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From mengxr <...@git.apache.org>
Subject [GitHub] spark pull request: [MLLIB][SPARK-5502] User guide for isotonic re...
Date Wed, 11 Feb 2015 21:02:50 GMT
Github user mengxr commented on a diff in the pull request:

    https://github.com/apache/spark/pull/4536#discussion_r24532966
  
    --- Diff: docs/mllib-regression.md ---
    @@ -0,0 +1,161 @@
    +---
    +layout: global
    +title: Naive Bayes - MLlib
    +displayTitle: <a href="mllib-guide.html">MLlib</a> - Regression
    +---
    +
    +## Regression
    +[Regression](http://en.wikipedia.org/wiki/Regression_analysis) is a statistical process
    +for estimating the relationships among variables. It includes many techniques for modeling
    +and analyzing several variables, when the focus is on the relationship between
    +a dependent variable and one or more independent variables.
    +
    +## 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.
    +
    +MLlib supports a
    +[pool adjacent violators algorithm](http://www.stat.cmu.edu/~ryantibs/papers/neariso.pdf)
    +which uses an approach to
    +[parallelizing isotonic regression](http://softlib.rice.edu/pub/CRPC-TRs/reports/CRPC-TR96640.pdf).
    --- End diff --
    
    Same here: http://doi.org/10.1007/978-3-642-99789-1_10


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