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<!DOCTYPE html>
<html>
<head><meta charset="utf-8" />
<title>boston_housing</title>
<script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
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<h1 id="Machine-Learning-Engineer-Nanodegree">Machine Learning Engineer Nanodegree<a class="anchor-link" href="#Machine-Learning-Engineer-Nanodegree">¶</a></h1><h2 id="Model-Evaluation-&-Validation">Model Evaluation & Validation<a class="anchor-link" href="#Model-Evaluation-&-Validation">¶</a></h2><h2 id="Project-1:-Predicting-Boston-Housing-Prices">Project 1: Predicting Boston Housing Prices<a class="anchor-link" href="#Project-1:-Predicting-Boston-Housing-Prices">¶</a></h2><p>Welcome to the first project of the Machine Learning Engineer Nanodegree! In this notebook, some template code has already been written. You will need to implement additional functionality to successfully answer all of the questions for this project. Unless it is requested, do not modify any of the code that has already been included. In this template code, there are four sections which you must complete to successfully produce a prediction with your model. Each section where you will write code is preceded by a <strong>STEP X</strong> header with comments describing what must be done. Please read the instructions carefully!</p>
<p>In addition to implementing code, there will be questions that you must answer that relate to the project and your implementation. Each section where you will answer a question is preceded by a <strong>QUESTION X</strong> header. Be sure that you have carefully read each question and provide thorough answers in the text boxes that begin with "<strong>Answer:</strong>". Your project submission will be evaluated based on your answers to each of the questions.</p>
<p>A description of the dataset can be found <a href="https://archive.ics.uci.edu/ml/datasets/Housing">here</a>, which is provided by the <strong>UCI Machine Learning Repository</strong>.</p>
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<h1 id="Getting-Started">Getting Started<a class="anchor-link" href="#Getting-Started">¶</a></h1><p>To familiarize yourself with an iPython Notebook, <strong>try double clicking on this cell</strong>. You will notice that the text changes so that all the formatting is removed. This allows you to make edits to the block of text you see here. This block of text (and mostly anything that's not code) is written using <a href="http://daringfireball.net/projects/markdown/syntax">Markdown</a>, which is a way to format text using headers, links, italics, and many other options! Whether you're editing a Markdown text block or a code block (like the one below), you can use the keyboard shortcut <strong>Shift + Enter</strong> or <strong>Shift + Return</strong> to execute the code or text block. In this case, it will show the formatted text.</p>
<p>Let's start by setting up some code we will need to get the rest of the project up and running. Use the keyboard shortcut mentioned above on the following code block to execute it. Alternatively, depending on your iPython Notebook program, you can press the <strong>Play</strong> button in the hotbar. You'll know the code block executes successfully if the message <em>"Boston Housing dataset loaded successfully!"</em> is printed.</p>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="c1"># Importing a few necessary libraries</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="kn">as</span> <span class="nn">pl</span>
<span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">datasets</span>
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">DecisionTreeRegressor</span>
<span class="c1"># Make matplotlib show our plots inline (nicely formatted in the notebook)</span>
<span class="o">%</span><span class="k">matplotlib</span> inline
<span class="c1"># Create our client's feature set for which we will be predicting a selling price</span>
<span class="n">CLIENT_FEATURES</span> <span class="o">=</span> <span class="p">[[</span><span class="mf">11.95</span><span class="p">,</span> <span class="mf">0.00</span><span class="p">,</span> <span class="mf">18.100</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mf">0.6590</span><span class="p">,</span> <span class="mf">5.6090</span><span class="p">,</span> <span class="mf">90.00</span><span class="p">,</span> <span class="mf">1.385</span><span class="p">,</span> <span class="mi">24</span><span class="p">,</span> <span class="mf">680.0</span><span class="p">,</span> <span class="mf">20.20</span><span class="p">,</span> <span class="mf">332.09</span><span class="p">,</span> <span class="mf">12.13</span><span class="p">]]</span>
<span class="c1"># Load the Boston Housing dataset into the city_data variable</span>
<span class="n">city_data</span> <span class="o">=</span> <span class="n">datasets</span><span class="o">.</span><span class="n">load_boston</span><span class="p">()</span>
<span class="c1"># Initialize the housing prices and housing features</span>
<span class="n">housing_prices</span> <span class="o">=</span> <span class="n">city_data</span><span class="o">.</span><span class="n">target</span>
<span class="n">housing_features</span> <span class="o">=</span> <span class="n">city_data</span><span class="o">.</span><span class="n">data</span>
<span class="k">print</span> <span class="s2">"Boston Housing dataset loaded successfully!"</span>
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<pre>Boston Housing dataset loaded successfully!
