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<h1 class="title">Feature Extraction and Transformation - RDD-based API</h1>
<ul id="markdown-toc">
<li><a href="#tf-idf" id="markdown-toc-tf-idf">TF-IDF</a></li>
<li><a href="#word2vec" id="markdown-toc-word2vec">Word2Vec</a> <ul>
<li><a href="#model" id="markdown-toc-model">Model</a></li>
<li><a href="#example" id="markdown-toc-example">Example</a></li>
</ul>
</li>
<li><a href="#standardscaler" id="markdown-toc-standardscaler">StandardScaler</a> <ul>
<li><a href="#model-fitting" id="markdown-toc-model-fitting">Model Fitting</a></li>
<li><a href="#example-1" id="markdown-toc-example-1">Example</a></li>
</ul>
</li>
<li><a href="#normalizer" id="markdown-toc-normalizer">Normalizer</a> <ul>
<li><a href="#example-2" id="markdown-toc-example-2">Example</a></li>
</ul>
</li>
<li><a href="#chisqselector" id="markdown-toc-chisqselector">ChiSqSelector</a> <ul>
<li><a href="#model-fitting-1" id="markdown-toc-model-fitting-1">Model Fitting</a></li>
<li><a href="#example-3" id="markdown-toc-example-3">Example</a></li>
</ul>
</li>
<li><a href="#elementwiseproduct" id="markdown-toc-elementwiseproduct">ElementwiseProduct</a> <ul>
<li><a href="#example-4" id="markdown-toc-example-4">Example</a></li>
</ul>
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<li><a href="#pca" id="markdown-toc-pca">PCA</a> <ul>
<li><a href="#example-5" id="markdown-toc-example-5">Example</a></li>
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</ul>
<h2 id="tf-idf">TF-IDF</h2>
<p><strong>Note</strong> We recommend using the DataFrame-based API, which is detailed in the <a href="ml-features.html#tf-idf">ML user guide on
TF-IDF</a>.</p>
<p><a href="http://en.wikipedia.org/wiki/Tf%E2%80%93idf">Term frequency-inverse document frequency (TF-IDF)</a> is a feature
vectorization method widely used in text mining to reflect the importance of a term to a document in the corpus.
Denote a term by <code>$t$</code>, a document by <code>$d$</code>, and the corpus by <code>$D$</code>.
Term frequency <code>$TF(t, d)$</code> is the number of times that term <code>$t$</code> appears in document <code>$d$</code>,
while document frequency <code>$DF(t, D)$</code> is the number of documents that contains term <code>$t$</code>.
If we only use term frequency to measure the importance, it is very easy to over-emphasize terms that
appear very often but carry little information about the document, e.g., &#8220;a&#8221;, &#8220;the&#8221;, and &#8220;of&#8221;.
If a term appears very often across the corpus, it means it doesn&#8217;t carry special information about
a particular document.
Inverse document frequency is a numerical measure of how much information a term provides:
<code>\[
IDF(t, D) = \log \frac{|D| + 1}{DF(t, D) + 1},
\]</code>
where <code>$|D|$</code> is the total number of documents in the corpus.
Since logarithm is used, if a term appears in all documents, its IDF value becomes 0.
Note that a smoothing term is applied to avoid dividing by zero for terms outside the corpus.
The TF-IDF measure is simply the product of TF and IDF:
<code>\[
TFIDF(t, d, D) = TF(t, d) \cdot IDF(t, D).
\]</code>
There are several variants on the definition of term frequency and document frequency.
In <code>spark.mllib</code>, we separate TF and IDF to make them flexible.</p>
<p>Our implementation of term frequency utilizes the
<a href="http://en.wikipedia.org/wiki/Feature_hashing">hashing trick</a>.
A raw feature is mapped into an index (term) by applying a hash function.
Then term frequencies are calculated based on the mapped indices.
This approach avoids the need to compute a global term-to-index map,
which can be expensive for a large corpus, but it suffers from potential hash collisions,
where different raw features may become the same term after hashing.
To reduce the chance of collision, we can increase the target feature dimension, i.e.,
the number of buckets of the hash table.
The default feature dimension is <code>$2^{20} = 1,048,576$</code>.</p>
<p><strong>Note:</strong> <code>spark.mllib</code> doesn&#8217;t provide tools for text segmentation.
We refer users to the <a href="http://nlp.stanford.edu/">Stanford NLP Group</a> and
<a href="https://github.com/scalanlp/chalk">scalanlp/chalk</a>.</p>
<div class="codetabs">
<div data-lang="scala">
<p>TF and IDF are implemented in <a href="api/scala/index.html#org.apache.spark.mllib.feature.HashingTF">HashingTF</a>
and <a href="api/scala/index.html#org.apache.spark.mllib.feature.IDF">IDF</a>.
<code>HashingTF</code> takes an <code>RDD[Iterable[_]]</code> as the input.
Each record could be an iterable of strings or other types.</p>
<p>Refer to the <a href="api/scala/index.html#org.apache.spark.mllib.feature.HashingTF"><code>HashingTF</code> Scala docs</a> for details on the API.</p>
<div class="highlight"><pre><span class="k">import</span> <span class="nn">org.apache.spark.mllib.feature.</span><span class="o">{</span><span class="nc">HashingTF</span><span class="o">,</span> <span class="nc">IDF</span><span class="o">}</span>
<span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg.Vector</span>
<span class="k">import</span> <span class="nn">org.apache.spark.rdd.RDD</span>
<span class="c1">// Load documents (one per line).</span>
<span class="k">val</span> <span class="n">documents</span><span class="k">:</span> <span class="kt">RDD</span><span class="o">[</span><span class="kt">Seq</span><span class="o">[</span><span class="kt">String</span><span class="o">]]</span> <span class="k">=</span> <span class="n">sc</span><span class="o">.</span><span class="n">textFile</span><span class="o">(</span><span class="s">&quot;data/mllib/kmeans_data.txt&quot;</span><span class="o">)</span>
<span class="o">.</span><span class="n">map</span><span class="o">(</span><span class="k">_</span><span class="o">.</span><span class="n">split</span><span class="o">(</span><span class="s">&quot; &quot;</span><span class="o">).</span><span class="n">toSeq</span><span class="o">)</span>
<span class="k">val</span> <span class="n">hashingTF</span> <span class="k">=</span> <span class="k">new</span> <span class="nc">HashingTF</span><span class="o">()</span>
<span class="k">val</span> <span class="n">tf</span><span class="k">:</span> <span class="kt">RDD</span><span class="o">[</span><span class="kt">Vector</span><span class="o">]</span> <span class="k">=</span> <span class="n">hashingTF</span><span class="o">.</span><span class="n">transform</span><span class="o">(</span><span class="n">documents</span><span class="o">)</span>
<span class="c1">// While applying HashingTF only needs a single pass to the data, applying IDF needs two passes:</span>
<span class="c1">// First to compute the IDF vector and second to scale the term frequencies by IDF.</span>
<span class="n">tf</span><span class="o">.</span><span class="n">cache</span><span class="o">()</span>
<span class="k">val</span> <span class="n">idf</span> <span class="k">=</span> <span class="k">new</span> <span class="nc">IDF</span><span class="o">().</span><span class="n">fit</span><span class="o">(</span><span class="n">tf</span><span class="o">)</span>
<span class="k">val</span> <span class="n">tfidf</span><span class="k">:</span> <span class="kt">RDD</span><span class="o">[</span><span class="kt">Vector</span><span class="o">]</span> <span class="k">=</span> <span class="n">idf</span><span class="o">.</span><span class="n">transform</span><span class="o">(</span><span class="n">tf</span><span class="o">)</span>
<span class="c1">// spark.mllib IDF implementation provides an option for ignoring terms which occur in less than</span>
<span class="c1">// a minimum number of documents. In such cases, the IDF for these terms is set to 0.</span>
<span class="c1">// This feature can be used by passing the minDocFreq value to the IDF constructor.</span>
<span class="k">val</span> <span class="n">idfIgnore</span> <span class="k">=</span> <span class="k">new</span> <span class="nc">IDF</span><span class="o">(</span><span class="n">minDocFreq</span> <span class="k">=</span> <span class="mi">2</span><span class="o">).</span><span class="n">fit</span><span class="o">(</span><span class="n">tf</span><span class="o">)</span>
<span class="k">val</span> <span class="n">tfidfIgnore</span><span class="k">:</span> <span class="kt">RDD</span><span class="o">[</span><span class="kt">Vector</span><span class="o">]</span> <span class="k">=</span> <span class="n">idfIgnore</span><span class="o">.</span><span class="n">transform</span><span class="o">(</span><span class="n">tf</span><span class="o">)</span>
</pre></div>
<div><small>Find full example code at "examples/src/main/scala/org/apache/spark/examples/mllib/TFIDFExample.scala" in the Spark repo.</small></div>
</div>
<div data-lang="python">
<p>TF and IDF are implemented in <a href="api/python/pyspark.mllib.html#pyspark.mllib.feature.HashingTF">HashingTF</a>
and <a href="api/python/pyspark.mllib.html#pyspark.mllib.feature.IDF">IDF</a>.
