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<!DOCTYPE html>
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<title>Machine Learning | Practical Data Science</title>
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<li class="chapter" data-level="" data-path="intro.html"><a href="intro.html#part-2-programming-basics"><i class="fa fa-check"></i>Part 2: Programming Basics</a></li>
<li class="chapter" data-level="" data-path="intro.html"><a href="intro.html#part-3-modeling"><i class="fa fa-check"></i>Part 3: Modeling</a></li>
<li class="chapter" data-level="" data-path="intro.html"><a href="intro.html#part-4-visualization"><i class="fa fa-check"></i>Part 4: Visualization</a></li>
<li class="chapter" data-level="" data-path="intro.html"><a href="intro.html#part-5-presentation"><i class="fa fa-check"></i>Part 5: Presentation</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="intro.html"><a href="intro.html#workshops"><i class="fa fa-check"></i>Workshops</a></li>
<li class="chapter" data-level="" data-path="intro.html"><a href="intro.html#other"><i class="fa fa-check"></i>Other</a>
<ul>
<li class="chapter" data-level="" data-path="intro.html"><a href="intro.html#python-notebooks"><i class="fa fa-check"></i>Python notebooks</a></li>
<li class="chapter" data-level="" data-path="intro.html"><a href="intro.html#other-r-packages"><i class="fa fa-check"></i>Other R packages</a></li>
<li class="chapter" data-level="" data-path="intro.html"><a href="intro.html#history"><i class="fa fa-check"></i>History</a></li>
<li class="chapter" data-level="" data-path="intro.html"><a href="intro.html#current-efforts"><i class="fa fa-check"></i>Current Efforts</a></li>
</ul></li>
</ul></li>
<li class="part"><span><b>Part I: Information Processing</b></span></li>
<li class="chapter" data-level="" data-path="data_structures.html"><a href="data_structures.html"><i class="fa fa-check"></i>Data Structures</a>
<ul>
<li class="chapter" data-level="" data-path="data_structures.html"><a href="data_structures.html#vectors"><i class="fa fa-check"></i>Vectors</a>
<ul>
<li class="chapter" data-level="" data-path="data_structures.html"><a href="data_structures.html#character-strings"><i class="fa fa-check"></i>Character strings</a></li>
<li class="chapter" data-level="" data-path="data_structures.html"><a href="data_structures.html#factors"><i class="fa fa-check"></i>Factors</a></li>
<li class="chapter" data-level="" data-path="data_structures.html"><a href="data_structures.html#logicals"><i class="fa fa-check"></i>Logicals</a></li>
<li class="chapter" data-level="" data-path="data_structures.html"><a href="data_structures.html#numeric-and-integer"><i class="fa fa-check"></i>Numeric and integer</a></li>
<li class="chapter" data-level="" data-path="data_structures.html"><a href="data_structures.html#dates"><i class="fa fa-check"></i>Dates</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="data_structures.html"><a href="data_structures.html#matrices"><i class="fa fa-check"></i>Matrices</a>
<ul>
<li class="chapter" data-level="" data-path="data_structures.html"><a href="data_structures.html#creating-a-matrix"><i class="fa fa-check"></i>Creating a matrix</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="data_structures.html"><a href="data_structures.html#lists"><i class="fa fa-check"></i>Lists</a></li>
<li class="chapter" data-level="" data-path="data_structures.html"><a href="data_structures.html#data-frames"><i class="fa fa-check"></i>Data Frames</a>
<ul>
<li class="chapter" data-level="" data-path="data_structures.html"><a href="data_structures.html#creating-a-data-frame"><i class="fa fa-check"></i>Creating a data frame</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="data_structures.html"><a href="data_structures.html#data-structure-exercises"><i class="fa fa-check"></i>Data Structure Exercises</a>
<ul>
<li class="chapter" data-level="" data-path="data_structures.html"><a href="data_structures.html#exercise-1"><i class="fa fa-check"></i>Exercise 1</a></li>
<li class="chapter" data-level="" data-path="data_structures.html"><a href="data_structures.html#exercise-2"><i class="fa fa-check"></i>Exercise 2</a></li>
<li class="chapter" data-level="" data-path="data_structures.html"><a href="data_structures.html#thinking-exercises"><i class="fa fa-check"></i>Thinking Exercises</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="data_structures.html"><a href="data_structures.html#python-data-structures-notebook"><i class="fa fa-check"></i>Python Data Structures Notebook</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="io.html"><a href="io.html"><i class="fa fa-check"></i>Input/Output</a>
<ul>
<li class="chapter" data-level="" data-path="io.html"><a href="io.html#better-faster-approaches"><i class="fa fa-check"></i>Better & Faster Approaches</a></li>
<li class="chapter" data-level="" data-path="io.html"><a href="io.html#r-specific-data"><i class="fa fa-check"></i>R-specific Data</a>
<ul>
<li class="chapter" data-level="" data-path="io.html"><a href="io.html#r-datasets"><i class="fa fa-check"></i>R Datasets</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="io.html"><a href="io.html#other-types-of-data"><i class="fa fa-check"></i>Other Types of Data</a></li>
<li class="chapter" data-level="" data-path="io.html"><a href="io.html#on-the-horizon"><i class="fa fa-check"></i>On the Horizon</a></li>
<li class="chapter" data-level="" data-path="io.html"><a href="io.html#big-data"><i class="fa fa-check"></i>Big Data</a></li>
<li class="chapter" data-level="" data-path="io.html"><a href="io.html#io-exercises"><i class="fa fa-check"></i>I/O Exercises</a>
<ul>
<li class="chapter" data-level="" data-path="io.html"><a href="io.html#exercise-1-1"><i class="fa fa-check"></i>Exercise 1</a></li>
<li class="chapter" data-level="" data-path="io.html"><a href="io.html#thinking-exercises-1"><i class="fa fa-check"></i>Thinking Exercises</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="io.html"><a href="io.html#python-io-notebook"><i class="fa fa-check"></i>Python I/O Notebook</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="indexing.html"><a href="indexing.html"><i class="fa fa-check"></i>Indexing</a>
<ul>
<li class="chapter" data-level="" data-path="indexing.html"><a href="indexing.html#slicing-vectors"><i class="fa fa-check"></i>Slicing Vectors</a></li>
<li class="chapter" data-level="" data-path="indexing.html"><a href="indexing.html#slicing-matricesdata.frames"><i class="fa fa-check"></i>Slicing Matrices/data.frames</a></li>
<li class="chapter" data-level="" data-path="indexing.html"><a href="indexing.html#label-based-indexing"><i class="fa fa-check"></i>Label-based Indexing</a></li>
<li class="chapter" data-level="" data-path="indexing.html"><a href="indexing.html#position-based-indexing"><i class="fa fa-check"></i>Position-based Indexing</a></li>
<li class="chapter" data-level="" data-path="indexing.html"><a href="indexing.html#mixed-indexing"><i class="fa fa-check"></i>Mixed Indexing</a></li>
<li class="chapter" data-level="" data-path="indexing.html"><a href="indexing.html#non-contiguous"><i class="fa fa-check"></i>Non-contiguous</a></li>
<li class="chapter" data-level="" data-path="indexing.html"><a href="indexing.html#boolean"><i class="fa fa-check"></i>Boolean</a></li>
<li class="chapter" data-level="" data-path="indexing.html"><a href="indexing.html#listdata.frame-extraction"><i class="fa fa-check"></i>List/data.frame Extraction</a></li>
<li class="chapter" data-level="" data-path="indexing.html"><a href="indexing.html#indexing-exercises"><i class="fa fa-check"></i>Indexing Exercises</a>
<ul>
<li class="chapter" data-level="" data-path="indexing.html"><a href="indexing.html#exercise-1-2"><i class="fa fa-check"></i>Exercise 1</a></li>
<li class="chapter" data-level="" data-path="indexing.html"><a href="indexing.html#exercise-2-1"><i class="fa fa-check"></i>Exercise 2</a></li>
<li class="chapter" data-level="" data-path="indexing.html"><a href="indexing.html#exercise-3"><i class="fa fa-check"></i>Exercise 3</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="indexing.html"><a href="indexing.html#python-indexing-notebook"><i class="fa fa-check"></i>Python Indexing Notebook</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="pipes.html"><a href="pipes.html"><i class="fa fa-check"></i>Pipes</a>
