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| <h1 class="title">Basic Statistics</h1> |
| |
| |
| <p><code class="language-plaintext highlighter-rouge">\[ |
| \newcommand{\R}{\mathbb{R}} |
| \newcommand{\E}{\mathbb{E}} |
| \newcommand{\x}{\mathbf{x}} |
| \newcommand{\y}{\mathbf{y}} |
| \newcommand{\wv}{\mathbf{w}} |
| \newcommand{\av}{\mathbf{\alpha}} |
| \newcommand{\bv}{\mathbf{b}} |
| \newcommand{\N}{\mathbb{N}} |
| \newcommand{\id}{\mathbf{I}} |
| \newcommand{\ind}{\mathbf{1}} |
| \newcommand{\0}{\mathbf{0}} |
| \newcommand{\unit}{\mathbf{e}} |
| \newcommand{\one}{\mathbf{1}} |
| \newcommand{\zero}{\mathbf{0}} |
| \]</code></p> |
| |
| <p><strong>Table of Contents</strong></p> |
| |
| <ul id="markdown-toc"> |
| <li><a href="#correlation" id="markdown-toc-correlation">Correlation</a></li> |
| <li><a href="#hypothesis-testing" id="markdown-toc-hypothesis-testing">Hypothesis testing</a> <ul> |
| <li><a href="#chisquaretest" id="markdown-toc-chisquaretest">ChiSquareTest</a></li> |
| </ul> |
| </li> |
| <li><a href="#summarizer" id="markdown-toc-summarizer">Summarizer</a></li> |
| </ul> |
| |
| <h2 id="correlation">Correlation</h2> |
| |
| <p>Calculating the correlation between two series of data is a common operation in Statistics. In <code class="language-plaintext highlighter-rouge">spark.ml</code> |
| we provide the flexibility to calculate pairwise correlations among many series. The supported |
| correlation methods are currently Pearson’s and Spearman’s correlation.</p> |
| |
| <div class="codetabs"> |
| <div data-lang="scala"> |
| <p><a href="api/scala/org/apache/spark/ml/stat/Correlation$.html"><code class="language-plaintext highlighter-rouge">Correlation</code></a> |
| computes the correlation matrix for the input Dataset of Vectors using the specified method. |
| The output will be a DataFrame that contains the correlation matrix of the column of vectors.</p> |
| |
| <div class="highlight"><pre class="codehilite"><code><span class="k">import</span> <span class="nn">org.apache.spark.ml.linalg.</span><span class="o">{</span><span class="nc">Matrix</span><span class="o">,</span> <span class="nc">Vectors</span><span class="o">}</span> |
| <span class="k">import</span> <span class="nn">org.apache.spark.ml.stat.Correlation</span> |
| <span class="k">import</span> <span class="nn">org.apache.spark.sql.Row</span> |
| |
| <span class="k">val</span> <span class="nv">data</span> <span class="k">=</span> <span class="nc">Seq</span><span class="o">(</span> |
| <span class="nv">Vectors</span><span class="o">.</span><span class="py">sparse</span><span class="o">(</span><span class="mi">4</span><span class="o">,</span> <span class="nc">Seq</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="mi">3</span><span class="o">,</span> <span class="o">-</span><span class="mf">2.0</span><span class="o">))),</span> |
| <span class="nv">Vectors</span><span class="o">.</span><span class="py">dense</span><span class="o">(</span><span class="mf">4.0</span><span class="o">,</span> <span class="mf">5.0</span><span class="o">,</span> <span class="mf">0.0</span><span class="o">,</span> <span class="mf">3.0</span><span class="o">),</span> |
| <span class="nv">Vectors</span><span class="o">.</span><span class="py">dense</span><span class="o">(</span><span class="mf">6.0</span><span class="o">,</span> <span class="mf">7.0</span><span class="o">,</span> <span class="mf">0.0</span><span class="o">,</span> <span class="mf">8.0</span><span class="o">),</span> |
| <span class="nv">Vectors</span><span class="o">.</span><span class="py">sparse</span><span class="o">(</span><span class="mi">4</span><span class="o">,</span> <span class="nc">Seq</span><span class="o">((</span><span class="mi">0</span><span class="o">,</span> <span class="mf">9.0</span><span class="o">),</span> <span class="o">(</span><span class="mi">3</span><span class="o">,</span> <span class="mf">1.0</span><span class="o">)))</span> |
| <span class="o">)</span> |
| |
| <span class="k">val</span> <span class="nv">df</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="nv">Tuple1</span><span class="o">.</span><span class="py">apply</span><span class="o">).</span><span class="py">toDF</span><span class="o">(</span><span class="s">"features"</span><span class="o">)</span> |
| <span class="k">val</span> <span class="nv">Row</span><span class="o">(</span><span class="n">coeff1</span><span class="k">:</span> <span class="kt">Matrix</span><span class="o">)</span> <span class="k">=</span> <span class="nv">Correlation</span><span class="o">.</span><span class="py">corr</span><span class="o">(</span><span class="n">df</span><span class="o">,</span> <span class="s">"features"</span><span class="o">).</span><span class="py">head</span> |
| <span class="nf">println</span><span class="o">(</span><span class="n">s</span><span class="s">"Pearson correlation matrix:\n $coeff1"</span><span class="o">)</span> |
| |
| <span class="k">val</span> <span class="nv">Row</span><span class="o">(</span><span class="n">coeff2</span><span class="k">:</span> <span class="kt">Matrix</span><span class="o">)</span> <span class="k">=</span> <span class="nv">Correlation</span><span class="o">.</span><span class="py">corr</span><span class="o">(</span><span class="n">df</span><span class="o">,</span> <span class="s">"features"</span><span class="o">,</span> <span class="s">"spearman"</span><span class="o">).</span><span class="py">head</span> |
| <span class="nf">println</span><span class="o">(</span><span class="n">s</span><span class="s">"Spearman correlation matrix:\n $coeff2"</span><span class="o">)</span></code></pre></div> |
| <div><small>Find full example code at "examples/src/main/scala/org/apache/spark/examples/ml/CorrelationExample.scala" in the Spark repo.</small></div> |
| </div> |
| |
| <div data-lang="java"> |
| <p><a href="api/java/org/apache/spark/ml/stat/Correlation.html"><code class="language-plaintext highlighter-rouge">Correlation</code></a> |
| computes the correlation matrix for the input Dataset of Vectors using the specified method. |
