docs/mllib-statistics.md - spark - Git at Google

 ---
 layout: global
 title: Basic Statistics - RDD-based API
 displayTitle: Basic Statistics - RDD-based API
 license: |
   Licensed to the Apache Software Foundation (ASF) under one or more
   contributor license agreements.  See the NOTICE file distributed with
   this work for additional information regarding copyright ownership.
   The ASF licenses this file to You under the Apache License, Version 2.0
   (the "License"); you may not use this file except in compliance with
   the License.  You may obtain a copy of the License at

      http://www.apache.org/licenses/LICENSE-2.0

   Unless required by applicable law or agreed to in writing, software
   distributed under the License is distributed on an "AS IS" BASIS,
   WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
   See the License for the specific language governing permissions and
   limitations under the License.
 ---

 * Table of contents
 {:toc}


 `\[
 \newcommand{\R}{\mathbb{R}}
 \newcommand{\E}{\mathbb{E}}
 \newcommand{\x}{\mathbf{x}}
 \newcommand{\y}{\mathbf{y}}
 \newcommand{\wv}{\mathbf{w}}
 \newcommand{\av}{\mathbf{\alpha}}
 \newcommand{\bv}{\mathbf{b}}
 \newcommand{\N}{\mathbb{N}}
 \newcommand{\id}{\mathbf{I}}
 \newcommand{\ind}{\mathbf{1}}
 \newcommand{\0}{\mathbf{0}}
 \newcommand{\unit}{\mathbf{e}}
 \newcommand{\one}{\mathbf{1}}
 \newcommand{\zero}{\mathbf{0}}
 \]`

 ## Summary statistics

 We provide column summary statistics for `RDD[Vector]` through the function `colStats`
 available in `Statistics`.

 <div class="codetabs">

 <div data-lang="python" markdown="1">
 [`colStats()`](api/python/reference/api/pyspark.mllib.stat.Statistics.html#pyspark.mllib.stat.Statistics.colStats) returns an instance of
 [`MultivariateStatisticalSummary`](api/python/reference/api/pyspark.mllib.stat.MultivariateStatisticalSummary.html),
 which contains the column-wise max, min, mean, variance, and number of nonzeros, as well as the
 total count.

 Refer to the [`MultivariateStatisticalSummary` Python docs](api/python/reference/api/pyspark.mllib.stat.MultivariateStatisticalSummary.html) for more details on the API.

 {% include_example python/mllib/summary_statistics_example.py %}
 </div>

 <div data-lang="scala" markdown="1">

 [`colStats()`](api/scala/org/apache/spark/mllib/stat/Statistics$.html) returns an instance of
 [`MultivariateStatisticalSummary`](api/scala/org/apache/spark/mllib/stat/MultivariateStatisticalSummary.html),
 which contains the column-wise max, min, mean, variance, and number of nonzeros, as well as the
 total count.

 Refer to the [`MultivariateStatisticalSummary` Scala docs](api/scala/org/apache/spark/mllib/stat/MultivariateStatisticalSummary.html) for details on the API.

 {% include_example scala/org/apache/spark/examples/mllib/SummaryStatisticsExample.scala %}
 </div>

 <div data-lang="java" markdown="1">

 [`colStats()`](api/java/org/apache/spark/mllib/stat/Statistics.html) returns an instance of
 [`MultivariateStatisticalSummary`](api/java/org/apache/spark/mllib/stat/MultivariateStatisticalSummary.html),
 which contains the column-wise max, min, mean, variance, and number of nonzeros, as well as the
 total count.

 Refer to the [`MultivariateStatisticalSummary` Java docs](api/java/org/apache/spark/mllib/stat/MultivariateStatisticalSummary.html) for details on the API.

 {% include_example java/org/apache/spark/examples/mllib/JavaSummaryStatisticsExample.java %}
 </div>

 </div>

 ## Correlations

 Calculating the correlation between two series of data is a common operation in Statistics. In `spark.mllib`
 we provide the flexibility to calculate pairwise correlations among many series. The supported
 correlation methods are currently Pearson's and Spearman's correlation.

 <div class="codetabs">

 <div data-lang="python" markdown="1">
 [`Statistics`](api/python/reference/api/pyspark.mllib.stat.Statistics.html) provides methods to
 calculate correlations between series. Depending on the type of input, two `RDD[Double]`s or
 an `RDD[Vector]`, the output will be a `Double` or the correlation `Matrix` respectively.

 Refer to the [`Statistics` Python docs](api/python/reference/api/pyspark.mllib.stat.Statistics.html) for more details on the API.

 {% include_example python/mllib/correlations_example.py %}
 </div>

 <div data-lang="scala" markdown="1">
 [`Statistics`](api/scala/org/apache/spark/mllib/stat/Statistics$.html) provides methods to
 calculate correlations between series. Depending on the type of input, two `RDD[Double]`s or
 an `RDD[Vector]`, the output will be a `Double` or the correlation `Matrix` respectively.