<ul>
<li class="chapter" data-level="" data-path="pipes.html"><a href="pipes.html#using-variables-as-they-are-created"><i class="fa fa-check"></i>Using Variables as They are Created</a></li>
<li class="chapter" data-level="" data-path="pipes.html"><a href="pipes.html#pipes-for-visualization-more-later"><i class="fa fa-check"></i>Pipes for Visualization (more later)</a></li>
<li class="chapter" data-level="" data-path="pipes.html"><a href="pipes.html#the-dot"><i class="fa fa-check"></i>The Dot</a></li>
<li class="chapter" data-level="" data-path="pipes.html"><a href="pipes.html#flexibility"><i class="fa fa-check"></i>Flexibility</a></li>
<li class="chapter" data-level="" data-path="pipes.html"><a href="pipes.html#pipes-summary"><i class="fa fa-check"></i>Pipes Summary</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="tidyverse.html"><a href="tidyverse.html"><i class="fa fa-check"></i>Tidyverse</a>
<ul>
<li class="chapter" data-level="" data-path="tidyverse.html"><a href="tidyverse.html#what-is-the-tidyverse"><i class="fa fa-check"></i>What is the Tidyverse?</a></li>
<li class="chapter" data-level="" data-path="tidyverse.html"><a href="tidyverse.html#what-is-tidy"><i class="fa fa-check"></i>What is Tidy?</a></li>
<li class="chapter" data-level="" data-path="tidyverse.html"><a href="tidyverse.html#dplyr"><i class="fa fa-check"></i>dplyr</a>
<ul>
<li class="chapter" data-level="" data-path="tidyverse.html"><a href="tidyverse.html#an-example"><i class="fa fa-check"></i>An example</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="tidyverse.html"><a href="tidyverse.html#running-example"><i class="fa fa-check"></i>Running Example</a></li>
<li class="chapter" data-level="" data-path="tidyverse.html"><a href="tidyverse.html#selecting-columns"><i class="fa fa-check"></i>Selecting Columns</a>
<ul>
<li class="chapter" data-level="" data-path="tidyverse.html"><a href="tidyverse.html#helper-functions"><i class="fa fa-check"></i>Helper functions</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="tidyverse.html"><a href="tidyverse.html#filtering-rows"><i class="fa fa-check"></i>Filtering Rows</a></li>
<li class="chapter" data-level="" data-path="tidyverse.html"><a href="tidyverse.html#generating-new-data"><i class="fa fa-check"></i>Generating New Data</a></li>
<li class="chapter" data-level="" data-path="tidyverse.html"><a href="tidyverse.html#grouping-and-summarizing-data"><i class="fa fa-check"></i>Grouping and Summarizing Data</a></li>
<li class="chapter" data-level="" data-path="tidyverse.html"><a href="tidyverse.html#renaming-columns"><i class="fa fa-check"></i>Renaming Columns</a></li>
<li class="chapter" data-level="" data-path="tidyverse.html"><a href="tidyverse.html#merging-data"><i class="fa fa-check"></i>Merging Data</a></li>
<li class="chapter" data-level="" data-path="tidyverse.html"><a href="tidyverse.html#pivoting-axes"><i class="fa fa-check"></i>Pivoting axes</a></li>
<li class="chapter" data-level="" data-path="tidyverse.html"><a href="tidyverse.html#more-tidyverse"><i class="fa fa-check"></i>More Tidyverse</a></li>
<li class="chapter" data-level="" data-path="tidyverse.html"><a href="tidyverse.html#personal-opinion"><i class="fa fa-check"></i>Personal Opinion</a></li>
<li class="chapter" data-level="" data-path="tidyverse.html"><a href="tidyverse.html#tidyverse-exercises"><i class="fa fa-check"></i>Tidyverse Exercises</a>
<ul>
<li class="chapter" data-level="" data-path="tidyverse.html"><a href="tidyverse.html#exercise-0"><i class="fa fa-check"></i>Exercise 0</a></li>
<li class="chapter" data-level="" data-path="tidyverse.html"><a href="tidyverse.html#exercise-1-3"><i class="fa fa-check"></i>Exercise 1</a></li>
<li class="chapter" data-level="" data-path="tidyverse.html"><a href="tidyverse.html#exercise-2-2"><i class="fa fa-check"></i>Exercise 2</a></li>
<li class="chapter" data-level="" data-path="tidyverse.html"><a href="tidyverse.html#exercise-3-1"><i class="fa fa-check"></i>Exercise 3</a></li>
<li class="chapter" data-level="" data-path="tidyverse.html"><a href="tidyverse.html#exercise-4"><i class="fa fa-check"></i>Exercise 4</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="tidyverse.html"><a href="tidyverse.html#python-pandas-notebook"><i class="fa fa-check"></i>Python Pandas Notebook</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="data_table.html"><a href="data_table.html"><i class="fa fa-check"></i>data.table</a>
<ul>
<li class="chapter" data-level="" data-path="data_table.html"><a href="data_table.html#data.table-basics"><i class="fa fa-check"></i>data.table Basics</a></li>
<li class="chapter" data-level="" data-path="data_table.html"><a href="data_table.html#grouped-operations"><i class="fa fa-check"></i>Grouped Operations</a></li>
<li class="chapter" data-level="" data-path="data_table.html"><a href="data_table.html#faster"><i class="fa fa-check"></i>Faster!</a>
<ul>
<li class="chapter" data-level="" data-path="data_table.html"><a href="data_table.html#joins"><i class="fa fa-check"></i>Joins</a></li>
<li class="chapter" data-level="" data-path="data_table.html"><a href="data_table.html#group-by"><i class="fa fa-check"></i>Group by</a></li>
<li class="chapter" data-level="" data-path="data_table.html"><a href="data_table.html#string-matching"><i class="fa fa-check"></i>String matching</a></li>
<li class="chapter" data-level="" data-path="data_table.html"><a href="data_table.html#reading-files"><i class="fa fa-check"></i>Reading files</a></li>
<li class="chapter" data-level="" data-path="data_table.html"><a href="data_table.html#more-speed"><i class="fa fa-check"></i>More speed</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="data_table.html"><a href="data_table.html#pipe-with-data.table"><i class="fa fa-check"></i>Pipe with data.table</a></li>
<li class="chapter" data-level="" data-path="data_table.html"><a href="data_table.html#data.table-summary"><i class="fa fa-check"></i>data.table Summary</a></li>
<li class="chapter" data-level="" data-path="data_table.html"><a href="data_table.html#faster-dplyr-alternatives"><i class="fa fa-check"></i>Faster dplyr Alternatives</a></li>
<li class="chapter" data-level="" data-path="data_table.html"><a href="data_table.html#data.table-exercises"><i class="fa fa-check"></i>data.table Exercises</a>
<ul>
<li class="chapter" data-level="" data-path="data_table.html"><a href="data_table.html#exercise-0-1"><i class="fa fa-check"></i>Exercise 0</a></li>
<li class="chapter" data-level="" data-path="data_table.html"><a href="data_table.html#exercise-1-4"><i class="fa fa-check"></i>Exercise 1</a></li>
<li class="chapter" data-level="" data-path="data_table.html"><a href="data_table.html#exercise-2-3"><i class="fa fa-check"></i>Exercise 2</a></li>
</ul></li>
</ul></li>
<li class="part"><span><b>Part II: Programming</b></span></li>
<li class="chapter" data-level="" data-path="programming.html"><a href="programming.html"><i class="fa fa-check"></i>Programming Basics</a>
<ul>
<li class="chapter" data-level="" data-path="programming.html"><a href="programming.html#r-objects"><i class="fa fa-check"></i>R Objects</a>
<ul>
<li class="chapter" data-level="" data-path="programming.html"><a href="programming.html#object-inspection-exploration"><i class="fa fa-check"></i>Object Inspection & Exploration</a></li>
<li class="chapter" data-level="" data-path="programming.html"><a href="programming.html#methods"><i class="fa fa-check"></i>Methods</a></li>
<li class="chapter" data-level="" data-path="programming.html"><a href="programming.html#s4-classes"><i class="fa fa-check"></i>S4 classes</a></li>
<li class="chapter" data-level="" data-path="programming.html"><a href="programming.html#others"><i class="fa fa-check"></i>Others</a></li>
<li class="chapter" data-level="" data-path="programming.html"><a href="programming.html#inspecting-functions"><i class="fa fa-check"></i>Inspecting Functions</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="programming.html"><a href="programming.html#documentation"><i class="fa fa-check"></i>Documentation</a></li>
