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diff --git a/docs/ml-classification-regression.md b/docs/ml-classification-regression.md index d7e5521cbc..3d6106b532 100644 --- a/docs/ml-classification-regression.md +++ b/docs/ml-classification-regression.md @@ -691,6 +691,76 @@ The implementation matches the result from R's survival function </div> +## Isotonic regression +[Isotonic regression](http://en.wikipedia.org/wiki/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](http://en.wikipedia.org/wiki/Monotonic_function) +best fitting the original data points. + +We implement a +[pool adjacent violators algorithm](http://doi.org/10.1198/TECH.2010.10111) +which uses an approach to +[parallelizing isotonic regression](http://doi.org/10.1007/978-3-642-99789-1_10). +The training input is a DataFrame which contains three columns +label, features and weight. 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: + +* If the prediction input exactly matches a training feature + then associated prediction is returned. In case there are multiple predictions with the same + feature then one of them is returned. Which one is undefined + (same as java.util.Arrays.binarySearch). +* If the prediction input is lower or higher than all training features + then prediction with lowest or highest feature is returned respectively. + In case there are multiple predictions with the same feature + then the lowest or highest is returned respectively. +* If the prediction input falls between two training features then prediction is treated + as piecewise linear function and interpolated value is calculated from the + predictions of the two closest features. In case there are multiple values + with the same feature then the same rules as in previous point are used. + +### Examples + +<div class="codetabs"> +<div data-lang="scala" markdown="1"> + +Refer to the [`IsotonicRegression` Scala docs](api/scala/index.html#org.apache.spark.ml.regression.IsotonicRegression) for details on the API. + +{% include_example scala/org/apache/spark/examples/ml/IsotonicRegressionExample.scala %} +</div> +<div data-lang="java" markdown="1"> + +Refer to the [`IsotonicRegression` Java docs](api/java/org/apache/spark/ml/regression/IsotonicRegression.html) for details on the API. + +{% include_example java/org/apache/spark/examples/ml/JavaIsotonicRegressionExample.java %} +</div> +<div data-lang="python" markdown="1"> + +Refer to the [`IsotonicRegression` Python docs](api/python/pyspark.ml.html#pyspark.ml.regression.IsotonicRegression) for more details on the API. + +{% include_example python/ml/isotonic_regression_example.py %} +</div> +</div> + + # Decision trees |