Isotonic regression belongs to the family of regression algorithms. Formally isotonic regression is a problem where given a finite set of real numbers $Y = {y_1, y_2, ..., y_n}$ representing observed responses and $X = {x_1, x_2, ..., x_n}$ the unknown response values to be fitted finding a function that minimises
\begin{equation} f(x) = \sum_{i=1}^n w_i (y_i - x_i)^2 \end{equation}
with respect to complete order subject to $x_1\le x_2\le ...\le x_n$ where $w_i$ are positive weights. The resulting function is called isotonic regression and it is unique. It can be viewed as least squares problem under order restriction. Essentially isotonic regression is a monotonic function best fitting the original data points.
MLlib supports a pool adjacent violators algorithm which uses an approach to parallelizing isotonic regression. The training input is a RDD of tuples of three double values that represent label, feature and weight in this order. Additionally IsotonicRegression algorithm has one optional parameter called $isotonic$ defaulting to true. This argument specifies if the isotonic regression is isotonic (monotonically increasing) or antitonic (monotonically decreasing).
Training returns an IsotonicRegressionModel that can be used to predict labels for both known and unknown features. The result of isotonic regression is treated as piecewise linear function. The rules for prediction therefore are:
{% highlight scala %} import org.apache.spark.mllib.regression.{IsotonicRegression, IsotonicRegressionModel}
val data = sc.textFile(“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) }
// 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)
// Save and load model model.save(sc, “myModelPath”) val sameModel = IsotonicRegressionModel.load(sc, “myModelPath”) {% endhighlight %}
{% highlight java %} import org.apache.spark.SparkConf; import org.apache.spark.api.java.JavaDoubleRDD; import org.apache.spark.api.java.JavaPairRDD; import org.apache.spark.api.java.JavaRDD; import org.apache.spark.api.java.JavaSparkContext; import org.apache.spark.api.java.function.Function; import org.apache.spark.api.java.function.PairFunction; import org.apache.spark.mllib.regression.IsotonicRegressionModel; import scala.Tuple2; import scala.Tuple3;
JavaRDD data = sc.textFile(“data/mllib/sample_isotonic_regression_data.txt”);
// Create label, feature, weight tuples from input data with weight set to default value 1.0. JavaRDD<Tuple3<Double, Double, Double>> parsedData = data.map( new Function<String, Tuple3<Double, Double, Double>>() { public Tuple3<Double, Double, Double> call(String line) { String[] parts = line.split(“,”); return new Tuple3<>(new Double(parts[0]), new Double(parts[1]), 1.0); } } );
// Split data into training (60%) and test (40%) sets. JavaRDD<Tuple3<Double, Double, Double>>[] splits = parsedData.randomSplit(new double[] {0.6, 0.4}, 11L); JavaRDD<Tuple3<Double, Double, Double>> training = splits[0]; JavaRDD<Tuple3<Double, Double, Double>> test = splits[1];
// Create isotonic regression model from training data. // Isotonic parameter defaults to true so it is only shown for demonstration IsotonicRegressionModel model = new IsotonicRegression().setIsotonic(true).run(training);
// Create tuples of predicted and real labels. JavaPairRDD<Double, Double> predictionAndLabel = test.mapToPair( new PairFunction<Tuple3<Double, Double, Double>, Double, Double>() { @Override public Tuple2<Double, Double> call(Tuple3<Double, Double, Double> point) { Double predictedLabel = model.predict(point._2()); return new Tuple2<Double, Double>(predictedLabel, point._1()); } } );
// Calculate mean squared error between predicted and real labels. Double meanSquaredError = new JavaDoubleRDD(predictionAndLabel.map( new Function<Tuple2<Double, Double>, Object>() { @Override public Object call(Tuple2<Double, Double> pl) { return Math.pow(pl._1() - pl._2(), 2); } } ).rdd()).mean();
System.out.println("Mean Squared Error = " + meanSquaredError);
// Save and load model model.save(sc.sc(), “myModelPath”); IsotonicRegressionModel sameModel = IsotonicRegressionModel.load(sc.sc(), “myModelPath”); {% endhighlight %}
{% highlight python %} import math from pyspark.mllib.regression import IsotonicRegression, IsotonicRegressionModel
data = sc.textFile(“data/mllib/sample_isotonic_regression_data.txt”)
parsedData = data.map(lambda line: tuple([float(x) for x in line.split(‘,’)]) + (1.0,))
training, test = parsedData.randomSplit([0.6, 0.4], 11)
model = IsotonicRegression.train(training)
predictionAndLabel = test.map(lambda p: (model.predict(p[1]), p[0]))
meanSquaredError = predictionAndLabel.map(lambda pl: math.pow((pl[0] - pl[1]), 2)).mean() print("Mean Squared Error = " + str(meanSquaredError))
model.save(sc, “myModelPath”) sameModel = IsotonicRegressionModel.load(sc, “myModelPath”) {% endhighlight %}