<code>HashingTF</code> takes an RDD of list as the input.
Each record could be an iterable of strings or other types.</p>
<p>Refer to the <a href="api/python/pyspark.mllib.html#pyspark.mllib.feature.HashingTF"><code>HashingTF</code> Python docs</a> for details on the API.</p>
<div class="highlight"><pre><span class="kn">from</span> <span class="nn">pyspark.mllib.feature</span> <span class="kn">import</span> <span class="n">HashingTF</span><span class="p">,</span> <span class="n">IDF</span>
<span class="c"># Load documents (one per line).</span>
<span class="n">documents</span> <span class="o">=</span> <span class="n">sc</span><span class="o">.</span><span class="n">textFile</span><span class="p">(</span><span class="s">&quot;data/mllib/kmeans_data.txt&quot;</span><span class="p">)</span><span class="o">.</span><span class="n">map</span><span class="p">(</span><span class="k">lambda</span> <span class="n">line</span><span class="p">:</span> <span class="n">line</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">&quot; &quot;</span><span class="p">))</span>
<span class="n">hashingTF</span> <span class="o">=</span> <span class="n">HashingTF</span><span class="p">()</span>
<span class="n">tf</span> <span class="o">=</span> <span class="n">hashingTF</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">documents</span><span class="p">)</span>
<span class="c"># While applying HashingTF only needs a single pass to the data, applying IDF needs two passes:</span>
<span class="c"># First to compute the IDF vector and second to scale the term frequencies by IDF.</span>
<span class="n">tf</span><span class="o">.</span><span class="n">cache</span><span class="p">()</span>
<span class="n">idf</span> <span class="o">=</span> <span class="n">IDF</span><span class="p">()</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">tf</span><span class="p">)</span>
<span class="n">tfidf</span> <span class="o">=</span> <span class="n">idf</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">tf</span><span class="p">)</span>
<span class="c"># spark.mllib&#39;s IDF implementation provides an option for ignoring terms</span>
<span class="c"># which occur in less than a minimum number of documents.</span>
<span class="c"># In such cases, the IDF for these terms is set to 0.</span>
<span class="c"># This feature can be used by passing the minDocFreq value to the IDF constructor.</span>
<span class="n">idfIgnore</span> <span class="o">=</span> <span class="n">IDF</span><span class="p">(</span><span class="n">minDocFreq</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">tf</span><span class="p">)</span>
<span class="n">tfidfIgnore</span> <span class="o">=</span> <span class="n">idfIgnore</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">tf</span><span class="p">)</span>
</pre></div>
<div><small>Find full example code at "examples/src/main/python/mllib/tf_idf_example.py" in the Spark repo.</small></div>
</div>
</div>
<h2 id="word2vec">Word2Vec</h2>
<p><a href="https://code.google.com/p/word2vec/">Word2Vec</a> computes distributed vector representation of words.
The main advantage of the distributed
representations is that similar words are close in the vector space, which makes generalization to
novel patterns easier and model estimation more robust. Distributed vector representation is
showed to be useful in many natural language processing applications such as named entity
recognition, disambiguation, parsing, tagging and machine translation.</p>
<h3 id="model">Model</h3>
<p>In our implementation of Word2Vec, we used skip-gram model. The training objective of skip-gram is
to learn word vector representations that are good at predicting its context in the same sentence.
Mathematically, given a sequence of training words <code>$w_1, w_2, \dots, w_T$</code>, the objective of the
skip-gram model is to maximize the average log-likelihood
<code>\[
\frac{1}{T} \sum_{t = 1}^{T}\sum_{j=-k}^{j=k} \log p(w_{t+j} | w_t)
\]</code>
where $k$ is the size of the training window.</p>
<p>In the skip-gram model, every word $w$ is associated with two vectors $u_w$ and $v_w$ which are
vector representations of $w$ as word and context respectively. The probability of correctly
predicting word $w_i$ given word $w_j$ is determined by the softmax model, which is
<code>\[
p(w_i | w_j ) = \frac{\exp(u_{w_i}^{\top}v_{w_j})}{\sum_{l=1}^{V} \exp(u_l^{\top}v_{w_j})}
\]</code>
where $V$ is the vocabulary size.</p>
<p>The skip-gram model with softmax is expensive because the cost of computing $\log p(w_i | w_j)$
is proportional to $V$, which can be easily in order of millions. To speed up training of Word2Vec,
we used hierarchical softmax, which reduced the complexity of computing of $\log p(w_i | w_j)$ to
$O(\log(V))$</p>
<h3 id="example">Example</h3>
<p>The example below demonstrates how to load a text file, parse it as an RDD of <code>Seq[String]</code>,
construct a <code>Word2Vec</code> instance and then fit a <code>Word2VecModel</code> with the input data. Finally,
we display the top 40 synonyms of the specified word. To run the example, first download
the <a href="http://mattmahoney.net/dc/text8.zip">text8</a> data and extract it to your preferred directory.