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<h1 id="Statistical-Analysis-and-Data-Exploration">Statistical Analysis and Data Exploration<a class="anchor-link" href="#Statistical-Analysis-and-Data-Exploration">¶</a></h1><p>In this first section of the project, you will quickly investigate a few basic statistics about the dataset you are working with. In addition, you'll look at the client's feature set in <code>CLIENT_FEATURES</code> and see how this particular sample relates to the features of the dataset. Familiarizing yourself with the data through an explorative process is a fundamental practice to help you better understand your results.</p>
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<h2 id="Step-1">Step 1<a class="anchor-link" href="#Step-1">¶</a></h2><p>In the code block below, use the imported <code>numpy</code> library to calculate the requested statistics. You will need to replace each <code>None</code> you find with the appropriate <code>numpy</code> coding for the proper statistic to be printed. Be sure to execute the code block each time to test if your implementation is working successfully. The print statements will show the statistics you calculate!</p>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="c1"># Number of houses in the dataset</span>
<span class="n">total_houses</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">housing_prices</span><span class="p">)</span>
<span class="c1"># Number of features in the dataset</span>
<span class="n">total_features</span> <span class="o">=</span> <span class="n">housing_features</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
<span class="c1"># Minimum housing value in the dataset</span>
<span class="n">minimum_price</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">amin</span><span class="p">(</span><span class="n">housing_prices</span><span class="p">)</span>
<span class="c1"># Maximum housing value in the dataset</span>
<span class="n">maximum_price</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">amax</span><span class="p">(</span><span class="n">housing_prices</span><span class="p">)</span>
<span class="c1"># Mean house value of the dataset</span>
<span class="n">mean_price</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">housing_prices</span><span class="p">)</span>
<span class="c1"># Median house value of the dataset</span>
<span class="n">median_price</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">median</span><span class="p">(</span><span class="n">housing_prices</span><span class="p">)</span>
<span class="c1"># Standard deviation of housing values of the dataset</span>
<span class="n">std_dev</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">housing_prices</span><span class="p">,</span><span class="n">ddof</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="c1"># Show the calculated statistics</span>
<span class="k">print</span> <span class="s2">"Boston Housing dataset statistics (in $1000's):</span><span class="se">\n</span><span class="s2">"</span>
<span class="k">print</span> <span class="s2">"Total number of houses:"</span><span class="p">,</span> <span class="n">total_houses</span>
<span class="k">print</span> <span class="s2">"Total number of features:"</span><span class="p">,</span> <span class="n">total_features</span>
<span class="k">print</span> <span class="s2">"Minimum house price:"</span><span class="p">,</span> <span class="n">minimum_price</span>
<span class="k">print</span> <span class="s2">"Maximum house price:"</span><span class="p">,</span> <span class="n">maximum_price</span>
<span class="k">print</span> <span class="s2">"Mean house price: {0:.3f}"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">mean_price</span><span class="p">)</span>
<span class="k">print</span> <span class="s2">"Median house price:"</span><span class="p">,</span> <span class="n">median_price</span>
<span class="k">print</span> <span class="s2">"Standard deviation of house price: {0:.3f}"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">std_dev</span><span class="p">)</span>
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<pre>Boston Housing dataset statistics (in $1000's):
Total number of houses: 506
Total number of features: 13
Minimum house price: 5.0
Maximum house price: 50.0
Mean house price: 22.533
Median house price: 21.2
Standard deviation of house price: 9.197
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<h2 id="Question-1">Question 1<a class="anchor-link" href="#Question-1">¶</a></h2><p>As a reminder, you can view a description of the Boston Housing dataset <a href="https://archive.ics.uci.edu/ml/datasets/Housing">here</a>, where you can find the different features under <strong>Attribute Information</strong>. The <code>MEDV</code> attribute relates to the values stored in our <code>housing_prices</code> variable, so we do not consider that a feature of the data.</p>