<li class="chapter" data-level="" data-path="programming.html"><a href="programming.html#objects-exercises"><i class="fa fa-check"></i>Objects Exercises</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="iterative.html"><a href="iterative.html"><i class="fa fa-check"></i>Iterative Programming</a>
<ul>
<li class="chapter" data-level="" data-path="iterative.html"><a href="iterative.html#for-loops"><i class="fa fa-check"></i>For Loops</a>
<ul>
<li class="chapter" data-level="" data-path="iterative.html"><a href="iterative.html#a-slight-speed-gain"><i class="fa fa-check"></i>A slight speed gain</a></li>
<li class="chapter" data-level="" data-path="iterative.html"><a href="iterative.html#while-alternative"><i class="fa fa-check"></i>While alternative</a></li>
<li class="chapter" data-level="" data-path="iterative.html"><a href="iterative.html#loops-summary"><i class="fa fa-check"></i>Loops summary</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="iterative.html"><a href="iterative.html#implicit-loops"><i class="fa fa-check"></i>Implicit Loops</a>
<ul>
<li class="chapter" data-level="" data-path="iterative.html"><a href="iterative.html#apply-family"><i class="fa fa-check"></i>apply family</a></li>
<li class="chapter" data-level="" data-path="iterative.html"><a href="iterative.html#apply-functions"><i class="fa fa-check"></i>Apply functions</a></li>
<li class="chapter" data-level="" data-path="iterative.html"><a href="iterative.html#purrr"><i class="fa fa-check"></i>purrr</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="iterative.html"><a href="iterative.html#looping-with-lists"><i class="fa fa-check"></i>Looping with Lists</a></li>
<li class="chapter" data-level="" data-path="iterative.html"><a href="iterative.html#iterative-programming-exercises"><i class="fa fa-check"></i>Iterative Programming Exercises</a>
<ul>
<li class="chapter" data-level="" data-path="iterative.html"><a href="iterative.html#exercise-1-5"><i class="fa fa-check"></i>Exercise 1</a></li>
<li class="chapter" data-level="" data-path="iterative.html"><a href="iterative.html#exercise-2-4"><i class="fa fa-check"></i>Exercise 2</a></li>
<li class="chapter" data-level="" data-path="iterative.html"><a href="iterative.html#exercise-3-2"><i class="fa fa-check"></i>Exercise 3</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="" data-path="functions.html"><a href="functions.html"><i class="fa fa-check"></i>Writing Functions</a>
<ul>
<li class="chapter" data-level="" data-path="functions.html"><a href="functions.html#a-starting-point"><i class="fa fa-check"></i>A Starting Point</a></li>
<li class="chapter" data-level="" data-path="functions.html"><a href="functions.html#dry"><i class="fa fa-check"></i>DRY</a></li>
<li class="chapter" data-level="" data-path="functions.html"><a href="functions.html#conditionals"><i class="fa fa-check"></i>Conditionals</a></li>
<li class="chapter" data-level="" data-path="functions.html"><a href="functions.html#anonymous-functions"><i class="fa fa-check"></i>Anonymous functions</a></li>
<li class="chapter" data-level="" data-path="functions.html"><a href="functions.html#writing-functions-exercises"><i class="fa fa-check"></i>Writing Functions Exercises</a>
<ul>
<li class="chapter" data-level="" data-path="functions.html"><a href="functions.html#excercise-1"><i class="fa fa-check"></i>Excercise 1</a></li>
<li class="chapter" data-level="" data-path="functions.html"><a href="functions.html#excercise-1b"><i class="fa fa-check"></i>Excercise 1b</a></li>
<li class="chapter" data-level="" data-path="functions.html"><a href="functions.html#exercise-2-5"><i class="fa fa-check"></i>Exercise 2</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="" data-path="more.html"><a href="more.html"><i class="fa fa-check"></i>More Programming</a>
<ul>
<li class="chapter" data-level="" data-path="more.html"><a href="more.html#code-style"><i class="fa fa-check"></i>Code Style</a>
<ul>
<li class="chapter" data-level="" data-path="more.html"><a href="more.html#why-does-your-code-exist"><i class="fa fa-check"></i>Why does your code exist?</a></li>
<li class="chapter" data-level="" data-path="more.html"><a href="more.html#assignment"><i class="fa fa-check"></i>Assignment</a></li>
<li class="chapter" data-level="" data-path="more.html"><a href="more.html#code-length"><i class="fa fa-check"></i>Code length</a></li>
<li class="chapter" data-level="" data-path="more.html"><a href="more.html#spacing"><i class="fa fa-check"></i>Spacing</a></li>
<li class="chapter" data-level="" data-path="more.html"><a href="more.html#naming-things"><i class="fa fa-check"></i>Naming things</a></li>
<li class="chapter" data-level="" data-path="more.html"><a href="more.html#other-1"><i class="fa fa-check"></i>Other</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="more.html"><a href="more.html#vectorization"><i class="fa fa-check"></i>Vectorization</a>
<ul>
<li class="chapter" data-level="" data-path="more.html"><a href="more.html#boolean-indexing"><i class="fa fa-check"></i>Boolean indexing</a></li>
<li class="chapter" data-level="" data-path="more.html"><a href="more.html#vectorized-operations"><i class="fa fa-check"></i>Vectorized operations</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="more.html"><a href="more.html#regular-expressions"><i class="fa fa-check"></i>Regular Expressions</a>
<ul>
<li class="chapter" data-level="" data-path="more.html"><a href="more.html#typical-uses"><i class="fa fa-check"></i>Typical uses</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="more.html"><a href="more.html#code-style-exercises"><i class="fa fa-check"></i>Code Style Exercises</a>
<ul>
<li class="chapter" data-level="" data-path="more.html"><a href="more.html#exercise-1-6"><i class="fa fa-check"></i>Exercise 1</a></li>
<li class="chapter" data-level="" data-path="more.html"><a href="more.html#exercise-2-6"><i class="fa fa-check"></i>Exercise 2</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="more.html"><a href="more.html#vectorization-exercises"><i class="fa fa-check"></i>Vectorization Exercises</a>
<ul>
<li class="chapter" data-level="" data-path="more.html"><a href="more.html#exercise-1-7"><i class="fa fa-check"></i>Exercise 1</a></li>
<li class="chapter" data-level="" data-path="more.html"><a href="more.html#exercise-2-7"><i class="fa fa-check"></i>Exercise 2</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="more.html"><a href="more.html#regex-exercises"><i class="fa fa-check"></i>Regex Exercises</a>
<ul>
<li class="chapter" data-level="" data-path="more.html"><a href="more.html#exercise-1-8"><i class="fa fa-check"></i>Exercise 1</a></li>
</ul></li>
</ul></li>
<li class="part"><span><b>Part III: Modeling</b></span></li>
<li class="chapter" data-level="" data-path="models.html"><a href="models.html"><i class="fa fa-check"></i>Model Exploration</a>
<ul>
<li class="chapter" data-level="" data-path="models.html"><a href="models.html#model-taxonomy"><i class="fa fa-check"></i>Model Taxonomy</a></li>
<li class="chapter" data-level="" data-path="models.html"><a href="models.html#linear-models"><i class="fa fa-check"></i>Linear models</a></li>
<li class="chapter" data-level="" data-path="models.html"><a href="models.html#estimation"><i class="fa fa-check"></i>Estimation</a>
<ul>
<li class="chapter" data-level="" data-path="models.html"><a href="models.html#minimizing-and-maximizing"><i class="fa fa-check"></i>Minimizing and maximizing</a></li>
<li class="chapter" data-level="" data-path="models.html"><a href="models.html#optimization"><i class="fa fa-check"></i>Optimization</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="models.html"><a href="models.html#fitting-models"><i class="fa fa-check"></i>Fitting Models</a>
<ul>
<li class="chapter" data-level="" data-path="models.html"><a href="models.html#using-matrices"><i class="fa fa-check"></i>Using matrices</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="models.html"><a href="models.html#summarizing-models"><i class="fa fa-check"></i>Summarizing Models</a></li>
<li class="chapter" data-level="" data-path="models.html"><a href="models.html#variable-transformations"><i class="fa fa-check"></i>Variable Transformations</a>
<ul>
