算法小白的第一次尝试---PCA（主成分分析）降维【适合各种纬度数据】

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笔者追求算法实现，不喜欢大篇幅叙述原理，有关PCA理论推荐查看该篇博客

http://www.cnblogs.com/pinard/p/6239403.html

特此说明：发现之前写的不是很好，特地从UCI上找了Wine葡萄酒的例子重新写了一次。

LDA降维欢迎前往笔者下一篇博客：https://blog.csdn.net/Java_Man_China/article/details/89504514

KPCA降维欢迎前往笔者下一篇博客：
https://blog.csdn.net/Java_Man_China/article/details/89677371
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import breeze.linalg.{DenseMatrix, eig}
import org.apache.log4j.{Level, Logger}
import org.apache.spark.ml.feature.{LabeledPoint,StandardScaler, VectorAssembler}
import org.apache.spark.ml.linalg.Vectors
import org.apache.spark.sql.types.{DoubleType, StructField, StructType}
import org.apache.spark.sql.{DataFrame, Row, SparkSession}
import scala.collection.mutable.ArrayBuffer

/** The method attempts to lower the dimensionality based on PCA
  * Data Source：http://archive.ics.uci.edu/ml/datasets/Wine
  * @author XiaoTangBao
  * @date 2019/4/16 9:16
  * @version 1.0
  */
object PCA {
  def main(args: Array[String]): Unit = {
    //屏蔽日志
    Logger.getLogger("org.apache.spark").setLevel(Level.ERROR)
    //spark初始化
    val spark = SparkSession.builder().master("local[4]").appName("LDA").getOrCreate()
    //获取数据源   http://archive.ics.uci.edu/ml/datasets/Wine
    val data = spark.sparkContext.textFile("G:\\mldata\\wine.data").map(line => line.split(","))
      .map(arr => arr.map(str => str.toDouble)).map(arr =>Row(arr(0),arr(1),arr(2),arr(3),arr(4),arr(5),
      arr(6),arr(7),arr(8),arr(9),arr(10),arr(11),arr(12),arr(13)))

    //设置featuresArr和schema,便于后期数据转化及生成dataFrame
    val featuresArr = Array("Alcohol","Malic acid","Ash","Alcalinity of ash","Magnesium",
      "Total phenols","Flavanoids","Nonflavanoid phenols","Proanthocyanins","Color intensity",
      "Hue","OD280/OD315 of diluted wines","Proline")
    val schema = StructType(List(StructField("label",DoubleType,true),StructField("Alcohol",DoubleType,true),StructField("Malic acid",DoubleType,true),
      StructField("Ash",DoubleType,true),StructField("Alcalinity of ash",DoubleType,true),StructField("Magnesium",DoubleType,true)
      ,StructField("Total phenols",DoubleType,true),StructField("Flavanoids",DoubleType,true),StructField("Nonflavanoid phenols",DoubleType,true)
      ,StructField("Proanthocyanins",DoubleType,true),StructField("Color intensity",DoubleType,true),StructField("Hue",DoubleType,true)
      ,StructField("OD280/OD315 of diluted wines",DoubleType,true),StructField("Proline",DoubleType,true)))
    val oridf = spark.createDataFrame(data,schema)

    //设置转化器
    val vectorAsb = new VectorAssembler().setInputCols(featuresArr).setOutputCol("features")
    //数据整理后传入run，启动PCA算法
    val newdf = vectorAsb.transform(oridf).select("label","features")
    //标准化处理数据,标准化后不再需要去中心化
    val std = new StandardScaler().setInputCol("features").setOutputCol("Scaledfeatures")
      .setWithMean(true).setWithStd(true).fit(newdf).transform(newdf)
      .select("label","Scaledfeatures")
      .withColumnRenamed("Scaledfeatures","features")

    val result = run(std,2)
    val arr = ArrayBuffer[(Double,Double)]()
    for(i<-0 until result.cols) arr.append((result(0,i),result(1,i)))
    arr.foreach(tp =>println(tp._1))
    println()
    arr.foreach(tp =>println(tp._2))
  }

