From 323e7390a5c123c48cc7d6d9be44bee3a7eecd99 Mon Sep 17 00:00:00 2001
From: Xiangrui Meng <meng@databricks.com>
Date: Wed, 13 Apr 2016 09:17:46 -0700
Subject: Revert "[SPARK-14154][MLLIB] Simplify the implementation for
 Kolmogorov–Smirnov test"
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This reverts commit d2a819a6363190b946986ebf6f8001d520098c3b.
---
 .../mllib/stat/test/KolmogorovSmirnovTest.scala    | 77 ++++++++++++++++++++--
 1 file changed, 73 insertions(+), 4 deletions(-)

diff --git a/mllib/src/main/scala/org/apache/spark/mllib/stat/test/KolmogorovSmirnovTest.scala b/mllib/src/main/scala/org/apache/spark/mllib/stat/test/KolmogorovSmirnovTest.scala
index ef284531c9..9748fbf2c9 100644
--- a/mllib/src/main/scala/org/apache/spark/mllib/stat/test/KolmogorovSmirnovTest.scala
+++ b/mllib/src/main/scala/org/apache/spark/mllib/stat/test/KolmogorovSmirnovTest.scala
@@ -64,10 +64,11 @@ private[stat] object KolmogorovSmirnovTest extends Logging {
    */
   def testOneSample(data: RDD[Double], cdf: Double => Double): KolmogorovSmirnovTestResult = {
     val n = data.count().toDouble
-    val ksStat = data.sortBy(x => x).zipWithIndex().map { case (v, i) =>
-      val f = cdf(v)
-      math.max(f - i / n, (i + 1) / n - f)
-    }.max()
+    val localData = data.sortBy(x => x).mapPartitions { part =>
+      val partDiffs = oneSampleDifferences(part, n, cdf) // local distances
+      searchOneSampleCandidates(partDiffs) // candidates: local extrema
+    }.collect()
+    val ksStat = searchOneSampleStatistic(localData, n) // result: global extreme
     evalOneSampleP(ksStat, n.toLong)
   }
 
@@ -83,6 +84,74 @@ private[stat] object KolmogorovSmirnovTest extends Logging {
     testOneSample(data, cdf)
   }
 
+  /**
+   * Calculate unadjusted distances between the empirical CDF and the theoretical CDF in a
+   * partition
+   * @param partData `Iterator[Double]` 1 partition of a sorted RDD
+   * @param n `Double` the total size of the RDD
+   * @param cdf `Double => Double` a function the calculates the theoretical CDF of a value
+   * @return `Iterator[(Double, Double)] `Unadjusted (ie. off by a constant) potential extrema
+   *        in a partition. The first element corresponds to the (empirical CDF - 1/N) - CDF,
+   *        the second element corresponds to empirical CDF - CDF.  We can then search the resulting
+   *        iterator for the minimum of the first and the maximum of the second element, and provide
+   *        this as a partition's candidate extrema
+   */
+  private def oneSampleDifferences(partData: Iterator[Double], n: Double, cdf: Double => Double)
+    : Iterator[(Double, Double)] = {
+    // zip data with index (within that partition)
+    // calculate local (unadjusted) empirical CDF and subtract CDF
+    partData.zipWithIndex.map { case (v, ix) =>
+      // dp and dl are later adjusted by constant, when global info is available
+      val dp = (ix + 1) / n
+      val dl = ix / n
+      val cdfVal = cdf(v)
+      (dl - cdfVal, dp - cdfVal)
+    }
+  }
+
+  /**
+   * Search the unadjusted differences in a partition and return the
+   * two extrema (furthest below and furthest above CDF), along with a count of elements in that
+   * partition
+   * @param partDiffs `Iterator[(Double, Double)]` the unadjusted differences between empirical CDF
+   *                 and CDFin a partition, which come as a tuple of
+   *                 (empirical CDF - 1/N - CDF, empirical CDF - CDF)
+   * @return `Iterator[(Double, Double, Double)]` the local extrema and a count of elements
+   */
+  private def searchOneSampleCandidates(partDiffs: Iterator[(Double, Double)])
+    : Iterator[(Double, Double, Double)] = {
+    val initAcc = (Double.MaxValue, Double.MinValue, 0.0)
+    val pResults = partDiffs.foldLeft(initAcc) { case ((pMin, pMax, pCt), (dl, dp)) =>
+      (math.min(pMin, dl), math.max(pMax, dp), pCt + 1)
+    }
+    val results = if (pResults == initAcc) Array[(Double, Double, Double)]() else Array(pResults)
+    results.iterator
+  }
+
+  /**
+   * Find the global maximum distance between empirical CDF and CDF (i.e. the KS statistic) after
+   * adjusting local extrema estimates from individual partitions with the amount of elements in
+   * preceding partitions
+   * @param localData `Array[(Double, Double, Double)]` A local array containing the collected
+   *                 results of `searchOneSampleCandidates` across all partitions
+   * @param n `Double`The size of the RDD
+   * @return The one-sample Kolmogorov Smirnov Statistic
+   */
+  private def searchOneSampleStatistic(localData: Array[(Double, Double, Double)], n: Double)
+    : Double = {
+    val initAcc = (Double.MinValue, 0.0)
+    // adjust differences based on the number of elements preceding it, which should provide
+    // the correct distance between empirical CDF and CDF
+    val results = localData.foldLeft(initAcc) { case ((prevMax, prevCt), (minCand, maxCand, ct)) =>
+      val adjConst = prevCt / n
+      val dist1 = math.abs(minCand + adjConst)
+      val dist2 = math.abs(maxCand + adjConst)
+      val maxVal = Array(prevMax, dist1, dist2).max
+      (maxVal, prevCt + ct)
+    }
+    results._1
+  }
+
   /**
    * A convenience function that allows running the KS test for 1 set of sample data against
    * a named distribution
-- 
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