SPARK-2272 [MLlib] Feature scaling which standardizes the range of independent variables or features of data

Feature scaling is a method used to standardize the range of independent variables or features of data. In data processing, it is generally performed during the data preprocessing step.

In this work, a trait called `VectorTransformer` is defined for generic transformation on a vector. It contains one method to be implemented, `transform` which applies transformation on a vector.

There are two implementations of `VectorTransformer` now, and they all can be easily extended with PMML transformation support.

1) `StandardScaler` - Standardizes features by removing the mean and scaling to unit variance using column summary statistics on the samples in the training set.

2) `Normalizer` - Normalizes samples individually to unit L^n norm

Author: DB Tsai <dbtsai@alpinenow.com>

Closes #1207 from dbtsai/dbtsai-feature-scaling and squashes the following commits:

78c15d3 [DB Tsai] Alpine Data Labs

6 files changed, 567 insertions, 1 deletions

diff --git a/mllib/src/main/scala/org/apache/spark/mllib/feature/Normalizer.scala b/mllib/src/main/scala/org/apache/spark/mllib/feature/Normalizer.scala
new file mode 100644
index 0000000000..ea9fd0a80d
--- /dev/null
+++ b/mllib/src/main/scala/org/apache/spark/mllib/feature/Normalizer.scala
@@ -0,0 +1,76 @@
+/*
+ * Licensed to the Apache Software Foundation (ASF) under one or more
+ * contributor license agreements.  See the NOTICE file distributed with
+ * this work for additional information regarding copyright ownership.
+ * The ASF licenses this file to You under the Apache License, Version 2.0
+ * (the "License"); you may not use this file except in compliance with
+ * the License.  You may obtain a copy of the License at
+ *
+ *    http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing, software
+ * distributed under the License is distributed on an "AS IS" BASIS,
+ * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ * See the License for the specific language governing permissions and
+ * limitations under the License.
+ */
+
+package org.apache.spark.mllib.feature
+
+import breeze.linalg.{DenseVector => BDV, SparseVector => BSV}
+
+import org.apache.spark.annotation.DeveloperApi
+import org.apache.spark.mllib.linalg.{Vector, Vectors}
+
+/**
+ * :: DeveloperApi ::
+ * Normalizes samples individually to unit L^p^ norm
+ *
+ * For any 1 <= p < Double.PositiveInfinity, normalizes samples using
+ * sum(abs(vector).^p^)^(1/p)^ as norm.
+ *
+ * For p = Double.PositiveInfinity, max(abs(vector)) will be used as norm for normalization.
+ *
+ * @param p Normalization in L^p^ space, p = 2 by default.
+ */
+@DeveloperApi
+class Normalizer(p: Double) extends VectorTransformer {
+
+  def this() = this(2)
+
+  require(p >= 1.0)
+
+  /**
+   * Applies unit length normalization on a vector.
+   *
+   * @param vector vector to be normalized.
+   * @return normalized vector. If the norm of the input is zero, it will return the input vector.
+   */
+  override def transform(vector: Vector): Vector = {
+    var norm = vector.toBreeze.norm(p)
+
+    if (norm != 0.0) {
+      // For dense vector, we've to allocate new memory for new output vector.
+      // However, for sparse vector, the `index` array will not be changed,
+      // so we can re-use it to save memory.
+      vector.toBreeze match {
+        case dv: BDV[Double] => Vectors.fromBreeze(dv :/ norm)
+        case sv: BSV[Double] =>
+          val output = new BSV[Double](sv.index, sv.data.clone(), sv.length)
+          var i = 0
+          while (i < output.data.length) {
+            output.data(i) /= norm
+            i += 1
+          }
+          Vectors.fromBreeze(output)
+        case v => throw new IllegalArgumentException("Do not support vector type " + v.getClass)
+      }
+    } else {
+      // Since the norm is zero, return the input vector object itself.
+      // Note that it's safe since we always assume that the data in RDD
+      // should be immutable.
