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     <h1><a href="../ml">FlinkML</a> - Multiple linear regression</h1>


 <ul id="markdown-toc">
   <li><a href="#description" id="markdown-toc-description">Description</a></li>
   <li><a href="#operations" id="markdown-toc-operations">Operations</a>    <ul>
       <li><a href="#fit" id="markdown-toc-fit">Fit</a></li>
       <li><a href="#predict" id="markdown-toc-predict">Predict</a></li>
     </ul>
   </li>
   <li><a href="#parameters" id="markdown-toc-parameters">Parameters</a></li>
   <li><a href="#examples" id="markdown-toc-examples">Examples</a></li>
 </ul>

 <h2 id="description">Description</h2>

 <p>Multiple linear regression tries to find a linear function which best fits the provided input data.
  Given a set of input data with its value $(\mathbf{x}, y)$, multiple linear regression finds
  a vector $\mathbf{w}$ such that the sum of the squared residuals is minimized:</p>

 <script type="math/tex; mode=display">S(\mathbf{w}) = \sum_{i=1} \left(y - \mathbf{w}^T\mathbf{x_i} \right)^2</script>

 <p>Written in matrix notation, we obtain the following formulation:</p>

 <script type="math/tex; mode=display">\mathbf{w}^* = \arg \min_{\mathbf{w}} (\mathbf{y} - X\mathbf{w})^2</script>

 <p>This problem has a closed form solution which is given by:</p>

 <script type="math/tex; mode=display">\mathbf{w}^* = \left(X^TX\right)^{-1}X^T\mathbf{y}</script>

 <p>However, in cases where the input data set is so huge that a complete parse over the whole data
   set is prohibitive, one can apply stochastic gradient descent (SGD) to approximate the solution.
   SGD first calculates for a random subset of the input data set the gradients. The gradient
   for a given point $\mathbf{x}_i$ is given by:</p>

 <script type="math/tex; mode=display">\nabla_{\mathbf{w}} S(\mathbf{w}, \mathbf{x_i}) = 2\left(\mathbf{w}^T\mathbf{x_i} -
     y\right)\mathbf{x_i}</script>

 <p>The gradients are averaged and scaled. The scaling is defined by $\gamma = \frac{s}{\sqrt{j}}$
   with $s$ being the initial step size and $j$ being the current iteration number. The resulting gradient is subtracted from the
   current weight vector giving the new weight vector for the next iteration:</p>

 <script type="math/tex; mode=display">\mathbf{w}_{t+1} = \mathbf{w}_t - \gamma \frac{1}{n}\sum_{i=1}^n \nabla_{\mathbf{w}} S(\mathbf{w}, \mathbf{x_i})</script>

 <p>The multiple linear regression algorithm computes either a fixed number of SGD iterations or terminates based on a dynamic convergence criterion.
   The convergence criterion is the relative change in the sum of squared residuals:</p>

 <script type="math/tex; mode=display">% <![CDATA[
 \frac{S_{k-1} - S_k}{S_{k-1}} < \rho %]]></script>

 <h2 id="operations">Operations</h2>

 <p><code>MultipleLinearRegression</code> is a <code>Predictor</code>.
 As such, it supports the <code>fit</code> and <code>predict</code> operation.</p>

 <h3 id="fit">Fit</h3>

 <p>MultipleLinearRegression is trained on a set of <code>LabeledVector</code>:</p>

 <ul>
   <li><code>fit: DataSet[LabeledVector] =&gt; Unit</code></li>
 </ul>

 <h3 id="predict">Predict</h3>

 <p>MultipleLinearRegression predicts for all subtypes of <code>Vector</code> the corresponding regression value:</p>

 <ul>
   <li><code>predict[T &lt;: Vector]: DataSet[T] =&gt; DataSet[LabeledVector]</code></li>
 </ul>

 <p>If we call predict with a <code>DataSet[LabeledVector]</code>, we make a prediction on the regression value
 for each example, and return a <code>DataSet[(Double, Double)]</code>. In each tuple the first element
 is the true value, as was provided from the input <code>DataSet[LabeledVector]</code> and the second element
 is the predicted value. You can then use these <code>(truth, prediction)</code> tuples to evaluate
 the algorithm’s performance.</p>

 <ul>
   <li><code>predict: DataSet[LabeledVector] =&gt; DataSet[(Double, Double)]</code></li>
 </ul>

 <h2 id="parameters">Parameters</h2>

 <p>The multiple linear regression implementation can be controlled by the following parameters:</p>

 <table class="table table-bordered">
     <thead>
       <tr>
         <th class="text-left" style="width: 20%">Parameters</th>
         <th class="text-center">Description</th>
       </tr>
     </thead>

