mllib-statistics.md

layout: global
title: Basic Statistics - MLlib
displayTitle: <a href="mllib-guide.html">MLlib</a> - Basic Statistics 

• 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.

colStats() returns an instance of MultivariateStatisticalSummary, which contains the column-wise max, min, mean, variance, and number of nonzeros, as well as the total count.

{% highlight scala %} import org.apache.spark.mllib.linalg.Vector import org.apache.spark.mllib.stat.{MultivariateStatisticalSummary, Statistics}

val observations: RDD[Vector] = ... // an RDD of Vectors

// Compute column summary statistics. val summary: MultivariateStatisticalSummary = Statistics.colStats(observations) println(summary.mean) // a dense vector containing the mean value for each column println(summary.variance) // column-wise variance println(summary.numNonzeros) // number of nonzeros in each column

{% endhighlight %}

colStats() returns an instance of MultivariateStatisticalSummary, which contains the column-wise max, min, mean, variance, and number of nonzeros, as well as the total count.

{% highlight java %} import org.apache.spark.api.java.JavaRDD; import org.apache.spark.api.java.JavaSparkContext; import org.apache.spark.mllib.linalg.Vector; import org.apache.spark.mllib.stat.MultivariateStatisticalSummary; import org.apache.spark.mllib.stat.Statistics;

JavaSparkContext jsc = ...

JavaRDD mat = ... // an RDD of Vectors

// Compute column summary statistics. MultivariateStatisticalSummary summary = Statistics.colStats(mat.rdd()); System.out.println(summary.mean()); // a dense vector containing the mean value for each column System.out.println(summary.variance()); // column-wise variance System.out.println(summary.numNonzeros()); // number of nonzeros in each column

{% endhighlight %}

colStats() returns an instance of MultivariateStatisticalSummary, which contains the column-wise max, min, mean, variance, and number of nonzeros, as well as the total count.

{% highlight python %} from pyspark.mllib.stat import Statistics

sc = ... # SparkContext

mat = ... # an RDD of Vectors

Compute column summary statistics.

summary = Statistics.colStats(mat) print summary.mean() print summary.variance() print summary.numNonzeros()

{% endhighlight %}

Correlations

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

Statistics 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.

{% highlight scala %} import org.apache.spark.SparkContext import org.apache.spark.mllib.linalg._ import org.apache.spark.mllib.stat.Statistics

val sc: SparkContext = ...

val seriesX: RDD[Double] = ... // a series val seriesY: RDD[Double] = ... // must have the same number of partitions and cardinality as seriesX

// compute the correlation using Pearson's method. Enter "spearman" for Spearman's method. If a // method is not specified, Pearson's method will be used by default. val correlation: Double = Statistics.corr(seriesX, seriesY, "pearson")

val data: RDD[Vector] = ... // note that each Vector is a row and not a column

// calculate the correlation matrix using Pearson's method. Use "spearman" for Spearman's method. // If a method is not specified, Pearson's method will be used by default. val correlMatrix: Matrix = Statistics.corr(data, "pearson")

{% endhighlight %}

Statistics provides methods to calculate correlations between series. Depending on the type of input, two `JavaDoubleRDD`s or a `JavaRDD`, the output will be a `Double` or the correlation `Matrix` respectively.

{% highlight java %} import org.apache.spark.api.java.JavaDoubleRDD; import org.apache.spark.api.java.JavaSparkContext; import org.apache.spark.mllib.linalg.*; import org.apache.spark.mllib.stat.Statistics;

JavaSparkContext jsc = ...

JavaDoubleRDD seriesX = ... // a series JavaDoubleRDD seriesY = ... // must have the same number of partitions and cardinality as seriesX

// compute the correlation using Pearson's method. Enter "spearman" for Spearman's method. If a // method is not specified, Pearson's method will be used by default. Double correlation = Statistics.corr(seriesX.srdd(), seriesY.srdd(), "pearson");

JavaRDD data = ... // note that each Vector is a row and not a column

// calculate the correlation matrix using Pearson's method. Use "spearman" for Spearman's method. // If a method is not specified, Pearson's method will be used by default. Matrix correlMatrix = Statistics.corr(data.rdd(), "pearson");

{% endhighlight %}

Statistics 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.

{% highlight python %} from pyspark.mllib.stat import Statistics

sc = ... # SparkContext

seriesX = ... # a series seriesY = ... # must have the same number of partitions and cardinality as seriesX

Compute the correlation using Pearson's method. Enter "spearman" for Spearman's method. If a

method is not specified, Pearson's method will be used by default.

print Statistics.corr(seriesX, seriesY, method="pearson")

data = ... # an RDD of Vectors

calculate the correlation matrix using Pearson's method. Use "spearman" for Spearman's method.

If a method is not specified, Pearson's method will be used by default.

print Statistics.corr(data, method="pearson")

{% endhighlight %}

Stratified sampling

Unlike the other statistics functions, which reside in 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.

sampleByKeyExact() 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.