+      vector
+    }
+  }
+
+}
diff --git a/mllib/src/main/scala/org/apache/spark/mllib/feature/StandardScaler.scala b/mllib/src/main/scala/org/apache/spark/mllib/feature/StandardScaler.scala
new file mode 100644
index 0000000000..cc2d7579c2
--- /dev/null
+++ b/mllib/src/main/scala/org/apache/spark/mllib/feature/StandardScaler.scala
@@ -0,0 +1,119 @@
+/*
+ * Licensed to the Apache Software Foundation (ASF) under one or more
+ * contributor license agreements.  See the NOTICE file distributed with
+ * this work for additional information regarding copyright ownership.
+ * The ASF licenses this file to You under the Apache License, Version 2.0
+ * (the "License"); you may not use this file except in compliance with
+ * the License.  You may obtain a copy of the License at
+ *
+ *    http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing, software
+ * distributed under the License is distributed on an "AS IS" BASIS,
+ * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ * See the License for the specific language governing permissions and
+ * limitations under the License.
+ */
+
+package org.apache.spark.mllib.feature
+
+import breeze.linalg.{DenseVector => BDV, SparseVector => BSV, Vector => BV}
+
+import org.apache.spark.annotation.DeveloperApi
+import org.apache.spark.mllib.linalg.{Vector, Vectors}
+import org.apache.spark.mllib.rdd.RDDFunctions._
+import org.apache.spark.mllib.stat.MultivariateOnlineSummarizer
+import org.apache.spark.rdd.RDD
+
+/**
+ * :: DeveloperApi ::
+ * Standardizes features by removing the mean and scaling to unit variance using column summary
+ * statistics on the samples in the training set.
+ *
+ * @param withMean False by default. Centers the data with mean before scaling. It will build a
+ *                 dense output, so this does not work on sparse input and will raise an exception.
+ * @param withStd True by default. Scales the data to unit standard deviation.
+ */
+@DeveloperApi
+class StandardScaler(withMean: Boolean, withStd: Boolean) extends VectorTransformer {
+
+  def this() = this(false, true)
+
+  require(withMean || withStd, s"withMean and withStd both equal to false. Doing nothing.")
+
+  private var mean: BV[Double] = _
+  private var factor: BV[Double] = _
+
+  /**
+   * Computes the mean and variance and stores as a model to be used for later scaling.
+   *
+   * @param data The data used to compute the mean and variance to build the transformation model.
+   * @return This StandardScalar object.
+   */
+  def fit(data: RDD[Vector]): this.type = {
+    val summary = data.treeAggregate(new MultivariateOnlineSummarizer)(
+      (aggregator, data) => aggregator.add(data),
+      (aggregator1, aggregator2) => aggregator1.merge(aggregator2))
+
+    mean = summary.mean.toBreeze
+    factor = summary.variance.toBreeze
+    require(mean.length == factor.length)
+
+    var i = 0
+    while (i < factor.length) {
+      factor(i) = if (factor(i) != 0.0) 1.0 / math.sqrt(factor(i)) else 0.0
+      i += 1
+    }
+
+    this
+  }
+
+  /**
+   * Applies standardization transformation on a vector.
+   *
+   * @param vector Vector to be standardized.
+   * @return Standardized vector. If the variance of a column is zero, it will return default `0.0`
+   *         for the column with zero variance.
+   */
+  override def transform(vector: Vector): Vector = {
+    if (mean == null || factor == null) {
+      throw new IllegalStateException(
+        "Haven't learned column summary statistics yet. Call fit first.")
+    }
+
+    require(vector.size == mean.length)
+
+    if (withMean) {
+      vector.toBreeze match {
+        case dv: BDV[Double] =>
+          val output = vector.toBreeze.copy
+          var i = 0
+          while (i < output.length) {
+            output(i) = (output(i) - mean(i)) * (if (withStd) factor(i) else 1.0)
+            i += 1
+          }
+          Vectors.fromBreeze(output)
+        case v => throw new IllegalArgumentException("Do not support vector type " + v.getClass)
+      }
+    } else if (withStd) {
+      vector.toBreeze match {
+        case dv: BDV[Double] => Vectors.fromBreeze(dv :* factor)
+        case sv: BSV[Double] =>
+          // For sparse vector, the `index` array inside sparse vector object will not be changed,
+          // so we can re-use it to save memory.
