Random forest

Decision trees are a popular method for various machine learning tasks. Tree learning "come[s] closest to meeting the requirements for serving as an off-the-shelf procedure for data mining", say Hastie et al., "because it is invariant under scaling and various other transformations of feature values, is robust to inclusion of irrelevant features, and produces inspectable models. However, they are seldom accurate".[3]^: 352

In particular, trees that are grown very deep tend to learn highly irregular patterns: they overfit their training sets, i.e. have low bias, but very high variance. Random forests are a way of averaging multiple deep decision trees, trained on different parts of the same training set, with the goal of reducing the variance.[3]^{: 587–588} This comes at the expense of a small increase in the bias and some loss of interpretability, but generally greatly boosts the performance in the final model.

Illustration of training a Random Forest model

The training algorithm for random forests applies the general technique of bootstrap aggregating, or bagging, to tree learners. Given a training set X = x₁, ..., x_n with responses Y = y₁, ..., y_n, bagging repeatedly (B times) selects a random sample with replacement of the training set and fits trees to these samples:

For b = 1, ..., B:

1. Sample, with replacement, n training examples from X, Y; call these X_b, Y_b.
2. Train a classification or regression tree f_b on X_b, Y_b.

After training, predictions for unseen samples x' can be made by averaging the predictions from all the individual regression trees on x':

${\displaystyle {\hat {f}}={\frac {1}{B}}\sum _{b=1}^{B}f_{b}(x')}$

or by taking the plurality vote in the case of classification trees.

This bootstrapping procedure leads to better model performance because it decreases the variance of the model, without increasing the bias. This means that while the predictions of a single tree are highly sensitive to noise in its training set, the average of many trees is not, as long as the trees are not correlated. Simply training many trees on a single training set would give strongly correlated trees (or even the same tree many times, if the training algorithm is deterministic); bootstrap sampling is a way of de-correlating the trees by showing them different training sets.

Additionally, an estimate of the uncertainty of the prediction can be made as the standard deviation of the predictions from all the individual regression trees on x':

${\displaystyle \sigma ={\sqrt {\frac {\sum _{b=1}^{B}(f_{b}(x')-{\hat {f}})^{2}}{B-1}}}.}$

The number of samples/trees, B, is a free parameter. Typically, a few hundred to several thousand trees are used, depending on the size and nature of the training set. An optimal number of trees B can be found using cross-validation, or by observing the out-of-bag error: the mean prediction error on each training sample x_i, using only the trees that did not have x_i in their bootstrap sample.[15] The training and test error tend to level off after some number of trees have been fit.

The above procedure describes the original bagging algorithm for trees. Random forests also include another type of bagging scheme: they use a modified tree learning algorithm that selects, at each candidate split in the learning process, a random subset of the features. This process is sometimes called "feature bagging". The reason for doing this is the correlation of the trees in an ordinary bootstrap sample: if one or a few features are very strong predictors for the response variable (target output), these features will be selected in many of the B trees, causing them to become correlated. An analysis of how bagging and random subspace projection contribute to accuracy gains under different conditions is given by Ho.[16]

Typically, for a classification problem with p features, √p (rounded down) features are used in each split.[3]^: 592 For regression problems the inventors recommend p/3 (rounded down) with a minimum node size of 5 as the default.[3]^: 592 In practice, the best values for these parameters should be tuned on a case-to-case basis for every problem.[3]^: 592

Adding one further step of randomization yields extremely randomized trees, or ExtraTrees. While similar to ordinary random forests in that they are an ensemble of individual trees, there are two main differences: first, each tree is trained using the whole learning sample (rather than a bootstrap sample), and second, the top-down splitting in the tree learner is randomized. Instead of computing the locally optimal cut-point for each feature under consideration (based on, e.g., information gain or the Gini impurity), a random cut-point is selected. This value is selected from a uniform distribution within the feature's empirical range (in the tree's training set). Then, of all the randomly generated splits, the split that yields the highest score is chosen to split the node. Similar to ordinary random forests, the number of randomly selected features to be considered at each node can be specified. Default values for this parameter are ${\displaystyle {\sqrt {p}}}$ for classification and ${\displaystyle p}$ for regression, where ${\displaystyle p}$ is the number of features in the model.[17]

