[SPARK-1506][MLLIB] Documentation improvements for MLlib 1.0

Preview: http://54.82.240.23:4000/mllib-guide.html

Table of contents:

* Basics
  * Data types
  * Summary statistics
* Classification and regression
  * linear support vector machine (SVM)
  * logistic regression
  * linear linear squares, Lasso, and ridge regression
  * decision tree
  * naive Bayes
* Collaborative Filtering
  * alternating least squares (ALS)
* Clustering
  * k-means
* Dimensionality reduction
  * singular value decomposition (SVD)
  * principal component analysis (PCA)
* Optimization
  * stochastic gradient descent
  * limited-memory BFGS (L-BFGS)

Author: Xiangrui Meng <meng@databricks.com>

Closes #422 from mengxr/mllib-doc and squashes the following commits:

944e3a9 [Xiangrui Meng] merge master
f9fda28 [Xiangrui Meng] minor
9474065 [Xiangrui Meng] add alpha to ALS examples
928e630 [Xiangrui Meng] initialization_mode -> initializationMode
5bbff49 [Xiangrui Meng] add imports to labeled point examples
c17440d [Xiangrui Meng] fix python nb example
28f40dc [Xiangrui Meng] remove localhost:4000
369a4d3 [Xiangrui Meng] Merge branch 'master' into mllib-doc
7dc95cc [Xiangrui Meng] update linear methods
053ad8a [Xiangrui Meng] add links to go back to the main page
abbbf7e [Xiangrui Meng] update ALS argument names
648283e [Xiangrui Meng] level down statistics
14e2287 [Xiangrui Meng] add sample libsvm data and use it in guide
8cd2441 [Xiangrui Meng] minor updates
186ab07 [Xiangrui Meng] update section names
6568d65 [Xiangrui Meng] update toc, level up lr and svm
162ee12 [Xiangrui Meng] rename section names
5c1e1b1 [Xiangrui Meng] minor
8aeaba1 [Xiangrui Meng] wrap long lines
6ce6a6f [Xiangrui Meng] add summary statistics to toc
5760045 [Xiangrui Meng] claim beta
cc604bf [Xiangrui Meng] remove classification and regression
92747b3 [Xiangrui Meng] make section titles consistent
e605dd6 [Xiangrui Meng] add LIBSVM loader
f639674 [Xiangrui Meng] add python section to migration guide
c82ffb4 [Xiangrui Meng] clean optimization
31660eb [Xiangrui Meng] update linear algebra and stat
0a40837 [Xiangrui Meng] first pass over linear methods
1fc8271 [Xiangrui Meng] update toc
906ed0a [Xiangrui Meng] add a python example to naive bayes
5f0a700 [Xiangrui Meng] update collaborative filtering
656d416 [Xiangrui Meng] update mllib-clustering
86e143a [Xiangrui Meng] remove data types section from main page
8d1a128 [Xiangrui Meng] move part of linear algebra to data types and add Java/Python examples
d1b5cbf [Xiangrui Meng] merge master
72e4804 [Xiangrui Meng] one pass over tree guide
64f8995 [Xiangrui Meng] move decision tree guide to a separate file
9fca001 [Xiangrui Meng] add first version of linear algebra guide
53c9552 [Xiangrui Meng] update dependencies
f316ec2 [Xiangrui Meng] add migration guide
f399f6c [Xiangrui Meng] move linear-algebra to dimensionality-reduction
182460f [Xiangrui Meng] add guide for naive Bayes
137fd1d [Xiangrui Meng] re-organize toc
a61e434 [Xiangrui Meng] update mllib's toc