Here we assume the extracted file is <code>text8</code> and in same directory as you run the spark shell.</p>
<div class="codetabs">
<div data-lang="scala">
<p>Refer to the <a href="api/scala/index.html#org.apache.spark.mllib.feature.Word2Vec"><code>Word2Vec</code> Scala docs</a> for details on the API.</p>
<div class="highlight"><pre><span class="k">import</span> <span class="nn">org.apache.spark.mllib.feature.</span><span class="o">{</span><span class="nc">Word2Vec</span><span class="o">,</span> <span class="nc">Word2VecModel</span><span class="o">}</span>
<span class="k">val</span> <span class="n">input</span> <span class="k">=</span> <span class="n">sc</span><span class="o">.</span><span class="n">textFile</span><span class="o">(</span><span class="s">&quot;data/mllib/sample_lda_data.txt&quot;</span><span class="o">).</span><span class="n">map</span><span class="o">(</span><span class="n">line</span> <span class="k">=&gt;</span> <span class="n">line</span><span class="o">.</span><span class="n">split</span><span class="o">(</span><span class="s">&quot; &quot;</span><span class="o">).</span><span class="n">toSeq</span><span class="o">)</span>
<span class="k">val</span> <span class="n">word2vec</span> <span class="k">=</span> <span class="k">new</span> <span class="nc">Word2Vec</span><span class="o">()</span>
<span class="k">val</span> <span class="n">model</span> <span class="k">=</span> <span class="n">word2vec</span><span class="o">.</span><span class="n">fit</span><span class="o">(</span><span class="n">input</span><span class="o">)</span>
<span class="k">val</span> <span class="n">synonyms</span> <span class="k">=</span> <span class="n">model</span><span class="o">.</span><span class="n">findSynonyms</span><span class="o">(</span><span class="s">&quot;1&quot;</span><span class="o">,</span> <span class="mi">5</span><span class="o">)</span>
<span class="k">for</span><span class="o">((</span><span class="n">synonym</span><span class="o">,</span> <span class="n">cosineSimilarity</span><span class="o">)</span> <span class="k">&lt;-</span> <span class="n">synonyms</span><span class="o">)</span> <span class="o">{</span>
<span class="n">println</span><span class="o">(</span><span class="n">s</span><span class="s">&quot;$synonym $cosineSimilarity&quot;</span><span class="o">)</span>
<span class="o">}</span>
<span class="c1">// Save and load model</span>
<span class="n">model</span><span class="o">.</span><span class="n">save</span><span class="o">(</span><span class="n">sc</span><span class="o">,</span> <span class="s">&quot;myModelPath&quot;</span><span class="o">)</span>
<span class="k">val</span> <span class="n">sameModel</span> <span class="k">=</span> <span class="nc">Word2VecModel</span><span class="o">.</span><span class="n">load</span><span class="o">(</span><span class="n">sc</span><span class="o">,</span> <span class="s">&quot;myModelPath&quot;</span><span class="o">)</span>
</pre></div>
<div><small>Find full example code at "examples/src/main/scala/org/apache/spark/examples/mllib/Word2VecExample.scala" in the Spark repo.</small></div>
</div>
<div data-lang="python">
<p>Refer to the <a href="api/python/pyspark.mllib.html#pyspark.mllib.feature.Word2Vec"><code>Word2Vec</code> Python docs</a> for more details on the API.</p>
<div class="highlight"><pre><span class="kn">from</span> <span class="nn">pyspark.mllib.feature</span> <span class="kn">import</span> <span class="n">Word2Vec</span>
<span class="n">inp</span> <span class="o">=</span> <span class="n">sc</span><span class="o">.</span><span class="n">textFile</span><span class="p">(</span><span class="s">&quot;data/mllib/sample_lda_data.txt&quot;</span><span class="p">)</span><span class="o">.</span><span class="n">map</span><span class="p">(</span><span class="k">lambda</span> <span class="n">row</span><span class="p">:</span> <span class="n">row</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">&quot; &quot;</span><span class="p">))</span>
<span class="n">word2vec</span> <span class="o">=</span> <span class="n">Word2Vec</span><span class="p">()</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">word2vec</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">inp</span><span class="p">)</span>
<span class="n">synonyms</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">findSynonyms</span><span class="p">(</span><span class="s">&#39;1&#39;</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
<span class="k">for</span> <span class="n">word</span><span class="p">,</span> <span class="n">cosine_distance</span> <span class="ow">in</span> <span class="n">synonyms</span><span class="p">:</span>
<span class="k">print</span><span class="p">(</span><span class="s">&quot;{}: {}&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">word</span><span class="p">,</span> <span class="n">cosine_distance</span><span class="p">))</span>
</pre></div>
<div><small>Find full example code at "examples/src/main/python/mllib/word2vec_example.py" in the Spark repo.</small></div>
</div>
</div>
<h2 id="standardscaler">StandardScaler</h2>
<p>Standardizes features by scaling to unit variance and/or removing the mean using column summary
statistics on the samples in the training set. This is a very common pre-processing step.</p>
<p>For example, RBF kernel of Support Vector Machines or the L1 and L2 regularized linear models
typically work better when all features have unit variance and/or zero mean.</p>
<p>Standardization can improve the convergence rate during the optimization process, and also prevents
against features with very large variances exerting an overly large influence during model training.</p>
<h3 id="model-fitting">Model Fitting</h3>
<p><a href="api/scala/index.html#org.apache.spark.mllib.feature.StandardScaler"><code>StandardScaler</code></a> has the
following parameters in the constructor:</p>
<ul>
<li><code>withMean</code> False by default. Centers the data with mean before scaling. It will build a dense
output, so take care when applying to sparse input.</li>
<li><code>withStd</code> True by default. Scales the data to unit standard deviation.</li>
</ul>
<p>We provide a <a href="api/scala/index.html#org.apache.spark.mllib.feature.StandardScaler"><code>fit</code></a> method in
<code>StandardScaler</code> which can take an input of <code>RDD[Vector]</code>, learn the summary statistics, and then
return a model which can transform the input dataset into unit standard deviation and/or zero mean features
depending how we configure the <code>StandardScaler</code>.</p>
<p>This model implements <a href="api/scala/index.html#org.apache.spark.mllib.feature.VectorTransformer"><code>VectorTransformer</code></a>
which can apply the standardization on a <code>Vector</code> to produce a transformed <code>Vector</code> or on
an <code>RDD[Vector]</code> to produce a transformed <code>RDD[Vector]</code>.</p>
<p>Note that if the variance of a feature is zero, it will return default <code>0.0</code> value in the <code>Vector</code>
for that feature.</p>
<h3 id="example-1">Example</h3>
<p>The example below demonstrates how to load a dataset in libsvm format, and standardize the features
so that the new features have unit standard deviation and/or zero mean.</p>
<div class="codetabs">
<div data-lang="scala">
<p>Refer to the <a href="api/scala/index.html#org.apache.spark.mllib.feature.StandardScaler"><code>StandardScaler</code> Scala docs</a> for details on the API.</p>
<div class="highlight"><pre><span class="k">import</span> <span class="nn">org.apache.spark.mllib.feature.</span><span class="o">{</span><span class="nc">StandardScaler</span><span class="o">,</span> <span class="nc">StandardScalerModel</span><span class="o">}</span>
<span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg.Vectors</span>
<span class="k">import</span> <span class="nn">org.apache.spark.mllib.util.MLUtils</span>
<span class="k">val</span> <span class="n">data</span> <span class="k">=</span> <span class="nc">MLUtils</span><span class="o">.</span><span class="n">loadLibSVMFile</span><span class="o">(</span><span class="n">sc</span><span class="o">,</span> <span class="s">&quot;data/mllib/sample_libsvm_data.txt&quot;</span><span class="o">)</span>
