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diff --git a/docs/mllib-statistics.md b/docs/mllib-statistics.md new file mode 100644 index 0000000000..c463241399 --- /dev/null +++ b/docs/mllib-statistics.md @@ -0,0 +1,457 @@ +--- +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`. + +<div class="codetabs"> +<div data-lang="scala" markdown="1"> + +[`colStats()`](api/scala/index.html#org.apache.spark.mllib.stat.Statistics$) returns an instance of +[`MultivariateStatisticalSummary`](api/scala/index.html#org.apache.spark.mllib.stat.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 %} +</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. + +{% 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<Vector> 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 %} +</div> + +<div data-lang="python" markdown="1"> +[`colStats()`](api/python/pyspark.mllib.stat.Statistics-class.html#colStats) returns an instance of +[`MultivariateStatisticalSummary`](api/python/pyspark.mllib.stat.MultivariateStatisticalSummary-class.html), +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 %} +</div> + +</div> + +## 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. + +<div class="codetabs"> +<div data-lang="scala" markdown="1"> +[`Statistics`](api/scala/index.html#org.apache.spark.mllib.stat.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 %} +</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. + +{% 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<Vector> 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 %} +</div> + +<div data-lang="python" markdown="1"> +[`Statistics`](api/python/pyspark.mllib.stat.Statistics-class.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. + +{% 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 %} +</div> + +</div> + +## 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. + +<div class="codetabs"> +<div data-lang="scala" markdown="1"> +[`sampleByKeyExact()`](api/scala/index.html#org.apache.spark.rdd.PairRDDFunctions) 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 %} +</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. + +{% 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 %} +</div> +<div data-lang="python" markdown="1"> +[`sampleByKey()`](api/python/pyspark.rdd.RDD-class.html#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 %} +</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. 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. + +<div class="codetabs"> +<div data-lang="scala" markdown="1"> +[`Statistics`](api/scala/index.html#org.apache.spark.mllib.stat.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 %} +</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. + +{% 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<LabeledPoint> 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 %} +</div> + +</div> + +## 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. + +<div class="codetabs"> +<div data-lang="scala" markdown="1"> +[`RandomRDDs`](api/scala/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)`. + +{% 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)`. + +{% 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 %} +</div> + +<div data-lang="python" markdown="1"> +[`RandomRDDs`](api/python/pyspark.mllib.random.RandomRDDs-class.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)`. + +{% 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 %} +</div> + +</div> |