--- layout: global title: Basic Statistics - MLlib displayTitle: MLlib - 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()`](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. Refer to the [`MultivariateStatisticalSummary` Scala docs](api/scala/index.html#org.apache.spark.mllib.stat.MultivariateStatisticalSummary) for details on the API. {% 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()`](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. {% 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()`](api/python/pyspark.mllib.html#pyspark.mllib.stat.Statistics.colStats) returns an instance of [`MultivariateStatisticalSummary`](api/python/pyspark.mllib.html#pyspark.mllib.stat.MultivariateStatisticalSummary), 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/pyspark.mllib.html#pyspark.mllib.stat.MultivariateStatisticalSummary) for more details on the API. {% 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`](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. Refer to the [`Statistics` Scala docs](api/scala/index.html#org.apache.spark.mllib.stat.Statistics) for details on the API. {% 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`](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`, 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. {% 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`](api/python/pyspark.mllib.html#pyspark.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. Refer to the [`Statistics` Python docs](api/python/pyspark.mllib.html#pyspark.mllib.stat.Statistics) for more details on the API. {% 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()`](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 %}
[`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 data = ... // an RDD of any key value pairs Map fractions = ... // specify the exact fraction desired from each key // Get an exact sample from each stratum JavaPairRDD approxSample = data.sampleByKey(false, fractions); JavaPairRDD exactSample = data.sampleByKeyExact(false, fractions); {% endhighlight %}
[`sampleByKey()`](api/python/pyspark.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. {% 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`](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 %}
[`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. {% 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 %}
[`Statistics`](api/python/index.html#pyspark.mllib.stat.Statistics$) 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/pyspark.mllib.html#pyspark.mllib.stat.Statistics) for more details on the API. {% highlight python %} from pyspark import SparkContext from pyspark.mllib.linalg import Vectors, Matrices from pyspark.mllib.regresssion import LabeledPoint from pyspark.mllib.stat import Statistics sc = SparkContext() vec = Vectors.dense(...) # 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. goodnessOfFitTestResult = Statistics.chiSqTest(vec) print(goodnessOfFitTestResult) # summary of the test including the p-value, degrees of freedom, # test statistic, the method used, and the null hypothesis. mat = Matrices.dense(...) # a contingency matrix # conduct Pearson's independence test on the input contingency matrix independenceTestResult = Statistics.chiSqTest(mat) print(independenceTestResult) # summary of the test including the p-value, degrees of freedom... obs = sc.parallelize(...) # LabeledPoint(feature, label) . # The contingency table is constructed from an RDD of LabeledPoint and used to conduct # the independence test. Returns an array containing the ChiSquaredTestResult for every feature # against the label. featureTestResults = Statistics.chiSqTest(obs) for i, result in enumerate(featureTestResults): print("Column $d:" % (i + 1)) print(result) {% endhighlight %}
Additionally, MLlib provides a 1-sample, 2-sided implementation of the Kolmogorov-Smirnov (KS) test for equality of probability distributions. By providing the name of a theoretical distribution (currently solely supported for the normal distribution) and its parameters, or a function to calculate the cumulative distribution according to a given theoretical distribution, the user can test the null hypothesis that their sample is drawn from that distribution. In the case that the user tests against the normal distribution (`distName="norm"`), but does not provide distribution parameters, the test initializes to the standard normal distribution and logs an appropriate message.