 Refer to the [`Statistics` Scala docs](api/scala/org/apache/spark/mllib/stat/Statistics$.html) for details on the API.

 {% include_example scala/org/apache/spark/examples/mllib/CorrelationsExample.scala %}
 </div>

 <div data-lang="java" markdown="1">
 [`Statistics`](api/java/org/apache/spark/mllib/stat/Statistics.html) provides methods to
 calculate correlations between series. Depending on the type of input, two `JavaDoubleRDD`s or
 a `JavaRDD<Vector>`, the output will be a `Double` or the correlation `Matrix` respectively.

 Refer to the [`Statistics` Java docs](api/java/org/apache/spark/mllib/stat/Statistics.html) for details on the API.

 {% include_example java/org/apache/spark/examples/mllib/JavaCorrelationsExample.java %}
 </div>

 </div>

 ## Stratified sampling

 Unlike the other statistics functions, which reside in `spark.mllib`, stratified sampling methods,
 `sampleByKey` and `sampleByKeyExact`, can be performed on RDD's of key-value pairs. For stratified
 sampling, the keys can be thought of as a label and the value as a specific attribute. For example
 the key can be man or woman, or document ids, and the respective values can be the list of ages
 of the people in the population or the list of words in the documents. The `sampleByKey` method
 will flip a coin to decide whether an observation will be sampled or not, therefore requires one
 pass over the data, and provides an *expected* sample size. `sampleByKeyExact` requires significant
 more resources than the per-stratum simple random sampling used in `sampleByKey`, but will provide
 the exact sampling size with 99.99% confidence. `sampleByKeyExact` is currently not supported in
 python.

 <div class="codetabs">

 <div data-lang="python" markdown="1">
 [`sampleByKey()`](api/python/reference/api/pyspark.RDD.sampleByKey.html#pyspark.RDD.sampleByKey) allows users to
 sample approximately $\lceil f_k \cdot n_k \rceil \, \forall k \in K$ items, where $f_k$ is the
 desired fraction for key $k$, $n_k$ is the number of key-value pairs for key $k$, and $K$ is the
 set of keys.

 *Note:* `sampleByKeyExact()` is currently not supported in Python.

 {% include_example python/mllib/stratified_sampling_example.py %}
 </div>

 <div data-lang="scala" markdown="1">
 [`sampleByKeyExact()`](api/scala/org/apache/spark/rdd/PairRDDFunctions.html) allows users to
 sample exactly $\lceil f_k \cdot n_k \rceil \, \forall k \in K$ items, where $f_k$ is the desired
 fraction for key $k$, $n_k$ is the number of key-value pairs for key $k$, and $K$ is the set of
 keys. Sampling without replacement requires one additional pass over the RDD to guarantee sample
 size, whereas sampling with replacement requires two additional passes.

 {% include_example scala/org/apache/spark/examples/mllib/StratifiedSamplingExample.scala %}
 </div>

 <div data-lang="java" markdown="1">
 [`sampleByKeyExact()`](api/java/org/apache/spark/api/java/JavaPairRDD.html) allows users to
 sample exactly $\lceil f_k \cdot n_k \rceil \, \forall k \in K$ items, where $f_k$ is the desired
 fraction for key $k$, $n_k$ is the number of key-value pairs for key $k$, and $K$ is the set of
 keys. Sampling without replacement requires one additional pass over the RDD to guarantee sample
 size, whereas sampling with replacement requires two additional passes.

 {% include_example java/org/apache/spark/examples/mllib/JavaStratifiedSamplingExample.java %}
 </div>

 </div>

 ## Hypothesis testing

 Hypothesis testing is a powerful tool in statistics to determine whether a result is statistically
 significant, whether this result occurred by chance or not. `spark.mllib` currently supports Pearson's
 chi-squared ( $\chi^2$) tests for goodness of fit and independence. The input data types determine
 whether the goodness of fit or the independence test is conducted. The goodness of fit test requires
 an input type of `Vector`, whereas the independence test requires a `Matrix` as input.

 `spark.mllib` also supports the input type `RDD[LabeledPoint]` to enable feature selection via chi-squared
 independence tests.

 <div class="codetabs">

 <div data-lang="python" markdown="1">
 [`Statistics`](api/python/reference/api/pyspark.mllib.stat.Statistics.html) provides methods to
 run Pearson's chi-squared tests. The following example demonstrates how to run and interpret
 hypothesis tests.

 Refer to the [`Statistics` Python docs](api/python/reference/api/pyspark.mllib.stat.Statistics.html) for more details on the API.

 {% include_example python/mllib/hypothesis_testing_example.py %}
 </div>

 <div data-lang="scala" markdown="1">
 [`Statistics`](api/scala/org/apache/spark/mllib/stat/Statistics$.html) provides methods to
 run Pearson's chi-squared tests. The following example demonstrates how to run and interpret
 hypothesis tests.