<li class="chapter" data-level="" data-path="models.html"><a href="models.html#numeric-variables"><i class="fa fa-check"></i>Numeric variables</a></li>
<li class="chapter" data-level="" data-path="models.html"><a href="models.html#categorical-variables"><i class="fa fa-check"></i>Categorical variables</a></li>
<li class="chapter" data-level="" data-path="models.html"><a href="models.html#scales-indices-and-dimension-reduction"><i class="fa fa-check"></i>Scales, indices, and dimension reduction</a></li>
<li class="chapter" data-level="" data-path="models.html"><a href="models.html#dont-discretize"><i class="fa fa-check"></i>Don’t discretize</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="models.html"><a href="models.html#variable-importance"><i class="fa fa-check"></i>Variable Importance</a></li>
<li class="chapter" data-level="" data-path="models.html"><a href="models.html#extracting-output"><i class="fa fa-check"></i>Extracting Output</a>
<ul>
<li class="chapter" data-level="" data-path="models.html"><a href="models.html#package-support"><i class="fa fa-check"></i>Package support</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="models.html"><a href="models.html#visualization"><i class="fa fa-check"></i>Visualization</a></li>
<li class="chapter" data-level="" data-path="models.html"><a href="models.html#extensions-to-the-standard-linear-model"><i class="fa fa-check"></i>Extensions to the Standard Linear Model</a>
<ul>
<li class="chapter" data-level="" data-path="models.html"><a href="models.html#different-types-of-targets"><i class="fa fa-check"></i>Different types of targets</a></li>
<li class="chapter" data-level="" data-path="models.html"><a href="models.html#correlated-data"><i class="fa fa-check"></i>Correlated data</a></li>
<li class="chapter" data-level="" data-path="models.html"><a href="models.html#other-extensions"><i class="fa fa-check"></i>Other extensions</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="models.html"><a href="models.html#model-exploration-summary"><i class="fa fa-check"></i>Model Exploration Summary</a></li>
<li class="chapter" data-level="" data-path="models.html"><a href="models.html#model-exploration-exercises"><i class="fa fa-check"></i>Model Exploration Exercises</a>
<ul>
<li class="chapter" data-level="" data-path="models.html"><a href="models.html#exercise-1-9"><i class="fa fa-check"></i>Exercise 1</a></li>
<li class="chapter" data-level="" data-path="models.html"><a href="models.html#exercise-2-8"><i class="fa fa-check"></i>Exercise 2</a></li>
<li class="chapter" data-level="" data-path="models.html"><a href="models.html#exercise-3-3"><i class="fa fa-check"></i>Exercise 3</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="models.html"><a href="models.html#python-model-exploration-notebook"><i class="fa fa-check"></i>Python Model Exploration Notebook</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="model_criticism.html"><a href="model_criticism.html"><i class="fa fa-check"></i>Model Criticism</a>
<ul>
<li class="chapter" data-level="" data-path="model_criticism.html"><a href="model_criticism.html#model-fit"><i class="fa fa-check"></i>Model Fit</a>
<ul>
<li class="chapter" data-level="" data-path="model_criticism.html"><a href="model_criticism.html#standard-linear-model"><i class="fa fa-check"></i>Standard linear model</a></li>
<li class="chapter" data-level="" data-path="model_criticism.html"><a href="model_criticism.html#beyond-ols"><i class="fa fa-check"></i>Beyond OLS</a></li>
<li class="chapter" data-level="" data-path="model_criticism.html"><a href="model_criticism.html#classification"><i class="fa fa-check"></i>Classification</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="model_criticism.html"><a href="model_criticism.html#model-assumptions"><i class="fa fa-check"></i>Model Assumptions</a></li>
<li class="chapter" data-level="" data-path="model_criticism.html"><a href="model_criticism.html#predictive-performance"><i class="fa fa-check"></i>Predictive Performance</a></li>
<li class="chapter" data-level="" data-path="model_criticism.html"><a href="model_criticism.html#model-comparison"><i class="fa fa-check"></i>Model Comparison</a>
<ul>
<li class="chapter" data-level="" data-path="model_criticism.html"><a href="model_criticism.html#example-additional-covariates"><i class="fa fa-check"></i>Example: Additional covariates</a></li>
<li class="chapter" data-level="" data-path="model_criticism.html"><a href="model_criticism.html#example-interactions"><i class="fa fa-check"></i>Example: Interactions</a></li>
<li class="chapter" data-level="" data-path="model_criticism.html"><a href="model_criticism.html#example-additive-models"><i class="fa fa-check"></i>Example: Additive models</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="model_criticism.html"><a href="model_criticism.html#model-averaging"><i class="fa fa-check"></i>Model Averaging</a></li>
<li class="chapter" data-level="" data-path="model_criticism.html"><a href="model_criticism.html#model-criticism-summary"><i class="fa fa-check"></i>Model Criticism Summary</a></li>
<li class="chapter" data-level="" data-path="model_criticism.html"><a href="model_criticism.html#model-criticism-exercises"><i class="fa fa-check"></i>Model Criticism Exercises</a>
<ul>
<li class="chapter" data-level="" data-path="model_criticism.html"><a href="model_criticism.html#exercise-0-2"><i class="fa fa-check"></i>Exercise 0</a></li>
<li class="chapter" data-level="" data-path="model_criticism.html"><a href="model_criticism.html#exercise-1-10"><i class="fa fa-check"></i>Exercise 1</a></li>
<li class="chapter" data-level="" data-path="model_criticism.html"><a href="model_criticism.html#exercise-2-9"><i class="fa fa-check"></i>Exercise 2</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="model_criticism.html"><a href="model_criticism.html#python-model-criticism-notebook"><i class="fa fa-check"></i>Python Model Criticism Notebook</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="ml.html"><a href="ml.html"><i class="fa fa-check"></i>Machine Learning</a>
<ul>
<li class="chapter" data-level="" data-path="ml.html"><a href="ml.html#concepts"><i class="fa fa-check"></i>Concepts</a>
<ul>
<li class="chapter" data-level="" data-path="ml.html"><a href="ml.html#loss"><i class="fa fa-check"></i>Loss</a></li>
<li class="chapter" data-level="" data-path="ml.html"><a href="ml.html#bias-variance-tradeoff"><i class="fa fa-check"></i>Bias-variance tradeoff</a></li>
<li class="chapter" data-level="" data-path="ml.html"><a href="ml.html#regularization"><i class="fa fa-check"></i>Regularization</a></li>
<li class="chapter" data-level="" data-path="ml.html"><a href="ml.html#cross-validation"><i class="fa fa-check"></i>Cross-validation</a></li>
<li class="chapter" data-level="" data-path="ml.html"><a href="ml.html#optimization-1"><i class="fa fa-check"></i>Optimization</a></li>
<li class="chapter" data-level="" data-path="ml.html"><a href="ml.html#tuning-parameters"><i class="fa fa-check"></i>Tuning parameters</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="ml.html"><a href="ml.html#techniques"><i class="fa fa-check"></i>Techniques</a>
<ul>
<li class="chapter" data-level="" data-path="ml.html"><a href="ml.html#regularized-regression"><i class="fa fa-check"></i>Regularized regression</a></li>
<li class="chapter" data-level="" data-path="ml.html"><a href="ml.html#random-forests"><i class="fa fa-check"></i>Random forests</a></li>
<li class="chapter" data-level="" data-path="ml.html"><a href="ml.html#neural-networks"><i class="fa fa-check"></i>Neural networks</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="ml.html"><a href="ml.html#interpreting-the-black-box"><i class="fa fa-check"></i>Interpreting the Black Box</a></li>
<li class="chapter" data-level="" data-path="ml.html"><a href="ml.html#machine-learning-summary"><i class="fa fa-check"></i>Machine Learning Summary</a></li>
<li class="chapter" data-level="" data-path="ml.html"><a href="ml.html#machine-learning-exercises"><i class="fa fa-check"></i>Machine Learning Exercises</a>
<ul>
<li class="chapter" data-level="" data-path="ml.html"><a href="ml.html#exercise-1-11"><i class="fa fa-check"></i>Exercise 1</a></li>