| The output will be a DataFrame that contains the correlation matrix of the column of vectors.</p> |
| |
| <div class="highlight"><pre class="codehilite"><code><span class="kn">import</span> <span class="nn">java.util.Arrays</span><span class="o">;</span> |
| <span class="kn">import</span> <span class="nn">java.util.List</span><span class="o">;</span> |
| |
| <span class="kn">import</span> <span class="nn">org.apache.spark.ml.linalg.Vectors</span><span class="o">;</span> |
| <span class="kn">import</span> <span class="nn">org.apache.spark.ml.linalg.VectorUDT</span><span class="o">;</span> |
| <span class="kn">import</span> <span class="nn">org.apache.spark.ml.stat.Correlation</span><span class="o">;</span> |
| <span class="kn">import</span> <span class="nn">org.apache.spark.sql.Dataset</span><span class="o">;</span> |
| <span class="kn">import</span> <span class="nn">org.apache.spark.sql.Row</span><span class="o">;</span> |
| <span class="kn">import</span> <span class="nn">org.apache.spark.sql.RowFactory</span><span class="o">;</span> |
| <span class="kn">import</span> <span class="nn">org.apache.spark.sql.types.*</span><span class="o">;</span> |
| |
| <span class="nc">List</span><span class="o"><</span><span class="nc">Row</span><span class="o">></span> <span class="n">data</span> <span class="o">=</span> <span class="nc">Arrays</span><span class="o">.</span><span class="na">asList</span><span class="o">(</span> |
| <span class="nc">RowFactory</span><span class="o">.</span><span class="na">create</span><span class="o">(</span><span class="nc">Vectors</span><span class="o">.</span><span class="na">sparse</span><span class="o">(</span><span class="mi">4</span><span class="o">,</span> <span class="k">new</span> <span class="kt">int</span><span class="o">[]{</span><span class="mi">0</span><span class="o">,</span> <span class="mi">3</span><span class="o">},</span> <span class="k">new</span> <span class="kt">double</span><span class="o">[]{</span><span class="mf">1.0</span><span class="o">,</span> <span class="o">-</span><span class="mf">2.0</span><span class="o">})),</span> |
| <span class="nc">RowFactory</span><span class="o">.</span><span class="na">create</span><span class="o">(</span><span class="nc">Vectors</span><span class="o">.</span><span class="na">dense</span><span class="o">(</span><span class="mf">4.0</span><span class="o">,</span> <span class="mf">5.0</span><span class="o">,</span> <span class="mf">0.0</span><span class="o">,</span> <span class="mf">3.0</span><span class="o">)),</span> |
| <span class="nc">RowFactory</span><span class="o">.</span><span class="na">create</span><span class="o">(</span><span class="nc">Vectors</span><span class="o">.</span><span class="na">dense</span><span class="o">(</span><span class="mf">6.0</span><span class="o">,</span> <span class="mf">7.0</span><span class="o">,</span> <span class="mf">0.0</span><span class="o">,</span> <span class="mf">8.0</span><span class="o">)),</span> |
| <span class="nc">RowFactory</span><span class="o">.</span><span class="na">create</span><span class="o">(</span><span class="nc">Vectors</span><span class="o">.</span><span class="na">sparse</span><span class="o">(</span><span class="mi">4</span><span class="o">,</span> <span class="k">new</span> <span class="kt">int</span><span class="o">[]{</span><span class="mi">0</span><span class="o">,</span> <span class="mi">3</span><span class="o">},</span> <span class="k">new</span> <span class="kt">double</span><span class="o">[]{</span><span class="mf">9.0</span><span class="o">,</span> <span class="mf">1.0</span><span class="o">}))</span> |
| <span class="o">);</span> |
| |
| <span class="nc">StructType</span> <span class="n">schema</span> <span class="o">=</span> <span class="k">new</span> <span class="nc">StructType</span><span class="o">(</span><span class="k">new</span> <span class="nc">StructField</span><span class="o">[]{</span> |
| <span class="k">new</span> <span class="nf">StructField</span><span class="o">(</span><span class="s">"features"</span><span class="o">,</span> <span class="k">new</span> <span class="nc">VectorUDT</span><span class="o">(),</span> <span class="kc">false</span><span class="o">,</span> <span class="nc">Metadata</span><span class="o">.</span><span class="na">empty</span><span class="o">()),</span> |
| <span class="o">});</span> |
| |
| <span class="nc">Dataset</span><span class="o"><</span><span class="nc">Row</span><span class="o">></span> <span class="n">df</span> <span class="o">=</span> <span class="n">spark</span><span class="o">.</span><span class="na">createDataFrame</span><span class="o">(</span><span class="n">data</span><span class="o">,</span> <span class="n">schema</span><span class="o">);</span> |
| <span class="nc">Row</span> <span class="n">r1</span> <span class="o">=</span> <span class="nc">Correlation</span><span class="o">.</span><span class="na">corr</span><span class="o">(</span><span class="n">df</span><span class="o">,</span> <span class="s">"features"</span><span class="o">).</span><span class="na">head</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">"Pearson correlation matrix:\n"</span> <span class="o">+</span> <span class="n">r1</span><span class="o">.</span><span class="na">get</span><span class="o">(</span><span class="mi">0</span><span class="o">).</span><span class="na">toString</span><span class="o">());</span> |
| |
| <span class="nc">Row</span> <span class="n">r2</span> <span class="o">=</span> <span class="nc">Correlation</span><span class="o">.</span><span class="na">corr</span><span class="o">(</span><span class="n">df</span><span class="o">,</span> <span class="s">"features"</span><span class="o">,</span> <span class="s">"spearman"</span><span class="o">).</span><span class="na">head</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">"Spearman correlation matrix:\n"</span> <span class="o">+</span> <span class="n">r2</span><span class="o">.</span><span class="na">get</span><span class="o">(</span><span class="mi">0</span><span class="o">).</span><span class="na">toString</span><span class="o">());</span></code></pre></div> |