<span class="k">val</span> <span class="n">scaler1</span> <span class="k">=</span> <span class="k">new</span> <span class="nc">StandardScaler</span><span class="o">().</span><span class="n">fit</span><span class="o">(</span><span class="n">data</span><span class="o">.</span><span class="n">map</span><span class="o">(</span><span class="n">x</span> <span class="k">=&gt;</span> <span class="n">x</span><span class="o">.</span><span class="n">features</span><span class="o">))</span>
<span class="k">val</span> <span class="n">scaler2</span> <span class="k">=</span> <span class="k">new</span> <span class="nc">StandardScaler</span><span class="o">(</span><span class="n">withMean</span> <span class="k">=</span> <span class="kc">true</span><span class="o">,</span> <span class="n">withStd</span> <span class="k">=</span> <span class="kc">true</span><span class="o">).</span><span class="n">fit</span><span class="o">(</span><span class="n">data</span><span class="o">.</span><span class="n">map</span><span class="o">(</span><span class="n">x</span> <span class="k">=&gt;</span> <span class="n">x</span><span class="o">.</span><span class="n">features</span><span class="o">))</span>
<span class="c1">// scaler3 is an identical model to scaler2, and will produce identical transformations</span>
<span class="k">val</span> <span class="n">scaler3</span> <span class="k">=</span> <span class="k">new</span> <span class="nc">StandardScalerModel</span><span class="o">(</span><span class="n">scaler2</span><span class="o">.</span><span class="n">std</span><span class="o">,</span> <span class="n">scaler2</span><span class="o">.</span><span class="n">mean</span><span class="o">)</span>
<span class="c1">// data1 will be unit variance.</span>
<span class="k">val</span> <span class="n">data1</span> <span class="k">=</span> <span class="n">data</span><span class="o">.</span><span class="n">map</span><span class="o">(</span><span class="n">x</span> <span class="k">=&gt;</span> <span class="o">(</span><span class="n">x</span><span class="o">.</span><span class="n">label</span><span class="o">,</span> <span class="n">scaler1</span><span class="o">.</span><span class="n">transform</span><span class="o">(</span><span class="n">x</span><span class="o">.</span><span class="n">features</span><span class="o">)))</span>
<span class="c1">// data2 will be unit variance and zero mean.</span>
<span class="k">val</span> <span class="n">data2</span> <span class="k">=</span> <span class="n">data</span><span class="o">.</span><span class="n">map</span><span class="o">(</span><span class="n">x</span> <span class="k">=&gt;</span> <span class="o">(</span><span class="n">x</span><span class="o">.</span><span class="n">label</span><span class="o">,</span> <span class="n">scaler2</span><span class="o">.</span><span class="n">transform</span><span class="o">(</span><span class="nc">Vectors</span><span class="o">.</span><span class="n">dense</span><span class="o">(</span><span class="n">x</span><span class="o">.</span><span class="n">features</span><span class="o">.</span><span class="n">toArray</span><span class="o">))))</span>
</pre></div>
<div><small>Find full example code at "examples/src/main/scala/org/apache/spark/examples/mllib/StandardScalerExample.scala" in the Spark repo.</small></div>
</div>
<div data-lang="python">
<p>Refer to the <a href="api/python/pyspark.mllib.html#pyspark.mllib.feature.StandardScaler"><code>StandardScaler</code> Python docs</a> for more details on the API.</p>
<div class="highlight"><pre><span class="kn">from</span> <span class="nn">pyspark.mllib.feature</span> <span class="kn">import</span> <span class="n">StandardScaler</span><span class="p">,</span> <span class="n">StandardScalerModel</span>
<span class="kn">from</span> <span class="nn">pyspark.mllib.linalg</span> <span class="kn">import</span> <span class="n">Vectors</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="n">data</span> <span class="o">=</span> <span class="n">MLUtils</span><span class="o">.</span><span class="n">loadLibSVMFile</span><span class="p">(</span><span class="n">sc</span><span class="p">,</span> <span class="s">&quot;data/mllib/sample_libsvm_data.txt&quot;</span><span class="p">)</span>
<span class="n">label</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="n">map</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">.</span><span class="n">label</span><span class="p">)</span>
<span class="n">features</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="n">map</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">.</span><span class="n">features</span><span class="p">)</span>
<span class="n">scaler1</span> <span class="o">=</span> <span class="n">StandardScaler</span><span class="p">()</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">features</span><span class="p">)</span>
<span class="n">scaler2</span> <span class="o">=</span> <span class="n">StandardScaler</span><span class="p">(</span><span class="n">withMean</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">withStd</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">features</span><span class="p">)</span>
<span class="c"># data1 will be unit variance.</span>
<span class="n">data1</span> <span class="o">=</span> <span class="n">label</span><span class="o">.</span><span class="n">zip</span><span class="p">(</span><span class="n">scaler1</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">features</span><span class="p">))</span>
<span class="c"># data2 will be unit variance and zero mean.</span>
<span class="n">data2</span> <span class="o">=</span> <span class="n">label</span><span class="o">.</span><span class="n">zip</span><span class="p">(</span><span class="n">scaler2</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">features</span><span class="o">.</span><span class="n">map</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">Vectors</span><span class="o">.</span><span class="n">dense</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">toArray</span><span class="p">()))))</span>
</pre></div>
<div><small>Find full example code at "examples/src/main/python/mllib/standard_scaler_example.py" in the Spark repo.</small></div>
</div>
</div>
<h2 id="normalizer">Normalizer</h2>
<p>Normalizer scales individual samples to have unit $L^p$ norm. This is a common operation for text
classification or clustering. For example, the dot product of two $L^2$ normalized TF-IDF vectors
is the cosine similarity of the vectors.</p>
<p><a href="api/scala/index.html#org.apache.spark.mllib.feature.Normalizer"><code>Normalizer</code></a> has the following
parameter in the constructor:</p>
<ul>
<li><code>p</code> Normalization in $L^p$ space, $p = 2$ by default.</li>
</ul>
<p><code>Normalizer</code> implements <a href="api/scala/index.html#org.apache.spark.mllib.feature.VectorTransformer"><code>VectorTransformer</code></a>
which can apply the normalization on a <code>Vector</code> to produce a transformed <code>Vector</code> or on
an <code>RDD[Vector]</code> to produce a transformed <code>RDD[Vector]</code>.</p>
<p>Note that if the norm of the input is zero, it will return the input vector.</p>
<h3 id="example-2">Example</h3>
<p>The example below demonstrates how to load a dataset in libsvm format, and normalizes the features
with $L^2$ norm, and $L^\infty$ norm.</p>
<div class="codetabs">
<div data-lang="scala">
<p>Refer to the <a href="api/scala/index.html#org.apache.spark.mllib.feature.Normalizer"><code>Normalizer</code> Scala docs</a> for details on the API.</p>
<div class="highlight"><pre><span class="k">import</span> <span class="nn">org.apache.spark.mllib.feature.Normalizer</span>
<span class="k">import</span> <span class="nn">org.apache.spark.mllib.util.MLUtils</span>
<span class="k">val</span> <span class="n">data</span> <span class="k">=</span> <span class="nc">MLUtils</span><span class="o">.</span><span class="n">loadLibSVMFile</span><span class="o">(</span><span class="n">sc</span><span class="o">,</span> <span class="s">&quot;data/mllib/sample_libsvm_data.txt&quot;</span><span class="o">)</span>
<span class="k">val</span> <span class="n">normalizer1</span> <span class="k">=</span> <span class="k">new</span> <span class="nc">Normalizer</span><span class="o">()</span>
<span class="k">val</span> <span class="n">normalizer2</span> <span class="k">=</span> <span class="k">new</span> <span class="nc">Normalizer</span><span class="o">(</span><span class="n">p</span> <span class="k">=</span> <span class="nc">Double</span><span class="o">.</span><span class="nc">PositiveInfinity</span><span class="o">)</span>