  /**
    * the method attempts to lower the dimensionality
    * @param data the ioriginal data which in high dimensions, include label and features cols
    * @param k the final dimensions
    */
  def run(df:DataFrame,k:Int)={
    val trainData = df.select("features").rdd.map(row => row.toString())
      .map(str => str.replace('[',' '))
      .map(str => str.replace(']',' '))
      .map(str => str.trim).map(str => str.split(','))
      .map(arr => arr.map(str => str.toDouble)).collect()

    val labels = df.select("label").rdd.map(row => row.toString())
      .map(str => str.replace('[',' '))
      .map(str => str.replace(']',' '))
      .map(str => str.trim).map(str => str.toDouble).collect()

    //特征列数
    val tz = trainData(0).length
    //生成新的带label的数据
    val labArr = ArrayBuffer[LabeledPoint]()
    for(i<-0 until trainData.length) labArr.append(LabeledPoint(labels(i),Vectors.dense(trainData(i))))
    //总样本组成的大型矩阵
    val allData = labArr.map(lab => lab.features).map(vec => vec.toArray).flatMap(x => x).toArray
    val big_Matrx =new DenseMatrix[Double](tz,trainData.length,allData)
    
//    //存放向量各维度的均值
//    val big_mean = sum(big_Matrx,Axis._1).*= (1.0 / big_Matrx.cols)
//    //样本中心化
//    for(i<-0 until big_Matrx.cols) big_Matrx(::,i) := big_Matrx(::,i) - big_mean

    //计算样本的协方差矩阵
    val covMatrix = (big_Matrx * big_Matrx.t) * (1.0 / (big_Matrx.cols-1))
    val eigValues = eig(covMatrix).eigenvalues
    val eigVectors = eig(covMatrix).eigenvectors

    //选取最大的k个特征值对应的特征向量
    val label_eig = DenseMatrix.horzcat(eigVectors.t,eigValues.toDenseMatrix.t)
    var strArr = ArrayBuffer[String]()
    for(i<-0 until label_eig.rows) strArr.append(label_eig.t(::,i).toString)
    for(i<-0 until strArr.length){
       strArr(i) = strArr(i).replace("DenseVector(","").replace(')',' ').trim()
    }
    val da = ArrayBuffer[LabeledPoint]()
    for(str <- strArr){
      val arr = str.split(',').map(string => string.toDouble)
      val lab = arr.takeRight(1)(0)
      val value = arr.take(arr.length -1)
      val labPoint = LabeledPoint(lab,Vectors.dense(value))
      da.append(labPoint)
    }

    val result = da.sortBy(labPoint => labPoint.label).reverse.take(k).map(lab => lab.features).map(vec => vec.toArray)
    var rt = DenseMatrix.zeros[Double](result.length,result(0).length)
    for(i<-0 until rt.rows){
      for(j<-0 until rt.cols){
        rt(i,j) = result(i)(j)
      }
    }

    //降维后的数据集
    val newData = rt * big_Matrx
    newData
  }
}

Wine 葡萄酒原始数据共13个特征，选取其中两列绘制二维图

数据未标准化（仅去中心化），采用PCA算法降至二维后的结果如下：

此时发现降维后几乎看不到效果，数据依旧混乱（无法区分）

下图为调用Python包实现PCA的结果，依旧没有对数据进行标准化

--------Python调包--------from sklearn.decomposition import PCA------------------

对比上面两幅图发现，似乎降维后两者结果不一样，这是由于Python调包求解的特征向量与笔者自己计算的特征向量不一致导致的，同一矩阵，相同的特征值下有n个不同的特征向量。为了拟合python结果，笔者将求出的某个特征向量乘以-1后，结果显示如图：

此时大家应该发现两者是一致的了。

数据标准化后，PCA降至二维的结果如下图所示：（对结果乘以了-1，方便对比Python库）

此时发现数据终于可分了。（干一杯葡萄酒）
下图为数据标注化后，调用python库的结果：（毫无意外的一样）

实验结果表明：笔者的算法是正确的。。。。。。。。。。。