 {% include_example scala/org/apache/spark/examples/mllib/HypothesisTestingExample.scala %}
 </div>

 <div data-lang="java" markdown="1">
 [`Statistics`](api/java/org/apache/spark/mllib/stat/Statistics.html) provides methods to
 run Pearson's chi-squared tests. The following example demonstrates how to run and interpret
 hypothesis tests.

 Refer to the [`ChiSqTestResult` Java docs](api/java/org/apache/spark/mllib/stat/test/ChiSqTestResult.html) for details on the API.

 {% include_example java/org/apache/spark/examples/mllib/JavaHypothesisTestingExample.java %}
 </div>

 </div>

 Additionally, `spark.mllib` provides a 1-sample, 2-sided implementation of the Kolmogorov-Smirnov (KS) test
 for equality of probability distributions. By providing the name of a theoretical distribution
 (currently solely supported for the normal distribution) and its parameters, or a function to
 calculate the cumulative distribution according to a given theoretical distribution, the user can
 test the null hypothesis that their sample is drawn from that distribution. In the case that the
 user tests against the normal distribution (`distName="norm"`), but does not provide distribution
 parameters, the test initializes to the standard normal distribution and logs an appropriate
 message.

 <div class="codetabs">

 <div data-lang="python" markdown="1">
 [`Statistics`](api/python/reference/api/pyspark.mllib.stat.Statistics.html) provides methods to
 run a 1-sample, 2-sided Kolmogorov-Smirnov test. The following example demonstrates how to run
 and interpret the hypothesis tests.

 Refer to the [`Statistics` Python docs](api/python/reference/api/pyspark.mllib.stat.Statistics.html) for more details on the API.

 {% include_example python/mllib/hypothesis_testing_kolmogorov_smirnov_test_example.py %}
 </div>

 <div data-lang="scala" markdown="1">
 [`Statistics`](api/scala/org/apache/spark/mllib/stat/Statistics$.html) provides methods to
 run a 1-sample, 2-sided Kolmogorov-Smirnov test. The following example demonstrates how to run
 and interpret the hypothesis tests.

 Refer to the [`Statistics` Scala docs](api/scala/org/apache/spark/mllib/stat/Statistics$.html) for details on the API.

 {% include_example scala/org/apache/spark/examples/mllib/HypothesisTestingKolmogorovSmirnovTestExample.scala %}
 </div>

 <div data-lang="java" markdown="1">
 [`Statistics`](api/java/org/apache/spark/mllib/stat/Statistics.html) provides methods to
 run a 1-sample, 2-sided Kolmogorov-Smirnov test. The following example demonstrates how to run
 and interpret the hypothesis tests.

 Refer to the [`Statistics` Java docs](api/java/org/apache/spark/mllib/stat/Statistics.html) for details on the API.

 {% include_example java/org/apache/spark/examples/mllib/JavaHypothesisTestingKolmogorovSmirnovTestExample.java %}
 </div>

 </div>

 ### Streaming Significance Testing
 `spark.mllib` provides online implementations of some tests to support use cases
 like A/B testing. These tests may be performed on a Spark Streaming
 `DStream[(Boolean, Double)]` where the first element of each tuple
 indicates control group (`false`) or treatment group (`true`) and the
 second element is the value of an observation.

 Streaming significance testing supports the following parameters:

 * `peacePeriod` - The number of initial data points from the stream to
 ignore, used to mitigate novelty effects.
 * `windowSize` - The number of past batches to perform hypothesis
 testing over. Setting to `0` will perform cumulative processing using
 all prior batches.


 <div class="codetabs">
 <div data-lang="scala" markdown="1">
 [`StreamingTest`](api/scala/org/apache/spark/mllib/stat/test/StreamingTest.html)
 provides streaming hypothesis testing.

 {% include_example scala/org/apache/spark/examples/mllib/StreamingTestExample.scala %}
 </div>

 <div data-lang="java" markdown="1">
 [`StreamingTest`](api/java/index.html#org.apache.spark.mllib.stat.test.StreamingTest)
 provides streaming hypothesis testing.

 {% include_example java/org/apache/spark/examples/mllib/JavaStreamingTestExample.java %}
 </div>
 </div>


 ## Random data generation

 Random data generation is useful for randomized algorithms, prototyping, and performance testing.
 `spark.mllib` supports generating random RDDs with i.i.d. values drawn from a given distribution:
 uniform, standard normal, or Poisson.

 <div class="codetabs">

 <div data-lang="python" markdown="1">
 [`RandomRDDs`](api/python/reference/api/pyspark.mllib.random.RandomRDDs.html) provides factory
 methods to generate random double RDDs or vector RDDs.
 The following example generates a random double RDD, whose values follows the standard normal
 distribution `N(0, 1)`, and then map it to `N(1, 4)`.

