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                         <h1 class="title">Regression - RDD-based API</h1>


                     <h2 id="isotonic-regression">Isotonic regression</h2>
 <p><a href="http://en.wikipedia.org/wiki/Isotonic_regression">Isotonic regression</a>
 belongs to the family of regression algorithms. Formally isotonic regression is a problem where
 given a finite set of real numbers <code class="language-plaintext highlighter-rouge">$Y = {y_1, y_2, ..., y_n}$</code> representing observed responses
 and <code class="language-plaintext highlighter-rouge">$X = {x_1, x_2, ..., x_n}$</code> the unknown response values to be fitted
 finding a function that minimizes</p>

 <p><code class="language-plaintext highlighter-rouge">\begin{equation}
   f(x) = \sum_{i=1}^n w_i (y_i - x_i)^2
 \end{equation}</code></p>

 <p>with respect to complete order subject to
 <code class="language-plaintext highlighter-rouge">$x_1\le x_2\le ...\le x_n$</code> where <code class="language-plaintext highlighter-rouge">$w_i$</code> are positive weights.
 The resulting function is called isotonic regression and it is unique.
 It can be viewed as least squares problem under order restriction.
 Essentially isotonic regression is a
 <a href="http://en.wikipedia.org/wiki/Monotonic_function">monotonic function</a>
 best fitting the original data points.</p>

 <p><code class="language-plaintext highlighter-rouge">spark.mllib</code> supports a
 <a href="https://doi.org/10.1198/TECH.2010.10111">pool adjacent violators algorithm</a>
 which uses an approach to
 <a href="https://doi.org/10.1007/978-3-642-99789-1_10">parallelizing isotonic regression</a>.
 The training input is an RDD of tuples of three double values that represent
 label, feature and weight in this order. In case there are multiple tuples with
 the same feature then these tuples are aggregated into a single tuple as follows:</p>

 <ul>
   <li>Aggregated label is the weighted average of all labels.</li>
   <li>Aggregated feature is the unique feature value.</li>
   <li>Aggregated weight is the sum of all weights.</li>
 </ul>

 <p>Additionally, IsotonicRegression algorithm has one
 optional parameter called $isotonic$ defaulting to true.
 This argument specifies if the isotonic regression is
 isotonic (monotonically increasing) or antitonic (monotonically decreasing).</p>

 <p>Training returns an IsotonicRegressionModel that can be used to predict
 labels for both known and unknown features. The result of isotonic regression
 is treated as piecewise linear function. The rules for prediction therefore are:</p>

 <ul>
   <li>If the prediction input exactly matches a training feature
 then associated prediction is returned.</li>
   <li>If the prediction input is lower or higher than all training features
 then prediction with lowest or highest feature is returned respectively.</li>
   <li>If the prediction input falls between two training features then prediction is treated
 as piecewise linear function and interpolated value is calculated from the
 predictions of the two closest features.</li>
 </ul>

 <h3 id="examples">Examples</h3>

 <div class="codetabs">

 <div data-lang="python">
     <p>Data are read from a file where each line has a format label,feature
 i.e. 4710.28,500.00. The data are split to training and testing set.
 Model is created using the training set and a mean squared error is calculated from the predicted
 labels and real labels in the test set.</p>

     <p>Refer to the <a href="api/python/reference/api/pyspark.mllib.regression.IsotonicRegression.html"><code class="language-plaintext highlighter-rouge">IsotonicRegression</code> Python docs</a> and <a href="api/python/reference/api/pyspark.mllib.regression.IsotonicRegressionModel.html"><code class="language-plaintext highlighter-rouge">IsotonicRegressionModel</code> Python docs</a> for more details on the API.</p>

     <div class="highlight"><pre class="codehilite"><code><span class="kn">import</span> <span class="nn">math</span>
 <span class="kn">from</span> <span class="nn">pyspark.mllib.regression</span> <span class="kn">import</span> <span class="n">IsotonicRegression</span><span class="p">,</span> <span class="n">IsotonicRegressionModel</span>
 <span class="kn">from</span> <span class="nn">pyspark.mllib.util</span> <span class="kn">import</span> <span class="n">MLUtils</span>

 <span class="c1"># Load and parse the data
 </span><span class="k">def</span> <span class="nf">parsePoint</span><span class="p">(</span><span class="n">labeledData</span><span class="p">):</span>
     <span class="k">return</span> <span class="p">(</span><span class="n">labeledData</span><span class="p">.</span><span class="n">label</span><span class="p">,</span> <span class="n">labeledData</span><span class="p">.</span><span class="n">features</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mf">1.0</span><span class="p">)</span>

