default method for <code>geom_smooth</code> because <code>geom_smooth()</code> uses <code>loess()</code> by default; the message even tells us that.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">plot_sim1_loess +
<span class="st"> </span><span class="kw">geom_smooth</span>(<span class="dt">colour =</span> <span class="st">"blue"</span>, <span class="dt">se =</span> <span class="ot">FALSE</span>, <span class="dt">alpha =</span> <span class="fl">0.20</span>)
<span class="co">#> `geom_smooth()` using method = 'loess'</span></code></pre></div>
<p><img src="model-basics_files/figure-html/unnamed-chunk-27-1.png" width="70%" style="display: block; margin: auto;" /></p>
<p>We can plot the residuals (red), and compare them to the residuals from lm (black). In general, the loess model has smaller residuals within the sample (out of sample is a different issue, and we haven’t considered the uncertainty of these estimates).</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">ggplot</span>(sim1, <span class="kw">aes</span>(<span class="dt">x =</span> x)) +
<span class="st"> </span><span class="kw">geom_ref_line</span>(<span class="dt">h =</span> <span class="dv">0</span>) +
<span class="st"> </span><span class="kw">geom_point</span>(<span class="kw">aes</span>(<span class="dt">y =</span> resid)) +
<span class="st"> </span><span class="kw">geom_point</span>(<span class="kw">aes</span>(<span class="dt">y =</span> resid_loess), <span class="dt">colour =</span> <span class="st">"red"</span>)</code></pre></div>
<p><img src="model-basics_files/figure-html/unnamed-chunk-28-1.png" width="70%" style="display: block; margin: auto;" /></p>
<ol start="2" style="list-style-type: decimal">
<li><code>add_predictions()</code> is paired with <code>gather_predictions()</code> and <code>spread_predictions()</code>. How do these three functions differ?</li>
</ol>
<p>The functions <code>gather_predictions</code> and <code>spread_predictions</code> allow for adding predictions from multiple models at once.</p>
<ol start="3" style="list-style-type: decimal">
<li>What does <code>geom_ref_line()</code> do? What package does it come from? Why is displaying a reference line in plots showing residuals useful and important?</li>
</ol>
<p>The geom <code>geom_ref_line()</code> adds as reference line to a plot. Even though it alters a <strong>ggplot2</strong> plot, it is in the <strong>modelr</strong> package. Putting a reference line at zero for residuals is important because good models (generally) should have residuals centered at zero, with approximately the same variance (or distribution) over the support of x, and no correlation. A zero reference line makes it easier to judge these characteristics visually.</p>
<ol start="4" style="list-style-type: decimal">
<li>Why might you want to look at a frequency polygon of absolute residuals? What are the pros and cons compared to looking at the raw residuals?</li>
</ol>
<p>The frequency polygon makes it easier to judge whether the variance and/or absolute size of the residuals varies with respect to x. This is called heteroskedasticity, and results in incorrect standard errors in inference. In prediction, this provides insight into where the model is working well and where it is not. What is lost, is that since the absolute values are shown, whether the model is over-predicting or underpredicting, or on average correctly predicting in different regions of x cannot be determined.</p>
</div>
</div>
<div id="formulas-and-model-families" class="section level2">
<h2><span class="header-section-number">18.4</span> Formulas and Model Families</h2>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">df <-<span class="st"> </span><span class="kw">tribble</span>(
~y, ~x1, ~x2,
<span class="dv">4</span>, <span class="dv">2</span>, <span class="dv">5</span>,
<span class="dv">5</span>, <span class="dv">1</span>, <span class="dv">6</span>
)</code></pre></div>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">model_matrix</span>(df, y ~<span class="st"> </span>x1)
<span class="co">#> # A tibble: 2 × 2</span>
<span class="co">#> `(Intercept)` x1</span>
<span class="co">#> <dbl> <dbl></span>
<span class="co">#> 1 1 2</span>
<span class="co">#> 2 1 1</span></code></pre></div>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">model_matrix</span>(df, y ~<span class="st"> </span>x1 -<span class="st"> </span><span class="dv">1</span>)
<span class="co">#> # A tibble: 2 × 1</span>
<span class="co">#> x1</span>
<span class="co">#> <dbl></span>
<span class="co">#> 1 2</span>
<span class="co">#> 2 1</span></code></pre></div>