The basic Random Forest procedure may not work well in situations where there are a large number of features but only a small proportion of these features are informative with respect to sample classification. This can be addressed by encouraging the procedure to focus mainly on features and trees that are informative. Some methods for accomplishing this are:

- Prefiltering: Eliminate features that are mostly just noise. [18] [19]

- Enriched Random Forest (ERF): Use weighted random sampling instead of simple random sampling at each node of each tree, giving greater weight to features that appear to be more informative. [20] [21]

- Tree Weighted Random Forest (TWRF): Weight trees so that trees exhibiting better accuracy are assigned higher weights. [22] [23]

Random forests can be used to rank the importance of variables in a regression or classification problem in a natural way. The following technique was described in Breiman's original paper[9] and is implemented in the R package randomForest.[10]

The first step in measuring the variable importance in a data set ${\displaystyle {\mathcal {D}}_{n}=\{(X_{i},Y_{i})\}_{i=1}^{n}}$ is to fit a random forest to the data. During the fitting process the out-of-bag error for each data point is recorded and averaged over the forest (errors on an independent test set can be substituted if bagging is not used during training).

To measure the importance of the ${\displaystyle j}$-th feature after training, the values of the ${\displaystyle j}$-th feature are permuted in the out-of-bag samples and the out-of-bag error is again computed on this perturbed data set. The importance score for the ${\displaystyle j}$-th feature is computed by averaging the difference in out-of-bag error before and after the permutation over all trees. The score is normalized by the standard deviation of these differences.

Features which produce large values for this score are ranked as more important than features which produce small values. The statistical definition of the variable importance measure was given and analyzed by Zhu et al.[24]

This method of determining variable importance has some drawbacks.

• For data including categorical variables with different number of levels, random forests are biased in favor of those attributes with more levels. Methods such as partial permutations[25][26][4] and growing unbiased trees[27][28] can be used to solve the problem.

• If the data contain groups of correlated features of similar relevance for the output, then smaller groups are favored over larger groups.[29]
• Additionally, the permutation procedure may fail to identify important features when there are collinear features. In this case permuting groups of correlated features together is a remedy.[30]

This feature importance for random forests is the default implementation in sci-kit learn and R. It is described in the book "Classification and Regression Trees" by Leo Breiman.[31] Variables which decrease the impurity during splits a lot are considered important:[32]

${\displaystyle {\text{unormalized average importance}}(x)={\frac {1}{n_{T}}}\sum _{i=1}^{n_{T}}\sum _{{\text{node }}j\in T_{i}|{\text{split variable}}(j)=x}p_{T_{i}}(j)\Delta i_{T_{i}}(j),}$

where ${\displaystyle x}$ indicates a feature, ${\displaystyle n_{T}}$ is the number of trees in the forest, ${\displaystyle T_{i}}$ indicates tree ${\displaystyle i}$, ${\displaystyle p_{T_{i}}(j)={\frac {n_{j}}{n}}}$ is the fraction of samples reaching node ${\displaystyle j}$, ${\displaystyle \Delta i_{T_{i}}(j)}$ is the change in impurity in tree ${\displaystyle t}$ at node ${\displaystyle j}$. As impurity measure for samples falling in a node e.g. the following statistics can be used:

• entropy
• gini coefficient
• mean squared error

The normalized importance is then obtained by normalizing over all features, so that the sum of normalized feature importances is 1.