+          val output = new BSV[Double](sv.index, sv.data.clone(), sv.length)
+          var i = 0
+          while (i < output.data.length) {
+            output.data(i) *= factor(output.index(i))
+            i += 1
+          }
+          Vectors.fromBreeze(output)
+        case v => throw new IllegalArgumentException("Do not support vector type " + v.getClass)
+      }
+    } else {
+      // Note that it's safe since we always assume that the data in RDD should be immutable.
+      vector
+    }
+  }
+
+}
diff --git a/mllib/src/main/scala/org/apache/spark/mllib/feature/VectorTransformer.scala b/mllib/src/main/scala/org/apache/spark/mllib/feature/VectorTransformer.scala
new file mode 100644
index 0000000000..415a845332
--- /dev/null
+++ b/mllib/src/main/scala/org/apache/spark/mllib/feature/VectorTransformer.scala
@@ -0,0 +1,51 @@
+/*
+ * Licensed to the Apache Software Foundation (ASF) under one or more
+ * contributor license agreements.  See the NOTICE file distributed with
+ * this work for additional information regarding copyright ownership.
+ * The ASF licenses this file to You under the Apache License, Version 2.0
+ * (the "License"); you may not use this file except in compliance with
+ * the License.  You may obtain a copy of the License at
+ *
+ *    http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing, software
+ * distributed under the License is distributed on an "AS IS" BASIS,
+ * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ * See the License for the specific language governing permissions and
+ * limitations under the License.
+ */
+
+package org.apache.spark.mllib.feature
+
+import org.apache.spark.annotation.DeveloperApi
+import org.apache.spark.mllib.linalg.Vector
+import org.apache.spark.rdd.RDD
+
+/**
+ * :: DeveloperApi ::
+ * Trait for transformation of a vector
+ */
+@DeveloperApi
+trait VectorTransformer extends Serializable {
+
+  /**
+   * Applies transformation on a vector.
+   *
+   * @param vector vector to be transformed.
+   * @return transformed vector.
+   */
+  def transform(vector: Vector): Vector
+
+  /**
+   * Applies transformation on an RDD[Vector].
+   *
+   * @param data RDD[Vector] to be transformed.
+   * @return transformed RDD[Vector].
+   */
+  def transform(data: RDD[Vector]): RDD[Vector] = {
+    // Later in #1498 , all RDD objects are sent via broadcasting instead of akka.
+    // So it should be no longer necessary to explicitly broadcast `this` object.
+    data.map(x => this.transform(x))
+  }
+
+}
diff --git a/mllib/src/main/scala/org/apache/spark/mllib/linalg/distributed/RowMatrix.scala b/mllib/src/main/scala/org/apache/spark/mllib/linalg/distributed/RowMatrix.scala
index 58c1322757..45486b2c7d 100644
--- a/mllib/src/main/scala/org/apache/spark/mllib/linalg/distributed/RowMatrix.scala
+++ b/mllib/src/main/scala/org/apache/spark/mllib/linalg/distributed/RowMatrix.scala
@@ -19,7 +19,7 @@ package org.apache.spark.mllib.linalg.distributed
 
 import java.util.Arrays
 
-import breeze.linalg.{Vector => BV, DenseMatrix => BDM, DenseVector => BDV, SparseVector => BSV}
+import breeze.linalg.{DenseMatrix => BDM, DenseVector => BDV, SparseVector => BSV}
 import breeze.linalg.{svd => brzSvd, axpy => brzAxpy}
 import breeze.numerics.{sqrt => brzSqrt}
 import com.github.fommil.netlib.BLAS.{getInstance => blas}
diff --git a/mllib/src/test/scala/org/apache/spark/mllib/feature/NormalizerSuite.scala b/mllib/src/test/scala/org/apache/spark/mllib/feature/NormalizerSuite.scala
new file mode 100644
index 0000000000..fb76dccfdf
--- /dev/null
+++ b/mllib/src/test/scala/org/apache/spark/mllib/feature/NormalizerSuite.scala
@@ -0,0 +1,120 @@
+/*
+ * Licensed to the Apache Software Foundation (ASF) under one or more
+ * contributor license agreements.  See the NOTICE file distributed with
+ * this work for additional information regarding copyright ownership.
+ * The ASF licenses this file to You under the Apache License, Version 2.0
+ * (the "License"); you may not use this file except in compliance with
+ * the License.  You may obtain a copy of the License at
+ *
+ *    http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing, software
+ * distributed under the License is distributed on an "AS IS" BASIS,
+ * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ * See the License for the specific language governing permissions and
+ * limitations under the License.
