GradientTreeBoost (Smile Core 1.0.0 API)

java.lang.Object
- smile.regression.GradientTreeBoost

All Implemented Interfaces:

Regression<double[]>
```
public class GradientTreeBoost
extends Object
implements Regression<double[]>
```
Gradient boosting for regression. Gradient boosting is typically used with decision trees (especially CART regression trees) of a fixed size as base learners. For this special case Friedman proposes a modification to gradient boosting method which improves the quality of fit of each base learner.
Generic gradient boosting at the t-th step would fit a regression tree to pseudo-residuals. Let J be the number of its leaves. The tree partitions the input space into J disjoint regions and predicts a constant value in each region. The parameter J controls the maximum allowed level of interaction between variables in the model. With J = 2 (decision stumps), no interaction between variables is allowed. With J = 3 the model may include effects of the interaction between up to two variables, and so on. Hastie et al. comment that typically 4 ≤ J ≤ 8 work well for boosting and results are fairly insensitive to the choice of in this range, J = 2 is insufficient for many applications, and J > 10 is unlikely to be required.
Fitting the training set too closely can lead to degradation of the model's generalization ability. Several so-called regularization techniques reduce this over-fitting effect by constraining the fitting procedure. One natural regularization parameter is the number of gradient boosting iterations T (i.e. the number of trees in the model when the base learner is a decision tree). Increasing T reduces the error on training set, but setting it too high may lead to over-fitting. An optimal value of T is often selected by monitoring prediction error on a separate validation data set.
Another regularization approach is the shrinkage which times a parameter η (called the "learning rate") to update term. Empirically it has been found that using small learning rates (such as η < 0.1) yields dramatic improvements in model's generalization ability over gradient boosting without shrinking (η = 1). However, it comes at the price of increasing computational time both during training and prediction: lower learning rate requires more iterations.
Soon after the introduction of gradient boosting Friedman proposed a minor modification to the algorithm, motivated by Breiman's bagging method. Specifically, he proposed that at each iteration of the algorithm, a base learner should be fit on a subsample of the training set drawn at random without replacement. Friedman observed a substantional improvement in gradient boosting's accuracy with this modification.
Subsample size is some constant fraction f of the size of the training set. When f = 1, the algorithm is deterministic and identical to the one described above. Smaller values of f introduce randomness into the algorithm and help prevent over-fitting, acting as a kind of regularization. The algorithm also becomes faster, because regression trees have to be fit to smaller datasets at each iteration. Typically, f is set to 0.5, meaning that one half of the training set is used to build each base learner.
Also, like in bagging, sub-sampling allows one to define an out-of-bag estimate of the prediction performance improvement by evaluating predictions on those observations which were not used in the building of the next base learner. Out-of-bag estimates help avoid the need for an independent validation dataset, but often underestimate actual performance improvement and the optimal number of iterations.
Gradient tree boosting implementations often also use regularization by limiting the minimum number of observations in trees' terminal nodes. It's used in the tree building process by ignoring any splits that lead to nodes containing fewer than this number of training set instances. Imposing this limit helps to reduce variance in predictions at leaves.
References
1. J. H. Friedman. Greedy Function Approximation: A Gradient Boosting Machine, 1999.
2. J. H. Friedman. Stochastic Gradient Boosting, 1999.
Author:

Haifeng Li

Nested Class Summary

Nested Classes
Modifier and Type	Class and Description
`static class`	`GradientTreeBoost.Loss` Regression loss function.
`static class`	`GradientTreeBoost.Trainer` Trainer for GradientTreeBoost regression.

Constructor Summary

Constructors
Constructor and Description
`GradientTreeBoost(Attribute[] attributes, double[][] x, double[] y, GradientTreeBoost.Loss loss, int T, int J, double shrinkage, double f)` Constructor.
`GradientTreeBoost(Attribute[] attributes, double[][] x, double[] y, int T)` Constructor.
`GradientTreeBoost(double[][] x, double[] y, GradientTreeBoost.Loss loss, int T, int J, double shrinkage, double f)` Constructor.
`GradientTreeBoost(double[][] x, double[] y, int T)` Constructor.

Method Summary

Methods
Modifier and Type	Method and Description
`GradientTreeBoost.Loss`	`getLossFunction()` Returns the loss function.
`int`	`getNumLeaves()` Returns the (maximum) number of leaves in decision tree.
`double`	`getSamplingRate()` Returns the sampling rate for stochastic gradient tree boosting.
`double[]`	`importance()` Returns the variable importance.
`double`	`predict(double[] x)` Predicts the dependent variable of an instance.
`int`	`size()` Returns the number of trees in the model.
`double[]`	`test(double[][] x, double[] y)` Test the model on a validation dataset.
`double[][]`	`test(double[][] x, double[] y, RegressionMeasure[] measures)` Test the model on a validation dataset.
`void`	`trim(int T)` Trims the tree model set to a smaller size in case of over-fitting.

Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

- Constructor Detail
  - GradientTreeBoost
```
public GradientTreeBoost(double[][] x,
                 double[] y,
                 int T)
```
    Constructor. Learns a gradient tree boosting for regression.
    
    Parameters:
    x - the training instances.
    y - the response variable.
    T - the number of iterations (trees).
  - GradientTreeBoost
```
public GradientTreeBoost(double[][] x,
                 double[] y,
                 GradientTreeBoost.Loss loss,
                 int T,
                 int J,
                 double shrinkage,
                 double f)
```
    Constructor. Learns a gradient tree boosting for regression.
    
    Parameters:
    x - the training instances.
    y - the response variable.
    loss - loss function for regression. By default, least absolute deviation is employed for robust regression.
    T - the number of iterations (trees).
    J - the number of leaves in each tree.
    shrinkage - the shrinkage parameter in (0, 1] controls the learning rate of procedure.
    f - the sampling rate for stochastic tree boosting.
  - GradientTreeBoost
```
public GradientTreeBoost(Attribute[] attributes,
                 double[][] x,
                 double[] y,
                 int T)
```
    Constructor. Learns a gradient tree boosting for regression.
    
    Parameters:
    attributes - the attribute properties.
    x - the training instances.
    y - the response variable.
    T - the number of iterations (trees).
  - GradientTreeBoost
```
public GradientTreeBoost(Attribute[] attributes,
                 double[][] x,
                 double[] y,
                 GradientTreeBoost.Loss loss,
                 int T,
                 int J,
                 double shrinkage,
                 double f)
```
    Constructor. Learns a gradient tree boosting for regression.
    
    Parameters:
    attributes - the attribute properties.
    x - the training instances.
    y - the response variable.
    loss - loss function for regression. By default, least absolute deviation is employed for robust regression.
    T - the number of iterations (trees).
    J - the number of leaves in each tree.
    shrinkage - the shrinkage parameter in (0, 1] controls the learning rate of procedure.
    f - the sampling fraction for stochastic tree boosting.
- Method Detail
  - importance
```
public double[] importance()
```
    Returns the variable importance. Every time a split of a node is made on variable the impurity criterion for the two descendant nodes is less than the parent node. Adding up the decreases for each individual variable over all trees in the forest gives a simple measure of variable importance.
    
    Returns:
    the variable importance
  - getSamplingRate
```
public double getSamplingRate()
```
    Returns the sampling rate for stochastic gradient tree boosting.
    
    Returns:
    the sampling rate for stochastic gradient tree boosting.
  - getNumLeaves
```
public int getNumLeaves()
```
    Returns the (maximum) number of leaves in decision tree.
    
    Returns:
    the (maximum) number of leaves in decision tree.
  - getLossFunction
```
public GradientTreeBoost.Loss getLossFunction()
```
    Returns the loss function.
    
    Returns:
    the loss function.
  - size
```
public int size()
```
    Returns the number of trees in the model.
    
    Returns:
    the number of trees in the model
  - trim
```
public void trim(int T)
```
    Trims the tree model set to a smaller size in case of over-fitting. Or if extra decision trees in the model don't improve the performance, we may remove them to reduce the model size and also improve the speed of prediction.
    
    Parameters:
    T - the new (smaller) size of tree model set.
  - predict
```
public double predict(double[] x)
```
    Description copied from interface: Regression
    
    Predicts the dependent variable of an instance.
    
    Specified by:
    
    predict in interface Regression<double[]>
    
    Parameters:
    x - the instance.
    
    Returns:
    the predicted value of dependent variable.
  - test
```
public double[] test(double[][] x,
            double[] y)
```
    Test the model on a validation dataset.
    
    Parameters:
    x - the test data set.
    y - the test data response values.
    
    Returns:
    RMSEs with first 1, 2, ..., regression trees.
  - test
```
public double[][] test(double[][] x,
              double[] y,
              RegressionMeasure[] measures)
```
    Test the model on a validation dataset.
    
    Parameters:
    x - the test data set.
    y - the test data labels.
    measures - the performance measures of regression.
    
    Returns:
    performance measures with first 1, 2, ..., regression trees.

Class GradientTreeBoost

References

Nested Class Summary

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Constructor Detail

GradientTreeBoost

GradientTreeBoost

GradientTreeBoost

GradientTreeBoost

Method Detail

importance

getSamplingRate

getNumLeaves

getLossFunction

size

trim

predict

test

test