<span class="c1">// Each sample in data1 will be normalized using $L^2$ norm.</span>
<span class="k">val</span> <span class="n">data1</span> <span class="k">=</span> <span class="n">data</span><span class="o">.</span><span class="n">map</span><span class="o">(</span><span class="n">x</span> <span class="k">=&gt;</span> <span class="o">(</span><span class="n">x</span><span class="o">.</span><span class="n">label</span><span class="o">,</span> <span class="n">normalizer1</span><span class="o">.</span><span class="n">transform</span><span class="o">(</span><span class="n">x</span><span class="o">.</span><span class="n">features</span><span class="o">)))</span>
<span class="c1">// Each sample in data2 will be normalized using $L^\infty$ norm.</span>
<span class="k">val</span> <span class="n">data2</span> <span class="k">=</span> <span class="n">data</span><span class="o">.</span><span class="n">map</span><span class="o">(</span><span class="n">x</span> <span class="k">=&gt;</span> <span class="o">(</span><span class="n">x</span><span class="o">.</span><span class="n">label</span><span class="o">,</span> <span class="n">normalizer2</span><span class="o">.</span><span class="n">transform</span><span class="o">(</span><span class="n">x</span><span class="o">.</span><span class="n">features</span><span class="o">)))</span>
</pre></div>
<div><small>Find full example code at "examples/src/main/scala/org/apache/spark/examples/mllib/NormalizerExample.scala" in the Spark repo.</small></div>
</div>
<div data-lang="python">
<p>Refer to the <a href="api/python/pyspark.mllib.html#pyspark.mllib.feature.Normalizer"><code>Normalizer</code> Python docs</a> for more details on the API.</p>
<div class="highlight"><pre><span class="kn">from</span> <span class="nn">pyspark.mllib.feature</span> <span class="kn">import</span> <span class="n">Normalizer</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="n">data</span> <span class="o">=</span> <span class="n">MLUtils</span><span class="o">.</span><span class="n">loadLibSVMFile</span><span class="p">(</span><span class="n">sc</span><span class="p">,</span> <span class="s">&quot;data/mllib/sample_libsvm_data.txt&quot;</span><span class="p">)</span>
<span class="n">labels</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="n">map</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">.</span><span class="n">label</span><span class="p">)</span>
<span class="n">features</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="n">map</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">.</span><span class="n">features</span><span class="p">)</span>
<span class="n">normalizer1</span> <span class="o">=</span> <span class="n">Normalizer</span><span class="p">()</span>
<span class="n">normalizer2</span> <span class="o">=</span> <span class="n">Normalizer</span><span class="p">(</span><span class="n">p</span><span class="o">=</span><span class="nb">float</span><span class="p">(</span><span class="s">&quot;inf&quot;</span><span class="p">))</span>
<span class="c"># Each sample in data1 will be normalized using $L^2$ norm.</span>
<span class="n">data1</span> <span class="o">=</span> <span class="n">labels</span><span class="o">.</span><span class="n">zip</span><span class="p">(</span><span class="n">normalizer1</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">features</span><span class="p">))</span>
<span class="c"># Each sample in data2 will be normalized using $L^\infty$ norm.</span>
<span class="n">data2</span> <span class="o">=</span> <span class="n">labels</span><span class="o">.</span><span class="n">zip</span><span class="p">(</span><span class="n">normalizer2</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">features</span><span class="p">))</span>
</pre></div>
<div><small>Find full example code at "examples/src/main/python/mllib/normalizer_example.py" in the Spark repo.</small></div>
</div>
</div>
<h2 id="chisqselector">ChiSqSelector</h2>
<p><a href="http://en.wikipedia.org/wiki/Feature_selection">Feature selection</a> tries to identify relevant
features for use in model construction. It reduces the size of the feature space, which can improve
both speed and statistical learning behavior.</p>
<p><a href="api/scala/index.html#org.apache.spark.mllib.feature.ChiSqSelector"><code>ChiSqSelector</code></a> implements
Chi-Squared feature selection. It operates on labeled data with categorical features. ChiSqSelector uses the
<a href="https://en.wikipedia.org/wiki/Chi-squared_test">Chi-Squared test of independence</a> to decide which
features to choose. It supports three selection methods: <code>numTopFeatures</code>, <code>percentile</code>, <code>fpr</code>:</p>
<ul>
<li><code>numTopFeatures</code> chooses a fixed number of top features according to a chi-squared test. This is akin to yielding the features with the most predictive power.</li>
<li><code>percentile</code> is similar to <code>numTopFeatures</code> but chooses a fraction of all features instead of a fixed number.</li>
<li><code>fpr</code> chooses all features whose p-value is below a threshold, thus controlling the false positive rate of selection.</li>
</ul>
<p>By default, the selection method is <code>numTopFeatures</code>, with the default number of top features set to 50.
The user can choose a selection method using <code>setSelectorType</code>.</p>
<p>The number of features to select can be tuned using a held-out validation set.</p>
<h3 id="model-fitting-1">Model Fitting</h3>
<p>The <a href="api/scala/index.html#org.apache.spark.mllib.feature.ChiSqSelector"><code>fit</code></a> method takes
an input of <code>RDD[LabeledPoint]</code> with categorical features, learns the summary statistics, and then
returns a <code>ChiSqSelectorModel</code> which can transform an input dataset into the reduced feature space.
The <code>ChiSqSelectorModel</code> can be applied either to a <code>Vector</code> to produce a reduced <code>Vector</code>, or to
an <code>RDD[Vector]</code> to produce a reduced <code>RDD[Vector]</code>.</p>
<p>Note that the user can also construct a <code>ChiSqSelectorModel</code> by hand by providing an array of selected feature indices (which must be sorted in ascending order).</p>
<h3 id="example-3">Example</h3>
<p>The following example shows the basic use of ChiSqSelector. The data set used has a feature matrix consisting of greyscale values that vary from 0 to 255 for each feature.</p>
<div class="codetabs">
<div data-lang="scala">
<p>Refer to the <a href="api/scala/index.html#org.apache.spark.mllib.feature.ChiSqSelector"><code>ChiSqSelector</code> Scala docs</a>
for details on the API.</p>
<div class="highlight"><pre><span class="k">import</span> <span class="nn">org.apache.spark.mllib.feature.ChiSqSelector</span>
<span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg.Vectors</span>
<span class="k">import</span> <span class="nn">org.apache.spark.mllib.regression.LabeledPoint</span>
<span class="k">import</span> <span class="nn">org.apache.spark.mllib.util.MLUtils</span>
<span class="c1">// Load some data in libsvm format</span>
<span class="k">val</span> <span class="n">data</span> <span class="k">=</span> <span class="nc">MLUtils</span><span class="o">.</span><span class="n">loadLibSVMFile</span><span class="o">(</span><span class="n">sc</span><span class="o">,</span> <span class="s">&quot;data/mllib/sample_libsvm_data.txt&quot;</span><span class="o">)</span>
<span class="c1">// Discretize data in 16 equal bins since ChiSqSelector requires categorical features</span>
<span class="c1">// Even though features are doubles, the ChiSqSelector treats each unique value as a category</span>
<span class="k">val</span> <span class="n">discretizedData</span> <span class="k">=</span> <span class="n">data</span><span class="o">.</span><span class="n">map</span> <span class="o">{</span> <span class="n">lp</span> <span class="k">=&gt;</span>
<span class="nc">LabeledPoint</span><span class="o">(</span><span class="n">lp</span><span class="o">.</span><span class="n">label</span><span class="o">,</span> <span class="nc">Vectors</span><span class="o">.</span><span class="n">dense</span><span class="o">(</span><span class="n">lp</span><span class="o">.</span><span class="n">features</span><span class="o">.</span><span class="n">toArray</span><span class="o">.</span><span class="n">map</span> <span class="o">{</span> <span class="n">x</span> <span class="k">=&gt;</span> <span class="o">(</span><span class="n">x</span> <span class="o">/</span> <span class="mi">16</span><span class="o">).</span><span class="n">floor</span> <span class="o">}))</span>