<p><em>Of the features available for each data point, choose three that you feel are significant and give a brief description for each of what they measure.</em></p>
<p>Remember, you can <strong>double click the text box below</strong> to add your answer!</p>
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<p><strong>Answer: </strong> CRIM: The crime rate of the town. <br> RM: Average number of rooms per dwelling. <br> INDUS: Proportion of non-retail business acres per town</p>
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<h2 id="Question-2">Question 2<a class="anchor-link" href="#Question-2">¶</a></h2><p><em>Using your client's feature set <code>CLIENT_FEATURES</code>, which values correspond with the features you've chosen above?</em><br>
<strong>Hint: </strong> Run the code block below to see the client's data.</p>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="k">print</span> <span class="n">CLIENT_FEATURES</span>
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<pre>[[11.95, 0.0, 18.1, 0, 0.659, 5.609, 90.0, 1.385, 24, 680.0, 20.2, 332.09, 12.13]]
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<h4 id="Answer:--The-features-for-CRIM-RM-and-INDUS-correspond-to-the-columns-0,-5,-and-2."><strong>Answer: </strong> The features for CRIM RM and INDUS correspond to the columns 0, 5, and 2.<a class="anchor-link" href="#Answer:--The-features-for-CRIM-RM-and-INDUS-correspond-to-the-columns-0,-5,-and-2.">¶</a></h4><p>CRIM - 11.95 incidences of crimes per 100,000 people
RM - 5.609 is the average number of rooms per dwelling
INDUS: 18.1 is the proportion of non-retail business acres in town.</p>
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<h1 id="Evaluating-Model-Performance">Evaluating Model Performance<a class="anchor-link" href="#Evaluating-Model-Performance">¶</a></h1><p>In this second section of the project, you will begin to develop the tools necessary for a model to make a prediction. Being able to accurately evaluate each model's performance through the use of these tools helps to greatly reinforce the confidence in your predictions.</p>
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<h2 id="Step-2">Step 2<a class="anchor-link" href="#Step-2">¶</a></h2><p>In the code block below, you will need to implement code so that the <code>shuffle_split_data</code> function does the following:</p>
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<li>Randomly shuffle the input data <code>X</code> and target labels (housing values) <code>y</code>.</li>
<li>Split the data into training and testing subsets, holding 30% of the data for testing.</li>
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<p>If you use any functions not already acessible from the imported libraries above, remember to include your import statement below as well!<br>
Ensure that you have executed the code block once you are done. You'll know the <code>shuffle_split_data</code> function is working if the statement <em>"Successfully shuffled and split the data!"</em> is printed.</p>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="c1"># Put any import statements you need for this code block here</span>
<span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">cross_validation</span>
<span class="k">def</span> <span class="nf">shuffle_split_data</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
<span class="sd">""" Shuffles and splits data into 70% training and 30% testing subsets,</span>
<span class="sd"> then returns the training and testing subsets. """</span>
<span class="c1"># Shuffle and split the data</span>
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">cross_validation</span><span class="o">.</span><span class="n">train_test_split</span><span class="p">(</span>
<span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">test_size</span><span class="o">=</span><span class="mf">0.3</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="c1"># Return the training and testing data subsets</span>
<span class="k">return</span> <span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_test</span>
<span class="c1"># Test shuffle_split_data</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">shuffle_split_data</span><span class="p">(</span><span class="n">housing_features</span><span class="p">,</span> <span class="n">housing_prices</span><span class="p">)</span>
<span class="k">print</span> <span class="s2">"Successfully shuffled and split the data!"</span>
<span class="k">except</span><span class="p">:</span>
<span class="k">print</span> <span class="s2">"Something went wrong with shuffling and splitting the data."</span>
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<pre>Successfully shuffled and split the data!