<li class="chapter" data-level="" data-path="ml.html"><a href="ml.html#exercise-2-10"><i class="fa fa-check"></i>Exercise 2</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="ml.html"><a href="ml.html#python-machine-learning-notebook"><i class="fa fa-check"></i>Python Machine Learning Notebook</a></li>
</ul></li>
<li class="part"><span><b>Part IV: Visualization</b></span></li>
<li class="chapter" data-level="" data-path="ggplot2.html"><a href="ggplot2.html"><i class="fa fa-check"></i>ggplot2</a>
<ul>
<li class="chapter" data-level="" data-path="ggplot2.html"><a href="ggplot2.html#layers"><i class="fa fa-check"></i>Layers</a></li>
<li class="chapter" data-level="" data-path="ggplot2.html"><a href="ggplot2.html#piping"><i class="fa fa-check"></i>Piping</a></li>
<li class="chapter" data-level="" data-path="ggplot2.html"><a href="ggplot2.html#aesthetics"><i class="fa fa-check"></i>Aesthetics</a></li>
<li class="chapter" data-level="" data-path="ggplot2.html"><a href="ggplot2.html#geoms"><i class="fa fa-check"></i>Geoms</a></li>
<li class="chapter" data-level="" data-path="ggplot2.html"><a href="ggplot2.html#examples"><i class="fa fa-check"></i>Examples</a></li>
<li class="chapter" data-level="" data-path="ggplot2.html"><a href="ggplot2.html#stats"><i class="fa fa-check"></i>Stats</a></li>
<li class="chapter" data-level="" data-path="ggplot2.html"><a href="ggplot2.html#scales"><i class="fa fa-check"></i>Scales</a></li>
<li class="chapter" data-level="" data-path="ggplot2.html"><a href="ggplot2.html#facets"><i class="fa fa-check"></i>Facets</a></li>
<li class="chapter" data-level="" data-path="ggplot2.html"><a href="ggplot2.html#multiple-plots"><i class="fa fa-check"></i>Multiple plots</a></li>
<li class="chapter" data-level="" data-path="ggplot2.html"><a href="ggplot2.html#fine-control"><i class="fa fa-check"></i>Fine control</a></li>
<li class="chapter" data-level="" data-path="ggplot2.html"><a href="ggplot2.html#themes"><i class="fa fa-check"></i>Themes</a></li>
<li class="chapter" data-level="" data-path="ggplot2.html"><a href="ggplot2.html#extensions"><i class="fa fa-check"></i>Extensions</a></li>
<li class="chapter" data-level="" data-path="ggplot2.html"><a href="ggplot2.html#ggplot2-summary"><i class="fa fa-check"></i>ggplot2 Summary</a></li>
<li class="chapter" data-level="" data-path="ggplot2.html"><a href="ggplot2.html#ggplot2-exercises"><i class="fa fa-check"></i>ggplot2 Exercises</a>
<ul>
<li class="chapter" data-level="" data-path="ggplot2.html"><a href="ggplot2.html#exercise-0-3"><i class="fa fa-check"></i>Exercise 0</a></li>
<li class="chapter" data-level="" data-path="ggplot2.html"><a href="ggplot2.html#exercise-1-12"><i class="fa fa-check"></i>Exercise 1</a></li>
<li class="chapter" data-level="" data-path="ggplot2.html"><a href="ggplot2.html#exercise-2-11"><i class="fa fa-check"></i>Exercise 2</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="ggplot2.html"><a href="ggplot2.html#python-plotnine-notebook"><i class="fa fa-check"></i>Python Plotnine Notebook</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="interactive.html"><a href="interactive.html"><i class="fa fa-check"></i>Interactive Visualization</a>
<ul>
<li class="chapter" data-level="" data-path="interactive.html"><a href="interactive.html#packages"><i class="fa fa-check"></i>Packages</a></li>
<li class="chapter" data-level="" data-path="interactive.html"><a href="interactive.html#piping-for-visualization"><i class="fa fa-check"></i>Piping for Visualization</a></li>
<li class="chapter" data-level="" data-path="interactive.html"><a href="interactive.html#htmlwidgets"><i class="fa fa-check"></i>htmlwidgets</a></li>
<li class="chapter" data-level="" data-path="interactive.html"><a href="interactive.html#plotly"><i class="fa fa-check"></i>Plotly</a>
<ul>
<li class="chapter" data-level="" data-path="interactive.html"><a href="interactive.html#modes"><i class="fa fa-check"></i>Modes</a></li>
<li class="chapter" data-level="" data-path="interactive.html"><a href="interactive.html#ggplotly"><i class="fa fa-check"></i>ggplotly</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="interactive.html"><a href="interactive.html#highcharter"><i class="fa fa-check"></i>Highcharter</a></li>
<li class="chapter" data-level="" data-path="interactive.html"><a href="interactive.html#graph-networks"><i class="fa fa-check"></i>Graph networks</a>
<ul>
<li class="chapter" data-level="" data-path="interactive.html"><a href="interactive.html#visnetwork"><i class="fa fa-check"></i>visNetwork</a></li>
<li class="chapter" data-level="" data-path="interactive.html"><a href="interactive.html#sigmajs"><i class="fa fa-check"></i>sigmajs</a></li>
<li class="chapter" data-level="" data-path="interactive.html"><a href="interactive.html#plotly-1"><i class="fa fa-check"></i>Plotly</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="interactive.html"><a href="interactive.html#leaflet"><i class="fa fa-check"></i>leaflet</a></li>
<li class="chapter" data-level="" data-path="interactive.html"><a href="interactive.html#dt"><i class="fa fa-check"></i>DT</a></li>
<li class="chapter" data-level="" data-path="interactive.html"><a href="interactive.html#shiny"><i class="fa fa-check"></i>Shiny</a>
<ul>
<li class="chapter" data-level="" data-path="interactive.html"><a href="interactive.html#dash"><i class="fa fa-check"></i>Dash</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="interactive.html"><a href="interactive.html#interactive-and-visual-data-exploration"><i class="fa fa-check"></i>Interactive and Visual Data Exploration</a></li>
<li class="chapter" data-level="" data-path="interactive.html"><a href="interactive.html#interactive-visualization-exercises"><i class="fa fa-check"></i>Interactive Visualization Exercises</a>
<ul>
<li class="chapter" data-level="" data-path="interactive.html"><a href="interactive.html#exercise-0-4"><i class="fa fa-check"></i>Exercise 0</a></li>
<li class="chapter" data-level="" data-path="interactive.html"><a href="interactive.html#exercise-1-13"><i class="fa fa-check"></i>Exercise 1</a></li>
<li class="chapter" data-level="" data-path="interactive.html"><a href="interactive.html#exercise-2-12"><i class="fa fa-check"></i>Exercise 2</a></li>
<li class="chapter" data-level="" data-path="interactive.html"><a href="interactive.html#exercise-3-4"><i class="fa fa-check"></i>Exercise 3</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="interactive.html"><a href="interactive.html#python-interactive-visualization-notebook"><i class="fa fa-check"></i>Python Interactive Visualization Notebook</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="thinking_vis.html"><a href="thinking_vis.html"><i class="fa fa-check"></i>Thinking Visually</a>
<ul>
<li class="chapter" data-level="" data-path="thinking_vis.html"><a href="thinking_vis.html#information"><i class="fa fa-check"></i>Information</a>
<ul>
<li class="chapter" data-level="" data-path="thinking_vis.html"><a href="thinking_vis.html#your-audience-isnt-dumb"><i class="fa fa-check"></i>Your audience isn’t dumb</a></li>
<li class="chapter" data-level="" data-path="thinking_vis.html"><a href="thinking_vis.html#clarity-is-key"><i class="fa fa-check"></i>Clarity is key</a></li>
<li class="chapter" data-level="" data-path="thinking_vis.html"><a href="thinking_vis.html#avoid-clutter"><i class="fa fa-check"></i>Avoid clutter</a></li>
<li class="chapter" data-level="" data-path="thinking_vis.html"><a href="thinking_vis.html#color-isnt-optional"><i class="fa fa-check"></i>Color isn’t optional</a></li>
<li class="chapter" data-level="" data-path="thinking_vis.html"><a href="thinking_vis.html#think-interactively"><i class="fa fa-check"></i>Think interactively</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="thinking_vis.html"><a href="thinking_vis.html#color"><i class="fa fa-check"></i>Color</a>
<ul>
<li class="chapter" data-level="" data-path="thinking_vis.html"><a href="thinking_vis.html#viridis"><i class="fa fa-check"></i>Viridis</a></li>
<li class="chapter" data-level="" data-path="thinking_vis.html"><a href="thinking_vis.html#scientific-colors"><i class="fa fa-check"></i>Scientific colors</a></li>
