[SPARK-9005] [MLLIB] Fix RegressionMetrics computation of explainedVariance

Fixes implementation of `explainedVariance` and `r2` to be consistent with their definitions as described in [SPARK-9005](https://issues.apache.org/jira/browse/SPARK-9005).

Author: Feynman Liang <fliang@databricks.com>

Closes #7361 from feynmanliang/SPARK-9005-RegressionMetrics-bugs and squashes the following commits:

f1112fc [Feynman Liang] Add explainedVariance formula
1a3d098 [Feynman Liang] SROwen code review comments
08a0e1b [Feynman Liang] Fix pyspark tests
db8605a [Feynman Liang] Style fix
bde9761 [Feynman Liang] Fix RegressionMetrics tests, relax assumption predictor is unbiased
c235de0 [Feynman Liang] Fix RegressionMetrics tests
4c4e56f [Feynman Liang] Fix RegressionMetrics computation of explainedVariance and r2

mllib/src/main/scala/org/apache/spark/mllib/evaluation/RegressionMetrics.scala

@@ -53,14 +53,22 @@ class RegressionMetrics(predictionAndObservations: RDD[(Double, Double)]) extend
       )
     summary
   }
+  private lazy val SSerr = math.pow(summary.normL2(1), 2)
+  private lazy val SStot = summary.variance(0) * (summary.count - 1)
+  private lazy val SSreg = {
+    val yMean = summary.mean(0)
+    predictionAndObservations.map {
+      case (prediction, _) => math.pow(prediction - yMean, 2)
+    }.sum()
+  }
 
   /**
-   * Returns the explained variance regression score.
-   * explainedVariance = 1 - variance(y - \hat{y}) / variance(y)
-   * Reference: [[http://en.wikipedia.org/wiki/Explained_variation]]
+   * Returns the variance explained by regression.
+   * explainedVariance = \sum_i (\hat{y_i} - \bar{y})^2 / n
+   * @see [[https://en.wikipedia.org/wiki/Fraction_of_variance_unexplained]]
    */
   def explainedVariance: Double = {
-    1 - summary.variance(1) / summary.variance(0)
+    SSreg / summary.count
   }
 
   /**
@@ -76,8 +84,7 @@ class RegressionMetrics(predictionAndObservations: RDD[(Double, Double)]) extend
    * expected value of the squared error loss or quadratic loss.
    */
   def meanSquaredError: Double = {
-    val rmse = summary.normL2(1) / math.sqrt(summary.count)
-    rmse * rmse
+    SSerr / summary.count
   }
 
   /**
@@ -85,14 +92,14 @@ class RegressionMetrics(predictionAndObservations: RDD[(Double, Double)]) extend
    * the mean squared error.
    */
   def rootMeanSquaredError: Double = {
-    summary.normL2(1) / math.sqrt(summary.count)
+    math.sqrt(this.meanSquaredError)
   }
 
   /**
-   * Returns R^2^, the coefficient of determination.
-   * Reference: [[http://en.wikipedia.org/wiki/Coefficient_of_determination]]
+   * Returns R^2^, the unadjusted coefficient of determination.
+   * @see [[http://en.wikipedia.org/wiki/Coefficient_of_determination]]
    */
   def r2: Double = {
-    1 - math.pow(summary.normL2(1), 2) / (summary.variance(0) * (summary.count - 1))
+    1 - SSerr / SStot
   }
 }

mllib/src/test/scala/org/apache/spark/mllib/evaluation/RegressionMetricsSuite.scala