1 files changed, 185 insertions, 0 deletions

diff --git a/docs/mllib-decision-tree.md b/docs/mllib-decision-tree.md
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+---
+layout: global
+title: <a href="mllib-guide.html">MLlib</a> - Decision Tree
+---
+
+* Table of contents
+{:toc}
+
+Decision trees and their ensembles are popular methods for the machine learning tasks of
+classification and regression. Decision trees are widely used since they are easy to interpret,
+handle categorical variables, extend to the multiclass classification setting, do not require
+feature scaling and are able to capture nonlinearities and feature interactions. Tree ensemble
+algorithms such as decision forest and boosting are among the top performers for classification and
+regression tasks.
+
+## Basic algorithm
+
+The decision tree is a greedy algorithm that performs a recursive binary partitioning of the feature
+space by choosing a single element from the *best split set* where each element of the set maximizes
+the information gain at a tree node. In other words, the split chosen at each tree node is chosen
+from the set `$\underset{s}{\operatorname{argmax}} IG(D,s)$` where `$IG(D,s)$` is the information
+gain when a split `$s$` is applied to a dataset `$D$`.
+
+### Node impurity and information gain
+
+The *node impurity* is a measure of the homogeneity of the labels at the node. The current
+implementation provides two impurity measures for classification (Gini impurity and entropy) and one
+impurity measure for regression (variance).
+
+<table class="table">
+  <thead>
+    <tr><th>Impurity</th><th>Task</th><th>Formula</th><th>Description</th></tr>
+  </thead>
+  <tbody>
+    <tr>
+      <td>Gini impurity</td>
+	  <td>Classification</td>
+	  <td>$\sum_{i=1}^{M} f_i(1-f_i)$</td><td>$f_i$ is the frequency of label $i$ at a node and $M$ is the number of unique labels.</td>
+    </tr>
+    <tr>
+      <td>Entropy</td>
+	  <td>Classification</td>
+	  <td>$\sum_{i=1}^{M} -f_ilog(f_i)$</td><td>$f_i$ is the frequency of label $i$ at a node and $M$ is the number of unique labels.</td>
+    </tr>
+    <tr>
+      <td>Variance</td>
+	  <td>Regression</td>
+     <td>$\frac{1}{n} \sum_{i=1}^{N} (x_i - \mu)^2$</td><td>$y_i$ is label for an instance,
+	  $N$ is the number of instances and $\mu$ is the mean given by $\frac{1}{N} \sum_{i=1}^n x_i$.</td>
+    </tr>
+  </tbody>
+</table>
+
+The *information gain* is the difference in the parent node impurity and the weighted sum of the two
+child node impurities. Assuming that a split $s$ partitions the dataset `$D$` of size `$N$` into two
+datasets `$D_{left}$` and `$D_{right}$` of sizes `$N_{left}$` and `$N_{right}$`, respectively:
+
+`$IG(D,s) = Impurity(D) - \frac{N_{left}}{N} Impurity(D_{left}) - \frac{N_{right}}{N} Impurity(D_{right})$`
+
+### Split candidates
+
+**Continuous features**
+
+For small datasets in single machine implementations, the split candidates for each continuous
+feature are typically the unique values for the feature. Some implementations sort the feature
+values and then use the ordered unique values as split candidates for faster tree calculations.
+
+Finding ordered unique feature values is computationally intensive for large distributed
+datasets. One can get an approximate set of split candidates by performing a quantile calculation
+over a sampled fraction of the data. The ordered splits create "bins" and the maximum number of such
+bins can be specified using the `maxBins` parameters.
+
+Note that the number of bins cannot be greater than the number of instances `$N$` (a rare scenario
+since the default `maxBins` value is 100). The tree algorithm automatically reduces the number of
+bins if the condition is not satisfied.
+
+**Categorical features**
+
+For `$M$` categorical features, one could come up with `$2^M-1$` split candidates. However, for
+binary classification, the number of split candidates can be reduced to `$M-1$` by ordering the
+categorical feature values by the proportion of labels falling in one of the two classes (see
+Section 9.2.4 in
+[Elements of Statistical Machine Learning](http://statweb.stanford.edu/~tibs/ElemStatLearn/) for