<div id="categorical-variables" class="section level3">
<h3><span class="header-section-number">18.4.1</span> Categorical Variables</h3>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">df <-<span class="st"> </span><span class="kw">tribble</span>(
~<span class="st"> </span>sex, ~<span class="st"> </span>response,
<span class="st">"male"</span>, <span class="dv">1</span>,
<span class="st">"female"</span>, <span class="dv">2</span>,
<span class="st">"male"</span>, <span class="dv">1</span>
)
<span class="kw">model_matrix</span>(df, response ~<span class="st"> </span>sex)
<span class="co">#> # A tibble: 3 × 2</span>
<span class="co">#> `(Intercept)` sexmale</span>
<span class="co">#> <dbl> <dbl></span>
<span class="co">#> 1 1 1</span>
<span class="co">#> 2 1 0</span>
<span class="co">#> 3 1 1</span></code></pre></div>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">ggplot</span>(sim2) +
<span class="st"> </span><span class="kw">geom_point</span>(<span class="kw">aes</span>(x, y))</code></pre></div>
<p><img src="model-basics_files/figure-html/unnamed-chunk-33-1.png" width="70%" style="display: block; margin: auto;" /></p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">mod2 <-<span class="st"> </span><span class="kw">lm</span>(y ~<span class="st"> </span>x, <span class="dt">data =</span> sim2)
grid <-<span class="st"> </span>sim2 %>%
<span class="st"> </span><span class="kw">data_grid</span>(x) %>%
<span class="st"> </span><span class="kw">add_predictions</span>(mod2)
grid
<span class="co">#> # A tibble: 4 × 2</span>
<span class="co">#> x pred</span>
<span class="co">#> <chr> <dbl></span>
<span class="co">#> 1 a 1.15</span>
<span class="co">#> 2 b 8.12</span>
<span class="co">#> 3 c 6.13</span>
<span class="co">#> 4 d 1.91</span></code></pre></div>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">ggplot</span>(sim3, <span class="kw">aes</span>(x1, y)) +<span class="st"> </span>
<span class="st"> </span><span class="kw">geom_point</span>(<span class="kw">aes</span>(<span class="dt">colour =</span> x2))</code></pre></div>
<p><img src="model-basics_files/figure-html/unnamed-chunk-35-1.png" width="70%" style="display: block; margin: auto;" /></p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">mod1 <-<span class="st"> </span><span class="kw">lm</span>(y ~<span class="st"> </span>x1 +<span class="st"> </span>x2, <span class="dt">data =</span> sim3)
mod2 <-<span class="st"> </span><span class="kw">lm</span>(y ~<span class="st"> </span>x1 *<span class="st"> </span>x2, <span class="dt">data =</span> sim3)</code></pre></div>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">grid <-<span class="st"> </span>sim3 %>%<span class="st"> </span>
<span class="st"> </span><span class="kw">data_grid</span>(x1, x2) %>%<span class="st"> </span>
<span class="st"> </span><span class="kw">gather_predictions</span>(mod1, mod2)
grid
<span class="co">#> # A tibble: 80 × 4</span>
<span class="co">#> model x1 x2 pred</span>
<span class="co">#> <chr> <int> <fctr> <dbl></span>
<span class="co">#> 1 mod1 1 a 1.67</span>
<span class="co">#> 2 mod1 1 b 4.56</span>
<span class="co">#> 3 mod1 1 c 6.48</span>
<span class="co">#> 4 mod1 1 d 4.03</span>
<span class="co">#> 5 mod1 2 a 1.48</span>
<span class="co">#> 6 mod1 2 b 4.37</span>
<span class="co">#> # ... with 74 more rows</span></code></pre></div>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">ggplot</span>(sim3, <span class="kw">aes</span>(x1, y, <span class="dt">colour =</span> x2)) +<span class="st"> </span>
<span class="st"> </span><span class="kw">geom_point</span>() +<span class="st"> </span>
<span class="st"> </span><span class="kw">geom_line</span>(<span class="dt">data =</span> grid, <span class="kw">aes</span>(<span class="dt">y =</span> pred)) +<span class="st"> </span>
<span class="st"> </span><span class="kw">facet_wrap</span>(~<span class="st"> </span>model)</code></pre></div>
<p><img src="model-basics_files/figure-html/unnamed-chunk-38-1.png" width="70%" style="display: block; margin: auto;" /></p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">sim3 <-<span class="st"> </span>sim3 %>%<span class="st"> </span>
<span class="st"> </span><span class="kw">gather_residuals</span>(mod1, mod2)
<span class="kw">ggplot</span>(sim3, <span class="kw">aes</span>(x1, resid, <span class="dt">colour =</span> x2)) +<span class="st"> </span>
<span class="st"> </span><span class="kw">geom_point</span>() +<span class="st"> </span>
<span class="st"> </span><span class="kw">facet_grid</span>(model ~<span class="st"> </span>x2)</code></pre></div>