<span class="o">}</span>
<span class="c1">// Create ChiSqSelector that will select top 50 of 692 features</span>
<span class="k">val</span> <span class="n">selector</span> <span class="k">=</span> <span class="k">new</span> <span class="nc">ChiSqSelector</span><span class="o">(</span><span class="mi">50</span><span class="o">)</span>
<span class="c1">// Create ChiSqSelector model (selecting features)</span>
<span class="k">val</span> <span class="n">transformer</span> <span class="k">=</span> <span class="n">selector</span><span class="o">.</span><span class="n">fit</span><span class="o">(</span><span class="n">discretizedData</span><span class="o">)</span>
<span class="c1">// Filter the top 50 features from each feature vector</span>
<span class="k">val</span> <span class="n">filteredData</span> <span class="k">=</span> <span class="n">discretizedData</span><span class="o">.</span><span class="n">map</span> <span class="o">{</span> <span class="n">lp</span> <span class="k">=&gt;</span>
<span class="nc">LabeledPoint</span><span class="o">(</span><span class="n">lp</span><span class="o">.</span><span class="n">label</span><span class="o">,</span> <span class="n">transformer</span><span class="o">.</span><span class="n">transform</span><span class="o">(</span><span class="n">lp</span><span class="o">.</span><span class="n">features</span><span class="o">))</span>
<span class="o">}</span>
</pre></div>
<div><small>Find full example code at "examples/src/main/scala/org/apache/spark/examples/mllib/ChiSqSelectorExample.scala" in the Spark repo.</small></div>
</div>
<div data-lang="java">
<p>Refer to the <a href="api/java/org/apache/spark/mllib/feature/ChiSqSelector.html"><code>ChiSqSelector</code> Java docs</a>
for details on the API.</p>
<div class="highlight"><pre><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.api.java.function.Function</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.feature.ChiSqSelector</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.feature.ChiSqSelectorModel</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.Vectors</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="n">JavaRDD</span><span class="o">&lt;</span><span class="n">LabeledPoint</span><span class="o">&gt;</span> <span class="n">points</span> <span class="o">=</span> <span class="n">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">&quot;data/mllib/sample_libsvm_data.txt&quot;</span><span class="o">).</span><span class="na">toJavaRDD</span><span class="o">().</span><span class="na">cache</span><span class="o">();</span>
<span class="c1">// Discretize data in 16 equal bins since ChiSqSelector requires categorical features</span>
<span class="c1">// Although features are doubles, the ChiSqSelector treats each unique value as a category</span>
<span class="n">JavaRDD</span><span class="o">&lt;</span><span class="n">LabeledPoint</span><span class="o">&gt;</span> <span class="n">discretizedData</span> <span class="o">=</span> <span class="n">points</span><span class="o">.</span><span class="na">map</span><span class="o">(</span>
<span class="k">new</span> <span class="n">Function</span><span class="o">&lt;</span><span class="n">LabeledPoint</span><span class="o">,</span> <span class="n">LabeledPoint</span><span class="o">&gt;()</span> <span class="o">{</span>
<span class="nd">@Override</span>
<span class="kd">public</span> <span class="n">LabeledPoint</span> <span class="nf">call</span><span class="o">(</span><span class="n">LabeledPoint</span> <span class="n">lp</span><span class="o">)</span> <span class="o">{</span>
<span class="kd">final</span> <span class="kt">double</span><span class="o">[]</span> <span class="n">discretizedFeatures</span> <span class="o">=</span> <span class="k">new</span> <span class="kt">double</span><span class="o">[</span><span class="n">lp</span><span class="o">.</span><span class="na">features</span><span class="o">().</span><span class="na">size</span><span class="o">()];</span>
<span class="k">for</span> <span class="o">(</span><span class="kt">int</span> <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span><span class="o">;</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">lp</span><span class="o">.</span><span class="na">features</span><span class="o">().</span><span class="na">size</span><span class="o">();</span> <span class="o">++</span><span class="n">i</span><span class="o">)</span> <span class="o">{</span>
<span class="n">discretizedFeatures</span><span class="o">[</span><span class="n">i</span><span class="o">]</span> <span class="o">=</span> <span class="n">Math</span><span class="o">.</span><span class="na">floor</span><span class="o">(</span><span class="n">lp</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="n">i</span><span class="o">)</span> <span class="o">/</span> <span class="mi">16</span><span class="o">);</span>
<span class="o">}</span>
<span class="k">return</span> <span class="k">new</span> <span class="nf">LabeledPoint</span><span class="o">(</span><span class="n">lp</span><span class="o">.</span><span class="na">label</span><span class="o">(),</span> <span class="n">Vectors</span><span class="o">.</span><span class="na">dense</span><span class="o">(</span><span class="n">discretizedFeatures</span><span class="o">));</span>
<span class="o">}</span>
<span class="o">}</span>
<span class="o">);</span>
<span class="c1">// Create ChiSqSelector that will select top 50 of 692 features</span>
<span class="n">ChiSqSelector</span> <span class="n">selector</span> <span class="o">=</span> <span class="k">new</span> <span class="nf">ChiSqSelector</span><span class="o">(</span><span class="mi">50</span><span class="o">);</span>
<span class="c1">// Create ChiSqSelector model (selecting features)</span>
<span class="kd">final</span> <span class="n">ChiSqSelectorModel</span> <span class="n">transformer</span> <span class="o">=</span> <span class="n">selector</span><span class="o">.</span><span class="na">fit</span><span class="o">(</span><span class="n">discretizedData</span><span class="o">.</span><span class="na">rdd</span><span class="o">());</span>
<span class="c1">// Filter the top 50 features from each feature vector</span>
<span class="n">JavaRDD</span><span class="o">&lt;</span><span class="n">LabeledPoint</span><span class="o">&gt;</span> <span class="n">filteredData</span> <span class="o">=</span> <span class="n">discretizedData</span><span class="o">.</span><span class="na">map</span><span class="o">(</span>
<span class="k">new</span> <span class="n">Function</span><span class="o">&lt;</span><span class="n">LabeledPoint</span><span class="o">,</span> <span class="n">LabeledPoint</span><span class="o">&gt;()</span> <span class="o">{</span>
<span class="nd">@Override</span>
<span class="kd">public</span> <span class="n">LabeledPoint</span> <span class="nf">call</span><span class="o">(</span><span class="n">LabeledPoint</span> <span class="n">lp</span><span class="o">)</span> <span class="o">{</span>
<span class="k">return</span> <span class="k">new</span> <span class="nf">LabeledPoint</span><span class="o">(</span><span class="n">lp</span><span class="o">.</span><span class="na">label</span><span class="o">(),</span> <span class="n">transformer</span><span class="o">.</span><span class="na">transform</span><span class="o">(</span><span class="n">lp</span><span class="o">.</span><span class="na">features</span><span class="o">()));</span>
<span class="o">}</span>
<span class="o">}</span>
<span class="o">);</span>
</pre></div>
<div><small>Find full example code at "examples/src/main/java/org/apache/spark/examples/mllib/JavaChiSqSelectorExample.java" in the Spark repo.</small></div>
</div>
</div>
<h2 id="elementwiseproduct">ElementwiseProduct</h2>
<p><code>ElementwiseProduct</code> multiplies each input vector by a provided &#8220;weight&#8221; vector, using element-wise
multiplication. In other words, it scales each column of the dataset by a scalar multiplier. This
represents the <a href="https://en.wikipedia.org/wiki/Hadamard_product_%28matrices%29">Hadamard product</a>
between the input vector, <code>v</code> and transforming vector, <code>scalingVec</code>, to yield a result vector.