<li class="chapter" data-level="" data-path="thinking_vis.html"><a href="thinking_vis.html#rcolorbrewer"><i class="fa fa-check"></i>RColorBrewer</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="thinking_vis.html"><a href="thinking_vis.html#contrast"><i class="fa fa-check"></i>Contrast</a></li>
<li class="chapter" data-level="" data-path="thinking_vis.html"><a href="thinking_vis.html#scaling-size"><i class="fa fa-check"></i>Scaling Size</a></li>
<li class="chapter" data-level="" data-path="thinking_vis.html"><a href="thinking_vis.html#transparency"><i class="fa fa-check"></i>Transparency</a></li>
<li class="chapter" data-level="" data-path="thinking_vis.html"><a href="thinking_vis.html#accessibility"><i class="fa fa-check"></i>Accessibility</a></li>
<li class="chapter" data-level="" data-path="thinking_vis.html"><a href="thinking_vis.html#file-types"><i class="fa fa-check"></i>File Types</a></li>
<li class="chapter" data-level="" data-path="thinking_vis.html"><a href="thinking_vis.html#summary-of-thinking-visually"><i class="fa fa-check"></i>Summary of Thinking Visually</a></li>
<li class="chapter" data-level="" data-path="thinking_vis.html"><a href="thinking_vis.html#a-casual-list-of-things-to-avoid"><i class="fa fa-check"></i>A casual list of things to avoid</a>
<ul>
<li class="chapter" data-level="" data-path="thinking_vis.html"><a href="thinking_vis.html#pie"><i class="fa fa-check"></i>Pie</a></li>
<li class="chapter" data-level="" data-path="thinking_vis.html"><a href="thinking_vis.html#histograms"><i class="fa fa-check"></i>Histograms</a></li>
<li class="chapter" data-level="" data-path="thinking_vis.html"><a href="thinking_vis.html#using-3d-without-adding-any-communicative-value"><i class="fa fa-check"></i>Using 3D without adding any communicative value</a></li>
<li class="chapter" data-level="" data-path="thinking_vis.html"><a href="thinking_vis.html#using-too-many-colors"><i class="fa fa-check"></i>Using too many colors</a></li>
<li class="chapter" data-level="" data-path="thinking_vis.html"><a href="thinking_vis.html#using-valenced-colors-when-data-isnt-applicable"><i class="fa fa-check"></i>Using valenced colors when data isn’t applicable</a></li>
<li class="chapter" data-level="" data-path="thinking_vis.html"><a href="thinking_vis.html#showing-maps-that-just-display-population"><i class="fa fa-check"></i>Showing maps that just display population</a></li>
<li class="chapter" data-level="" data-path="thinking_vis.html"><a href="thinking_vis.html#biplots"><i class="fa fa-check"></i>Biplots</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="thinking_vis.html"><a href="thinking_vis.html#thinking-visually-exercises"><i class="fa fa-check"></i>Thinking Visually Exercises</a>
<ul>
<li class="chapter" data-level="" data-path="thinking_vis.html"><a href="thinking_vis.html#exercise-1-14"><i class="fa fa-check"></i>Exercise 1</a></li>
<li class="chapter" data-level="" data-path="thinking_vis.html"><a href="thinking_vis.html#exercise-2-13"><i class="fa fa-check"></i>Exercise 2</a></li>
<li class="chapter" data-level="" data-path="thinking_vis.html"><a href="thinking_vis.html#thinking-exercises-2"><i class="fa fa-check"></i>Thinking exercises</a></li>
</ul></li>
</ul></li>
<li class="part"><span><b>Part V: Presentation</b></span></li>
<li class="chapter" data-level="" data-path="reproducibility.html"><a href="reproducibility.html"><i class="fa fa-check"></i>Building Better Data-Driven Products</a>
<ul>
<li class="chapter" data-level="" data-path="reproducibility.html"><a href="reproducibility.html#rep-analysis"><i class="fa fa-check"></i>Rep* Analysis</a>
<ul>
<li class="chapter" data-level="" data-path="reproducibility.html"><a href="reproducibility.html#example"><i class="fa fa-check"></i>Example</a></li>
<li class="chapter" data-level="" data-path="reproducibility.html"><a href="reproducibility.html#repeatable"><i class="fa fa-check"></i>Repeatable</a></li>
<li class="chapter" data-level="" data-path="reproducibility.html"><a href="reproducibility.html#reproducible"><i class="fa fa-check"></i>Reproducible</a></li>
<li class="chapter" data-level="" data-path="reproducibility.html"><a href="reproducibility.html#replicable"><i class="fa fa-check"></i>Replicable</a></li>
<li class="chapter" data-level="" data-path="reproducibility.html"><a href="reproducibility.html#summary-of-rep-analysis"><i class="fa fa-check"></i>Summary of rep* analysis</a></li>
</ul></li>
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<h1>
<i class="fa fa-circle-o-notch fa-spin"></i><a href="./">Practical Data Science</a>
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<section class="normal" id="section-">
<div id="machine-learning" class="section level1">
<h1>Machine Learning</h1>
<div style="text-align: center">
<i class="fas fa-magic fa-5x " style="color:#990024;"></i>
</div>
<p><em>Machine learning</em> (ML) encompasses a wide variety of techniques, from standard regression models to almost impenetrably complex modeling tools. While it may seem like magic to the uninitiated, the main thing that distinguishes it from standard statistical methods discussed thus far is an approach that heavily favors prediction over inference and explanatory power, and which takes the necessary steps to gain any predictive advantage<a href="#fn38" class="footnote-ref" id="fnref38"><sup>38</sup></a>.</p>
<p>ML could potentially be applied in any setting, but typically works best with data sets much larger than classical statistical methods are usually applied to. However, nowadays even complex regression models can be applied to extremely large data sets, and properly applied ML may even work in simpler data settings, so this distinction is muddier than it used to be. The main distinguishing factor is mostly one of focus.</p>
<p>The following only very briefly provides a demonstration of concepts and approaches. I have more <a href="https://m-clark.github.io/introduction-to-machine-learning/">in-depth document available</a> for more details.</p>
<div id="concepts" class="section level2">
<h2>Concepts</h2>
<div id="loss" class="section level3">
<h3>Loss</h3>
<p>We discussed loss functions <a href="models.html#estimation">before</a>, and there was a reason I went more in depth there, mainly because I feel, unlike with ML, loss is not explicitly focused on as much in applied research, leaving the results produced to come across as more magical than it should be. In ML however, we are explicitly concerned with loss functions and, more specifically, evaluating loss on test data. This loss is evaluated over successive iterations of a particular technique, or averaged over several test sets via cross-validation. Typical loss functions are <em>Root Mean Squared Error</em> for numeric targets (essentially the same as for a standard linear model), and <em>cross-entropy</em> for categorical outcomes. There are robust alternatives, such as mean absolute error and hinge loss functions respectively, and many other options besides. You will come across others that might be used for specific scenarios.</p>
<p>The following image, typically called a <em>learning curve</em>, shows an example of loss on a test set as a function of model complexity. In this case, models with more complexity perform better, but only to a point, before test error begins to rise again.</p>
<p><img src="img/learningcurve.svg" width="50%" style="display: block; margin: auto;" /></p>
</div>
<div id="bias-variance-tradeoff" class="section level3">
<h3>Bias-variance tradeoff</h3>
<p>Prediction error, i.e. loss, is composed of several sources. One part is <em>measurement error</em>, which we can’t do anything about, and two components of specific interest: <em>bias</em>, the difference in the observed value and our average predicted value, and <em>variance</em> how much that prediction would change had we trained on different data. More generally we can think of this as a problem of <em>underfitting</em> vs. <em>overfitting</em>. With a model that is too simple, we underfit, and bias is high. If we overfit, the model is too close to the training data, and likely will do poorly with new observations. ML techniques trade some increased bias for even greater reduced variance, which often means less overfitting to the training data, leading to increased performance on new data.</p>
