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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
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| layout | title |
|---|---|
| global | <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).
| Impurity | Task | Formula | Description |
|---|---|---|---|
| Gini impurity | Classification | $\sum_{i=1}^{M} f_i(1-f_i)$ | $f_i$ is the frequency of label $i$ at a node and $M$ is the number of unique labels. |
| Entropy | Classification | $\sum_{i=1}^{M} -f_ilog(f_i)$ | $f_i$ is the frequency of label $i$ at a node and $M$ is the number of unique labels. |
| Variance | Regression | $\frac{1}{n} \sum_{i=1}^{N} (x_i - \mu)^2$ | $y_i$ is label for an instance, $N$ is the number of instances and $\mu$ is the mean given by $\frac{1}{N} \sum_{i=1}^n x_i$. |
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 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:
- The node depth is equal to the
maxDepthtraining parammeter - No split candidate leads to an information gain at the node.
Practical limitations
- 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.
- The implemented algorithm reads both sparse and dense data. However, it is not optimized for sparse input.
- 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.
// 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 %}
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.
// 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 %}