+details). For example, for a binary classification problem with one categorical feature with three
+categories A, B and C with corresponding proportion of label 1 as 0.2, 0.6 and 0.4, the categorical
+features are orded as A followed by C followed B or A, B, C. The two split candidates are A \| C, B
+and A , B \| C where \| denotes the split.
+
+### Stopping rule
+
+The recursive tree construction is stopped at a node when one of the two conditions is met:
+
+1. The node depth is equal to the `maxDepth` training parammeter
+2. No split candidate leads to an information gain at the node.
+
+### Practical limitations
+
+1. The tree implementation stores an Array[Double] of size *O(#features \* #splits \* 2^maxDepth)*
+   in memory for aggregating histograms over partitions. The current implementation might not scale
+   to very deep trees since the memory requirement grows exponentially with tree depth.
+2. The implemented algorithm reads both sparse and dense data. However, it is not optimized for
+   sparse input.
+3. Python is not supported in this release.
+ 
+We are planning to solve these problems in the near future. Please drop us a line if you encounter
+any issues.
+
+## Examples
+
+### Classification
+
+The example below demonstrates how to load a CSV file, parse it as an RDD of `LabeledPoint` and then
+perform classification using a decision tree using Gini impurity as an impurity measure and a
+maximum tree depth of 5. The training error is calculated to measure the algorithm accuracy.
+
+<div class="codetabs">
+<div data-lang="scala">
+{% highlight scala %}
+import org.apache.spark.SparkContext
+import org.apache.spark.mllib.tree.DecisionTree
+import org.apache.spark.mllib.regression.LabeledPoint
+import org.apache.spark.mllib.linalg.Vectors
+import org.apache.spark.mllib.tree.configuration.Algo._
+import org.apache.spark.mllib.tree.impurity.Gini
+
+// Load and parse the data file
+val data = sc.textFile("mllib/data/sample_tree_data.csv")
+val parsedData = data.map { line =>
+  val parts = line.split(',').map(_.toDouble)
+  LabeledPoint(parts(0), Vectors.dense(parts.tail))
+}
+
+// Run training algorithm to build the model
+val maxDepth = 5
+val model = DecisionTree.train(parsedData, Classification, Gini, maxDepth)
+
+// Evaluate model on training examples and compute training error
+val labelAndPreds = parsedData.map { point =>
+  val prediction = model.predict(point.features)
+  (point.label, prediction)
+}
+val trainErr = labelAndPreds.filter(r => r._1 != r._2).count.toDouble / parsedData.count
+println("Training Error = " + trainErr)
+{% endhighlight %}
+</div>
+</div>
+
+### Regression
+
+The example below demonstrates how to load a CSV file, parse it as an RDD of `LabeledPoint` and then
+perform regression using a decision tree using variance as an impurity measure and a maximum tree
+depth of 5. The Mean Squared Error (MSE) is computed at the end to evaluate
+[goodness of fit](http://en.wikipedia.org/wiki/Goodness_of_fit).
+
+<div class="codetabs">
+<div data-lang="scala">
+{% highlight scala %}
+import org.apache.spark.SparkContext
+import org.apache.spark.mllib.tree.DecisionTree
+import org.apache.spark.mllib.regression.LabeledPoint
+import org.apache.spark.mllib.linalg.Vectors
+import org.apache.spark.mllib.tree.configuration.Algo._
+import org.apache.spark.mllib.tree.impurity.Variance
+
+// Load and parse the data file
+val data = sc.textFile("mllib/data/sample_tree_data.csv")
+val parsedData = data.map { line =>
+  val parts = line.split(',').map(_.toDouble)
+  LabeledPoint(parts(0), Vectors.dense(parts.tail))
+}
+
+// Run training algorithm to build the model
+val maxDepth = 5
+val model = DecisionTree.train(parsedData, Regression, Variance, maxDepth)
+
+// Evaluate model on training examples and compute training error
+val valuesAndPreds = parsedData.map { point =>
+  val prediction = model.predict(point.features)
+  (point.label, prediction)
+}
+val MSE = valuesAndPreds.map{ case(v, p) => math.pow((v - p), 2)}.reduce(_ + _)/valuesAndPreds.count
+println("training Mean Squared Error = " + MSE)
+{% endhighlight %}
+</div>
+</div>