 Refer to the [`RandomRDDs` Python docs](api/python/reference/api/pyspark.mllib.random.RandomRDDs.html) for more details on the API.

 {% highlight python %}
 from pyspark.mllib.random import RandomRDDs

 sc = ... # SparkContext

 # Generate a random double RDD that contains 1 million i.i.d. values drawn from the
 # standard normal distribution `N(0, 1)`, evenly distributed in 10 partitions.
 u = RandomRDDs.normalRDD(sc, 1000000L, 10)
 # Apply a transform to get a random double RDD following `N(1, 4)`.
 v = u.map(lambda x: 1.0 + 2.0 * x)
 {% endhighlight %}
 </div>

 <div data-lang="scala" markdown="1">
 [`RandomRDDs`](api/scala/org/apache/spark/mllib/random/RandomRDDs$.html) provides factory
 methods to generate random double RDDs or vector RDDs.
 The following example generates a random double RDD, whose values follows the standard normal
 distribution `N(0, 1)`, and then map it to `N(1, 4)`.

 Refer to the [`RandomRDDs` Scala docs](api/scala/org/apache/spark/mllib/random/RandomRDDs$.html) for details on the API.

 {% highlight scala %}
 import org.apache.spark.SparkContext
 import org.apache.spark.mllib.random.RandomRDDs._

 val sc: SparkContext = ...

 // Generate a random double RDD that contains 1 million i.i.d. values drawn from the
 // standard normal distribution `N(0, 1)`, evenly distributed in 10 partitions.
 val u = normalRDD(sc, 1000000L, 10)
 // Apply a transform to get a random double RDD following `N(1, 4)`.
 val v = u.map(x => 1.0 + 2.0 * x)
 {% endhighlight %}
 </div>

 <div data-lang="java" markdown="1">
 [`RandomRDDs`](api/java/index.html#org.apache.spark.mllib.random.RandomRDDs) provides factory
 methods to generate random double RDDs or vector RDDs.
 The following example generates a random double RDD, whose values follows the standard normal
 distribution `N(0, 1)`, and then map it to `N(1, 4)`.

 Refer to the [`RandomRDDs` Java docs](api/java/org/apache/spark/mllib/random/RandomRDDs) for details on the API.

 {% highlight java %}
 import org.apache.spark.SparkContext;
 import org.apache.spark.api.JavaDoubleRDD;
 import static org.apache.spark.mllib.random.RandomRDDs.*;

 JavaSparkContext jsc = ...

 // Generate a random double RDD that contains 1 million i.i.d. values drawn from the
 // standard normal distribution `N(0, 1)`, evenly distributed in 10 partitions.
 JavaDoubleRDD u = normalJavaRDD(jsc, 1000000L, 10);
 // Apply a transform to get a random double RDD following `N(1, 4)`.
 JavaDoubleRDD v = u.mapToDouble(x -> 1.0 + 2.0 * x);
 {% endhighlight %}
 </div>

 </div>

 ## Kernel density estimation

 [Kernel density estimation](https://en.wikipedia.org/wiki/Kernel_density_estimation) is a technique
 useful for visualizing empirical probability distributions without requiring assumptions about the
 particular distribution that the observed samples are drawn from. It computes an estimate of the
 probability density function of a random variables, evaluated at a given set of points. It achieves
 this estimate by expressing the PDF of the empirical distribution at a particular point as the
 mean of PDFs of normal distributions centered around each of the samples.

 <div class="codetabs">

 <div data-lang="python" markdown="1">
 [`KernelDensity`](api/python/reference/api/pyspark.mllib.stat.KernelDensity.html) provides methods
 to compute kernel density estimates from an RDD of samples. The following example demonstrates how
 to do so.

 Refer to the [`KernelDensity` Python docs](api/python/reference/api/pyspark.mllib.stat.KernelDensity.html) for more details on the API.

 {% include_example python/mllib/kernel_density_estimation_example.py %}
 </div>

 <div data-lang="scala" markdown="1">
 [`KernelDensity`](api/scala/org/apache/spark/mllib/stat/KernelDensity.html) provides methods
 to compute kernel density estimates from an RDD of samples. The following example demonstrates how
 to do so.

 Refer to the [`KernelDensity` Scala docs](api/scala/org/apache/spark/mllib/stat/KernelDensity.html) for details on the API.

 {% include_example scala/org/apache/spark/examples/mllib/KernelDensityEstimationExample.scala %}
 </div>

 <div data-lang="java" markdown="1">
 [`KernelDensity`](api/java/index.html#org.apache.spark.mllib.stat.KernelDensity) provides methods
 to compute kernel density estimates from an RDD of samples. The following example demonstrates how
 to do so.

 Refer to the [`KernelDensity` Java docs](api/java/org/apache/spark/mllib/stat/KernelDensity.html) for details on the API.