+ */
+
+package org.apache.spark.mllib.feature
+
+import org.scalatest.FunSuite
+
+import org.apache.spark.mllib.linalg.{DenseVector, SparseVector, Vectors}
+import org.apache.spark.mllib.util.LocalSparkContext
+import org.apache.spark.mllib.util.TestingUtils._
+
+class NormalizerSuite extends FunSuite with LocalSparkContext {
+
+  val data = Array(
+    Vectors.sparse(3, Seq((0, -2.0), (1, 2.3))),
+    Vectors.dense(0.0, 0.0, 0.0),
+    Vectors.dense(0.6, -1.1, -3.0),
+    Vectors.sparse(3, Seq((1, 0.91), (2, 3.2))),
+    Vectors.sparse(3, Seq((0, 5.7), (1, 0.72), (2, 2.7))),
+    Vectors.sparse(3, Seq())
+  )
+
+  lazy val dataRDD = sc.parallelize(data, 3)
+
+  test("Normalization using L1 distance") {
+    val l1Normalizer = new Normalizer(1)
+
+    val data1 = data.map(l1Normalizer.transform)
+    val data1RDD = l1Normalizer.transform(dataRDD)
+
+    assert((data, data1, data1RDD.collect()).zipped.forall {
+      case (v1: DenseVector, v2: DenseVector, v3: DenseVector) => true
+      case (v1: SparseVector, v2: SparseVector, v3: SparseVector) => true
+      case _ => false
+    }, "The vector type should be preserved after normalization.")
+
+    assert((data1, data1RDD.collect()).zipped.forall((v1, v2) => v1 ~== v2 absTol 1E-5))
+
+    assert(data1(0).toBreeze.norm(1) ~== 1.0 absTol 1E-5)
+    assert(data1(2).toBreeze.norm(1) ~== 1.0 absTol 1E-5)
+    assert(data1(3).toBreeze.norm(1) ~== 1.0 absTol 1E-5)
+    assert(data1(4).toBreeze.norm(1) ~== 1.0 absTol 1E-5)
+
+    assert(data1(0) ~== Vectors.sparse(3, Seq((0, -0.465116279), (1, 0.53488372))) absTol 1E-5)
+    assert(data1(1) ~== Vectors.dense(0.0, 0.0, 0.0) absTol 1E-5)
+    assert(data1(2) ~== Vectors.dense(0.12765957, -0.23404255, -0.63829787) absTol 1E-5)
+    assert(data1(3) ~== Vectors.sparse(3, Seq((1, 0.22141119), (2, 0.7785888))) absTol 1E-5)
+    assert(data1(4) ~== Vectors.dense(0.625, 0.07894737, 0.29605263) absTol 1E-5)
+    assert(data1(5) ~== Vectors.sparse(3, Seq()) absTol 1E-5)
+  }
+
+  test("Normalization using L2 distance") {
+    val l2Normalizer = new Normalizer()
+
+    val data2 = data.map(l2Normalizer.transform)
+    val data2RDD = l2Normalizer.transform(dataRDD)
+
+    assert((data, data2, data2RDD.collect()).zipped.forall {
+      case (v1: DenseVector, v2: DenseVector, v3: DenseVector) => true
+      case (v1: SparseVector, v2: SparseVector, v3: SparseVector) => true
+      case _ => false
+    }, "The vector type should be preserved after normalization.")