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<h2 id="Question-3">Question 3<a class="anchor-link" href="#Question-3">¶</a></h2><p><em>Why do we split the data into training and testing subsets for our model?</em></p>
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<p><strong>Answer: </strong> We split the data into training and testing subsets so that our model can be tested on data that it has not seen before. Otherwise the model would just repeat labels it has seen from the training set and fail to predict anything useful from un-seen data - aka overfitting.</p>
<p>Random shuffling is done to ensure that if the data is ordered, we can still get an equal distribution of the classes in our testing and training data sets.</p>
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<h2 id="Step-3">Step 3<a class="anchor-link" href="#Step-3">¶</a></h2><p>In the code block below, you will need to implement code so that the <code>performance_metric</code> function does the following:</p>
<ul>
<li>Perform a total error calculation between the true values of the <code>y</code> labels <code>y_true</code> and the predicted values of the <code>y</code> labels <code>y_predict</code>.</li>
</ul>
<p>You will need to first choose an appropriate performance metric for this problem. See <a href="http://scikit-learn.org/stable/modules/classes.html#sklearn-metrics-metrics">the sklearn metrics documentation</a> to view a list of available metric functions. <strong>Hint: </strong> Look at the question below to see a list of the metrics that were covered in the supporting course for this project.</p>
<p>Once you have determined which metric you will use, remember to include the necessary import statement as well!<br>
Ensure that you have executed the code block once you are done. You'll know the <code>performance_metric</code> function is working if the statement <em>"Successfully performed a metric calculation!"</em> is printed.</p>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="c1"># Put any import statements you need for this code block here</span>
<span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">metrics</span>
<span class="k">def</span> <span class="nf">performance_metric</span><span class="p">(</span><span class="n">y_true</span><span class="p">,</span> <span class="n">y_predict</span><span class="p">):</span>
<span class="sd">""" Calculates and returns the total error between true and predicted values</span>
<span class="sd"> based on a performance metric chosen by the student. """</span>
<span class="n">error</span> <span class="o">=</span> <span class="n">metrics</span><span class="o">.</span><span class="n">mean_squared_error</span><span class="p">(</span><span class="n">y_true</span><span class="p">,</span> <span class="n">y_predict</span><span class="p">)</span>
<span class="k">return</span> <span class="n">error</span>
<span class="c1"># Test performance_metric</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">total_error</span> <span class="o">=</span> <span class="n">performance_metric</span><span class="p">(</span><span class="n">y_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
<span class="k">print</span> <span class="s2">"Successfully performed a metric calculation!"</span>
<span class="k">except</span><span class="p">:</span>
<span class="k">print</span> <span class="s2">"Something went wrong with performing a metric calculation."</span>
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<pre>Successfully performed a metric calculation!