 {% include_example java/org/apache/spark/examples/mllib/JavaKernelDensityEstimationExample.java %}
 </div>

 </div>
	---
	layout: global
	title: Basic Statistics - RDD-based API
	displayTitle: Basic Statistics - RDD-based API
	license: \|
	Licensed to the Apache Software Foundation (ASF) under one or more
	contributor license agreements. See the NOTICE file distributed with
	this work for additional information regarding copyright ownership.
	The ASF licenses this file to You under the Apache License, Version 2.0
	(the "License"); you may not use this file except in compliance with
	the License. You may obtain a copy of the License at

	http://www.apache.org/licenses/LICENSE-2.0

	Unless required by applicable law or agreed to in writing, software
	distributed under the License is distributed on an "AS IS" BASIS,
	WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
	See the License for the specific language governing permissions and
	limitations under the License.
	---

	* Table of contents
	{:toc}


	`\[
	\newcommand{\R}{\mathbb{R}}
	\newcommand{\E}{\mathbb{E}}
	\newcommand{\x}{\mathbf{x}}
	\newcommand{\y}{\mathbf{y}}
	\newcommand{\wv}{\mathbf{w}}
	\newcommand{\av}{\mathbf{\alpha}}
	\newcommand{\bv}{\mathbf{b}}
	\newcommand{\N}{\mathbb{N}}
	\newcommand{\id}{\mathbf{I}}
	\newcommand{\ind}{\mathbf{1}}
	\newcommand{\0}{\mathbf{0}}
	\newcommand{\unit}{\mathbf{e}}
	\newcommand{\one}{\mathbf{1}}
	\newcommand{\zero}{\mathbf{0}}
	\]`

	## Summary statistics

	We provide column summary statistics for `RDD[Vector]` through the function `colStats`
	available in `Statistics`.

	<div class="codetabs">

	<div data-lang="python" markdown="1">
	[`colStats()`](api/python/reference/api/pyspark.mllib.stat.Statistics.html#pyspark.mllib.stat.Statistics.colStats) returns an instance of
	[`MultivariateStatisticalSummary`](api/python/reference/api/pyspark.mllib.stat.MultivariateStatisticalSummary.html),
	which contains the column-wise max, min, mean, variance, and number of nonzeros, as well as the
	total count.

	Refer to the [`MultivariateStatisticalSummary` Python docs](api/python/reference/api/pyspark.mllib.stat.MultivariateStatisticalSummary.html) for more details on the API.

	{% include_example python/mllib/summary_statistics_example.py %}
	</div>

	<div data-lang="scala" markdown="1">

	[`colStats()`](api/scala/org/apache/spark/mllib/stat/Statistics$.html) returns an instance of
	[`MultivariateStatisticalSummary`](api/scala/org/apache/spark/mllib/stat/MultivariateStatisticalSummary.html),
	which contains the column-wise max, min, mean, variance, and number of nonzeros, as well as the
	total count.

	Refer to the [`MultivariateStatisticalSummary` Scala docs](api/scala/org/apache/spark/mllib/stat/MultivariateStatisticalSummary.html) for details on the API.

	{% include_example scala/org/apache/spark/examples/mllib/SummaryStatisticsExample.scala %}
	</div>

	<div data-lang="java" markdown="1">

	[`colStats()`](api/java/org/apache/spark/mllib/stat/Statistics.html) returns an instance of
	[`MultivariateStatisticalSummary`](api/java/org/apache/spark/mllib/stat/MultivariateStatisticalSummary.html),
	which contains the column-wise max, min, mean, variance, and number of nonzeros, as well as the
	total count.

	Refer to the [`MultivariateStatisticalSummary` Java docs](api/java/org/apache/spark/mllib/stat/MultivariateStatisticalSummary.html) for details on the API.

	{% include_example java/org/apache/spark/examples/mllib/JavaSummaryStatisticsExample.java %}
	</div>

	</div>

	## Correlations

	Calculating the correlation between two series of data is a common operation in Statistics. In `spark.mllib`
	we provide the flexibility to calculate pairwise correlations among many series. The supported
	correlation methods are currently Pearson's and Spearman's correlation.

	<div class="codetabs">

	<div data-lang="python" markdown="1">
	[`Statistics`](api/python/reference/api/pyspark.mllib.stat.Statistics.html) provides methods to
	calculate correlations between series. Depending on the type of input, two `RDD[Double]`s or
	an `RDD[Vector]`, the output will be a `Double` or the correlation `Matrix` respectively.

	Refer to the [`Statistics` Python docs](api/python/reference/api/pyspark.mllib.stat.Statistics.html) for more details on the API.

	{% include_example python/mllib/correlations_example.py %}
	</div>

	<div data-lang="scala" markdown="1">
	[`Statistics`](api/scala/org/apache/spark/mllib/stat/Statistics$.html) provides methods to
	calculate correlations between series. Depending on the type of input, two `RDD[Double]`s or
	an `RDD[Vector]`, the output will be a `Double` or the correlation `Matrix` respectively.