+
+    assert((data2, data2RDD.collect()).zipped.forall((v1, v2) => v1 ~== v2 absTol 1E-5))
+
+    assert(data2(0).toBreeze.norm(2) ~== 1.0 absTol 1E-5)
+    assert(data2(2).toBreeze.norm(2) ~== 1.0 absTol 1E-5)
+    assert(data2(3).toBreeze.norm(2) ~== 1.0 absTol 1E-5)
+    assert(data2(4).toBreeze.norm(2) ~== 1.0 absTol 1E-5)
+
+    assert(data2(0) ~== Vectors.sparse(3, Seq((0, -0.65617871), (1, 0.75460552))) absTol 1E-5)
+    assert(data2(1) ~== Vectors.dense(0.0, 0.0, 0.0) absTol 1E-5)
+    assert(data2(2) ~== Vectors.dense(0.184549876, -0.3383414, -0.922749378) absTol 1E-5)
+    assert(data2(3) ~== Vectors.sparse(3, Seq((1, 0.27352993), (2, 0.96186349))) absTol 1E-5)
+    assert(data2(4) ~== Vectors.dense(0.897906166, 0.113419726, 0.42532397) absTol 1E-5)
+    assert(data2(5) ~== Vectors.sparse(3, Seq()) absTol 1E-5)
+  }
+
+  test("Normalization using L^Inf distance.") {
+    val lInfNormalizer = new Normalizer(Double.PositiveInfinity)
+
+    val dataInf = data.map(lInfNormalizer.transform)
+    val dataInfRDD = lInfNormalizer.transform(dataRDD)
+
+    assert((data, dataInf, dataInfRDD.collect()).zipped.forall {
+      case (v1: DenseVector, v2: DenseVector, v3: DenseVector) => true
+      case (v1: SparseVector, v2: SparseVector, v3: SparseVector) => true
+      case _ => false
+    }, "The vector type should be preserved after normalization.")
+
+    assert((dataInf, dataInfRDD.collect()).zipped.forall((v1, v2) => v1 ~== v2 absTol 1E-5))
+
+    assert(dataInf(0).toArray.map(Math.abs).max ~== 1.0 absTol 1E-5)
+    assert(dataInf(2).toArray.map(Math.abs).max ~== 1.0 absTol 1E-5)
+    assert(dataInf(3).toArray.map(Math.abs).max ~== 1.0 absTol 1E-5)
+    assert(dataInf(4).toArray.map(Math.abs).max ~== 1.0 absTol 1E-5)
+
+    assert(dataInf(0) ~== Vectors.sparse(3, Seq((0, -0.86956522), (1, 1.0))) absTol 1E-5)
+    assert(dataInf(1) ~== Vectors.dense(0.0, 0.0, 0.0) absTol 1E-5)
+    assert(dataInf(2) ~== Vectors.dense(0.2, -0.36666667, -1.0) absTol 1E-5)
+    assert(dataInf(3) ~== Vectors.sparse(3, Seq((1, 0.284375), (2, 1.0))) absTol 1E-5)
+    assert(dataInf(4) ~== Vectors.dense(1.0, 0.12631579, 0.473684211) absTol 1E-5)
+    assert(dataInf(5) ~== Vectors.sparse(3, Seq()) absTol 1E-5)
+  }
+
+}
diff --git a/mllib/src/test/scala/org/apache/spark/mllib/feature/StandardScalerSuite.scala b/mllib/src/test/scala/org/apache/spark/mllib/feature/StandardScalerSuite.scala
new file mode 100644
index 0000000000..5a9be923a8
--- /dev/null
+++ b/mllib/src/test/scala/org/apache/spark/mllib/feature/StandardScalerSuite.scala
@@ -0,0 +1,200 @@
+/*
+ * Licensed to the Apache Software Foundation (ASF) under one or more
+ * contributor license agreements.  See the NOTICE file distributed with
+ * this work for additional information regarding copyright ownership.
+ * The ASF licenses this file to You under the Apache License, Version 2.0
+ * (the "License"); you may not use this file except in compliance with
+ * the License.  You may obtain a copy of the License at
+ *
+ *    http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing, software
+ * distributed under the License is distributed on an "AS IS" BASIS,
+ * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ * See the License for the specific language governing permissions and
+ * limitations under the License.