 <span class="n">data</span> <span class="o">=</span> <span class="n">MLUtils</span><span class="p">.</span><span class="n">loadLibSVMFile</span><span class="p">(</span><span class="n">sc</span><span class="p">,</span> <span class="s">"data/mllib/sample_isotonic_regression_libsvm_data.txt"</span><span class="p">)</span>

 <span class="c1"># Create label, feature, weight tuples from input data with weight set to default value 1.0.
 </span><span class="n">parsedData</span> <span class="o">=</span> <span class="n">data</span><span class="p">.</span><span class="nb">map</span><span class="p">(</span><span class="n">parsePoint</span><span class="p">)</span>

 <span class="c1"># Split data into training (60%) and test (40%) sets.
 </span><span class="n">training</span><span class="p">,</span> <span class="n">test</span> <span class="o">=</span> <span class="n">parsedData</span><span class="p">.</span><span class="n">randomSplit</span><span class="p">([</span><span class="mf">0.6</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">],</span> <span class="mi">11</span><span class="p">)</span>

 <span class="c1"># Create isotonic regression model from training data.
 # Isotonic parameter defaults to true so it is only shown for demonstration
 </span><span class="n">model</span> <span class="o">=</span> <span class="n">IsotonicRegression</span><span class="p">.</span><span class="n">train</span><span class="p">(</span><span class="n">training</span><span class="p">)</span>

 <span class="c1"># Create tuples of predicted and real labels.
 </span><span class="n">predictionAndLabel</span> <span class="o">=</span> <span class="n">test</span><span class="p">.</span><span class="nb">map</span><span class="p">(</span><span class="k">lambda</span> <span class="n">p</span><span class="p">:</span> <span class="p">(</span><span class="n">model</span><span class="p">.</span><span class="n">predict</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span> <span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>

 <span class="c1"># Calculate mean squared error between predicted and real labels.
 </span><span class="n">meanSquaredError</span> <span class="o">=</span> <span class="n">predictionAndLabel</span><span class="p">.</span><span class="nb">map</span><span class="p">(</span><span class="k">lambda</span> <span class="n">pl</span><span class="p">:</span> <span class="n">math</span><span class="p">.</span><span class="nb">pow</span><span class="p">((</span><span class="n">pl</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="n">pl</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="n">mean</span><span class="p">()</span>
 <span class="k">print</span><span class="p">(</span><span class="s">"Mean Squared Error = "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">meanSquaredError</span><span class="p">))</span>

 <span class="c1"># Save and load model
 </span><span class="n">model</span><span class="p">.</span><span class="n">save</span><span class="p">(</span><span class="n">sc</span><span class="p">,</span> <span class="s">"target/tmp/myIsotonicRegressionModel"</span><span class="p">)</span>
 <span class="n">sameModel</span> <span class="o">=</span> <span class="n">IsotonicRegressionModel</span><span class="p">.</span><span class="n">load</span><span class="p">(</span><span class="n">sc</span><span class="p">,</span> <span class="s">"target/tmp/myIsotonicRegressionModel"</span><span class="p">)</span></code></pre></div>
     <div><small>Find full example code at "examples/src/main/python/mllib/isotonic_regression_example.py" in the Spark repo.</small></div>
   </div>

 <div data-lang="scala">
     <p>Data are read from a file where each line has a format label,feature
 i.e. 4710.28,500.00. The data are split to training and testing set.
 Model is created using the training set and a mean squared error is calculated from the predicted
 labels and real labels in the test set.</p>

     <p>Refer to the <a href="api/scala/org/apache/spark/mllib/regression/IsotonicRegression.html"><code class="language-plaintext highlighter-rouge">IsotonicRegression</code> Scala docs</a> and <a href="api/scala/org/apache/spark/mllib/regression/IsotonicRegressionModel.html"><code class="language-plaintext highlighter-rouge">IsotonicRegressionModel</code> Scala docs</a> for details on the API.</p>

     <div class="highlight"><pre class="codehilite"><code><span class="k">import</span> <span class="nn">org.apache.spark.mllib.regression.</span><span class="o">{</span><span class="nc">IsotonicRegression</span><span class="o">,</span> <span class="nc">IsotonicRegressionModel</span><span class="o">}</span>
 <span class="k">import</span> <span class="nn">org.apache.spark.mllib.util.MLUtils</span>

 <span class="k">val</span> <span class="nv">data</span> <span class="k">=</span> <span class="nv">MLUtils</span><span class="o">.</span><span class="py">loadLibSVMFile</span><span class="o">(</span><span class="n">sc</span><span class="o">,</span>
   <span class="s">"data/mllib/sample_isotonic_regression_libsvm_data.txt"</span><span class="o">).</span><span class="py">cache</span><span class="o">()</span>