     <tbody>
       <tr>
         <td><strong>Iterations</strong></td>
         <td>
           <p>
             The maximum number of iterations. (Default value: <strong>10</strong>)
           </p>
         </td>
       </tr>
       <tr>
         <td><strong>Stepsize</strong></td>
         <td>
           <p>
             Initial step size for the gradient descent method.
             This value controls how far the gradient descent method moves in the opposite direction of the gradient.
             Tuning this parameter might be crucial to make it stable and to obtain a better performance.
             (Default value: <strong>0.1</strong>)
           </p>
         </td>
       </tr>
       <tr>
         <td><strong>ConvergenceThreshold</strong></td>
         <td>
           <p>
             Threshold for relative change of the sum of squared residuals until the iteration is stopped.
             (Default value: <strong>None</strong>)
           </p>
         </td>
       </tr>
     </tbody>
   </table>

 <h2 id="examples">Examples</h2>

 <div class="highlight"><pre><code class="language-scala" data-lang="scala"><span class="c1">// Create multiple linear regression learner</span>
 <span class="k">val</span> <span class="n">mlr</span> <span class="k">=</span> <span class="nc">MultipleLinearRegression</span><span class="o">()</span>
 <span class="o">.</span><span class="n">setIterations</span><span class="o">(</span><span class="mi">10</span><span class="o">)</span>
 <span class="o">.</span><span class="n">setStepsize</span><span class="o">(</span><span class="mf">0.5</span><span class="o">)</span>
 <span class="o">.</span><span class="n">setConvergenceThreshold</span><span class="o">(</span><span class="mf">0.001</span><span class="o">)</span>

 <span class="c1">// Obtain training and testing data set</span>
 <span class="k">val</span> <span class="n">trainingDS</span><span class="k">:</span> <span class="kt">DataSet</span><span class="o">[</span><span class="kt">LabeledVector</span><span class="o">]</span> <span class="k">=</span> <span class="o">...</span>
 <span class="k">val</span> <span class="n">testingDS</span><span class="k">:</span> <span class="kt">DataSet</span><span class="o">[</span><span class="kt">Vector</span><span class="o">]</span> <span class="k">=</span> <span class="o">...</span>

 <span class="c1">// Fit the linear model to the provided data</span>
 <span class="n">mlr</span><span class="o">.</span><span class="n">fit</span><span class="o">(</span><span class="n">trainingDS</span><span class="o">)</span>

 <span class="c1">// Calculate the predictions for the test data</span>
 <span class="k">val</span> <span class="n">predictions</span> <span class="k">=</span> <span class="n">mlr</span><span class="o">.</span><span class="n">predict</span><span class="o">(</span><span class="n">testingDS</span><span class="o">)</span></code></pre></div>


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	<h1><a href="../ml">FlinkML</a> - Multiple linear regression</h1>



	<ul id="markdown-toc">
	<li><a href="#description" id="markdown-toc-description">Description</a></li>
	<li><a href="#operations" id="markdown-toc-operations">Operations</a> <ul>
	<li><a href="#fit" id="markdown-toc-fit">Fit</a></li>
	<li><a href="#predict" id="markdown-toc-predict">Predict</a></li>
	</ul>
	</li>
	<li><a href="#parameters" id="markdown-toc-parameters">Parameters</a></li>
	<li><a href="#examples" id="markdown-toc-examples">Examples</a></li>
	</ul>

	<h2 id="description">Description</h2>

	<p>Multiple linear regression tries to find a linear function which best fits the provided input data.
	Given a set of input data with its value $(\mathbf{x}, y)$, multiple linear regression finds
	a vector $\mathbf{w}$ such that the sum of the squared residuals is minimized:</p>

	<script type="math/tex; mode=display">S(\mathbf{w}) = \sum_{i=1} \left(y - \mathbf{w}^T\mathbf{x_i} \right)^2</script>

	<p>Written in matrix notation, we obtain the following formulation:</p>

	<script type="math/tex; mode=display">\mathbf{w}^* = \arg \min_{\mathbf{w}} (\mathbf{y} - X\mathbf{w})^2</script>

	<p>This problem has a closed form solution which is given by:</p>

	<script type="math/tex; mode=display">\mathbf{w}^* = \left(X^TX\right)^{-1}X^T\mathbf{y}</script>

	<p>However, in cases where the input data set is so huge that a complete parse over the whole data
	set is prohibitive, one can apply stochastic gradient descent (SGD) to approximate the solution.
	SGD first calculates for a random subset of the input data set the gradients. The gradient
	for a given point $\mathbf{x}_i$ is given by:</p>

	<script type="math/tex; mode=display">\nabla_{\mathbf{w}} S(\mathbf{w}, \mathbf{x_i}) = 2\left(\mathbf{w}^T\mathbf{x_i} -
	y\right)\mathbf{x_i}</script>

	<p>The gradients are averaged and scaled. The scaling is defined by $\gamma = \frac{s}{\sqrt{j}}$
	with $s$ being the initial step size and $j$ being the current iteration number. The resulting gradient is subtracted from the
	current weight vector giving the new weight vector for the next iteration:</p>