| <div><small>Find full example code at "examples/src/main/java/org/apache/spark/examples/ml/JavaCorrelationExample.java" in the Spark repo.</small></div> |
| </div> |
| |
| <div data-lang="python"> |
| <p><a href="api/python/reference/api/pyspark.ml.stat.Correlation.html"><code class="language-plaintext highlighter-rouge">Correlation</code></a> |
| computes the correlation matrix for the input Dataset of Vectors using the specified method. |
| The output will be a DataFrame that contains the correlation matrix of the column of vectors.</p> |
| |
| <div class="highlight"><pre class="codehilite"><code><span class="kn">from</span> <span class="nn">pyspark.ml.linalg</span> <span class="kn">import</span> <span class="n">Vectors</span> |
| <span class="kn">from</span> <span class="nn">pyspark.ml.stat</span> <span class="kn">import</span> <span class="n">Correlation</span> |
| |
| <span class="n">data</span> <span class="o">=</span> <span class="p">[(</span><span class="n">Vectors</span><span class="p">.</span><span class="n">sparse</span><span class="p">(</span><span class="mi">4</span><span class="p">,</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="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.0</span><span class="p">)]),),</span> |
| <span class="p">(</span><span class="n">Vectors</span><span class="p">.</span><span class="n">dense</span><span class="p">([</span><span class="mf">4.0</span><span class="p">,</span> <span class="mf">5.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">3.0</span><span class="p">]),),</span> |
| <span class="p">(</span><span class="n">Vectors</span><span class="p">.</span><span class="n">dense</span><span class="p">([</span><span class="mf">6.0</span><span class="p">,</span> <span class="mf">7.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">8.0</span><span class="p">]),),</span> |
| <span class="p">(</span><span class="n">Vectors</span><span class="p">.</span><span class="n">sparse</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="p">[(</span><span class="mi">0</span><span class="p">,</span> <span class="mf">9.0</span><span class="p">),</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">)]),)]</span> |
| <span class="n">df</span> <span class="o">=</span> <span class="n">spark</span><span class="p">.</span><span class="n">createDataFrame</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="p">[</span><span class="s">"features"</span><span class="p">])</span> |
| |
| <span class="n">r1</span> <span class="o">=</span> <span class="n">Correlation</span><span class="p">.</span><span class="n">corr</span><span class="p">(</span><span class="n">df</span><span class="p">,</span> <span class="s">"features"</span><span class="p">).</span><span class="n">head</span><span class="p">()</span> |
| <span class="k">print</span><span class="p">(</span><span class="s">"Pearson correlation matrix:</span><span class="se">\n</span><span class="s">"</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">r1</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span> |
| |
| <span class="n">r2</span> <span class="o">=</span> <span class="n">Correlation</span><span class="p">.</span><span class="n">corr</span><span class="p">(</span><span class="n">df</span><span class="p">,</span> <span class="s">"features"</span><span class="p">,</span> <span class="s">"spearman"</span><span class="p">).</span><span class="n">head</span><span class="p">()</span> |
| <span class="k">print</span><span class="p">(</span><span class="s">"Spearman correlation matrix:</span><span class="se">\n</span><span class="s">"</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">r2</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span></code></pre></div> |
| <div><small>Find full example code at "examples/src/main/python/ml/correlation_example.py" in the Spark repo.</small></div> |
| </div> |
| |
| </div> |
| |
| <h2 id="hypothesis-testing">Hypothesis testing</h2> |
| |
| <p>Hypothesis testing is a powerful tool in statistics to determine whether a result is statistically |
| significant, whether this result occurred by chance or not. <code class="language-plaintext highlighter-rouge">spark.ml</code> currently supports Pearson’s |
| Chi-squared ( $\chi^2$) tests for independence.</p> |
| |
| <h3 id="chisquaretest">ChiSquareTest</h3> |
| |
| <p><code class="language-plaintext highlighter-rouge">ChiSquareTest</code> conducts Pearson’s independence test for every feature against the label. |
| For each feature, the (feature, label) pairs are converted into a contingency matrix for which |
| the Chi-squared statistic is computed. All label and feature values must be categorical.</p> |
| |
| <div class="codetabs"> |
| <div data-lang="scala"> |
| <p>Refer to the <a href="api/scala/org/apache/spark/ml/stat/ChiSquareTest$.html"><code class="language-plaintext highlighter-rouge">ChiSquareTest</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.ml.linalg.</span><span class="o">{</span><span class="nc">Vector</span><span class="o">,</span> <span class="nc">Vectors</span><span class="o">}</span> |
| <span class="k">import</span> <span class="nn">org.apache.spark.ml.stat.ChiSquareTest</span> |
| |
| <span class="k">val</span> <span class="nv">data</span> <span class="k">=</span> <span class="nc">Seq</span><span class="o">(</span> |
| <span class="o">(</span><span class="mf">0.0</span><span class="o">,</span> <span class="nv">Vectors</span><span class="o">.</span><span class="py">dense</span><span class="o">(</span><span class="mf">0.5</span><span class="o">,</span> <span class="mf">10.0</span><span class="o">)),</span> |
| <span class="o">(</span><span class="mf">0.0</span><span class="o">,</span> <span class="nv">Vectors</span><span class="o">.</span><span class="py">dense</span><span class="o">(</span><span class="mf">1.5</span><span class="o">,</span> <span class="mf">20.0</span><span class="o">)),</span> |
