SparkML之回归(三)保序回归

在写這篇博客的时候，翻阅了一些互联网上的资料，发现文献[1]写的比较系统。所以推荐大家读读文献[1].但是出现了一些错误，所以我在此简述一些。如果推理不过去了。可以看看我的简述。

------------------------------------前言

背景：

（1）在医学领域药物剂量反应中，随着药物剂量的增加，疗效和副作用会呈现一定趋势。比如剂量越高，疗效越

高，剂量越高，毒性越大等

（2）评估药物在不同剂量水平下的毒性，并且建议一个对病人既安全又有效的剂量称为最大耐受剂量（Maximum Tolerated Dose）简称 MTD。

（3）随着药物的增加，药物的毒性是非减的。MTD被定义为毒性概率不超过毒性靶水平的最高剂量水平

（4）基于每个剂量水平下病人的毒性反应的比率估计不同，剂量水平下的毒性概率可能不是剂量水平的非减函

数，于是我们可以采用保序回归的方法

L2保序回归

L2保序回归算法

一些具体的定义和命题查看文献[1]

Spark源码分析(大图见附录)

/** * 保序回归模型 * * @param boundaries 用于预测的边界数组，它必须是排好顺序的。（分段函数的分段点数组） * @param predictions 保序回归的结果，即分段点x对应的预测值 * @param isotonic 升序还是降序（true为升） */@Since("1.3.0")class IsotonicRegressionModel @Since("1.3.0") (    @Since("1.3.0") val boundaries: Array[Double],    @Since("1.3.0") val predictions: Array[Double],    @Since("1.3.0") val isotonic: Boolean) extends Serializable with Saveable {  private val predictionOrd = if (isotonic) Ordering[Double] else Ordering[Double].reverse  require(boundaries.length == predictions.length)  assertOrdered(boundaries)  assertOrdered(predictions)(predictionOrd)  /**   * A Java-friendly constructor that takes two Iterable parameters and one Boolean parameter.   */  @Since("1.4.0")  def this(boundaries: java.lang.Iterable[Double],      predictions: java.lang.Iterable[Double],      isotonic: java.lang.Boolean) = {    this(boundaries.asScala.toArray, predictions.asScala.toArray, isotonic)  }  /** 序列顺序的检测 */  private def assertOrdered(xs: Array[Double])(implicit ord: Ordering[Double]): Unit = {    var i = 1    val len = xs.length    while (i < len) {      require(ord.compare(xs(i - 1), xs(i)) <= 0,        s"Elements (${xs(i - 1)}, ${xs(i)}) are not ordered.")      i += 1    }  }  /**   * 利用分段函数的线性函数，输入feature进行预测   *   * @param testData Features to be labeled.   * @return Predicted labels.   *   */  @Since("1.3.0")  def predict(testData: RDD[Double]): RDD[Double] = {    testData.map(predict)  }  /**   * 利用分段函数的线性函数，输入feature进行预测   *   * @param testData Features to be labeled.   * @return Predicted labels.   *   */  @Since("1.3.0")  def predict(testData: JavaDoubleRDD): JavaDoubleRDD = {    JavaDoubleRDD.fromRDD(predict(testData.rdd.retag.asInstanceOf[RDD[Double]]))  }  /**   * 利用分段函数的线性函数，输入feature进行预测   *   * @param testData Feature to be labeled.   * @return Predicted label.   *         1) 如果testdata可以精确匹配到一个边界数组，那么就返回对应的数值，如果多个，那么随机返回一个   *         2) 如果testdata 低于或者高于所有的边界数组，那么返回第一个或者最后一个If testData is lower or higher than all boundaries then first or last prediction   *         3) 如果testdat在两个边界数组之间，那么采用分段函数的线性插值方法得到的数值   *   */  @Since("1.3.0")  def predict(testData: Double): Double = {    def linearInterpolation(x1: Double, y1: Double, x2: Double, y2: Double, x: Double): Double = {      y1 + (y2 - y1) * (x - x1) / (x2 - x1)    }    val foundIndex = binarySearch(boundaries, testData)    val insertIndex = -foundIndex - 1    // Find if the index was lower than all values,    // higher than all values, in between two values or exact match.    if (insertIndex == 0) {      predictions.head    } else if (insertIndex == boundaries.length) {      predictions.last    } else if (foundIndex < 0) {      linearInterpolation(        boundaries(insertIndex - 1),        predictions(insertIndex - 1),        boundaries(insertIndex),        predictions(insertIndex),        testData)    } else {      predictions(foundIndex)    }  }  /** A convenient method for boundaries called by the Python API. */  private[mllib] def boundaryVector: Vector = Vectors.dense(boundaries)  /** A convenient method for boundaries called by the Python API. */  private[mllib] def predictionVector: Vector = Vectors.dense(predictions)  @Since("1.4.0")  override def save(sc: SparkContext, path: String): Unit = {    IsotonicRegressionModel.SaveLoadV1_0.save(sc, path, boundaries, predictions, isotonic)  }  override protected def formatVersion: String = "1.0"}@Since("1.4.0")object IsotonicRegressionModel