[`Statistics`](api/scala/index.html#org.apache.spark.mllib.stat.Statistics$) 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/index.html#org.apache.spark.mllib.stat.Statistics) for details on the API. {% highlight scala %} import org.apache.spark.mllib.stat.Statistics val data: RDD[Double] = ... // an RDD of sample data // run a KS test for the sample versus a standard normal distribution val testResult = Statistics.kolmogorovSmirnovTest(data, "norm", 0, 1) println(testResult) // summary of the test including the p-value, test statistic, // and null hypothesis // if our p-value indicates significance, we can reject the null hypothesis // perform a KS test using a cumulative distribution function of our making val myCDF: Double => Double = ... val testResult2 = Statistics.kolmogorovSmirnovTest(data, myCDF) {% endhighlight %}
[`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. {% highlight java %} import java.util.Arrays; import org.apache.spark.api.java.JavaDoubleRDD; import org.apache.spark.api.java.JavaSparkContext; import org.apache.spark.mllib.stat.Statistics; import org.apache.spark.mllib.stat.test.KolmogorovSmirnovTestResult; JavaSparkContext jsc = ... JavaDoubleRDD data = jsc.parallelizeDoubles(Arrays.asList(0.2, 1.0, ...)); KolmogorovSmirnovTestResult testResult = Statistics.kolmogorovSmirnovTest(data, "norm", 0.0, 1.0); // summary of the test including the p-value, test statistic, // and null hypothesis // if our p-value indicates significance, we can reject the null hypothesis System.out.println(testResult); {% endhighlight %}
[`Statistics`](api/python/pyspark.mllib.html#pyspark.mllib.stat.Statistics) 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/pyspark.mllib.html#pyspark.mllib.stat.Statistics) for more details on the API. {% highlight python %} from pyspark.mllib.stat import Statistics parallelData = sc.parallelize([1.0, 2.0, ... ]) # run a KS test for the sample versus a standard normal distribution testResult = Statistics.kolmogorovSmirnovTest(parallelData, "norm", 0, 1) print(testResult) # summary of the test including the p-value, test statistic, # and null hypothesis # if our p-value indicates significance, we can reject the null hypothesis # Note that the Scala functionality of calling Statistics.kolmogorovSmirnovTest with # a lambda to calculate the CDF is not made available in the Python API {% 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`](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)`. Refer to the [`RandomRDDs` Scala docs](api/scala/index.html#org.apache.spark.mllib.random.RandomRDDs) 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 %}
[`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.map( new Function() { public Double call(Double x) { return 1.0 + 2.0 * x; } }); {% endhighlight %}
[`RandomRDDs`](api/python/pyspark.mllib.html#pyspark.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` Python docs](api/python/pyspark.mllib.html#pyspark.mllib.random.RandomRDDs) 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 %}
## 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 the mean of PDFs of normal distributions centered around each of the samples.
[`KernelDensity`](api/scala/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` Scala docs](api/scala/index.html#org.apache.spark.mllib.stat.KernelDensity) for details on the API. {% highlight scala %} import org.apache.spark.mllib.stat.KernelDensity import org.apache.spark.rdd.RDD val data: RDD[Double] = ... // an RDD of sample data // Construct the density estimator with the sample data and a standard deviation for the Gaussian // kernels val kd = new KernelDensity() .setSample(data) .setBandwidth(3.0) // Find density estimates for the given values val densities = kd.estimate(Array(-1.0, 2.0, 5.0)) {% endhighlight %}
[`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. {% highlight java %} import org.apache.spark.mllib.stat.KernelDensity; import org.apache.spark.rdd.RDD; RDD data = ... // an RDD of sample data // Construct the density estimator with the sample data and a standard deviation for the Gaussian // kernels KernelDensity kd = new KernelDensity() .setSample(data) .setBandwidth(3.0); // Find density estimates for the given values double[] densities = kd.estimate(new double[] {-1.0, 2.0, 5.0}); {% endhighlight %}
[`KernelDensity`](api/python/pyspark.mllib.html#pyspark.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` Python docs](api/python/pyspark.mllib.html#pyspark.mllib.stat.KernelDensity) for more details on the API. {% highlight python %} from pyspark.mllib.stat import KernelDensity data = ... # an RDD of sample data # Construct the density estimator with the sample data and a standard deviation for the Gaussian # kernels kd = KernelDensity() kd.setSample(data) kd.setBandwidth(3.0) # Find density estimates for the given values densities = kd.estimate([-1.0, 2.0, 5.0]) {% endhighlight %}