| <span class="o">(</span><span class="mf">1.0</span><span class="o">,</span> <span class="nv">Vectors</span><span class="o">.</span><span class="py">dense</span><span class="o">(</span><span class="mf">1.5</span><span class="o">,</span> <span class="mf">30.0</span><span class="o">)),</span> |
| <span class="o">(</span><span class="mf">0.0</span><span class="o">,</span> <span class="nv">Vectors</span><span class="o">.</span><span class="py">dense</span><span class="o">(</span><span class="mf">3.5</span><span class="o">,</span> <span class="mf">30.0</span><span class="o">)),</span> |
| <span class="o">(</span><span class="mf">0.0</span><span class="o">,</span> <span class="nv">Vectors</span><span class="o">.</span><span class="py">dense</span><span class="o">(</span><span class="mf">3.5</span><span class="o">,</span> <span class="mf">40.0</span><span class="o">)),</span> |
| <span class="o">(</span><span class="mf">1.0</span><span class="o">,</span> <span class="nv">Vectors</span><span class="o">.</span><span class="py">dense</span><span class="o">(</span><span class="mf">3.5</span><span class="o">,</span> <span class="mf">40.0</span><span class="o">))</span> |
| <span class="o">)</span> |
| |
| <span class="k">val</span> <span class="nv">df</span> <span class="k">=</span> <span class="nv">data</span><span class="o">.</span><span class="py">toDF</span><span class="o">(</span><span class="s">"label"</span><span class="o">,</span> <span class="s">"features"</span><span class="o">)</span> |
| <span class="k">val</span> <span class="nv">chi</span> <span class="k">=</span> <span class="nv">ChiSquareTest</span><span class="o">.</span><span class="py">test</span><span class="o">(</span><span class="n">df</span><span class="o">,</span> <span class="s">"features"</span><span class="o">,</span> <span class="s">"label"</span><span class="o">).</span><span class="py">head</span> |
| <span class="nf">println</span><span class="o">(</span><span class="n">s</span><span class="s">"pValues = ${chi.getAs[Vector](0)}"</span><span class="o">)</span> |
| <span class="nf">println</span><span class="o">(</span><span class="n">s</span><span class="s">"degreesOfFreedom ${chi.getSeq[Int](1).mkString("</span><span class="o">[</span><span class="err">"</span>, <span class="err">"</span>,<span class="err">"</span>, <span class="err">"</span><span class="o">]</span><span class="s">")}"</span><span class="o">)</span> |
| <span class="nf">println</span><span class="o">(</span><span class="n">s</span><span class="s">"statistics ${chi.getAs[Vector](2)}"</span><span class="o">)</span></code></pre></div> |
| <div><small>Find full example code at "examples/src/main/scala/org/apache/spark/examples/ml/ChiSquareTestExample.scala" in the Spark repo.</small></div> |
| </div> |
| |
| <div data-lang="java"> |
| <p>Refer to the <a href="api/java/org/apache/spark/ml/stat/ChiSquareTest.html"><code class="language-plaintext highlighter-rouge">ChiSquareTest</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">java.util.Arrays</span><span class="o">;</span> |
| <span class="kn">import</span> <span class="nn">java.util.List</span><span class="o">;</span> |
| |
| <span class="kn">import</span> <span class="nn">org.apache.spark.ml.linalg.Vectors</span><span class="o">;</span> |
| <span class="kn">import</span> <span class="nn">org.apache.spark.ml.linalg.VectorUDT</span><span class="o">;</span> |
| <span class="kn">import</span> <span class="nn">org.apache.spark.ml.stat.ChiSquareTest</span><span class="o">;</span> |
| <span class="kn">import</span> <span class="nn">org.apache.spark.sql.Dataset</span><span class="o">;</span> |
| <span class="kn">import</span> <span class="nn">org.apache.spark.sql.Row</span><span class="o">;</span> |
| <span class="kn">import</span> <span class="nn">org.apache.spark.sql.RowFactory</span><span class="o">;</span> |
| <span class="kn">import</span> <span class="nn">org.apache.spark.sql.types.*</span><span class="o">;</span> |
| |
| <span class="nc">List</span><span class="o"><</span><span class="nc">Row</span><span class="o">></span> <span class="n">data</span> <span class="o">=</span> <span class="nc">Arrays</span><span class="o">.</span><span class="na">asList</span><span class="o">(</span> |
| <span class="nc">RowFactory</span><span class="o">.</span><span class="na">create</span><span class="o">(</span><span class="mf">0.0</span><span class="o">,</span> <span class="nc">Vectors</span><span class="o">.</span><span class="na">dense</span><span class="o">(</span><span class="mf">0.5</span><span class="o">,</span> <span class="mf">10.0</span><span class="o">)),</span> |
| <span class="nc">RowFactory</span><span class="o">.</span><span class="na">create</span><span class="o">(</span><span class="mf">0.0</span><span class="o">,</span> <span class="nc">Vectors</span><span class="o">.</span><span class="na">dense</span><span class="o">(</span><span class="mf">1.5</span><span class="o">,</span> <span class="mf">20.0</span><span class="o">)),</span> |
| <span class="nc">RowFactory</span><span class="o">.</span><span class="na">create</span><span class="o">(</span><span class="mf">1.0</span><span class="o">,</span> <span class="nc">Vectors</span><span class="o">.</span><span class="na">dense</span><span class="o">(</span><span class="mf">1.5</span><span class="o">,</span> <span class="mf">30.0</span><span class="o">)),</span> |
| <span class="nc">RowFactory</span><span class="o">.</span><span class="na">create</span><span class="o">(</span><span class="mf">0.0</span><span class="o">,</span> <span class="nc">Vectors</span><span class="o">.</span><span class="na">dense</span><span class="o">(</span><span class="mf">3.5</span><span class="o">,</span> <span class="mf">30.0</span><span class="o">)),</span> |
| <span class="nc">RowFactory</span><span class="o">.</span><span class="na">create</span><span class="o">(</span><span class="mf">0.0</span><span class="o">,</span> <span class="nc">Vectors</span><span class="o">.</span><span class="na">dense</span><span class="o">(</span><span class="mf">3.5</span><span class="o">,</span> <span class="mf">40.0</span><span class="o">)),</span> |