 <span class="c1">// Create label, feature, weight tuples from input data with weight set to default value 1.0.</span>
 <span class="k">val</span> <span class="nv">parsedData</span> <span class="k">=</span> <span class="nv">data</span><span class="o">.</span><span class="py">map</span> <span class="o">{</span> <span class="n">labeledPoint</span> <span class="k">=&gt;</span>
   <span class="o">(</span><span class="nv">labeledPoint</span><span class="o">.</span><span class="py">label</span><span class="o">,</span> <span class="nv">labeledPoint</span><span class="o">.</span><span class="py">features</span><span class="o">(</span><span class="mi">0</span><span class="o">),</span> <span class="mf">1.0</span><span class="o">)</span>
 <span class="o">}</span>

 <span class="c1">// Split data into training (60%) and test (40%) sets.</span>
 <span class="k">val</span> <span class="nv">splits</span> <span class="k">=</span> <span class="nv">parsedData</span><span class="o">.</span><span class="py">randomSplit</span><span class="o">(</span><span class="nc">Array</span><span class="o">(</span><span class="mf">0.6</span><span class="o">,</span> <span class="mf">0.4</span><span class="o">),</span> <span class="n">seed</span> <span class="k">=</span> <span class="mi">11L</span><span class="o">)</span>
 <span class="k">val</span> <span class="nv">training</span> <span class="k">=</span> <span class="nf">splits</span><span class="o">(</span><span class="mi">0</span><span class="o">)</span>
 <span class="k">val</span> <span class="nv">test</span> <span class="k">=</span> <span class="nf">splits</span><span class="o">(</span><span class="mi">1</span><span class="o">)</span>

 <span class="c1">// Create isotonic regression model from training data.</span>
 <span class="c1">// Isotonic parameter defaults to true so it is only shown for demonstration</span>
 <span class="k">val</span> <span class="nv">model</span> <span class="k">=</span> <span class="k">new</span> <span class="nc">IsotonicRegression</span><span class="o">().</span><span class="py">setIsotonic</span><span class="o">(</span><span class="kc">true</span><span class="o">).</span><span class="py">run</span><span class="o">(</span><span class="n">training</span><span class="o">)</span>

 <span class="c1">// Create tuples of predicted and real labels.</span>
 <span class="k">val</span> <span class="nv">predictionAndLabel</span> <span class="k">=</span> <span class="nv">test</span><span class="o">.</span><span class="py">map</span> <span class="o">{</span> <span class="n">point</span> <span class="k">=&gt;</span>
   <span class="k">val</span> <span class="nv">predictedLabel</span> <span class="k">=</span> <span class="nv">model</span><span class="o">.</span><span class="py">predict</span><span class="o">(</span><span class="nv">point</span><span class="o">.</span><span class="py">_2</span><span class="o">)</span>
   <span class="o">(</span><span class="n">predictedLabel</span><span class="o">,</span> <span class="nv">point</span><span class="o">.</span><span class="py">_1</span><span class="o">)</span>
 <span class="o">}</span>

 <span class="c1">// Calculate mean squared error between predicted and real labels.</span>
 <span class="k">val</span> <span class="nv">meanSquaredError</span> <span class="k">=</span> <span class="nv">predictionAndLabel</span><span class="o">.</span><span class="py">map</span> <span class="o">{</span> <span class="nf">case</span> <span class="o">(</span><span class="n">p</span><span class="o">,</span> <span class="n">l</span><span class="o">)</span> <span class="k">=&gt;</span> <span class="nv">math</span><span class="o">.</span><span class="py">pow</span><span class="o">((</span><span class="n">p</span> <span class="o">-</span> <span class="n">l</span><span class="o">),</span> <span class="mi">2</span><span class="o">)</span> <span class="o">}.</span><span class="py">mean</span><span class="o">()</span>
 <span class="nf">println</span><span class="o">(</span><span class="n">s</span><span class="s">"Mean Squared Error = $meanSquaredError"</span><span class="o">)</span>