<p>In the following<a href="#fn39" class="footnote-ref" id="fnref39"><sup>39</sup></a>, the blue line represents models applied to training data, while the red line regards performance on the test set. We can see that for the data we train the model to, error will always go down with increased complexity. However, we can see that at some point, the test error will increase as we have started to overfit to the training data.</p>
<p><img src="img/biasvar2.svg" width="50%" style="display: block; margin: auto;" /></p>
</div>
<div id="regularization" class="section level3">
<h3>Regularization</h3>
<p>As we have noted, a model fit to a single data set might do very well with the data at hand, but then suffer when predicting independent data. Also, oftentimes we are interested in a ‘best’ subset of predictors among a great many, and typically the estimated coefficients from standard approaches are overly optimistic unless dealing with sufficiently large sample sizes. This general issue can be improved by shrinking estimates toward zero, such that some of the performance in the initial fit is sacrificed for improvement with regard to prediction. The basic idea in terms of the tradeoff is that we are trading some bias for notably reduced variance. We demonstrate regularized regression below.</p>
</div>
<div id="cross-validation" class="section level3">
<h3>Cross-validation</h3>
<p><em>Cross-validation</em> is widely used for validation and/or testing. With validation, we are usually concerned with picking parameter settings for the model, while the testing is used for ultimate assessment of model performance. Conceptually there is nothing new beyond what was <a href="model_criticism.html#predictive-performance">discussed previously</a> regarding holding out data for assessing predictive performance, we just do more of it.</p>
<p>As an example, let’s say we split our data into three parts. We use two parts (combined) as our training data, then the third part as test. At this point this is identical to our demonstration before. But then, we switch which part is test and which two are training, and do the whole thing over again. And finally once more, so that each of our three parts has taken a turn as a test set. Our estimated error is the average loss across the three times.</p>
<p><img src="img/kfold.svg" width="50%" style="display: block; margin: auto;" /></p>
<p>Typically we do it more than three times, usually 10, and there are fancier methods of <em>k-fold cross-validation</em>, though they typically don’t serve to add much value. In any case, let’s try it with our previous example. The following uses the <span class="pack" style="">tidymodels</span> approach to be consistent with early chapters use of the <span class="pack" style="">tidyverse</span><a href="#fn40" class="footnote-ref" id="fnref40"><sup>40</sup></a>. With it we can employ k-fold cross validation to evaluate the loss.</p>
<div class="sourceCode" id="cb502"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb502-1"><a href="ml.html#cb502-1"></a><span class="co"># install.packages(tidymodels) # if needed</span></span>
<span id="cb502-2"><a href="ml.html#cb502-2"></a><span class="kw">library</span>(tidymodels) </span>
<span id="cb502-3"><a href="ml.html#cb502-3"></a></span>
<span id="cb502-4"><a href="ml.html#cb502-4"></a><span class="kw">load</span>(<span class="st">'data/world_happiness.RData'</span>)</span>
<span id="cb502-5"><a href="ml.html#cb502-5"></a></span>
<span id="cb502-6"><a href="ml.html#cb502-6"></a><span class="kw">set.seed</span>(<span class="dv">1212</span>)</span>
<span id="cb502-7"><a href="ml.html#cb502-7"></a></span>
<span id="cb502-8"><a href="ml.html#cb502-8"></a><span class="co"># specify the model</span></span>
<span id="cb502-9"><a href="ml.html#cb502-9"></a>happy_base_spec =<span class="st"> </span><span class="kw">linear_reg</span>() <span class="op">%>%</span></span>
<span id="cb502-10"><a href="ml.html#cb502-10"></a><span class="st"> </span><span class="kw">set_engine</span>(<span class="dt">engine =</span> <span class="st">"lm"</span>)</span>
<span id="cb502-11"><a href="ml.html#cb502-11"></a></span>
<span id="cb502-12"><a href="ml.html#cb502-12"></a><span class="co"># by default 10-folds</span></span>
<span id="cb502-13"><a href="ml.html#cb502-13"></a>happy_folds =<span class="st"> </span><span class="kw">vfold_cv</span>(happy)</span>
<span id="cb502-14"><a href="ml.html#cb502-14"></a></span>
<span id="cb502-15"><a href="ml.html#cb502-15"></a><span class="kw">library</span>(tune)</span>
<span id="cb502-16"><a href="ml.html#cb502-16"></a></span>
<span id="cb502-17"><a href="ml.html#cb502-17"></a>happy_base_results =<span class="st"> </span><span class="kw">fit_resamples</span>(</span>
<span id="cb502-18"><a href="ml.html#cb502-18"></a> happy_base_spec,</span>
<span id="cb502-19"><a href="ml.html#cb502-19"></a> happiness_score <span class="op">~</span><span class="st"> </span>democratic_quality <span class="op">+</span><span class="st"> </span>generosity <span class="op">+</span><span class="st"> </span>log_gdp_per_capita,</span>
<span id="cb502-20"><a href="ml.html#cb502-20"></a> happy_folds,</span>
<span id="cb502-21"><a href="ml.html#cb502-21"></a> <span class="dt">control =</span> <span class="kw">control_resamples</span>(<span class="dt">save_pred =</span> <span class="ot">TRUE</span>)</span>
<span id="cb502-22"><a href="ml.html#cb502-22"></a>)</span>
<span id="cb502-23"><a href="ml.html#cb502-23"></a></span>
<span id="cb502-24"><a href="ml.html#cb502-24"></a>cv_res =<span class="st"> </span>happy_base_results <span class="op">%>%</span></span>
<span id="cb502-25"><a href="ml.html#cb502-25"></a><span class="st"> </span><span class="kw">collect_metrics</span>()</span></code></pre></div>
<table class="table" style="width: auto !important; margin-left: auto; margin-right: auto;">
<thead>
<tr>
<th style="text-align:left;">
.metric
</th>
<th style="text-align:left;">
.estimator
</th>
<th style="text-align:right;">
mean
</th>
<th style="text-align:right;">
n
</th>
<th style="text-align:right;">
std_err
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
rmse
</td>
<td style="text-align:left;">
standard
</td>
<td style="text-align:right;">
0.629
</td>
<td style="text-align:right;">
10
</td>
<td style="text-align:right;">
0.022
</td>
</tr>
<tr>
<td style="text-align:left;">
rsq
</td>
<td style="text-align:left;">
standard
</td>
<td style="text-align:right;">
0.697
</td>
<td style="text-align:right;">
10
</td>
<td style="text-align:right;">
0.022
</td>
</tr>
</tbody>
</table>
<p>We now see that our average test error is 0.629. It also gives the average R2.</p>
</div>
<div id="optimization-1" class="section level3">
<h3>Optimization</h3>
<p>With ML, much more attention is paid to different optimizers, but the vast majority for deep learning and other many other methods are some flavor of <em>stochastic gradient descent</em>. Often due to the sheer volume of data/parameters, this optimization is done on chunks of the data and in parallel. In general, some optimization approaches may work better in some situations or for some models, where ‘better’ means quicker convergence, or perhaps a smoother ride toward convergence. It is not the case that you would come to incorrect conclusions using one method vs. another per se, just that you might reach those conclusions in a more efficient fashion. The following graphic displays SGD versus several variants<a href="#fn41" class="footnote-ref" id="fnref41"><sup>41</sup></a>. The x and y axes represent the potential values two parameters might take, with the best selection of those values based on a loss function somewhere toward the bottom right. We can see that they all would get there eventually, but some might do so more quickly. This may or may not be the case for some other data situation.</p>
<p><img src="img/opt_vis_rad.gif" width="50%" style="display: block; margin: auto;" /></p>
</div>
<div id="tuning-parameters" class="section level3">
<h3>Tuning parameters</h3>