| <span class="nc">RowFactory</span><span class="o">.</span><span class="na">create</span><span class="o">(</span><span class="mf">1.0</span><span class="o">,</span> <span class="nc">Vectors</span><span class="o">.</span><span class="na">dense</span><span class="o">(</span><span class="mf">3.5</span><span class="o">,</span> <span class="mf">40.0</span><span class="o">))</span> |
| <span class="o">);</span> |
| |
| <span class="nc">StructType</span> <span class="n">schema</span> <span class="o">=</span> <span class="k">new</span> <span class="nc">StructType</span><span class="o">(</span><span class="k">new</span> <span class="nc">StructField</span><span class="o">[]{</span> |
| <span class="k">new</span> <span class="nf">StructField</span><span class="o">(</span><span class="s">"label"</span><span class="o">,</span> <span class="nc">DataTypes</span><span class="o">.</span><span class="na">DoubleType</span><span class="o">,</span> <span class="kc">false</span><span class="o">,</span> <span class="nc">Metadata</span><span class="o">.</span><span class="na">empty</span><span class="o">()),</span> |
| <span class="k">new</span> <span class="nf">StructField</span><span class="o">(</span><span class="s">"features"</span><span class="o">,</span> <span class="k">new</span> <span class="nc">VectorUDT</span><span class="o">(),</span> <span class="kc">false</span><span class="o">,</span> <span class="nc">Metadata</span><span class="o">.</span><span class="na">empty</span><span class="o">()),</span> |
| <span class="o">});</span> |
| |
| <span class="nc">Dataset</span><span class="o"><</span><span class="nc">Row</span><span class="o">></span> <span class="n">df</span> <span class="o">=</span> <span class="n">spark</span><span class="o">.</span><span class="na">createDataFrame</span><span class="o">(</span><span class="n">data</span><span class="o">,</span> <span class="n">schema</span><span class="o">);</span> |
| <span class="nc">Row</span> <span class="n">r</span> <span class="o">=</span> <span class="nc">ChiSquareTest</span><span class="o">.</span><span class="na">test</span><span class="o">(</span><span class="n">df</span><span class="o">,</span> <span class="s">"features"</span><span class="o">,</span> <span class="s">"label"</span><span class="o">).</span><span class="na">head</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">"pValues: "</span> <span class="o">+</span> <span class="n">r</span><span class="o">.</span><span class="na">get</span><span class="o">(</span><span class="mi">0</span><span class="o">).</span><span class="na">toString</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">"degreesOfFreedom: "</span> <span class="o">+</span> <span class="n">r</span><span class="o">.</span><span class="na">getList</span><span class="o">(</span><span class="mi">1</span><span class="o">).</span><span class="na">toString</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">"statistics: "</span> <span class="o">+</span> <span class="n">r</span><span class="o">.</span><span class="na">get</span><span class="o">(</span><span class="mi">2</span><span class="o">).</span><span class="na">toString</span><span class="o">());</span></code></pre></div> |
| <div><small>Find full example code at "examples/src/main/java/org/apache/spark/examples/ml/JavaChiSquareTestExample.java" in the Spark repo.</small></div> |
| </div> |
| |
| <div data-lang="python"> |
| <p>Refer to the <a href="api/python/reference/api/pyspark.ml.stat.ChiSquareTest.html"><code class="language-plaintext highlighter-rouge">ChiSquareTest</code> Python docs</a> for details on the API.</p> |
| |
| <div class="highlight"><pre class="codehilite"><code><span class="kn">from</span> <span class="nn">pyspark.ml.linalg</span> <span class="kn">import</span> <span class="n">Vectors</span> |
| <span class="kn">from</span> <span class="nn">pyspark.ml.stat</span> <span class="kn">import</span> <span class="n">ChiSquareTest</span> |
| |
| <span class="n">data</span> <span class="o">=</span> <span class="p">[(</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">Vectors</span><span class="p">.</span><span class="n">dense</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">10.0</span><span class="p">)),</span> |
| <span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">Vectors</span><span class="p">.</span><span class="n">dense</span><span class="p">(</span><span class="mf">1.5</span><span class="p">,</span> <span class="mf">20.0</span><span class="p">)),</span> |
| <span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">Vectors</span><span class="p">.</span><span class="n">dense</span><span class="p">(</span><span class="mf">1.5</span><span class="p">,</span> <span class="mf">30.0</span><span class="p">)),</span> |
| <span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">Vectors</span><span class="p">.</span><span class="n">dense</span><span class="p">(</span><span class="mf">3.5</span><span class="p">,</span> <span class="mf">30.0</span><span class="p">)),</span> |
| <span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">Vectors</span><span class="p">.</span><span class="n">dense</span><span class="p">(</span><span class="mf">3.5</span><span class="p">,</span> <span class="mf">40.0</span><span class="p">)),</span> |
| <span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">Vectors</span><span class="p">.</span><span class="n">dense</span><span class="p">(</span><span class="mf">3.5</span><span class="p">,</span> <span class="mf">40.0</span><span class="p">))]</span> |
| <span class="n">df</span> <span class="o">=</span> <span class="n">spark</span><span class="p">.</span><span class="n">createDataFrame</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="p">[</span><span class="s">"label"</span><span class="p">,</span> <span class="s">"features"</span><span class="p">])</span> |
| |
| <span class="n">r</span> <span class="o">=</span> <span class="n">ChiSquareTest</span><span class="p">.</span><span class="n">test</span><span class="p">(</span><span class="n">df</span><span class="p">,</span> <span class="s">"features"</span><span class="p">,</span> <span class="s">"label"</span><span class="p">).</span><span class="n">head</span><span class="p">()</span> |