	<script type="math/tex; mode=display">\mathbf{w}_{t+1} = \mathbf{w}_t - \gamma \frac{1}{n}\sum_{i=1}^n \nabla_{\mathbf{w}} S(\mathbf{w}, \mathbf{x_i})</script>

	<p>The multiple linear regression algorithm computes either a fixed number of SGD iterations or terminates based on a dynamic convergence criterion.
	The convergence criterion is the relative change in the sum of squared residuals:</p>

	<script type="math/tex; mode=display">% <![CDATA[
	\frac{S_{k-1} - S_k}{S_{k-1}} < \rho %]]></script>

	<h2 id="operations">Operations</h2>

	<p><code>MultipleLinearRegression</code> is a <code>Predictor</code>.
	As such, it supports the <code>fit</code> and <code>predict</code> operation.</p>

	<h3 id="fit">Fit</h3>

	<p>MultipleLinearRegression is trained on a set of <code>LabeledVector</code>:</p>

	<ul>
	<li><code>fit: DataSet[LabeledVector] => Unit</code></li>
	</ul>

	<h3 id="predict">Predict</h3>

	<p>MultipleLinearRegression predicts for all subtypes of <code>Vector</code> the corresponding regression value:</p>

	<ul>
	<li><code>predict[T <: Vector]: DataSet[T] => DataSet[LabeledVector]</code></li>
	</ul>

	<p>If we call predict with a <code>DataSet[LabeledVector]</code>, we make a prediction on the regression value
	for each example, and return a <code>DataSet[(Double, Double)]</code>. In each tuple the first element
	is the true value, as was provided from the input <code>DataSet[LabeledVector]</code> and the second element
	is the predicted value. You can then use these <code>(truth, prediction)</code> tuples to evaluate
	the algorithm’s performance.</p>

	<ul>
	<li><code>predict: DataSet[LabeledVector] => DataSet[(Double, Double)]</code></li>
	</ul>

	<h2 id="parameters">Parameters</h2>

	<p>The multiple linear regression implementation can be controlled by the following parameters:</p>

	<table class="table table-bordered">
	<thead>
	<tr>
	<th class="text-left" style="width: 20%">Parameters</th>
	<th class="text-center">Description</th>
	</tr>
	</thead>

	<tbody>
	<tr>
	<td><strong>Iterations</strong></td>
	<td>
	<p>
	The maximum number of iterations. (Default value: <strong>10</strong>)
	</p>
	</td>
	</tr>
	<tr>
	<td><strong>Stepsize</strong></td>
	<td>
	<p>
	Initial step size for the gradient descent method.
	This value controls how far the gradient descent method moves in the opposite direction of the gradient.
	Tuning this parameter might be crucial to make it stable and to obtain a better performance.
	(Default value: <strong>0.1</strong>)
	</p>
	</td>
	</tr>
	<tr>
	<td><strong>ConvergenceThreshold</strong></td>
	<td>
	<p>
	Threshold for relative change of the sum of squared residuals until the iteration is stopped.
	(Default value: <strong>None</strong>)
	</p>
	</td>
	</tr>
	</tbody>
	</table>

	<h2 id="examples">Examples</h2>

	<div class="highlight"><pre><code class="language-scala" data-lang="scala"><span class="c1">// Create multiple linear regression learner</span>
	<span class="k">val</span> <span class="n">mlr</span> <span class="k">=</span> <span class="nc">MultipleLinearRegression</span><span class="o">()</span>
	<span class="o">.</span><span class="n">setIterations</span><span class="o">(</span><span class="mi">10</span><span class="o">)</span>
	<span class="o">.</span><span class="n">setStepsize</span><span class="o">(</span><span class="mf">0.5</span><span class="o">)</span>
	<span class="o">.</span><span class="n">setConvergenceThreshold</span><span class="o">(</span><span class="mf">0.001</span><span class="o">)</span>

	<span class="c1">// Obtain training and testing data set</span>
	<span class="k">val</span> <span class="n">trainingDS</span><span class="k">:</span> <span class="kt">DataSet</span><span class="o">[</span><span class="kt">LabeledVector</span><span class="o">]</span> <span class="k">=</span> <span class="o">...</span>
	<span class="k">val</span> <span class="n">testingDS</span><span class="k">:</span> <span class="kt">DataSet</span><span class="o">[</span><span class="kt">Vector</span><span class="o">]</span> <span class="k">=</span> <span class="o">...</span>

	<span class="c1">// Fit the linear model to the provided data</span>
	<span class="n">mlr</span><span class="o">.</span><span class="n">fit</span><span class="o">(</span><span class="n">trainingDS</span><span class="o">)</span>

	<span class="c1">// Calculate the predictions for the test data</span>
	<span class="k">val</span> <span class="n">predictions</span> <span class="k">=</span> <span class="n">mlr</span><span class="o">.</span><span class="n">predict</span><span class="o">(</span><span class="n">testingDS</span><span class="o">)</span></code></pre></div>


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