 <span class="c1">// Save and load model</span>
 <span class="nv">model</span><span class="o">.</span><span class="py">save</span><span class="o">(</span><span class="n">sc</span><span class="o">,</span> <span class="s">"target/tmp/myIsotonicRegressionModel"</span><span class="o">)</span>
 <span class="k">val</span> <span class="nv">sameModel</span> <span class="k">=</span> <span class="nv">IsotonicRegressionModel</span><span class="o">.</span><span class="py">load</span><span class="o">(</span><span class="n">sc</span><span class="o">,</span> <span class="s">"target/tmp/myIsotonicRegressionModel"</span><span class="o">)</span></code></pre></div>
     <div><small>Find full example code at "examples/src/main/scala/org/apache/spark/examples/mllib/IsotonicRegressionExample.scala" in the Spark repo.</small></div>
   </div>
 <div data-lang="java">
     <p>Data are read from a file where each line has a format label,feature
 i.e. 4710.28,500.00. The data are split to training and testing set.
 Model is created using the training set and a mean squared error is calculated from the predicted
 labels and real labels in the test set.</p>

     <p>Refer to the <a href="api/java/org/apache/spark/mllib/regression/IsotonicRegression.html"><code class="language-plaintext highlighter-rouge">IsotonicRegression</code> Java docs</a> and <a href="api/java/org/apache/spark/mllib/regression/IsotonicRegressionModel.html"><code class="language-plaintext highlighter-rouge">IsotonicRegressionModel</code> Java docs</a> for details on the API.</p>

     <div class="highlight"><pre class="codehilite"><code><span class="kn">import</span> <span class="nn">scala.Tuple2</span><span class="o">;</span>
 <span class="kn">import</span> <span class="nn">scala.Tuple3</span><span class="o">;</span>
 <span class="kn">import</span> <span class="nn">org.apache.spark.api.java.JavaPairRDD</span><span class="o">;</span>
 <span class="kn">import</span> <span class="nn">org.apache.spark.api.java.JavaSparkContext</span><span class="o">;</span>
 <span class="kn">import</span> <span class="nn">org.apache.spark.api.java.JavaRDD</span><span class="o">;</span>
 <span class="kn">import</span> <span class="nn">org.apache.spark.mllib.regression.IsotonicRegression</span><span class="o">;</span>
 <span class="kn">import</span> <span class="nn">org.apache.spark.mllib.regression.IsotonicRegressionModel</span><span class="o">;</span>
 <span class="kn">import</span> <span class="nn">org.apache.spark.mllib.regression.LabeledPoint</span><span class="o">;</span>
 <span class="kn">import</span> <span class="nn">org.apache.spark.mllib.util.MLUtils</span><span class="o">;</span>

 <span class="nc">JavaRDD</span><span class="o">&lt;</span><span class="nc">LabeledPoint</span><span class="o">&gt;</span> <span class="n">data</span> <span class="o">=</span> <span class="nc">MLUtils</span><span class="o">.</span><span class="na">loadLibSVMFile</span><span class="o">(</span>
   <span class="n">jsc</span><span class="o">.</span><span class="na">sc</span><span class="o">(),</span> <span class="s">"data/mllib/sample_isotonic_regression_libsvm_data.txt"</span><span class="o">).</span><span class="na">toJavaRDD</span><span class="o">();</span>

 <span class="c1">// Create label, feature, weight tuples from input data with weight set to default value 1.0.</span>
 <span class="nc">JavaRDD</span><span class="o">&lt;</span><span class="nc">Tuple3</span><span class="o">&lt;</span><span class="nc">Double</span><span class="o">,</span> <span class="nc">Double</span><span class="o">,</span> <span class="nc">Double</span><span class="o">&gt;&gt;</span> <span class="n">parsedData</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="na">map</span><span class="o">(</span><span class="n">point</span> <span class="o">-&gt;</span>
   <span class="k">new</span> <span class="nc">Tuple3</span><span class="o">&lt;&gt;(</span><span class="n">point</span><span class="o">.</span><span class="na">label</span><span class="o">(),</span> <span class="n">point</span><span class="o">.</span><span class="na">features</span><span class="o">().</span><span class="na">apply</span><span class="o">(</span><span class="mi">0</span><span class="o">),</span> <span class="mf">1.0</span><span class="o">));</span>