| <span class="k">print</span><span class="p">(</span><span class="s">"pValues: "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">r</span><span class="p">.</span><span class="n">pValues</span><span class="p">))</span> |
| <span class="k">print</span><span class="p">(</span><span class="s">"degreesOfFreedom: "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">r</span><span class="p">.</span><span class="n">degreesOfFreedom</span><span class="p">))</span> |
| <span class="k">print</span><span class="p">(</span><span class="s">"statistics: "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">r</span><span class="p">.</span><span class="n">statistics</span><span class="p">))</span></code></pre></div> |
| <div><small>Find full example code at "examples/src/main/python/ml/chi_square_test_example.py" in the Spark repo.</small></div> |
| </div> |
| |
| </div> |
| |
| <h2 id="summarizer">Summarizer</h2> |
| |
| <p>We provide vector column summary statistics for <code class="language-plaintext highlighter-rouge">Dataframe</code> through <code class="language-plaintext highlighter-rouge">Summarizer</code>. |
| Available metrics are the column-wise max, min, mean, sum, variance, std, and number of nonzeros, |
| as well as the total count.</p> |
| |
| <div class="codetabs"> |
| <div data-lang="scala"> |
| <p>The following example demonstrates using <a href="api/scala/org/apache/spark/ml/stat/Summarizer$.html"><code class="language-plaintext highlighter-rouge">Summarizer</code></a> |
| to compute the mean and variance for a vector column of the input dataframe, with and without a weight column.</p> |
| |
| <div class="highlight"><pre class="codehilite"><code><span class="k">import</span> <span class="nn">org.apache.spark.ml.linalg.</span><span class="o">{</span><span class="nc">Vector</span><span class="o">,</span> <span class="nc">Vectors</span><span class="o">}</span> |
| <span class="k">import</span> <span class="nn">org.apache.spark.ml.stat.Summarizer</span> |
| |
| <span class="k">val</span> <span class="nv">data</span> <span class="k">=</span> <span class="nc">Seq</span><span class="o">(</span> |
| <span class="o">(</span><span class="nv">Vectors</span><span class="o">.</span><span class="py">dense</span><span class="o">(</span><span class="mf">2.0</span><span class="o">,</span> <span class="mf">3.0</span><span class="o">,</span> <span class="mf">5.0</span><span class="o">),</span> <span class="mf">1.0</span><span class="o">),</span> |
| <span class="o">(</span><span class="nv">Vectors</span><span class="o">.</span><span class="py">dense</span><span class="o">(</span><span class="mf">4.0</span><span class="o">,</span> <span class="mf">6.0</span><span class="o">,</span> <span class="mf">7.0</span><span class="o">),</span> <span class="mf">2.0</span><span class="o">)</span> |
| <span class="o">)</span> |
| |
| <span class="k">val</span> <span class="nv">df</span> <span class="k">=</span> <span class="nv">data</span><span class="o">.</span><span class="py">toDF</span><span class="o">(</span><span class="s">"features"</span><span class="o">,</span> <span class="s">"weight"</span><span class="o">)</span> |
| |
| <span class="nf">val</span> <span class="o">(</span><span class="n">meanVal</span><span class="o">,</span> <span class="n">varianceVal</span><span class="o">)</span> <span class="k">=</span> <span class="nv">df</span><span class="o">.</span><span class="py">select</span><span class="o">(</span><span class="nf">metrics</span><span class="o">(</span><span class="s">"mean"</span><span class="o">,</span> <span class="s">"variance"</span><span class="o">)</span> |
| <span class="o">.</span><span class="py">summary</span><span class="o">(</span><span class="n">$</span><span class="s">"features"</span><span class="o">,</span> <span class="n">$</span><span class="s">"weight"</span><span class="o">).</span><span class="py">as</span><span class="o">(</span><span class="s">"summary"</span><span class="o">))</span> |
| <span class="o">.</span><span class="py">select</span><span class="o">(</span><span class="s">"summary.mean"</span><span class="o">,</span> <span class="s">"summary.variance"</span><span class="o">)</span> |
| <span class="o">.</span><span class="py">as</span><span class="o">[(</span><span class="kt">Vector</span>, <span class="kt">Vector</span><span class="o">)].</span><span class="py">first</span><span class="o">()</span> |
| |
| <span class="nf">println</span><span class="o">(</span><span class="n">s</span><span class="s">"with weight: mean = ${meanVal}, variance = ${varianceVal}"</span><span class="o">)</span> |
| |
| <span class="nf">val</span> <span class="o">(</span><span class="n">meanVal2</span><span class="o">,</span> <span class="n">varianceVal2</span><span class="o">)</span> <span class="k">=</span> <span class="nv">df</span><span class="o">.</span><span class="py">select</span><span class="o">(</span><span class="nf">mean</span><span class="o">(</span><span class="n">$</span><span class="s">"features"</span><span class="o">),</span> <span class="nf">variance</span><span class="o">(</span><span class="n">$</span><span class="s">"features"</span><span class="o">))</span> |
| <span class="o">.</span><span class="py">as</span><span class="o">[(</span><span class="kt">Vector</span>, <span class="kt">Vector</span><span class="o">)].</span><span class="py">first</span><span class="o">()</span> |
| |
| <span class="nf">println</span><span class="o">(</span><span class="n">s</span><span class="s">"without weight: mean = ${meanVal2}, sum = ${varianceVal2}"</span><span class="o">)</span></code></pre></div> |
| <div><small>Find full example code at "examples/src/main/scala/org/apache/spark/examples/ml/SummarizerExample.scala" in the Spark repo.</small></div> |
| </div> |
| |
| <div data-lang="java"> |
| <p>The following example demonstrates using <a href="api/java/org/apache/spark/ml/stat/Summarizer.html"><code class="language-plaintext highlighter-rouge">Summarizer</code></a> |