extends Loader[IsotonicRegressionModel] {  import org.apache.spark.mllib.util.Loader._  private object SaveLoadV1_0 {    def thisFormatVersion: String = "1.0"    /** Hard-code class name string in case it changes in the future */    def thisClassName: String = "org.apache.spark.mllib.regression.IsotonicRegressionModel"    /** Model data for model import/export */    case class Data(boundary: Double, prediction: Double)    def save(        sc: SparkContext,        path: String,        boundaries: Array[Double],        predictions: Array[Double],        isotonic: Boolean): Unit = {      val sqlContext = SQLContext.getOrCreate(sc)      val metadata = compact(render(        ("class" -> thisClassName) ~ ("version" -> thisFormatVersion) ~          ("isotonic" -> isotonic)))      sc.parallelize(Seq(metadata), 1).saveAsTextFile(metadataPath(path))      sqlContext.createDataFrame(        boundaries.toSeq.zip(predictions).map { case (b, p) => Data(b, p) }      ).write.parquet(dataPath(path))    }    def load(sc: SparkContext, path: String): (Array[Double], Array[Double]) = {      val sqlContext = SQLContext.getOrCreate(sc)      val dataRDD = sqlContext.read.parquet(dataPath(path))      checkSchema[Data](dataRDD.schema)      val dataArray = dataRDD.select("boundary", "prediction").collect()      val (boundaries, predictions) = dataArray.map { x =>        (x.getDouble(0), x.getDouble(1))      }.toList.sortBy(_._1).unzip      (boundaries.toArray, predictions.toArray)    }  }  @Since("1.4.0")  override def load(sc: SparkContext, path: String): IsotonicRegressionModel = {    implicit val formats = DefaultFormats    val (loadedClassName, version, metadata) = loadMetadata(sc, path)    val isotonic = (metadata \ "isotonic").extract[Boolean]    val classNameV1_0 = SaveLoadV1_0.thisClassName    (loadedClassName, version) match {      case (className, "1.0") if className == classNameV1_0 =>        val (boundaries, predictions) = SaveLoadV1_0.load(sc, path)        new IsotonicRegressionModel(boundaries, predictions, isotonic)      case _ => throw new Exception(        s"IsotonicRegressionModel.load did not recognize model with (className, format version):" +        s"($loadedClassName, $version).  Supported:\n" +        s"  ($classNameV1_0, 1.0)"      )    }  }}/** * Isotonic regression. * Currently implemented using parallelized pool adjacent violators algorithm. * Only univariate (single feature) algorithm supported. * * Sequential PAV implementation based on: * Tibshirani, Ryan J., Holger Hoefling, and Robert Tibshirani. *   "Nearly-isotonic regression." Technometrics 53.1 (2011): 54-61. *   Available from [[http://www.stat.cmu.edu/~ryantibs/papers/neariso.pdf]] * * Sequential PAV parallelization based on: * Kearsley, Anthony J., Richard A. Tapia, and Michael W. Trosset. *   "An approach to parallelizing isotonic regression." *   Applied Mathematics and Parallel Computing. Physica-Verlag HD, 1996. 141-147. *   Available from [[http://softlib.rice.edu/pub/CRPC-TRs/reports/CRPC-TR96640.pdf]] * * @see [[http://en.wikipedia.org/wiki/Isotonic_regression Isotonic regression (Wikipedia)]] */@Since("1.3.0")class IsotonicRegression private (private var isotonic: Boolean) extends Serializable {  /**   * 构建IsotonicRegression实例的默认参数：isotonic = true   *   * @return New instance of IsotonicRegression.   */  @Since("1.3.0")  def this() = this(true)  /**   * 设置序列的参数（Sets the isotonic parameter）.   *   * @param isotonic 序列是递增的还是递减的   * @return This instance of IsotonicRegression.   */  @Since("1.3.0")  def setIsotonic(isotonic: Boolean): this.type = {    this.isotonic = isotonic    this  }  /**   * 运行保序回归算法，来构建保序回归模型   * @param input 输入一个 RDD 内部数据形式为 tuples (label, feature, weight) ，其中，label 是对每次计算都会改变   *feature 特征变量 你weight 权重（默认为1）           * @return Isotonic regression model.   */  @Since("1.3.0")  def run(input: RDD[(Double, Double, Double)]): IsotonicRegressionModel = {    val preprocessedInput = if (isotonic) {      input    } else {      input.map(x => (-x._1, x._2, x._3))    }    val pooled = parallelPoolAdjacentViolators(preprocessedInput)    val predictions = if (isotonic) pooled.map(_._1) else pooled.map(-_._1)    val boundaries = pooled.map(_._2)    new IsotonicRegressionModel(boundaries, predictions, isotonic)  }  /**   * Run pool adjacent violators algorithm to obtain isotonic regression model.   *   * @param input JavaRDD of tuples (label, feature, weight) where label is dependent variable   *              for which we calculate isotonic regression, feature is independent variable   *              and weight represents number of measures with default 1.   *              If multiple labels share the same feature value then they are ordered before   *              the algorithm is executed.   * @return Isotonic regression model.   */  @Since("1.3.0")  def run(input: JavaRDD[(JDouble, JDouble, JDouble)]): IsotonicRegressionModel = {    run(input.rdd.retag.asInstanceOf[RDD[(Double, Double, Double)]])  }  /**   * Performs a pool adjacent violators algorithm (PAV算法).   * @param input 输入的数据  形式为： (label, feature, weight).   * @return 按照保序回归的定义，返回一个有序的序列   */  private def poolAdjacentViolators(      input: Array[(Double, Double, Double)]): Array[(Double, Double, Double)] = {    if (input.isEmpty) {      return Array.empty    }    // Pools sub array within given bounds assigning weighted average value to all elements.    def pool(input: Array[(Double, Double, Double)], start: Int, end: Int): Unit = {      val poolSubArray = input.slice(start, end + 1)      val weightedSum = poolSubArray.map(lp => lp._1 * lp._3).sum      val weight = poolSubArray.map(_._3).sum      var i = start      while (i <= end) {        input(i) = (weightedSum / weight, input(i)._2, input(i)._3)        i = i + 1      }    }    var i = 0    val len = input.length    while (i < len) {      var j = i      // Find monotonicity violating sequence, if any.      while (j < len - 1 && input(j)._1 > input(j + 1)._1) {        j = j + 1      }      // If monotonicity was not violated, move to next data point.      if (i == j) {        i = i + 1      } else {        // Otherwise pool the violating sequence        // and check if pooling caused monotonicity violation in previously processed points.        while (i >= 0 && input(i)._1 > input(i + 1)._1) {          pool(input, i, j)          i = i - 1        }        i = j      }    }    // For points having the same prediction, we only keep two boundary points.    val compressed = ArrayBuffer.empty[(Double, Double, Double)]    var (curLabel, curFeature, curWeight) = input.head    var rightBound = curFeature    def merge(): Unit = {      compressed += ((curLabel, curFeature, curWeight))      if (rightBound > curFeature) {        compressed += ((curLabel, rightBound, 0.0))      }    }    i = 1    while (i < input.length) {      val (label, feature, weight) = input(i)      if (label == curLabel) {        curWeight += weight        rightBound = feature      } else {        merge()        curLabel = label        curFeature = feature        curWeight = weight        rightBound = curFeature      }      i += 1    }    merge()    compressed.toArray  }  /**   * Performs并行PAV算法实现   * 将pav应用在每个分区，之后再进行合并。   * @param input Input data of tuples (label, feature, weight).   * @return Result tuples (label, feature, weight) where labels were updated   *         to form a monotone sequence as per isotonic regression definition.   */  private def parallelPoolAdjacentViolators(      input: RDD[(Double, Double, Double)]): Array[(Double, Double, Double)] = {    val parallelStepResult = input      .sortBy(x => (x._2, x._1))      .glom()      .flatMap(poolAdjacentViolators)      .collect()      .sortBy(x => (x._2, x._1)) // Sort again because collect() doesn't promise ordering.    poolAdjacentViolators(parallelStepResult)  }}