+ */
+
+package org.apache.spark.mllib.feature
+
+import org.scalatest.FunSuite
+
+import org.apache.spark.mllib.linalg.{DenseVector, SparseVector, Vector, Vectors}
+import org.apache.spark.mllib.util.LocalSparkContext
+import org.apache.spark.mllib.util.TestingUtils._
+import org.apache.spark.mllib.rdd.RDDFunctions._
+import org.apache.spark.mllib.stat.{MultivariateStatisticalSummary, MultivariateOnlineSummarizer}
+import org.apache.spark.rdd.RDD
+
+class StandardScalerSuite extends FunSuite with LocalSparkContext {
+
+  private def computeSummary(data: RDD[Vector]): MultivariateStatisticalSummary = {
+    data.treeAggregate(new MultivariateOnlineSummarizer)(
+      (aggregator, data) => aggregator.add(data),
+      (aggregator1, aggregator2) => aggregator1.merge(aggregator2))
+  }
+
+  test("Standardization with dense input") {
+    val data = Array(
+      Vectors.dense(-2.0, 2.3, 0),
+      Vectors.dense(0.0, -1.0, -3.0),
+      Vectors.dense(0.0, -5.1, 0.0),
+      Vectors.dense(3.8, 0.0, 1.9),
+      Vectors.dense(1.7, -0.6, 0.0),
+      Vectors.dense(0.0, 1.9, 0.0)
+    )
+
+    val dataRDD = sc.parallelize(data, 3)
+
+    val standardizer1 = new StandardScaler(withMean = true, withStd = true)
+    val standardizer2 = new StandardScaler()
+    val standardizer3 = new StandardScaler(withMean = true, withStd = false)
+
+    withClue("Using a standardizer before fitting the model should throw exception.") {
+      intercept[IllegalStateException] {
+        data.map(standardizer1.transform)
+      }
+    }
+
+    standardizer1.fit(dataRDD)
+    standardizer2.fit(dataRDD)
+    standardizer3.fit(dataRDD)
+
+    val data1 = data.map(standardizer1.transform)
+    val data2 = data.map(standardizer2.transform)
+    val data3 = data.map(standardizer3.transform)
+
+    val data1RDD = standardizer1.transform(dataRDD)
+    val data2RDD = standardizer2.transform(dataRDD)
+    val data3RDD = standardizer3.transform(dataRDD)
+
+    val summary = computeSummary(dataRDD)
+    val summary1 = computeSummary(data1RDD)
+    val summary2 = computeSummary(data2RDD)
+    val summary3 = computeSummary(data3RDD)
+
+    assert((data, data1, data1RDD.collect()).zipped.forall {
+      case (v1: DenseVector, v2: DenseVector, v3: DenseVector) => true
+      case (v1: SparseVector, v2: SparseVector, v3: SparseVector) => true
+      case _ => false
+    }, "The vector type should be preserved after standardization.")
+
+    assert((data, data2, data2RDD.collect()).zipped.forall {
+      case (v1: DenseVector, v2: DenseVector, v3: DenseVector) => true
+      case (v1: SparseVector, v2: SparseVector, v3: SparseVector) => true
+      case _ => false
+    }, "The vector type should be preserved after standardization.")
+
+    assert((data, data3, data3RDD.collect()).zipped.forall {
+      case (v1: DenseVector, v2: DenseVector, v3: DenseVector) => true
+      case (v1: SparseVector, v2: SparseVector, v3: SparseVector) => true
+      case _ => false
+    }, "The vector type should be preserved after standardization.")
+
+    assert((data1, data1RDD.collect()).zipped.forall((v1, v2) => v1 ~== v2 absTol 1E-5))
+    assert((data2, data2RDD.collect()).zipped.forall((v1, v2) => v1 ~== v2 absTol 1E-5))
+    assert((data3, data3RDD.collect()).zipped.forall((v1, v2) => v1 ~== v2 absTol 1E-5))
+
+    assert(summary1.mean ~== Vectors.dense(0.0, 0.0, 0.0) absTol 1E-5)
+    assert(summary1.variance ~== Vectors.dense(1.0, 1.0, 1.0) absTol 1E-5)
+
+    assert(summary2.mean !~== Vectors.dense(0.0, 0.0, 0.0) absTol 1E-5)
+    assert(summary2.variance ~== Vectors.dense(1.0, 1.0, 1.0) absTol 1E-5)
+
+    assert(summary3.mean ~== Vectors.dense(0.0, 0.0, 0.0) absTol 1E-5)
+    assert(summary3.variance ~== summary.variance absTol 1E-5)
+