<p><img src="model-basics_files/figure-html/unnamed-chunk-39-1.png" width="70%" style="display: block; margin: auto;" /></p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">mod1 <-<span class="st"> </span><span class="kw">lm</span>(y ~<span class="st"> </span>x1 +<span class="st"> </span>x2, <span class="dt">data =</span> sim4)
mod2 <-<span class="st"> </span><span class="kw">lm</span>(y ~<span class="st"> </span>x1 *<span class="st"> </span>x2, <span class="dt">data =</span> sim4)
grid <-<span class="st"> </span>sim4 %>%<span class="st"> </span>
<span class="st"> </span><span class="kw">data_grid</span>(
<span class="dt">x1 =</span> <span class="kw">seq_range</span>(x1, <span class="dv">5</span>),
<span class="dt">x2 =</span> <span class="kw">seq_range</span>(x2, <span class="dv">5</span>)
) %>%<span class="st"> </span>
<span class="st"> </span><span class="kw">gather_predictions</span>(mod1, mod2)
grid
<span class="co">#> # A tibble: 50 × 4</span>
<span class="co">#> model x1 x2 pred</span>
<span class="co">#> <chr> <dbl> <dbl> <dbl></span>
<span class="co">#> 1 mod1 -1.0 -1.0 0.996</span>
<span class="co">#> 2 mod1 -1.0 -0.5 -0.395</span>
<span class="co">#> 3 mod1 -1.0 0.0 -1.786</span>
<span class="co">#> 4 mod1 -1.0 0.5 -3.177</span>
<span class="co">#> 5 mod1 -1.0 1.0 -4.569</span>
<span class="co">#> 6 mod1 -0.5 -1.0 1.907</span>
<span class="co">#> # ... with 44 more rows</span></code></pre></div>
<p>Function <code>seq_range</code> is useful.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">ggplot</span>(grid, <span class="kw">aes</span>(x1, x2)) +
<span class="st"> </span><span class="kw">geom_tile</span>(<span class="kw">aes</span>(<span class="dt">fill =</span> pred)) +
<span class="st"> </span><span class="kw">facet_wrap</span>(~<span class="st"> </span>model)</code></pre></div>
<p><img src="model-basics_files/figure-html/unnamed-chunk-41-1.png" width="70%" style="display: block; margin: auto;" /></p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">ggplot</span>(grid, <span class="kw">aes</span>(x1, pred, <span class="dt">colour =</span> x2, <span class="dt">group =</span> x2)) +
<span class="st"> </span><span class="kw">geom_line</span>() +
<span class="st"> </span><span class="kw">facet_wrap</span>(~<span class="st"> </span>model)
<span class="kw">ggplot</span>(grid, <span class="kw">aes</span>(x2, pred, <span class="dt">colour =</span> x1, <span class="dt">group =</span> x1)) +<span class="st"> </span>
<span class="st"> </span><span class="kw">geom_line</span>() +
<span class="st"> </span><span class="kw">facet_wrap</span>(~<span class="st"> </span>model)</code></pre></div>
<p><img src="model-basics_files/figure-html/unnamed-chunk-42-1.png" width="70%" style="display: block; margin: auto;" /><img src="model-basics_files/figure-html/unnamed-chunk-42-2.png" width="70%" style="display: block; margin: auto;" /></p>
<p><strong>TODO</strong> We should visualize interactions with plotly</p>
</div>
<div id="exercises-50" class="section level3">
<h3><span class="header-section-number">18.4.2</span> Exercises</h3>
</div>
</div>
<div id="missing-values-2" class="section level2">
<h2><span class="header-section-number">18.5</span> Missing values</h2>
<p><strong>TODO</strong> Need to write a tidyverse compliant na.omit function.</p>
</div>
<div id="other-model-families" class="section level2">
<h2><span class="header-section-number">18.6</span> Other model families</h2>
<p><strong>NOTE</strong> It’s worth mentioning these as more general models. Though they don’t appear as much in social science work. I should try to explain that. I can think of several reasons</p>
<ul>
<li>preference for easy to explain models (though I think that’s wrong–most people can’t visualize high-dimensional space well, and interpret results marginally even though they are conditional)</li>
<li>status-quo bias and path dependence combined with lack of knowledge of work outside the field and median lack of technical ability to understand or use these models.</li>
<li>the most principled reason is that those modre complicated models really excel in prediction. If we take an agnostic approach to regression, as in the Angrist and Pischke books, then regression isn’t being used to fit <span class="math inline">\(f(y | x)\)</span>, its being used to fit <span class="math inline">\(E(f(y | x))\)</span>, and more specifically to get some sort of average effect for a change in a specific variable.</li>
</ul>
</div>
</div>
</section>
</div>
</div>
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