| to compute the mean and variance for a vector column of the input dataframe, with and without a weight column.</p> |
| |
| <div class="highlight"><pre class="codehilite"><code><span class="kn">import</span> <span class="nn">java.util.Arrays</span><span class="o">;</span> |
| <span class="kn">import</span> <span class="nn">java.util.List</span><span class="o">;</span> |
| |
| <span class="kn">import</span> <span class="nn">org.apache.spark.ml.linalg.Vector</span><span class="o">;</span> |
| <span class="kn">import</span> <span class="nn">org.apache.spark.ml.linalg.Vectors</span><span class="o">;</span> |
| <span class="kn">import</span> <span class="nn">org.apache.spark.ml.linalg.VectorUDT</span><span class="o">;</span> |
| <span class="kn">import</span> <span class="nn">org.apache.spark.ml.stat.Summarizer</span><span class="o">;</span> |
| <span class="kn">import</span> <span class="nn">org.apache.spark.sql.types.DataTypes</span><span class="o">;</span> |
| <span class="kn">import</span> <span class="nn">org.apache.spark.sql.types.Metadata</span><span class="o">;</span> |
| <span class="kn">import</span> <span class="nn">org.apache.spark.sql.types.StructField</span><span class="o">;</span> |
| <span class="kn">import</span> <span class="nn">org.apache.spark.sql.types.StructType</span><span class="o">;</span> |
| |
| <span class="nc">List</span><span class="o"><</span><span class="nc">Row</span><span class="o">></span> <span class="n">data</span> <span class="o">=</span> <span class="nc">Arrays</span><span class="o">.</span><span class="na">asList</span><span class="o">(</span> |
| <span class="nc">RowFactory</span><span class="o">.</span><span class="na">create</span><span class="o">(</span><span class="nc">Vectors</span><span class="o">.</span><span class="na">dense</span><span class="o">(</span><span class="mf">2.0</span><span class="o">,</span> <span class="mf">3.0</span><span class="o">,</span> <span class="mf">5.0</span><span class="o">),</span> <span class="mf">1.0</span><span class="o">),</span> |
| <span class="nc">RowFactory</span><span class="o">.</span><span class="na">create</span><span class="o">(</span><span class="nc">Vectors</span><span class="o">.</span><span class="na">dense</span><span class="o">(</span><span class="mf">4.0</span><span class="o">,</span> <span class="mf">6.0</span><span class="o">,</span> <span class="mf">7.0</span><span class="o">),</span> <span class="mf">2.0</span><span class="o">)</span> |
| <span class="o">);</span> |
| |
| <span class="nc">StructType</span> <span class="n">schema</span> <span class="o">=</span> <span class="k">new</span> <span class="nc">StructType</span><span class="o">(</span><span class="k">new</span> <span class="nc">StructField</span><span class="o">[]{</span> |
| <span class="k">new</span> <span class="nf">StructField</span><span class="o">(</span><span class="s">"features"</span><span class="o">,</span> <span class="k">new</span> <span class="nc">VectorUDT</span><span class="o">(),</span> <span class="kc">false</span><span class="o">,</span> <span class="nc">Metadata</span><span class="o">.</span><span class="na">empty</span><span class="o">()),</span> |
| <span class="k">new</span> <span class="nf">StructField</span><span class="o">(</span><span class="s">"weight"</span><span class="o">,</span> <span class="nc">DataTypes</span><span class="o">.</span><span class="na">DoubleType</span><span class="o">,</span> <span class="kc">false</span><span class="o">,</span> <span class="nc">Metadata</span><span class="o">.</span><span class="na">empty</span><span class="o">())</span> |
| <span class="o">});</span> |
| |
| <span class="nc">Dataset</span><span class="o"><</span><span class="nc">Row</span><span class="o">></span> <span class="n">df</span> <span class="o">=</span> <span class="n">spark</span><span class="o">.</span><span class="na">createDataFrame</span><span class="o">(</span><span class="n">data</span><span class="o">,</span> <span class="n">schema</span><span class="o">);</span> |
| |
| <span class="nc">Row</span> <span class="n">result1</span> <span class="o">=</span> <span class="n">df</span><span class="o">.</span><span class="na">select</span><span class="o">(</span><span class="nc">Summarizer</span><span class="o">.</span><span class="na">metrics</span><span class="o">(</span><span class="s">"mean"</span><span class="o">,</span> <span class="s">"variance"</span><span class="o">)</span> |
| <span class="o">.</span><span class="na">summary</span><span class="o">(</span><span class="k">new</span> <span class="nc">Column</span><span class="o">(</span><span class="s">"features"</span><span class="o">),</span> <span class="k">new</span> <span class="nc">Column</span><span class="o">(</span><span class="s">"weight"</span><span class="o">)).</span><span class="na">as</span><span class="o">(</span><span class="s">"summary"</span><span class="o">))</span> |
| <span class="o">.</span><span class="na">select</span><span class="o">(</span><span class="s">"summary.mean"</span><span class="o">,</span> <span class="s">"summary.variance"</span><span class="o">).</span><span class="na">first</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">"with weight: mean = "</span> <span class="o">+</span> <span class="n">result1</span><span class="o">.<</span><span class="nc">Vector</span><span class="o">></span><span class="n">getAs</span><span class="o">(</span><span class="mi">0</span><span class="o">).</span><span class="na">toString</span><span class="o">()</span> <span class="o">+</span> |
| <span class="s">", variance = "</span> <span class="o">+</span> <span class="n">result1</span><span class="o">.<</span><span class="nc">Vector</span><span class="o">></span><span class="n">getAs</span><span class="o">(</span><span class="mi">1</span><span class="o">).</span><span class="na">toString</span><span class="o">());</span> |
| |
| <span class="nc">Row</span> <span class="n">result2</span> <span class="o">=</span> <span class="n">df</span><span class="o">.</span><span class="na">select</span><span class="o">(</span> |