+    assert(data1(0) ~== Vectors.dense(-1.31527964, 1.023470449, 0.11637768424) absTol 1E-5)
+    assert(data1(3) ~== Vectors.dense(1.637735298, 0.156973995, 1.32247368462) absTol 1E-5)
+    assert(data2(4) ~== Vectors.dense(0.865538862, -0.22604255, 0.0) absTol 1E-5)
+    assert(data2(5) ~== Vectors.dense(0.0, 0.71580142, 0.0) absTol 1E-5)
+    assert(data3(1) ~== Vectors.dense(-0.58333333, -0.58333333, -2.8166666666) absTol 1E-5)
+    assert(data3(5) ~== Vectors.dense(-0.58333333, 2.316666666, 0.18333333333) absTol 1E-5)
+  }
+
+
+  test("Standardization with sparse input") {
+    val data = Array(
+      Vectors.sparse(3, Seq((0, -2.0), (1, 2.3))),
+      Vectors.sparse(3, Seq((1, -1.0), (2, -3.0))),
+      Vectors.sparse(3, Seq((1, -5.1))),
+      Vectors.sparse(3, Seq((0, 3.8), (2, 1.9))),
+      Vectors.sparse(3, Seq((0, 1.7), (1, -0.6))),
+      Vectors.sparse(3, Seq((1, 1.9)))
+    )
+
+    val dataRDD = sc.parallelize(data, 3)
+
+    val standardizer1 = new StandardScaler(withMean = true, withStd = true)
+    val standardizer2 = new StandardScaler()
+    val standardizer3 = new StandardScaler(withMean = true, withStd = false)
+
+    standardizer1.fit(dataRDD)
+    standardizer2.fit(dataRDD)
+    standardizer3.fit(dataRDD)
+
+    val data2 = data.map(standardizer2.transform)
+
+    withClue("Standardization with mean can not be applied on sparse input.") {
+      intercept[IllegalArgumentException] {
+        data.map(standardizer1.transform)
+      }
+    }
+
+    withClue("Standardization with mean can not be applied on sparse input.") {
+      intercept[IllegalArgumentException] {
+        data.map(standardizer3.transform)
+      }
+    }
+
+    val data2RDD = standardizer2.transform(dataRDD)
+
+    val summary2 = computeSummary(data2RDD)
+
+    assert((data, data2, data2RDD.collect()).zipped.forall {
+      case (v1: DenseVector, v2: DenseVector, v3: DenseVector) => true
+      case (v1: SparseVector, v2: SparseVector, v3: SparseVector) => true
+      case _ => false
+    }, "The vector type should be preserved after standardization.")
+
+    assert((data2, data2RDD.collect()).zipped.forall((v1, v2) => v1 ~== v2 absTol 1E-5))
+
+    assert(summary2.mean !~== Vectors.dense(0.0, 0.0, 0.0) absTol 1E-5)
+    assert(summary2.variance ~== Vectors.dense(1.0, 1.0, 1.0) absTol 1E-5)
+
+    assert(data2(4) ~== Vectors.sparse(3, Seq((0, 0.865538862), (1, -0.22604255))) absTol 1E-5)
+    assert(data2(5) ~== Vectors.sparse(3, Seq((1, 0.71580142))) absTol 1E-5)
+  }
+
+  test("Standardization with constant input") {
+    // When the input data is all constant, the variance is zero. The standardization against
+    // zero variance is not well-defined, but we decide to just set it into zero here.
+    val data = Array(
+      Vectors.dense(2.0),
+      Vectors.dense(2.0),
+      Vectors.dense(2.0)
+    )
+
+    val dataRDD = sc.parallelize(data, 2)
+
+    val standardizer1 = new StandardScaler(withMean = true, withStd = true)
+    val standardizer2 = new StandardScaler(withMean = true, withStd = false)
+    val standardizer3 = new StandardScaler(withMean = false, withStd = true)
+
+    standardizer1.fit(dataRDD)
+    standardizer2.fit(dataRDD)
+    standardizer3.fit(dataRDD)
+
+    val data1 = data.map(standardizer1.transform)
+    val data2 = data.map(standardizer2.transform)
+    val data3 = data.map(standardizer3.transform)
+
+    assert(data1.forall(_.toArray.forall(_ == 0.0)),
+      "The variance is zero, so the transformed result should be 0.0")
+    assert(data2.forall(_.toArray.forall(_ == 0.0)),
+      "The variance is zero, so the transformed result should be 0.0")
+    assert(data3.forall(_.toArray.forall(_ == 0.0)),
+      "The variance is zero, so the transformed result should be 0.0")
+  }
+
+}