@@ -23,24 +23,85 @@ import org.apache.spark.mllib.util.TestingUtils._
 
 class RegressionMetricsSuite extends SparkFunSuite with MLlibTestSparkContext {
 
-  test("regression metrics") {
+  test("regression metrics for unbiased (includes intercept term) predictor") {
+    /* Verify results in R:
+       preds = c(2.25, -0.25, 1.75, 7.75)
+       obs = c(3.0, -0.5, 2.0, 7.0)
+
+       SStot = sum((obs - mean(obs))^2)
+       SSreg = sum((preds - mean(obs))^2)
+       SSerr = sum((obs - preds)^2)
+
+       explainedVariance = SSreg / length(obs)
+       explainedVariance
+       > [1] 8.796875
+       meanAbsoluteError = mean(abs(preds - obs))
+       meanAbsoluteError
+       > [1] 0.5
+       meanSquaredError = mean((preds - obs)^2)
+       meanSquaredError
+       > [1] 0.3125
+       rmse = sqrt(meanSquaredError)
+       rmse
+       > [1] 0.559017
+       r2 = 1 - SSerr / SStot
+       r2
+       > [1] 0.9571734
+     */
+    val predictionAndObservations = sc.parallelize(
+      Seq((2.25, 3.0), (-0.25, -0.5), (1.75, 2.0), (7.75, 7.0)), 2)
+    val metrics = new RegressionMetrics(predictionAndObservations)
+    assert(metrics.explainedVariance ~== 8.79687 absTol 1E-5,
+      "explained variance regression score mismatch")
+    assert(metrics.meanAbsoluteError ~== 0.5 absTol 1E-5, "mean absolute error mismatch")
+    assert(metrics.meanSquaredError ~== 0.3125 absTol 1E-5, "mean squared error mismatch")
+    assert(metrics.rootMeanSquaredError ~== 0.55901 absTol 1E-5,
+      "root mean squared error mismatch")
+    assert(metrics.r2 ~== 0.95717 absTol 1E-5, "r2 score mismatch")
+  }
+
+  test("regression metrics for biased (no intercept term) predictor") {
+    /* Verify results in R:
+       preds = c(2.5, 0.0, 2.0, 8.0)
+       obs = c(3.0, -0.5, 2.0, 7.0)
+
+       SStot = sum((obs - mean(obs))^2)
+       SSreg = sum((preds - mean(obs))^2)
+       SSerr = sum((obs - preds)^2)
+
+       explainedVariance = SSreg / length(obs)
+       explainedVariance
+       > [1] 8.859375
+       meanAbsoluteError = mean(abs(preds - obs))
+       meanAbsoluteError
+       > [1] 0.5
+       meanSquaredError = mean((preds - obs)^2)
+       meanSquaredError
+       > [1] 0.375
+       rmse = sqrt(meanSquaredError)
+       rmse
+       > [1] 0.6123724
+       r2 = 1 - SSerr / SStot
+       r2
+       > [1] 0.9486081
+     */
     val predictionAndObservations = sc.parallelize(
       Seq((2.5, 3.0), (0.0, -0.5), (2.0, 2.0), (8.0, 7.0)), 2)
     val metrics = new RegressionMetrics(predictionAndObservations)
-    assert(metrics.explainedVariance ~== 0.95717 absTol 1E-5,
+    assert(metrics.explainedVariance ~== 8.85937 absTol 1E-5,
       "explained variance regression score mismatch")
     assert(metrics.meanAbsoluteError ~== 0.5 absTol 1E-5, "mean absolute error mismatch")
     assert(metrics.meanSquaredError ~== 0.375 absTol 1E-5, "mean squared error mismatch")
     assert(metrics.rootMeanSquaredError ~== 0.61237 absTol 1E-5,
       "root mean squared error mismatch")
-    assert(metrics.r2 ~== 0.94861 absTol 1E-5, "r2 score mismatch")
+    assert(metrics.r2 ~== 0.94860 absTol 1E-5, "r2 score mismatch")
   }
 
   test("regression metrics with complete fitting") {
     val predictionAndObservations = sc.parallelize(
       Seq((3.0, 3.0), (0.0, 0.0), (2.0, 2.0), (8.0, 8.0)), 2)
     val metrics = new RegressionMetrics(predictionAndObservations)
-    assert(metrics.explainedVariance ~== 1.0 absTol 1E-5,
+    assert(metrics.explainedVariance ~== 8.6875 absTol 1E-5,
       "explained variance regression score mismatch")
     assert(metrics.meanAbsoluteError ~== 0.0 absTol 1E-5, "mean absolute error mismatch")
     assert(metrics.meanSquaredError ~== 0.0 absTol 1E-5, "mean squared error mismatch")

python/pyspark/mllib/evaluation.py

@@ -82,7 +82,7 @@ class RegressionMetrics(JavaModelWrapper):
     ...     (2.5, 3.0), (0.0, -0.5), (2.0, 2.0), (8.0, 7.0)])
     >>> metrics = RegressionMetrics(predictionAndObservations)
     >>> metrics.explainedVariance
-    0.95...
+    8.859...
     >>> metrics.meanAbsoluteError
     0.5...
     >>> metrics.meanSquaredError