	Refer to the [`Statistics` Scala docs](api/scala/org/apache/spark/mllib/stat/Statistics$.html) for details on the API.

	{% include_example scala/org/apache/spark/examples/mllib/CorrelationsExample.scala %}
	</div>

	<div data-lang="java" markdown="1">
	[`Statistics`](api/java/org/apache/spark/mllib/stat/Statistics.html) provides methods to
	calculate correlations between series. Depending on the type of input, two `JavaDoubleRDD`s or
	a `JavaRDD<Vector>`, the output will be a `Double` or the correlation `Matrix` respectively.

	Refer to the [`Statistics` Java docs](api/java/org/apache/spark/mllib/stat/Statistics.html) for details on the API.

	{% include_example java/org/apache/spark/examples/mllib/JavaCorrelationsExample.java %}
	</div>

	</div>

	## Stratified sampling

	Unlike the other statistics functions, which reside in `spark.mllib`, stratified sampling methods,
	`sampleByKey` and `sampleByKeyExact`, can be performed on RDD's of key-value pairs. For stratified
	sampling, the keys can be thought of as a label and the value as a specific attribute. For example
	the key can be man or woman, or document ids, and the respective values can be the list of ages
	of the people in the population or the list of words in the documents. The `sampleByKey` method
	will flip a coin to decide whether an observation will be sampled or not, therefore requires one
	pass over the data, and provides an expected sample size. `sampleByKeyExact` requires significant
	more resources than the per-stratum simple random sampling used in `sampleByKey`, but will provide
	the exact sampling size with 99.99% confidence. `sampleByKeyExact` is currently not supported in
	python.

	<div class="codetabs">

	<div data-lang="python" markdown="1">
	[`sampleByKey()`](api/python/reference/api/pyspark.RDD.sampleByKey.html#pyspark.RDD.sampleByKey) allows users to
	sample approximately $\lceil f_k \cdot n_k \rceil \, \forall k \in K$ items, where $f_k$ is the
	desired fraction for key $k$, $n_k$ is the number of key-value pairs for key $k$, and $K$ is the
	set of keys.

	Note: `sampleByKeyExact()` is currently not supported in Python.

	{% include_example python/mllib/stratified_sampling_example.py %}
	</div>

	<div data-lang="scala" markdown="1">
	[`sampleByKeyExact()`](api/scala/org/apache/spark/rdd/PairRDDFunctions.html) allows users to
	sample exactly $\lceil f_k \cdot n_k \rceil \, \forall k \in K$ items, where $f_k$ is the desired
	fraction for key $k$, $n_k$ is the number of key-value pairs for key $k$, and $K$ is the set of
	keys. Sampling without replacement requires one additional pass over the RDD to guarantee sample
	size, whereas sampling with replacement requires two additional passes.

	{% include_example scala/org/apache/spark/examples/mllib/StratifiedSamplingExample.scala %}
	</div>

	<div data-lang="java" markdown="1">
	[`sampleByKeyExact()`](api/java/org/apache/spark/api/java/JavaPairRDD.html) allows users to
	sample exactly $\lceil f_k \cdot n_k \rceil \, \forall k \in K$ items, where $f_k$ is the desired
	fraction for key $k$, $n_k$ is the number of key-value pairs for key $k$, and $K$ is the set of
	keys. Sampling without replacement requires one additional pass over the RDD to guarantee sample
	size, whereas sampling with replacement requires two additional passes.

	{% include_example java/org/apache/spark/examples/mllib/JavaStratifiedSamplingExample.java %}
	</div>

	</div>

	## Hypothesis testing

	Hypothesis testing is a powerful tool in statistics to determine whether a result is statistically
	significant, whether this result occurred by chance or not. `spark.mllib` currently supports Pearson's
	chi-squared ( $\chi^2$) tests for goodness of fit and independence. The input data types determine
	whether the goodness of fit or the independence test is conducted. The goodness of fit test requires
	an input type of `Vector`, whereas the independence test requires a `Matrix` as input.

	`spark.mllib` also supports the input type `RDD[LabeledPoint]` to enable feature selection via chi-squared
	independence tests.

	<div class="codetabs">

	<div data-lang="python" markdown="1">
	[`Statistics`](api/python/reference/api/pyspark.mllib.stat.Statistics.html) provides methods to
	run Pearson's chi-squared tests. The following example demonstrates how to run and interpret
	hypothesis tests.

	Refer to the [`Statistics` Python docs](api/python/reference/api/pyspark.mllib.stat.Statistics.html) for more details on the API.

	{% include_example python/mllib/hypothesis_testing_example.py %}
	</div>

	<div data-lang="scala" markdown="1">
	[`Statistics`](api/scala/org/apache/spark/mllib/stat/Statistics$.html) provides methods to
	run Pearson's chi-squared tests. The following example demonstrates how to run and interpret
	hypothesis tests.