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<h2 id="Question-4">Question 4<a class="anchor-link" href="#Question-4">¶</a></h2><p><em>Which performance metric below did you find was most appropriate for predicting housing prices and analyzing the total error. Why?</em></p>
<ul>
<li><em>Accuracy</em></li>
<li><em>Precision</em></li>
<li><em>Recall</em></li>
<li><em>F1 Score</em></li>
<li><em>Mean Squared Error (MSE)</em></li>
<li><em>Mean Absolute Error (MAE)</em></li>
</ul>
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<p><strong>Answer: </strong> The act of predicting housing prices is a regression problem so it is most appropriate to using metrics related to regression - either MSE or MAE. The other four metrics are for classification problems. In this case I chose MSE as we want the model to be sensitive to large errors in predicting the prices rather than a scaled weight which plays down errors when overfitting.</p>
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<h2 id="Step-4-(Final-Step)">Step 4 (Final Step)<a class="anchor-link" href="#Step-4-(Final-Step)">¶</a></h2><p>In the code block below, you will need to implement code so that the <code>fit_model</code> function does the following:</p>
<ul>
<li>Create a scoring function using the same performance metric as in <strong>Step 2</strong>. See the <a href="http://scikit-learn.org/stable/modules/generated/sklearn.metrics.make_scorer.html">sklearn <code>make_scorer</code> documentation</a>.</li>
<li>Build a GridSearchCV object using <code>regressor</code>, <code>parameters</code>, and <code>scoring_function</code>. See the <a href="http://scikit-learn.org/stable/modules/generated/sklearn.grid_search.GridSearchCV.html">sklearn documentation on GridSearchCV</a>.</li>
</ul>
<p>When building the scoring function and GridSearchCV object, <em>be sure that you read the parameters documentation thoroughly.</em> It is not always the case that a default parameter for a function is the appropriate setting for the problem you are working on.</p>
<p>Since you are using <code>sklearn</code> functions, remember to include the necessary import statements below as well!<br>
Ensure that you have executed the code block once you are done. You'll know the <code>fit_model</code> function is working if the statement <em>"Successfully fit a model to the data!"</em> is printed.</p>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="c1"># Put any import statements you need for this code block</span>
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">DecisionTreeRegressor</span>
<span class="kn">from</span> <span class="nn">sklearn.grid_search</span> <span class="kn">import</span> <span class="n">GridSearchCV</span>
<span class="k">def</span> <span class="nf">fit_model</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
<span class="sd">""" Tunes a decision tree regressor model using GridSearchCV on the input data X </span>
<span class="sd"> and target labels y and returns this optimal model. """</span>
<span class="c1"># Create a decision tree regressor object</span>
<span class="n">regressor</span> <span class="o">=</span> <span class="n">DecisionTreeRegressor</span><span class="p">(</span><span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
<span class="c1"># Set up the parameters we wish to tune</span>
<span class="n">parameters</span> <span class="o">=</span> <span class="p">{</span><span class="s1">'max_depth'</span><span class="p">:(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">9</span><span class="p">,</span><span class="mi">10</span><span class="p">)}</span>
<span class="c1"># Make an appropriate scoring function</span>
<span class="n">scoring_function</span> <span class="o">=</span> <span class="n">metrics</span><span class="o">.</span><span class="n">make_scorer</span><span class="p">(</span><span class="n">metrics</span><span class="o">.</span><span class="n">mean_absolute_error</span><span class="p">,</span> <span class="n">greater_is_better</span> <span class="o">=</span> <span class="bp">False</span><span class="p">)</span>
<span class="c1"># Make the GridSearchCV object</span>
<span class="n">reg</span> <span class="o">=</span> <span class="n">GridSearchCV</span><span class="p">(</span><span class="n">regressor</span><span class="p">,</span><span class="n">parameters</span><span class="p">,</span> <span class="n">scoring</span><span class="o">=</span><span class="n">scoring_function</span><span class="p">)</span>
<span class="c1"># Fit the learner to the data to obtain the optimal model with tuned parameters</span>
<span class="n">reg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="c1"># Return the optimal model</span>
<span class="k">return</span> <span class="n">reg</span>
<span class="c1"># Test fit_model on entire dataset</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">reg</span> <span class="o">=</span> <span class="n">fit_model</span><span class="p">(</span><span class="n">housing_features</span><span class="p">,</span> <span class="n">housing_prices</span><span class="p">)</span>
<span class="k">print</span> <span class="s2">"Successfully fit a model!"</span>
<span class="k">except</span><span class="p">:</span>
<span class="k">print</span> <span class="s2">"Something went wrong with fitting a model."</span>
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<pre>Successfully fit a model!
