[SPARK-11473][ML] R-like summary statistics with intercept for OLS via normal equation solver

Follow up [SPARK-9836](https://issues.apache.org/jira/browse/SPARK-9836), we should also support summary statistics for ```intercept```.

Author: Yanbo Liang <ybliang8@gmail.com>

Closes #9485 from yanboliang/spark-11473.

2 files changed, 32 insertions, 25 deletions

diff --git a/mllib/src/main/scala/org/apache/spark/ml/optim/WeightedLeastSquares.scala b/mllib/src/main/scala/org/apache/spark/ml/optim/WeightedLeastSquares.scala
index e612a2122e..8617722ae5 100644
--- a/mllib/src/main/scala/org/apache/spark/ml/optim/WeightedLeastSquares.scala
+++ b/mllib/src/main/scala/org/apache/spark/ml/optim/WeightedLeastSquares.scala
@@ -75,7 +75,7 @@ private[ml] class WeightedLeastSquares(
     val summary = instances.treeAggregate(new Aggregator)(_.add(_), _.merge(_))
     summary.validate()
     logInfo(s"Number of instances: ${summary.count}.")
-    val k = summary.k
+    val k = if (fitIntercept) summary.k + 1 else summary.k
     val triK = summary.triK
     val wSum = summary.wSum
     val bBar = summary.bBar
@@ -86,14 +86,6 @@ private[ml] class WeightedLeastSquares(
     val aaBar = summary.aaBar
     val aaValues = aaBar.values
 
-    if (fitIntercept) {
-      // shift centers
-      // A^T A - aBar aBar^T
-      BLAS.spr(-1.0, aBar, aaValues)
-      // A^T b - bBar aBar
-      BLAS.axpy(-bBar, aBar, abBar)
-    }
-
     // add regularization to diagonals
     var i = 0
     var j = 2
@@ -111,21 +103,32 @@ private[ml] class WeightedLeastSquares(
       j += 1
     }
 
-    val x = new DenseVector(CholeskyDecomposition.solve(aaBar.values, abBar.values))
+    val aa = if (fitIntercept) {
+      Array.concat(aaBar.values, aBar.values, Array(1.0))
+    } else {
+      aaBar.values
+    }
+    val ab = if (fitIntercept) {
+      Array.concat(abBar.values, Array(bBar))
+    } else {
+      abBar.values
+    }
+
+    val x = CholeskyDecomposition.solve(aa, ab)
+
+    val aaInv = CholeskyDecomposition.inverse(aa, k)
 
-    val aaInv = CholeskyDecomposition.inverse(aaBar.values, k)
     // aaInv is a packed upper triangular matrix, here we get all elements on diagonal
     val diagInvAtWA = new DenseVector((1 to k).map { i =>
       aaInv(i + (i - 1) * i / 2 - 1) / wSum }.toArray)
 
-    // compute intercept
-    val intercept = if (fitIntercept) {
-      bBar - BLAS.dot(aBar, x)
+    val (coefficients, intercept) = if (fitIntercept) {
+      (new DenseVector(x.slice(0, x.length - 1)), x.last)
     } else {
-      0.0
+      (new DenseVector(x), 0.0)
     }
 
-    new WeightedLeastSquaresModel(x, intercept, diagInvAtWA)
+    new WeightedLeastSquaresModel(coefficients, intercept, diagInvAtWA)
   }
 }
 
diff --git a/mllib/src/main/scala/org/apache/spark/ml/regression/LinearRegression.scala b/mllib/src/main/scala/org/apache/spark/ml/regression/LinearRegression.scala
index c51e30483a..6638313818 100644
--- a/mllib/src/main/scala/org/apache/spark/ml/regression/LinearRegression.scala
+++ b/mllib/src/main/scala/org/apache/spark/ml/regression/LinearRegression.scala
@@ -511,8 +511,7 @@ class LinearRegressionSummary private[regression] (
   }
 
   /**
-   * Standard error of estimated coefficients.
-   * Note that standard error of estimated intercept is not supported currently.
+   * Standard error of estimated coefficients and intercept.
    */
   lazy val coefficientStandardErrors: Array[Double] = {
     if (diagInvAtWA.length == 1 && diagInvAtWA(0) == 0) {
@@ -532,21 +531,26 @@ class LinearRegressionSummary private[regression] (
     }
   }
 
-  /** T-statistic of estimated coefficients.
-    * Note that t-statistic of estimated intercept is not supported currently.
-    */
+  /**
+   * T-statistic of estimated coefficients and intercept.
+   */
   lazy val tValues: Array[Double] = {
     if (diagInvAtWA.length == 1 && diagInvAtWA(0) == 0) {
       throw new UnsupportedOperationException(
         "No t-statistic available for this LinearRegressionModel")
     } else {
-      model.coefficients.toArray.zip(coefficientStandardErrors).map { x => x._1 / x._2 }
+      val estimate = if (model.getFitIntercept) {
+        Array.concat(model.coefficients.toArray, Array(model.intercept))
+      } else {
+        model.coefficients.toArray
+      }
+      estimate.zip(coefficientStandardErrors).map { x => x._1 / x._2 }
     }
   }
 
-  /** Two-sided p-value of estimated coefficients.
-    * Note that p-value of estimated intercept is not supported currently.
-    */
+  /**
+   * Two-sided p-value of estimated coefficients and intercept.
+   */
   lazy val pValues: Array[Double] = {
     if (diagInvAtWA.length == 1 && diagInvAtWA(0) == 0) {
       throw new UnsupportedOperationException(