	{% include_example scala/org/apache/spark/examples/mllib/HypothesisTestingExample.scala %}
	</div>

	<div data-lang="java" markdown="1">
	[`Statistics`](api/java/org/apache/spark/mllib/stat/Statistics.html) provides methods to
	run Pearson's chi-squared tests. The following example demonstrates how to run and interpret
	hypothesis tests.

	Refer to the [`ChiSqTestResult` Java docs](api/java/org/apache/spark/mllib/stat/test/ChiSqTestResult.html) for details on the API.

	{% include_example java/org/apache/spark/examples/mllib/JavaHypothesisTestingExample.java %}
	</div>

	</div>

	Additionally, `spark.mllib` provides a 1-sample, 2-sided implementation of the Kolmogorov-Smirnov (KS) test
	for equality of probability distributions. By providing the name of a theoretical distribution
	(currently solely supported for the normal distribution) and its parameters, or a function to
	calculate the cumulative distribution according to a given theoretical distribution, the user can
	test the null hypothesis that their sample is drawn from that distribution. In the case that the
	user tests against the normal distribution (`distName="norm"`), but does not provide distribution
	parameters, the test initializes to the standard normal distribution and logs an appropriate
	message.

	<div class="codetabs">

	<div data-lang="python" markdown="1">
	[`Statistics`](api/python/reference/api/pyspark.mllib.stat.Statistics.html) provides methods to
	run a 1-sample, 2-sided Kolmogorov-Smirnov test. The following example demonstrates how to run
	and interpret the hypothesis tests.

	Refer to the [`Statistics` Python docs](api/python/reference/api/pyspark.mllib.stat.Statistics.html) for more details on the API.

	{% include_example python/mllib/hypothesis_testing_kolmogorov_smirnov_test_example.py %}
	</div>

	<div data-lang="scala" markdown="1">
	[`Statistics`](api/scala/org/apache/spark/mllib/stat/Statistics$.html) provides methods to
	run a 1-sample, 2-sided Kolmogorov-Smirnov test. The following example demonstrates how to run
	and interpret the hypothesis tests.

	Refer to the [`Statistics` Scala docs](api/scala/org/apache/spark/mllib/stat/Statistics$.html) for details on the API.

	{% include_example scala/org/apache/spark/examples/mllib/HypothesisTestingKolmogorovSmirnovTestExample.scala %}
	</div>

	<div data-lang="java" markdown="1">
	[`Statistics`](api/java/org/apache/spark/mllib/stat/Statistics.html) provides methods to
	run a 1-sample, 2-sided Kolmogorov-Smirnov test. The following example demonstrates how to run
	and interpret the hypothesis tests.

	Refer to the [`Statistics` Java docs](api/java/org/apache/spark/mllib/stat/Statistics.html) for details on the API.

	{% include_example java/org/apache/spark/examples/mllib/JavaHypothesisTestingKolmogorovSmirnovTestExample.java %}
	</div>

	</div>

	### Streaming Significance Testing
	`spark.mllib` provides online implementations of some tests to support use cases
	like A/B testing. These tests may be performed on a Spark Streaming
	`DStream[(Boolean, Double)]` where the first element of each tuple
	indicates control group (`false`) or treatment group (`true`) and the
	second element is the value of an observation.

	Streaming significance testing supports the following parameters:

	* `peacePeriod` - The number of initial data points from the stream to
	ignore, used to mitigate novelty effects.
	* `windowSize` - The number of past batches to perform hypothesis
	testing over. Setting to `0` will perform cumulative processing using
	all prior batches.


	<div class="codetabs">
	<div data-lang="scala" markdown="1">
	[`StreamingTest`](api/scala/org/apache/spark/mllib/stat/test/StreamingTest.html)
	provides streaming hypothesis testing.

	{% include_example scala/org/apache/spark/examples/mllib/StreamingTestExample.scala %}
	</div>

	<div data-lang="java" markdown="1">
	[`StreamingTest`](api/java/index.html#org.apache.spark.mllib.stat.test.StreamingTest)
	provides streaming hypothesis testing.

	{% include_example java/org/apache/spark/examples/mllib/JavaStreamingTestExample.java %}
	</div>
	</div>


	## Random data generation

	Random data generation is useful for randomized algorithms, prototyping, and performance testing.
	`spark.mllib` supports generating random RDDs with i.i.d. values drawn from a given distribution:
	uniform, standard normal, or Poisson.

	<div class="codetabs">

	<div data-lang="python" markdown="1">
	[`RandomRDDs`](api/python/reference/api/pyspark.mllib.random.RandomRDDs.html) provides factory
	methods to generate random double RDDs or vector RDDs.
	The following example generates a random double RDD, whose values follows the standard normal
	distribution `N(0, 1)`, and then map it to `N(1, 4)`.