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<h2 id="Question-5">Question 5<a class="anchor-link" href="#Question-5">¶</a></h2><p><em>What is the grid search algorithm and when is it applicable?</em></p>
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<p><strong>Answer: </strong> The grid search algorithm is used to search for optimal hyperparameters which will give the highest score for when the model fits the data. This score is based on metrics we decide on. The algorithm is applicable when our regressor has many hyperparameters we can tune on.</p>
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<h2 id="Question-6">Question 6<a class="anchor-link" href="#Question-6">¶</a></h2><p><em>What is cross-validation, and how is it performed on a model? Why would cross-validation be helpful when using grid search?</em></p>
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<p><strong>Answer: </strong> Cross-validation is used to evaluate a models performance by partitioning the training into k bins of equal size and then declare one of these sets as the validation set. We then use the remaining k-1 bins as training sets to train the model. Afterwards, the model's performance can be evaluated using the validation set we set aside.</p>
<p>Cross-validation is performed k times. The bins are all used as a validation set once to train the model k-1 times. It then takes the average of the k performance metric values that were computed the validation sets.</p>
<p>Cross-validation is useful for grid search because it allows for a model to be trained on a large set of observations. This is benefitial for grid search as it will try to tune a model's hyperparameters.</p>
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<h1 id="Checkpoint!">Checkpoint!<a class="anchor-link" href="#Checkpoint!">¶</a></h1><p>You have now successfully completed your last code implementation section. Pat yourself on the back! All of your functions written above will be executed in the remaining sections below, and questions will be asked about various results for you to analyze. To prepare the <strong>Analysis</strong> and <strong>Prediction</strong> sections, you will need to intialize the two functions below. Remember, there's no need to implement any more code, so sit back and execute the code blocks! Some code comments are provided if you find yourself interested in the functionality.</p>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="k">def</span> <span class="nf">learning_curves</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_test</span><span class="p">):</span>
<span class="sd">""" Calculates the performance of several models with varying sizes of training data.</span>
<span class="sd"> The learning and testing error rates for each model are then plotted. """</span>
<span class="k">print</span> <span class="s2">"Creating learning curve graphs for max_depths of 1, 3, 6, and 10. . ."</span>
<span class="c1"># Create the figure window</span>
<span class="n">fig</span> <span class="o">=</span> <span class="n">pl</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">8</span><span class="p">))</span>
<span class="c1"># We will vary the training set size so that we have 50 different sizes</span>
<span class="n">sizes</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">rint</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">X_train</span><span class="p">),</span> <span class="mi">50</span><span class="p">))</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span>
<span class="n">train_err</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">sizes</span><span class="p">))</span>
<span class="n">test_err</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">sizes</span><span class="p">))</span>
<span class="c1"># Create four different models based on max_depth</span>
<span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">depth</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">10</span><span class="p">]):</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">s</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">sizes</span><span class="p">):</span>
<span class="c1"># Setup a decision tree regressor so that it learns a tree with max_depth = depth</span>
<span class="n">regressor</span> <span class="o">=</span> <span class="n">DecisionTreeRegressor</span><span class="p">(</span><span class="n">max_depth</span> <span class="o">=</span> <span class="n">depth</span><span class="p">)</span>
<span class="c1"># Fit the learner to the training data</span>
<span class="n">regressor</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">[:</span><span class="n">s</span><span class="p">],</span> <span class="n">y_train</span><span class="p">[:</span><span class="n">s</span><span class="p">])</span>
<span class="c1"># Find the performance on the training set</span>
<span class="n">train_err</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">performance_metric</span><span class="p">(</span><span class="n">y_train</span><span class="p">[:</span><span class="n">s</span><span class="p">],</span> <span class="n">regressor</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_train</span><span class="p">[:</span><span class="n">s</span><span class="p">]))</span>
<span class="c1"># Find the performance on the testing set</span>
<span class="n">test_err</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">performance_metric</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">regressor</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">))</span>
<span class="c1"># Subplot the learning curve graph</span>