Qu8T948*1#
Denoting the <code>scalingVec</code> as &#8220;<code>w</code>,&#8221; this transformation may be written as:</p>
<p><code>\[ \begin{pmatrix}
v_1 \\
\vdots \\
v_N
\end{pmatrix} \circ \begin{pmatrix}
w_1 \\
\vdots \\
w_N
\end{pmatrix}
= \begin{pmatrix}
v_1 w_1 \\
\vdots \\
v_N w_N
\end{pmatrix}
\]</code></p>
<p><a href="api/scala/index.html#org.apache.spark.mllib.feature.ElementwiseProduct"><code>ElementwiseProduct</code></a> has the following parameter in the constructor:</p>
<ul>
<li><code>scalingVec</code>: the transforming vector.</li>
</ul>
<p><code>ElementwiseProduct</code> implements <a href="api/scala/index.html#org.apache.spark.mllib.feature.VectorTransformer"><code>VectorTransformer</code></a> which can apply the weighting on a <code>Vector</code> to produce a transformed <code>Vector</code> or on an <code>RDD[Vector]</code> to produce a transformed <code>RDD[Vector]</code>.</p>
<h3 id="example-4">Example</h3>
<p>This example below demonstrates how to transform vectors using a transforming vector value.</p>
<div class="codetabs">
<div data-lang="scala">
<p>Refer to the <a href="api/scala/index.html#org.apache.spark.mllib.feature.ElementwiseProduct"><code>ElementwiseProduct</code> Scala docs</a> for details on the API.</p>
<div class="highlight"><pre><span class="k">import</span> <span class="nn">org.apache.spark.mllib.feature.ElementwiseProduct</span>
<span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg.Vectors</span>
<span class="c1">// Create some vector data; also works for sparse vectors</span>
<span class="k">val</span> <span class="n">data</span> <span class="k">=</span> <span class="n">sc</span><span class="o">.</span><span class="n">parallelize</span><span class="o">(</span><span class="nc">Array</span><span class="o">(</span><span class="nc">Vectors</span><span class="o">.</span><span class="n">dense</span><span class="o">(</span><span class="mf">1.0</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="nc">Vectors</span><span class="o">.</span><span class="n">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">6.0</span><span class="o">)))</span>
<span class="k">val</span> <span class="n">transformingVector</span> <span class="k">=</span> <span class="nc">Vectors</span><span class="o">.</span><span class="n">dense</span><span class="o">(</span><span class="mf">0.0</span><span class="o">,</span> <span class="mf">1.0</span><span class="o">,</span> <span class="mf">2.0</span><span class="o">)</span>
<span class="k">val</span> <span class="n">transformer</span> <span class="k">=</span> <span class="k">new</span> <span class="nc">ElementwiseProduct</span><span class="o">(</span><span class="n">transformingVector</span><span class="o">)</span>
<span class="c1">// Batch transform and per-row transform give the same results:</span>
<span class="k">val</span> <span class="n">transformedData</span> <span class="k">=</span> <span class="n">transformer</span><span class="o">.</span><span class="n">transform</span><span class="o">(</span><span class="n">data</span><span class="o">)</span>
<span class="k">val</span> <span class="n">transformedData2</span> <span class="k">=</span> <span class="n">data</span><span class="o">.</span><span class="n">map</span><span class="o">(</span><span class="n">x</span> <span class="k">=&gt;</span> <span class="n">transformer</span><span class="o">.</span><span class="n">transform</span><span class="o">(</span><span class="n">x</span><span class="o">))</span>
</pre></div>
<div><small>Find full example code at "examples/src/main/scala/org/apache/spark/examples/mllib/ElementwiseProductExample.scala" in the Spark repo.</small></div>
</div>
<div data-lang="java">
<p>Refer to the <a href="api/java/org/apache/spark/mllib/feature/ElementwiseProduct.html"><code>ElementwiseProduct</code> Java docs</a> for details on the API.</p>
<div class="highlight"><pre><span class="kn">import</span> <span class="nn">java.util.Arrays</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.api.java.function.Function</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.feature.ElementwiseProduct</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.Vector</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.Vectors</span><span class="o">;</span>
<span class="c1">// Create some vector data; also works for sparse vectors</span>
<span class="n">JavaRDD</span><span class="o">&lt;</span><span class="n">Vector</span><span class="o">&gt;</span> <span class="n">data</span> <span class="o">=</span> <span class="n">jsc</span><span class="o">.</span><span class="na">parallelize</span><span class="o">(</span><span class="n">Arrays</span><span class="o">.</span><span class="na">asList</span><span class="o">(</span>
<span class="n">Vectors</span><span class="o">.</span><span class="na">dense</span><span class="o">(</span><span class="mf">1.0</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="n">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">6.0</span><span class="o">)));</span>
<span class="n">Vector</span> <span class="n">transformingVector</span> <span class="o">=</span> <span class="n">Vectors</span><span class="o">.</span><span class="na">dense</span><span class="o">(</span><span class="mf">0.0</span><span class="o">,</span> <span class="mf">1.0</span><span class="o">,</span> <span class="mf">2.0</span><span class="o">);</span>
<span class="kd">final</span> <span class="n">ElementwiseProduct</span> <span class="n">transformer</span> <span class="o">=</span> <span class="k">new</span> <span class="nf">ElementwiseProduct</span><span class="o">(</span><span class="n">transformingVector</span><span class="o">);</span>
<span class="c1">// Batch transform and per-row transform give the same results:</span>
<span class="n">JavaRDD</span><span class="o">&lt;</span><span class="n">Vector</span><span class="o">&gt;</span> <span class="n">transformedData</span> <span class="o">=</span> <span class="n">transformer</span><span class="o">.</span><span class="na">transform</span><span class="o">(</span><span class="n">data</span><span class="o">);</span>
<span class="n">JavaRDD</span><span class="o">&lt;</span><span class="n">Vector</span><span class="o">&gt;</span> <span class="n">transformedData2</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="k">new</span> <span class="n">Function</span><span class="o">&lt;</span><span class="n">Vector</span><span class="o">,</span> <span class="n">Vector</span><span class="o">&gt;()</span> <span class="o">{</span>
<span class="nd">@Override</span>
<span class="kd">public</span> <span class="n">Vector</span> <span class="nf">call</span><span class="o">(</span><span class="n">Vector</span> <span class="n">v</span><span class="o">)</span> <span class="o">{</span>
<span class="k">return</span> <span class="n">transformer</span><span class="o">.</span><span class="na">transform</span><span class="o">(</span><span class="n">v</span><span class="o">);</span>
<span class="o">}</span>
<span class="o">}</span>
<span class="o">);</span>
</pre></div>
<div><small>Find full example code at "examples/src/main/java/org/apache/spark/examples/mllib/JavaElementwiseProductExample.java" in the Spark repo.</small></div>
</div>
<div data-lang="python">
<p>Refer to the <a href="api/python/pyspark.mllib.html#pyspark.mllib.feature.ElementwiseProduct"><code>ElementwiseProduct</code> Python docs</a> for more details on the API.</p>
<div class="highlight"><pre><span class="kn">from</span> <span class="nn">pyspark.mllib.feature</span> <span class="kn">import</span> <span class="n">ElementwiseProduct</span>
<span class="kn">from</span> <span class="nn">pyspark.mllib.linalg</span> <span class="kn">import</span> <span class="n">Vectors</span>
<span class="n">data</span> <span class="o">=</span> <span class="n">sc</span><span class="o">.</span><span class="n">textFile</span><span class="p">(</span><span class="s">&quot;data/mllib/kmeans_data.txt&quot;</span><span class="p">)</span>
<span class="n">parsedData</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="n">map</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="p">[</span><span class="nb">float</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">x</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">&quot; &quot;</span><span class="p">)])</span>
<span class="c"># Create weight vector.</span>
<span class="n">transformingVector</span> <span class="o">=</span> <span class="n">Vectors</span><span class="o">.</span><span class="n">dense</span><span class="p">([</span><span class="mf">0.0</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="n">transformer</span> <span class="o">=</span> <span class="n">ElementwiseProduct</span><span class="p">(</span><span class="n">transformingVector</span><span class="p">)</span>
<span class="c"># Batch transform</span>
<span class="n">transformedData</span> <span class="o">=</span> <span class="n">transformer</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">parsedData</span><span class="p">)</span>
<span class="c"># Single-row transform</span>
<span class="n">transformedData2</span> <span class="o">=</span> <span class="n">transformer</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">parsedData</span><span class="o">.</span><span class="n">first</span><span class="p">())</span>
</pre></div>
<div><small>Find full example code at "examples/src/main/python/mllib/elementwise_product_example.py" in the Spark repo.</small></div>
</div>
</div>
<h2 id="pca">PCA</h2>
<p>A feature transformer that projects vectors to a low-dimensional space using PCA.