	Refer to the [`RandomRDDs` Python docs](api/python/reference/api/pyspark.mllib.random.RandomRDDs.html) for more details on the API.

	{% highlight python %}
	from pyspark.mllib.random import RandomRDDs

	sc = ... # SparkContext

	# Generate a random double RDD that contains 1 million i.i.d. values drawn from the
	# standard normal distribution `N(0, 1)`, evenly distributed in 10 partitions.
	u = RandomRDDs.normalRDD(sc, 1000000L, 10)
	# Apply a transform to get a random double RDD following `N(1, 4)`.
	v = u.map(lambda x: 1.0 + 2.0 * x)
	{% endhighlight %}
	</div>

	<div data-lang="scala" markdown="1">
	[`RandomRDDs`](api/scala/org/apache/spark/mllib/random/RandomRDDs$.html) provides factory
	methods to generate random double RDDs or vector RDDs.
	The following example generates a random double RDD, whose values follows the standard normal
	distribution `N(0, 1)`, and then map it to `N(1, 4)`.

	Refer to the [`RandomRDDs` Scala docs](api/scala/org/apache/spark/mllib/random/RandomRDDs$.html) for details on the API.

	{% highlight scala %}
	import org.apache.spark.SparkContext
	import org.apache.spark.mllib.random.RandomRDDs._

	val sc: SparkContext = ...

	// Generate a random double RDD that contains 1 million i.i.d. values drawn from the
	// standard normal distribution `N(0, 1)`, evenly distributed in 10 partitions.
	val u = normalRDD(sc, 1000000L, 10)
	// Apply a transform to get a random double RDD following `N(1, 4)`.
	val v = u.map(x => 1.0 + 2.0 * x)
	{% endhighlight %}
	</div>

	<div data-lang="java" markdown="1">
	[`RandomRDDs`](api/java/index.html#org.apache.spark.mllib.random.RandomRDDs) provides factory
	methods to generate random double RDDs or vector RDDs.
	The following example generates a random double RDD, whose values follows the standard normal
	distribution `N(0, 1)`, and then map it to `N(1, 4)`.

	Refer to the [`RandomRDDs` Java docs](api/java/org/apache/spark/mllib/random/RandomRDDs) for details on the API.

	{% highlight java %}
	import org.apache.spark.SparkContext;
	import org.apache.spark.api.JavaDoubleRDD;
	import static org.apache.spark.mllib.random.RandomRDDs.*;

	JavaSparkContext jsc = ...

	// Generate a random double RDD that contains 1 million i.i.d. values drawn from the
	// standard normal distribution `N(0, 1)`, evenly distributed in 10 partitions.
	JavaDoubleRDD u = normalJavaRDD(jsc, 1000000L, 10);
	// Apply a transform to get a random double RDD following `N(1, 4)`.
	JavaDoubleRDD v = u.mapToDouble(x -> 1.0 + 2.0 * x);
	{% endhighlight %}
	</div>

	</div>

	## Kernel density estimation

	[Kernel density estimation](https://en.wikipedia.org/wiki/Kernel_density_estimation) is a technique
	useful for visualizing empirical probability distributions without requiring assumptions about the
	particular distribution that the observed samples are drawn from. It computes an estimate of the
	probability density function of a random variables, evaluated at a given set of points. It achieves
	this estimate by expressing the PDF of the empirical distribution at a particular point as the
	mean of PDFs of normal distributions centered around each of the samples.

	<div class="codetabs">

	<div data-lang="python" markdown="1">
	[`KernelDensity`](api/python/reference/api/pyspark.mllib.stat.KernelDensity.html) provides methods
	to compute kernel density estimates from an RDD of samples. The following example demonstrates how
	to do so.

	Refer to the [`KernelDensity` Python docs](api/python/reference/api/pyspark.mllib.stat.KernelDensity.html) for more details on the API.

	{% include_example python/mllib/kernel_density_estimation_example.py %}
	</div>

	<div data-lang="scala" markdown="1">
	[`KernelDensity`](api/scala/org/apache/spark/mllib/stat/KernelDensity.html) provides methods
	to compute kernel density estimates from an RDD of samples. The following example demonstrates how
	to do so.

	Refer to the [`KernelDensity` Scala docs](api/scala/org/apache/spark/mllib/stat/KernelDensity.html) for details on the API.

	{% include_example scala/org/apache/spark/examples/mllib/KernelDensityEstimationExample.scala %}
	</div>

	<div data-lang="java" markdown="1">
	[`KernelDensity`](api/java/index.html#org.apache.spark.mllib.stat.KernelDensity) provides methods
	to compute kernel density estimates from an RDD of samples. The following example demonstrates how
	to do so.

	Refer to the [`KernelDensity` Java docs](api/java/org/apache/spark/mllib/stat/KernelDensity.html) for details on the API.

	{% include_example java/org/apache/spark/examples/mllib/JavaKernelDensityEstimationExample.java %}
	</div>

	</div>