spark实验

import org.apache.spark.mllib.regression.{IsotonicRegression, IsotonicRegressionModel}import org.apache.spark.{SparkConf, SparkContext}object IsotonicRegressionExample {  def main(args: Array[String]) {    val conf = new SparkConf().setAppName("IsotonicRegressionExample").setMaster("local")    val sc = new SparkContext(conf)    val data = sc.textFile("C:\\Users\\alienware\\IdeaProjects\\sparkCore\\data\\mllib\\sample_isotonic_regression_data.txt")    // Create label, feature, weight tuples from input data with weight set to default value 1.0.    val parsedData = data.map { line =>      val parts = line.split(',').map(_.toDouble)      (parts(0), parts(1), 1.0)    }    // Split data into training (60%) and test (40%) sets.    val splits = parsedData.randomSplit(Array(0.6, 0.4), seed = 11L)    val training = splits(0)    val test = splits(1)    // Create isotonic regression model from training data.    // Isotonic parameter defaults to true so it is only shown for demonstration    val model = new IsotonicRegression().setIsotonic(true).run(training)    // Create tuples of predicted and real labels.    val predictionAndLabel = test.map { point =>      val predictedLabel = model.predict(point._2)      (predictedLabel, point._1)    }    //predictionAndLabel.foreach(println)    /**      * (0.16868944399999988,0.31208567)(0.16868944399999988,0.35900051)(0.16868944399999988,0.03926568)(0.16868944399999988,0.12952575)(0.16868944399999988,0.0)(0.16868944399999988,0.01376849)(0.16868944399999988,0.13105558)(0.19545421571428565,0.13717491)(0.19545421571428565,0.19020908)(0.19545421571428565,0.19581846)(0.31718510999999966,0.29576747)(0.5322114566666667,0.4854666)(0.5368859433333334,0.49209587)(0.5602243760000001,0.5017848)(0.5701674724126985,0.58286588)(0.5801105688253968,0.64660887)(0.5900536652380952,0.65782764)(0.5900536652380952,0.63029067)(0.5900536652380952,0.63233044)(0.5900536652380952,0.33299337)(0.5900536652380952,0.36206017)(0.5900536652380952,0.56348802)(0.5900536652380952,0.48393677)(0.5900536652380952,0.46965834)(0.5900536652380952,0.45843957)(0.5900536652380952,0.47118817)(0.5900536652380952,0.51555329)(0.5900536652380952,0.56297807)(0.6881693,0.65119837)(0.7135390099999999,0.66598674)(0.861295255,0.91330954)(0.903875573,0.90719021)(0.9275879659999999,0.93115757)(0.9275879659999999,0.91942886)      */    // Calculate mean squared error between predicted and real labels.    val meanSquaredError = predictionAndLabel.map { case (p, l) => math.pow((p - l), 2) }.mean()    println("Mean Squared Error = " + meanSquaredError)    //Mean Squared Error = 0.010049744711808193    // Save and load model    model.save(sc, "target/tmp/myIsotonicRegressionModel")    val sameModel = IsotonicRegressionModel.load(sc, "target/tmp/myIsotonicRegressionModel")  }}

参考文献

1、http://wenku.baidu.com/link?url=rbcbI3L7M83F62Aey_kyGZk7kwuJxr5ZW61EqFH5T45umsdZOCrAbfpl8a1yuMyzObd1_kG-kQ9DPcSTl7wnoX6UyNN_gT5bBYh_p1yMgD7url=rbcbI3L7M83F62Aey_kyGZk7kwuJxr5ZW61EqFH5T45umsdZOCrAbfpl8a1yuMyzObd1_kG-kQ9DPcSTl7wnoX6UyNN_gT5bBYh_p1yMgD7

附录

链接：http://pan.baidu.com/s/1i4DwQs1 密码：moor