 <span class="c1">// Split data into training (60%) and test (40%) sets.</span>
 <span class="nc">JavaRDD</span><span class="o">&lt;</span><span class="nc">Tuple3</span><span class="o">&lt;</span><span class="nc">Double</span><span class="o">,</span> <span class="nc">Double</span><span class="o">,</span> <span class="nc">Double</span><span class="o">&gt;&gt;[]</span> <span class="n">splits</span> <span class="o">=</span>
   <span class="n">parsedData</span><span class="o">.</span><span class="na">randomSplit</span><span class="o">(</span><span class="k">new</span> <span class="kt">double</span><span class="o">[]{</span><span class="mf">0.6</span><span class="o">,</span> <span class="mf">0.4</span><span class="o">},</span> <span class="mi">11L</span><span class="o">);</span>
 <span class="nc">JavaRDD</span><span class="o">&lt;</span><span class="nc">Tuple3</span><span class="o">&lt;</span><span class="nc">Double</span><span class="o">,</span> <span class="nc">Double</span><span class="o">,</span> <span class="nc">Double</span><span class="o">&gt;&gt;</span> <span class="n">training</span> <span class="o">=</span> <span class="n">splits</span><span class="o">[</span><span class="mi">0</span><span class="o">];</span>
 <span class="nc">JavaRDD</span><span class="o">&lt;</span><span class="nc">Tuple3</span><span class="o">&lt;</span><span class="nc">Double</span><span class="o">,</span> <span class="nc">Double</span><span class="o">,</span> <span class="nc">Double</span><span class="o">&gt;&gt;</span> <span class="n">test</span> <span class="o">=</span> <span class="n">splits</span><span class="o">[</span><span class="mi">1</span><span class="o">];</span>

 <span class="c1">// Create isotonic regression model from training data.</span>
 <span class="c1">// Isotonic parameter defaults to true so it is only shown for demonstration</span>
 <span class="nc">IsotonicRegressionModel</span> <span class="n">model</span> <span class="o">=</span> <span class="k">new</span> <span class="nc">IsotonicRegression</span><span class="o">().</span><span class="na">setIsotonic</span><span class="o">(</span><span class="kc">true</span><span class="o">).</span><span class="na">run</span><span class="o">(</span><span class="n">training</span><span class="o">);</span>

 <span class="c1">// Create tuples of predicted and real labels.</span>
 <span class="nc">JavaPairRDD</span><span class="o">&lt;</span><span class="nc">Double</span><span class="o">,</span> <span class="nc">Double</span><span class="o">&gt;</span> <span class="n">predictionAndLabel</span> <span class="o">=</span> <span class="n">test</span><span class="o">.</span><span class="na">mapToPair</span><span class="o">(</span><span class="n">point</span> <span class="o">-&gt;</span>
   <span class="k">new</span> <span class="nc">Tuple2</span><span class="o">&lt;&gt;(</span><span class="n">model</span><span class="o">.</span><span class="na">predict</span><span class="o">(</span><span class="n">point</span><span class="o">.</span><span class="na">_2</span><span class="o">()),</span> <span class="n">point</span><span class="o">.</span><span class="na">_1</span><span class="o">()));</span>

 <span class="c1">// Calculate mean squared error between predicted and real labels.</span>
 <span class="kt">double</span> <span class="n">meanSquaredError</span> <span class="o">=</span> <span class="n">predictionAndLabel</span><span class="o">.</span><span class="na">mapToDouble</span><span class="o">(</span><span class="n">pl</span> <span class="o">-&gt;</span> <span class="o">{</span>
   <span class="kt">double</span> <span class="n">diff</span> <span class="o">=</span> <span class="n">pl</span><span class="o">.</span><span class="na">_1</span><span class="o">()</span> <span class="o">-</span> <span class="n">pl</span><span class="o">.</span><span class="na">_2</span><span class="o">();</span>
   <span class="k">return</span> <span class="n">diff</span> <span class="o">*</span> <span class="n">diff</span><span class="o">;</span>
 <span class="o">}).</span><span class="na">mean</span><span class="o">();</span>
 <span class="nc">System</span><span class="o">.</span><span class="na">out</span><span class="o">.</span><span class="na">println</span><span class="o">(</span><span class="s">"Mean Squared Error = "</span> <span class="o">+</span> <span class="n">meanSquaredError</span><span class="o">);</span>

 <span class="c1">// Save and load model</span>
 <span class="n">model</span><span class="o">.</span><span class="na">save</span><span class="o">(</span><span class="n">jsc</span><span class="o">.</span><span class="na">sc</span><span class="o">(),</span> <span class="s">"target/tmp/myIsotonicRegressionModel"</span><span class="o">);</span>
 <span class="nc">IsotonicRegressionModel</span> <span class="n">sameModel</span> <span class="o">=</span>
   <span class="nc">IsotonicRegressionModel</span><span class="o">.</span><span class="na">load</span><span class="o">(</span><span class="n">jsc</span><span class="o">.</span><span class="na">sc</span><span class="o">(),</span> <span class="s">"target/tmp/myIsotonicRegressionModel"</span><span class="o">);</span></code></pre></div>
     <div><small>Find full example code at "examples/src/main/java/org/apache/spark/examples/mllib/JavaIsotonicRegressionExample.java" in the Spark repo.</small></div>
   </div>

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