{% highlight scala %} import org.apache.spark.SparkContext import org.apache.spark.SparkContext._ import org.apache.spark.rdd.PairRDDFunctions

val sc: SparkContext = ...

val data = ... // an RDD[(K, V)] of any key value pairs val fractions: Map[K, Double] = ... // specify the exact fraction desired from each key

// Get an exact sample from each stratum val approxSample = data.sampleByKey(withReplacement = false, fractions) val exactSample = data.sampleByKeyExact(withReplacement = false, fractions)

{% endhighlight %}

sampleByKeyExact() 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.

{% highlight java %} import java.util.Map;

import org.apache.spark.api.java.JavaPairRDD; import org.apache.spark.api.java.JavaSparkContext;

JavaSparkContext jsc = ...

JavaPairRDD<K, V> data = ... // an RDD of any key value pairs Map<K, Object> fractions = ... // specify the exact fraction desired from each key

// Get an exact sample from each stratum JavaPairRDD<K, V> approxSample = data.sampleByKey(false, fractions); JavaPairRDD<K, V> exactSample = data.sampleByKeyExact(false, fractions);

{% endhighlight %}

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.

{% highlight python %}

sc = ... # SparkContext

data = ... # an RDD of any key value pairs fractions = ... # specify the exact fraction desired from each key as a dictionary

approxSample = data.sampleByKey(False, fractions);

{% endhighlight %}

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. 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.

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

Statistics provides methods to run Pearson's chi-squared tests. The following example demonstrates how to run and interpret hypothesis tests.

{% highlight scala %} import org.apache.spark.SparkContext import org.apache.spark.mllib.linalg._ import org.apache.spark.mllib.regression.LabeledPoint import org.apache.spark.mllib.stat.Statistics._

val sc: SparkContext = ...

val vec: Vector = ... // a vector composed of the frequencies of events

// compute the goodness of fit. If a second vector to test against is not supplied as a parameter, // the test runs against a uniform distribution.
val goodnessOfFitTestResult = Statistics.chiSqTest(vec) println(goodnessOfFitTestResult) // summary of the test including the p-value, degrees of freedom, // test statistic, the method used, and the null hypothesis.

val mat: Matrix = ... // a contingency matrix

// conduct Pearson's independence test on the input contingency matrix val independenceTestResult = Statistics.chiSqTest(mat) println(independenceTestResult) // summary of the test including the p-value, degrees of freedom...

val obs: RDD[LabeledPoint] = ... // (feature, label) pairs.

// The contingency table is constructed from the raw (feature, label) pairs and used to conduct // the independence test. Returns an array containing the ChiSquaredTestResult for every feature // against the label. val featureTestResults: Array[ChiSqTestResult] = Statistics.chiSqTest(obs) var i = 1 featureTestResults.foreach { result => println(s"Column

i:\n

result") i += 1 } // summary of the test

{% endhighlight %}

Statistics provides methods to run Pearson's chi-squared tests. The following example demonstrates how to run and interpret hypothesis tests.

{% highlight java %} import org.apache.spark.api.java.JavaRDD; import org.apache.spark.api.java.JavaSparkContext; import org.apache.spark.mllib.linalg.*; import org.apache.spark.mllib.regression.LabeledPoint; import org.apache.spark.mllib.stat.Statistics; import org.apache.spark.mllib.stat.test.ChiSqTestResult;

JavaSparkContext jsc = ...

Vector vec = ... // a vector composed of the frequencies of events

// compute the goodness of fit. If a second vector to test against is not supplied as a parameter, // the test runs against a uniform distribution.
ChiSqTestResult goodnessOfFitTestResult = Statistics.chiSqTest(vec); // summary of the test including the p-value, degrees of freedom, test statistic, the method used, // and the null hypothesis. System.out.println(goodnessOfFitTestResult);

Matrix mat = ... // a contingency matrix

// conduct Pearson's independence test on the input contingency matrix ChiSqTestResult independenceTestResult = Statistics.chiSqTest(mat); // summary of the test including the p-value, degrees of freedom... System.out.println(independenceTestResult);

JavaRDD obs = ... // an RDD of labeled points

// The contingency table is constructed from the raw (feature, label) pairs and used to conduct // the independence test. Returns an array containing the ChiSquaredTestResult for every feature // against the label. ChiSqTestResult[] featureTestResults = Statistics.chiSqTest(obs.rdd()); int i = 1; for (ChiSqTestResult result : featureTestResults) { System.out.println("Column " + i + ":"); System.out.println(result); // summary of the test i++; }

{% endhighlight %}

Random data generation

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

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)`.

{% 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 %}

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)`.

{% 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.map( new Function<Double, Double>() { public Double call(Double x) { return 1.0 + 2.0 * x; } }); {% endhighlight %}

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)`.

{% 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.uniformRDD(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 %}