Details you can read at <a href="mllib-dimensionality-reduction.html">dimensionality reduction</a>.</p>
<h3 id="example-5">Example</h3>
<p>The following code demonstrates how to compute principal components on a <code>Vector</code>
and use them to project the vectors into a low-dimensional space while keeping associated labels
for calculation a <a href="mllib-linear-methods.html">Linear Regression</a></p>
<div class="codetabs">
<div data-lang="scala">
<p>Refer to the <a href="api/scala/index.html#org.apache.spark.mllib.feature.PCA"><code>PCA</code> Scala docs</a> for details on the API.</p>
<div class="highlight"><pre><span class="k">import</span> <span class="nn">org.apache.spark.mllib.feature.PCA</span>
<span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg.Vectors</span>
<span class="k">import</span> <span class="nn">org.apache.spark.mllib.regression.</span><span class="o">{</span><span class="nc">LabeledPoint</span><span class="o">,</span> <span class="nc">LinearRegressionWithSGD</span><span class="o">}</span>
<span class="k">val</span> <span class="n">data</span> <span class="k">=</span> <span class="n">sc</span><span class="o">.</span><span class="n">textFile</span><span class="o">(</span><span class="s">&quot;data/mllib/ridge-data/lpsa.data&quot;</span><span class="o">).</span><span class="n">map</span> <span class="o">{</span> <span class="n">line</span> <span class="k">=&gt;</span>
<span class="k">val</span> <span class="n">parts</span> <span class="k">=</span> <span class="n">line</span><span class="o">.</span><span class="n">split</span><span class="o">(</span><span class="sc">&#39;,&#39;</span><span class="o">)</span>
<span class="nc">LabeledPoint</span><span class="o">(</span><span class="n">parts</span><span class="o">(</span><span class="mi">0</span><span class="o">).</span><span class="n">toDouble</span><span class="o">,</span> <span class="nc">Vectors</span><span class="o">.</span><span class="n">dense</span><span class="o">(</span><span class="n">parts</span><span class="o">(</span><span class="mi">1</span><span class="o">).</span><span class="n">split</span><span class="o">(</span><span class="sc">&#39; &#39;</span><span class="o">).</span><span class="n">map</span><span class="o">(</span><span class="k">_</span><span class="o">.</span><span class="n">toDouble</span><span class="o">)))</span>
<span class="o">}.</span><span class="n">cache</span><span class="o">()</span>
<span class="k">val</span> <span class="n">splits</span> <span class="k">=</span> <span class="n">data</span><span class="o">.</span><span class="n">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="n">training</span> <span class="k">=</span> <span class="n">splits</span><span class="o">(</span><span class="mi">0</span><span class="o">).</span><span class="n">cache</span><span class="o">()</span>
<span class="k">val</span> <span class="n">test</span> <span class="k">=</span> <span class="n">splits</span><span class="o">(</span><span class="mi">1</span><span class="o">)</span>
<span class="k">val</span> <span class="n">pca</span> <span class="k">=</span> <span class="k">new</span> <span class="nc">PCA</span><span class="o">(</span><span class="n">training</span><span class="o">.</span><span class="n">first</span><span class="o">().</span><span class="n">features</span><span class="o">.</span><span class="n">size</span> <span class="o">/</span> <span class="mi">2</span><span class="o">).</span><span class="n">fit</span><span class="o">(</span><span class="n">data</span><span class="o">.</span><span class="n">map</span><span class="o">(</span><span class="k">_</span><span class="o">.</span><span class="n">features</span><span class="o">))</span>
<span class="k">val</span> <span class="n">training_pca</span> <span class="k">=</span> <span class="n">training</span><span class="o">.</span><span class="n">map</span><span class="o">(</span><span class="n">p</span> <span class="k">=&gt;</span> <span class="n">p</span><span class="o">.</span><span class="n">copy</span><span class="o">(</span><span class="n">features</span> <span class="k">=</span> <span class="n">pca</span><span class="o">.</span><span class="n">transform</span><span class="o">(</span><span class="n">p</span><span class="o">.</span><span class="n">features</span><span class="o">)))</span>
<span class="k">val</span> <span class="n">test_pca</span> <span class="k">=</span> <span class="n">test</span><span class="o">.</span><span class="n">map</span><span class="o">(</span><span class="n">p</span> <span class="k">=&gt;</span> <span class="n">p</span><span class="o">.</span><span class="n">copy</span><span class="o">(</span><span class="n">features</span> <span class="k">=</span> <span class="n">pca</span><span class="o">.</span><span class="n">transform</span><span class="o">(</span><span class="n">p</span><span class="o">.</span><span class="n">features</span><span class="o">)))</span>
<span class="k">val</span> <span class="n">numIterations</span> <span class="k">=</span> <span class="mi">100</span>
<span class="k">val</span> <span class="n">model</span> <span class="k">=</span> <span class="nc">LinearRegressionWithSGD</span><span class="o">.</span><span class="n">train</span><span class="o">(</span><span class="n">training</span><span class="o">,</span> <span class="n">numIterations</span><span class="o">)</span>
<span class="k">val</span> <span class="n">model_pca</span> <span class="k">=</span> <span class="nc">LinearRegressionWithSGD</span><span class="o">.</span><span class="n">train</span><span class="o">(</span><span class="n">training_pca</span><span class="o">,</span> <span class="n">numIterations</span><span class="o">)</span>
<span class="k">val</span> <span class="n">valuesAndPreds</span> <span class="k">=</span> <span class="n">test</span><span class="o">.</span><span class="n">map</span> <span class="o">{</span> <span class="n">point</span> <span class="k">=&gt;</span>
<span class="k">val</span> <span class="n">score</span> <span class="k">=</span> <span class="n">model</span><span class="o">.</span><span class="n">predict</span><span class="o">(</span><span class="n">point</span><span class="o">.</span><span class="n">features</span><span class="o">)</span>
<span class="o">(</span><span class="n">score</span><span class="o">,</span> <span class="n">point</span><span class="o">.</span><span class="n">label</span><span class="o">)</span>
<span class="o">}</span>
<span class="k">val</span> <span class="n">valuesAndPreds_pca</span> <span class="k">=</span> <span class="n">test_pca</span><span class="o">.</span><span class="n">map</span> <span class="o">{</span> <span class="n">point</span> <span class="k">=&gt;</span>
<span class="k">val</span> <span class="n">score</span> <span class="k">=</span> <span class="n">model_pca</span><span class="o">.</span><span class="n">predict</span><span class="o">(</span><span class="n">point</span><span class="o">.</span><span class="n">features</span><span class="o">)</span>
<span class="o">(</span><span class="n">score</span><span class="o">,</span> <span class="n">point</span><span class="o">.</span><span class="n">label</span><span class="o">)</span>
<span class="o">}</span>
<span class="k">val</span> <span class="nc">MSE</span> <span class="k">=</span> <span class="n">valuesAndPreds</span><span class="o">.</span><span class="n">map</span> <span class="o">{</span> <span class="k">case</span> <span class="o">(</span><span class="n">v</span><span class="o">,</span> <span class="n">p</span><span class="o">)</span> <span class="k">=&gt;</span> <span class="n">math</span><span class="o">.</span><span class="n">pow</span><span class="o">((</span><span class="n">v</span> <span class="o">-</span> <span class="n">p</span><span class="o">),</span> <span class="mi">2</span><span class="o">)</span> <span class="o">}.</span><span class="n">mean</span><span class="o">()</span>
<span class="k">val</span> <span class="nc">MSE_pca</span> <span class="k">=</span> <span class="n">valuesAndPreds_pca</span><span class="o">.</span><span class="n">map</span> <span class="o">{</span> <span class="k">case</span> <span class="o">(</span><span class="n">v</span><span class="o">,</span> <span class="n">p</span><span class="o">)</span> <span class="k">=&gt;</span> <span class="n">math</span><span class="o">.</span><span class="n">pow</span><span class="o">((</span><span class="n">v</span> <span class="o">-</span> <span class="n">p</span><span class="o">),</span> <span class="mi">2</span><span class="o">)</span> <span class="o">}.</span><span class="n">mean</span><span class="o">()</span>
<span class="n">println</span><span class="o">(</span><span class="s">&quot;Mean Squared Error = &quot;</span> <span class="o">+</span> <span class="nc">MSE</span><span class="o">)</span>
<span class="n">println</span><span class="o">(</span><span class="s">&quot;PCA Mean Squared Error = &quot;</span> <span class="o">+</span> <span class="nc">MSE_pca</span><span class="o">)</span>
</pre></div>
<div><small>Find full example code at "examples/src/main/scala/org/apache/spark/examples/mllib/PCAExample.scala" in the Spark repo.</small></div>
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