Target audience: Intermediate
Estimated reading time: 4'
This article introduces a very powerful, non-parametric method to extract an empirical continuous probability density function from a dataset: Multivariate Kernel Density Estimation (KDE), also known as the Parzen’s window. At its core the KDE is a smooth approximation of an histogram [ref 1].
Table of contents
Notes:
- This article requires a basic knowledge in Apache spark MLlib framework and understanding of statistics and/or machine learning.
- This implementation relies on Spark 2.3.1 and Scala 2.12.4
Introduction
KDE is one of the most well-known approaches to estimate the underlying probability density function of a dataset. KDE will learn the shape of the density from the data automatically. This flexibility arising from its non- parametric nature makes KDE a very popular approach for data drawn from a complicated distribution.
The KDE algorithm takes a parameter, bandwidth, that affects how “smooth” the resulting curve is. For a set of observations y, and given a kernel function K and a bandwidth, the estimation of the density function f, can be expressed as.
This post addresses the limitations of the current implementation of KDE in Apache Spark for the multi-variate features.
Spark implementation
Apache Spark is a fast and general-purpose cluster computing solution that provides high-level APIs in Java, Scala, Python and R, and an optimized engine that supports general execution graphs.The Apache Spark ecosystems includes a machine learning library, MLlib.
The implementation of the kernel density estimation in the current version of Apache Spark MLlib library, 2.3.1 [ref 2], org.apache.spark.mllib.stats.KernelDensity has two important limitations:
- It is a univariate estimation
- The estimation is performed on a sequence of observations, not an RDD or data set, putting computation load on the Spark driver.
val sample = sparkSession.sparkContext.parallelize(data) val kd = new KernelDensity().setSample(sample).setBandwidth(3.0)
val densities = kd.estimate(Array(-2.0, 5.0))
The method setSample specifies the training set but the KDE is actually trained when the method estimate is invoked on the driver.
Multivariate KDE
The purpose of this post is to extend the current functionality of the KDE by supporting multi-dimensional features and allows the developers to apply the estimation to a dataset. This implementation is restricted to the Normal distribution although it can easily be extended to other kernel functions.
We assume
- The reference to the current Spark session is implicit (line 1)
- The encoding of a row for serialization of the task is provided (line 1)
The method estimate has 3 arguments
- TrainingDS training dataset (line 9)
- Validation validation set (line 10)
- bandwidth size of the Parzen window
The validation set has to be broadcast to each worker nodes (line 14). This should not be a problem as the size of the validation set is expected of reasonable size.
The training set is passed to each partitions as iterator through a mapPartitions (line 17). The probability densities and count are computed through a Scala aggregate method with a zero function of type, (Array[Double], Long) (line 23). The sequence operator invokes the multinomial normal distribution (line 29).
The combiner (3rd argument of the aggregate) relies on the BLAS vectorization z = <- a.x+y dxapy (line 38). BLAS library [ref 3] has 3 levels (1D, 2D and 3D arrays). Blas library
The vector of densities is scaled with invCount using the decal BLAS level 1 method (line 45).
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50 | final class KDE(implicit sparkSession: SparkSession,
encoder: Encoder[Row]) {
/**
* Applied the trained KDE to a set of validation data
* @param trainingDS Training data sets
* @param validationRdd Validation data sets
* @return Datasets of probability densities
*/
def estimate(
trainingDS: Dataset[Obs],
validationDS: Dataset[Obs],
bandwidth: Double = 1.0): Dataset[Double] = {
import math._, sparkSession.implicits._
val validation_brdcast = sparkSession.sparkContext
.broadcast[Array[Obs]](validationDS.collect)
trainingDS.mapPartitions((iter: Iterator[Obs]) => {
val seqObs = iter.toArray
val scale = 0.5 * seqObs.size* log(2 * Pi)
val validation = validation_brdcast.value
val (densities, count) = seqObs.aggregate(
(new Array[Double](validation.length), 0L) ) (
{ // seqOp (U, T) => U
case ((x, z), y) => {
var i = 0
while (i < validation.length) {
// Call the pdf function for the normal distribution
x(i) += multiNorm(y, bandwidth, scale, validation(i))
i += 1
}
(x, z + 1) // Update count & validation values
}
},
{ // combOp: (U, U) => U
case ((u, z), (v, t)) => {
// Combiner calls vectorization z <- a.x + y
blas.daxpy(validation.length, 1.0, v, 1, u, 1)
(u, z + t)
}
}
)
val invCount: Double = 1.0 / count
blas.dscal(validation.length, invCount, densities, 1)
// Rescale the density using LINPACK z <- a.x
densities.iterator
})
}
}
|
The companion singleton is used to define the multinomial normal distribution (line 5). The type of observations (feature) is Array[Double].
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21 | final object KDE {
import math._
type Obs = Array[Double]
@throws(classOf[IllegalArgumentException])
def multiNorm(
means: Obs,
bandWidth: Double,
scale: Double,
x: Obs): Double = {
require(x.length == means.length,
"Dimension of means and observations differs")
exp(
-scale - (0 until means.length).map( n => {
val sx = (means(n) - x(n)) / bandWidth
-0.5 * sx * sx
} ).sum
)
}
}
|
Application
This simple application requires that the spark context (SparkSession) to be defined as well as an explicit encoding of Row using Kryo serializer. The implicit conversion are made available by importing sparkSession.implicits.The training set is a sequence of key-value pairs (lines 3-14). The validation set is synthetically generated by multiplying the data in the training value with 2.0 (line 17).
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30 | implicit val sparkSession: SparkSession =
confToSessionFromFile(LocalSparkConf)
implicit val encoder = Encoders.kryo[Row]
import sparkSession.implicits._
val trainingData = Seq[(String, Array[Double])](
("A", Array[Double](1.0, 0.6)), ("B", Array[Double](2.0, 0.6)),
("C", Array[Double](1.5, 9.5)), ("D", Array[Double](4.5, 0.7)),
("E", Array[Double](0.4, 1.5)), ("F", Array[Double](2.1, 0.6)),
("G", Array[Double](0.5, 6.3)), ("H", Array[Double](1.5, 0.1)),
("I", Array[Double](1.2, 3.0)), ("B", Array[Double](3.1, 1.1))
).toDS
val validationData = trainingData
.map { case (key, values) => values.map(_ *2.0) }
val kde = new KDE
val result = kde.estimate(trainingData.map(_._2),validationData)
println(s"result: ${result.collect.mkString(", ")}")
sparkSession.close
val data = Seq[Double](1.0, 5.6)
val sample = sparkSession.sparkContext.parallelize(data)
val kd = new KernelDensity().setSample(sample) .setBandwidth(3.0)
val densities = kd.estimate(Array(-2.0, 5.0))
|
Note: There are excellent research papers highlighting the statistical foundation behind KDE as well as recent advances [ref 4].
Thank you for reading this article. For more information ...
References
[3] BLAS
Environment Scala: 2.12.4, Java JDK 1.8, Apache Spark 2.3.1, OpenBLAS 0.3.4
---------------------------
Patrick Nicolas has over 25 years of experience in software and data engineering, architecture design and end-to-end deployment and support with extensive knowledge in machine learning.
He has been director of data engineering at Aideo Technologies since 2017 and he is the author of "Scala for Machine Learning" Packt Publishing ISBN 978-1-78712-238-3
He has been director of data engineering at Aideo Technologies since 2017 and he is the author of "Scala for Machine Learning" Packt Publishing ISBN 978-1-78712-238-3
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