| <span class="nc">Summarizer</span><span class="o">.</span><span class="na">mean</span><span class="o">(</span><span class="k">new</span> <span class="nc">Column</span><span class="o">(</span><span class="s">"features"</span><span class="o">)),</span> |
| <span class="nc">Summarizer</span><span class="o">.</span><span class="na">variance</span><span class="o">(</span><span class="k">new</span> <span class="nc">Column</span><span class="o">(</span><span class="s">"features"</span><span class="o">))</span> |
| <span class="o">).</span><span class="na">first</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">"without weight: mean = "</span> <span class="o">+</span> <span class="n">result2</span><span class="o">.<</span><span class="nc">Vector</span><span class="o">></span><span class="n">getAs</span><span class="o">(</span><span class="mi">0</span><span class="o">).</span><span class="na">toString</span><span class="o">()</span> <span class="o">+</span> |
| <span class="s">", variance = "</span> <span class="o">+</span> <span class="n">result2</span><span class="o">.<</span><span class="nc">Vector</span><span class="o">></span><span class="n">getAs</span><span class="o">(</span><span class="mi">1</span><span class="o">).</span><span class="na">toString</span><span class="o">());</span></code></pre></div> |
| <div><small>Find full example code at "examples/src/main/java/org/apache/spark/examples/ml/JavaSummarizerExample.java" in the Spark repo.</small></div> |
| </div> |
| |
| <div data-lang="python"> |
| <p>Refer to the <a href="api/python/reference/api/pyspark.ml.stat.Summarizer.html"><code class="language-plaintext highlighter-rouge">Summarizer</code> Python docs</a> for details on the API.</p> |
| |
| <div class="highlight"><pre class="codehilite"><code><span class="kn">from</span> <span class="nn">pyspark.ml.stat</span> <span class="kn">import</span> <span class="n">Summarizer</span> |
| <span class="kn">from</span> <span class="nn">pyspark.sql</span> <span class="kn">import</span> <span class="n">Row</span> |
| <span class="kn">from</span> <span class="nn">pyspark.ml.linalg</span> <span class="kn">import</span> <span class="n">Vectors</span> |
| |
| <span class="n">df</span> <span class="o">=</span> <span class="n">sc</span><span class="p">.</span><span class="n">parallelize</span><span class="p">([</span><span class="n">Row</span><span class="p">(</span><span class="n">weight</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">features</span><span class="o">=</span><span class="n">Vectors</span><span class="p">.</span><span class="n">dense</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">)),</span> |
| <span class="n">Row</span><span class="p">(</span><span class="n">weight</span><span class="o">=</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">features</span><span class="o">=</span><span class="n">Vectors</span><span class="p">.</span><span class="n">dense</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">,</span> <span class="mf">3.0</span><span class="p">))]).</span><span class="n">toDF</span><span class="p">()</span> |
| |
| <span class="c1"># create summarizer for multiple metrics "mean" and "count" |
| </span><span class="n">summarizer</span> <span class="o">=</span> <span class="n">Summarizer</span><span class="p">.</span><span class="n">metrics</span><span class="p">(</span><span class="s">"mean"</span><span class="p">,</span> <span class="s">"count"</span><span class="p">)</span> |
| |
| <span class="c1"># compute statistics for multiple metrics with weight |
| </span><span class="n">df</span><span class="p">.</span><span class="n">select</span><span class="p">(</span><span class="n">summarizer</span><span class="p">.</span><span class="n">summary</span><span class="p">(</span><span class="n">df</span><span class="p">.</span><span class="n">features</span><span class="p">,</span> <span class="n">df</span><span class="p">.</span><span class="n">weight</span><span class="p">)).</span><span class="n">show</span><span class="p">(</span><span class="n">truncate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span> |
| |
| <span class="c1"># compute statistics for multiple metrics without weight |
| </span><span class="n">df</span><span class="p">.</span><span class="n">select</span><span class="p">(</span><span class="n">summarizer</span><span class="p">.</span><span class="n">summary</span><span class="p">(</span><span class="n">df</span><span class="p">.</span><span class="n">features</span><span class="p">)).</span><span class="n">show</span><span class="p">(</span><span class="n">truncate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span> |
| |
| <span class="c1"># compute statistics for single metric "mean" with weight |
| </span><span class="n">df</span><span class="p">.</span><span class="n">select</span><span class="p">(</span><span class="n">Summarizer</span><span class="p">.</span><span class="n">mean</span><span class="p">(</span><span class="n">df</span><span class="p">.</span><span class="n">features</span><span class="p">,</span> <span class="n">df</span><span class="p">.</span><span class="n">weight</span><span class="p">)).</span><span class="n">show</span><span class="p">(</span><span class="n">truncate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span> |
| |
| <span class="c1"># compute statistics for single metric "mean" without weight |
| </span><span class="n">df</span><span class="p">.</span><span class="n">select</span><span class="p">(</span><span class="n">Summarizer</span><span class="p">.</span><span class="n">mean</span><span class="p">(</span><span class="n">df</span><span class="p">.</span><span class="n">features</span><span class="p">)).</span><span class="n">show</span><span class="p">(</span><span class="n">truncate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span></code></pre></div> |
| <div><small>Find full example code at "examples/src/main/python/ml/summarizer_example.py" in the Spark repo.</small></div> |
| </div> |
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