<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">sizes</span><span class="p">,</span> <span class="n">test_err</span><span class="p">,</span> <span class="n">lw</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="n">label</span> <span class="o">=</span> <span class="s1">'Testing Error'</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">sizes</span><span class="p">,</span> <span class="n">train_err</span><span class="p">,</span> <span class="n">lw</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="n">label</span> <span class="o">=</span> <span class="s1">'Training Error'</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'max_depth = </span><span class="si">%s</span><span class="s1">'</span><span class="o">%</span><span class="p">(</span><span class="n">depth</span><span class="p">))</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'Number of Data Points in Training Set'</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Total Error'</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">X_train</span><span class="p">)])</span>
<span class="c1"># Visual aesthetics</span>
<span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s1">'Decision Tree Regressor Learning Performances'</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">18</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="mf">1.03</span><span class="p">)</span>
<span class="n">fig</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
<span class="n">fig</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="k">def</span> <span class="nf">model_complexity</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_test</span><span class="p">):</span>
<span class="sd">""" Calculates the performance of the model as model complexity increases.</span>
<span class="sd"> The learning and testing errors rates are then plotted. """</span>
<span class="k">print</span> <span class="s2">"Creating a model complexity graph. . . "</span>
<span class="c1"># We will vary the max_depth of a decision tree model from 1 to 14</span>
<span class="n">max_depth</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">14</span><span class="p">)</span>
<span class="n">train_err</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">max_depth</span><span class="p">))</span>
<span class="n">test_err</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">max_depth</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">max_depth</span><span class="p">):</span>
<span class="c1"># Setup a Decision Tree Regressor so that it learns a tree with depth d</span>
<span class="n">regressor</span> <span class="o">=</span> <span class="n">DecisionTreeRegressor</span><span class="p">(</span><span class="n">max_depth</span> <span class="o">=</span> <span class="n">d</span><span class="p">)</span>
<span class="c1"># Fit the learner to the training data</span>
<span class="n">regressor</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
<span class="c1"># Find the performance on the training set</span>
<span class="n">train_err</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">performance_metric</span><span class="p">(</span><span class="n">y_train</span><span class="p">,</span> <span class="n">regressor</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_train</span><span class="p">))</span>
<span class="c1"># Find the performance on the testing set</span>
<span class="n">test_err</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">performance_metric</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">regressor</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">))</span>
<span class="c1"># Plot the model complexity graph</span>
<span class="n">pl</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
<span class="n">pl</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'Decision Tree Regressor Complexity Performance'</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">max_depth</span><span class="p">,</span> <span class="n">test_err</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span> <span class="o">=</span> <span class="s1">'Testing Error'</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">max_depth</span><span class="p">,</span> <span class="n">train_err</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span> <span class="o">=</span> <span class="s1">'Training Error'</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">pl</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">'Maximum Depth'</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">'Total Error'</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<h1 id="Analyzing-Model-Performance">Analyzing Model Performance<a class="anchor-link" href="#Analyzing-Model-Performance">¶</a></h1><p>In this third section of the project, you'll take a look at several models' learning and testing error rates on various subsets of training data. Additionally, you'll investigate one particular algorithm with an increasing <code>max_depth</code> parameter on the full training set to observe how model complexity affects learning and testing errors. Graphing your model's performance based on varying criteria can be beneficial in the analysis process, such as visualizing behavior that may not have been apparent from the results alone.</p>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">learning_curves</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_test</span><span class="p">)</span>
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<pre>Creating learning curve graphs for max_depths of 1, 3, 6, and 10. . .
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