<p>In any ML setting there are parameters that need to set in order to even run the model. In regularized regression this may be the penalty parameter, for random forests the tree depth, for neural nets, how many hidden units, and many other things. None of these <em>tuning parameters</em> is known beforehand, and so must be tuned, or learned, just like any other. This is usually done with through validation process like k-fold cross validation. The ‘best’ settings are then used to make final predictions on the test set.</p>
<p>The usual workflow is something like the following:</p>
<ul>
<li><p><strong>Tuning</strong>: With the <strong>training data</strong>, use a cross-validation approach to run models with different values for tuning parameters.</p></li>
<li><p><strong>Model Selection</strong>: Select the ‘best’ model as that which minimizes or maximizes the objective function estimated during cross-validation (e.g. RMSE, accuracy, etc.). The test data in this setting are typically referred to as <em>validation sets</em>.</p></li>
<li><p><strong>Prediction</strong>: Use the best model to make predictions on the <strong>test set</strong>.</p></li>
</ul>
</div>
</div>
<div id="techniques" class="section level2">
<h2>Techniques</h2>
<div id="regularized-regression" class="section level3">
<h3>Regularized regression</h3>
<p>A starting point for getting into ML from the more inferential methods is to use <em>regularized regression</em>. These are conceptually no different than standard LM/GLM types of approaches, but they add something to the loss function.</p>
<p><span class="math display">\[\mathcal{Loss} = \Sigma(y - \hat{y})^2 + \lambda\cdot\Sigma\beta^2\]</span>
In the above, this is the same squared error loss function as before, but we add a penalty that is based on the size of the coefficients. So, while initially our loss goes down with some set of estimates, the penalty based on their size might be such that the estimated loss actually increases. This has the effect of shrinking the estimates toward zero. Well, <a href="https://stats.stackexchange.com/questions/179864/why-does-shrinkage-work">why would we want that</a>? This introduces <a href="https://stats.stackexchange.com/questions/207760/when-is-a-biased-estimator-preferable-to-unbiased-one">bias in the coefficients</a>, but the end result is a model that will do better on test set prediction, which is the goal of the ML approach. The way this works regards the bias-variance tradeoff we discussed previously.</p>
<p>The following demonstrates regularized regression using the <span class="pack" style="">glmnet</span> package. It actually uses <em>elastic net</em>, which has a mixture of two penalties, one of which is the squared sum of coefficients (typically called <em>ridge regression</em>) and the other is the sum of their absolute values (the so-called <em>lasso</em>).</p>
<div class="sourceCode" id="cb503"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb503-1"><a href="ml.html#cb503-1"></a><span class="kw">library</span>(tidymodels)</span>
<span id="cb503-2"><a href="ml.html#cb503-2"></a></span>
<span id="cb503-3"><a href="ml.html#cb503-3"></a>happy_prepped =<span class="st"> </span>happy <span class="op">%>%</span><span class="st"> </span></span>
<span id="cb503-4"><a href="ml.html#cb503-4"></a><span class="st"> </span><span class="kw">select</span>(<span class="op">-</span>country, <span class="op">-</span>gini_index_world_bank_estimate, <span class="op">-</span>dystopia_residual) <span class="op">%>%</span><span class="st"> </span></span>
<span id="cb503-5"><a href="ml.html#cb503-5"></a><span class="st"> </span><span class="kw">recipe</span>(happiness_score <span class="op">~</span><span class="st"> </span>.) <span class="op">%>%</span><span class="st"> </span></span>
<span id="cb503-6"><a href="ml.html#cb503-6"></a><span class="st"> </span><span class="kw">step_scale</span>(<span class="kw">everything</span>()) <span class="op">%>%</span><span class="st"> </span></span>
<span id="cb503-7"><a href="ml.html#cb503-7"></a><span class="st"> </span><span class="kw">step_naomit</span>(happiness_score) <span class="op">%>%</span></span>
<span id="cb503-8"><a href="ml.html#cb503-8"></a><span class="st"> </span><span class="kw">prep</span>() <span class="op">%>%</span><span class="st"> </span></span>
<span id="cb503-9"><a href="ml.html#cb503-9"></a><span class="st"> </span><span class="kw">bake</span>(happy) </span>
<span id="cb503-10"><a href="ml.html#cb503-10"></a></span>
<span id="cb503-11"><a href="ml.html#cb503-11"></a>happy_folds =<span class="st"> </span>happy_prepped <span class="op">%>%</span><span class="st"> </span></span>
<span id="cb503-12"><a href="ml.html#cb503-12"></a><span class="st"> </span><span class="kw">drop_na</span>() <span class="op">%>%</span></span>
<span id="cb503-13"><a href="ml.html#cb503-13"></a><span class="st"> </span><span class="kw">vfold_cv</span>() </span>
<span id="cb503-14"><a href="ml.html#cb503-14"></a></span>
<span id="cb503-15"><a href="ml.html#cb503-15"></a><span class="kw">library</span>(tune)</span>
<span id="cb503-16"><a href="ml.html#cb503-16"></a></span>
<span id="cb503-17"><a href="ml.html#cb503-17"></a>happy_regLM_spec =<span class="st"> </span><span class="kw">linear_reg</span>(<span class="dt">penalty =</span> <span class="fl">1e-3</span>, <span class="dt">mixture =</span> <span class="fl">.5</span>) <span class="op">%>%</span></span>
<span id="cb503-18"><a href="ml.html#cb503-18"></a><span class="st"> </span><span class="kw">set_engine</span>(<span class="dt">engine =</span> <span class="st">"glmnet"</span>)</span>
<span id="cb503-19"><a href="ml.html#cb503-19"></a></span>
<span id="cb503-20"><a href="ml.html#cb503-20"></a>happy_regLM_results =<span class="st"> </span><span class="kw">fit_resamples</span>(</span>
<span id="cb503-21"><a href="ml.html#cb503-21"></a> happy_regLM_spec,</span>
<span id="cb503-22"><a href="ml.html#cb503-22"></a> happiness_score <span class="op">~</span><span class="st"> </span>.,</span>
<span id="cb503-23"><a href="ml.html#cb503-23"></a> happy_folds,</span>
<span id="cb503-24"><a href="ml.html#cb503-24"></a> <span class="dt">control =</span> <span class="kw">control_resamples</span>(<span class="dt">save_pred =</span> <span class="ot">TRUE</span>)</span>
<span id="cb503-25"><a href="ml.html#cb503-25"></a>)</span>
<span id="cb503-26"><a href="ml.html#cb503-26"></a></span>
<span id="cb503-27"><a href="ml.html#cb503-27"></a>cv_regLM_res =<span class="st"> </span>happy_regLM_results <span class="op">%>%</span></span>
<span id="cb503-28"><a href="ml.html#cb503-28"></a><span class="st"> </span><span class="kw">collect_metrics</span>()</span></code></pre></div>
<table class="table" style="width: auto !important; margin-left: auto; margin-right: auto;">
<thead>
<tr>
<th style="text-align:left;">
.metric
</th>
<th style="text-align:left;">
.estimator
</th>
<th style="text-align:right;">
mean
</th>
<th style="text-align:right;">
n
</th>
<th style="text-align:right;">
std_err
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
rmse
</td>
<td style="text-align:left;">
standard
</td>
<td style="text-align:right;">
0.335
</td>
<td style="text-align:right;">
10
</td>
<td style="text-align:right;">
0.018
</td>
</tr>
<tr>
<td style="text-align:left;">
rsq
</td>
<td style="text-align:left;">
standard
</td>
<td style="text-align:right;">
0.897
</td>
<td style="text-align:right;">
10
</td>
<td style="text-align:right;">
0.013
</td>
</tr>
</tbody>
</table>
<div id="tuning-parameters-for-regularized-regression" class="section level4">
<h4>Tuning parameters for regularized regression</h4>
<p>For the previous model setting, we wouldn’t know what the penalty or the mixing parameter should be. This is where we can use cross validation to choose those. We’ll redo our model spec, create a set of values to search over, and pass that to the tuning function for cross-validation. Our ultimate model will then be applied to the test data.</p>
<p>First we create our training-test split.</p>
<div class="sourceCode" id="cb504"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb504-1"><a href="ml.html#cb504-1"></a><span class="co"># removing some variables with lots of missing values</span></span>
<span id="cb504-2"><a href="ml.html#cb504-2"></a>happy_split =<span class="st"> </span>happy <span class="op">%>%</span><span class="st"> </span></span>