The sci-kit learn default implementation of Mean Decrease in Impurity Feature Importance is susceptible to misleading feature importances:[30]

• the importance measure prefers high cardinality features
• it uses training statistics and therefore does not "reflect the ability of feature to be useful to make predictions that generalize to the test set"[33]

A relationship between random forests and the k-nearest neighbor algorithm (k-NN) was pointed out by Lin and Jeon in 2002.[34] It turns out that both can be viewed as so-called weighted neighborhoods schemes. These are models built from a training set ${\displaystyle \{(x_{i},y_{i})\}_{i=1}^{n}}$ that make predictions ${\displaystyle {\hat {y}}}$ for new points x' by looking at the "neighborhood" of the point, formalized by a weight function W:

${\displaystyle {\hat {y}}=\sum _{i=1}^{n}W(x_{i},x')\,y_{i}.}$

Here, ${\displaystyle W(x_{i},x')}$ is the non-negative weight of the i'th training point relative to the new point x' in the same tree. For any particular x', the weights for points ${\displaystyle x_{i}}$ must sum to one. Weight functions are given as follows:

• In k-NN, the weights are ${\displaystyle W(x_{i},x')={\frac {1}{k}}}$ if x_i is one of the k points closest to x', and zero otherwise.
• In a tree, ${\displaystyle W(x_{i},x')={\frac {1}{k'}}}$ if x_i is one of the k' points in the same leaf as x', and zero otherwise.

Since a forest averages the predictions of a set of m trees with individual weight functions ${\displaystyle W_{j}}$, its predictions are

${\displaystyle {\hat {y}}={\frac {1}{m}}\sum _{j=1}^{m}\sum _{i=1}^{n}W_{j}(x_{i},x')\,y_{i}=\sum _{i=1}^{n}\left({\frac {1}{m}}\sum _{j=1}^{m}W_{j}(x_{i},x')\right)\,y_{i}.}$

This shows that the whole forest is again a weighted neighborhood scheme, with weights that average those of the individual trees. The neighbors of x' in this interpretation are the points ${\displaystyle x_{i}}$ sharing the same leaf in any tree ${\displaystyle j}$. In this way, the neighborhood of x' depends in a complex way on the structure of the trees, and thus on the structure of the training set. Lin and Jeon show that the shape of the neighborhood used by a random forest adapts to the local importance of each feature.[34]

Leo Breiman[41] was the first person to notice the link between random forest and kernel methods. He pointed out that random forests which are grown using i.i.d. random vectors in the tree construction are equivalent to a kernel acting on the true margin. Lin and Jeon[42] established the connection between random forests and adaptive nearest neighbor, implying that random forests can be seen as adaptive kernel estimates. Davies and Ghahramani[43] proposed Random Forest Kernel and show that it can empirically outperform state-of-art kernel methods. Scornet[40] first defined KeRF estimates and gave the explicit link between KeRF estimates and random forest. He also gave explicit expressions for kernels based on centered random forest[44] and uniform random forest,[45] two simplified models of random forest. He named these two KeRFs Centered KeRF and Uniform KeRF, and proved upper bounds on their rates of consistency.

Centered forest[44] is a simplified model for Breiman's original random forest, which uniformly selects an attribute among all attributes and performs splits at the center of the cell along the pre-chosen attribute. The algorithm stops when a fully binary tree of level ${\displaystyle k}$ is built, where ${\displaystyle k\in \mathbb {N} }$ is a parameter of the algorithm.

Uniform forest[45] is another simplified model for Breiman's original random forest, which uniformly selects a feature among all features and performs splits at a point uniformly drawn on the side of the cell, along the preselected feature.

Given a training sample ${\displaystyle {\mathcal {D}}_{n}=\{(\mathbf {X} _{i},Y_{i})\}_{i=1}^{n}}$ of ${\displaystyle [0,1]^{p}\times \mathbb {R} }$-valued independent random variables distributed as the independent prototype pair ${\displaystyle (\mathbf {X} ,Y)}$, where ${\displaystyle \operatorname {E} [Y^{2}]<\infty }$. We aim at predicting the response ${\displaystyle Y}$, associated with the random variable ${\displaystyle \mathbf {X} }$, by estimating the regression function ${\displaystyle m(\mathbf {x} )=\operatorname {E} [Y\mid \mathbf {X} =\mathbf {x} ]}$. A random regression forest is an ensemble of ${\displaystyle M}$ randomized regression trees. Denote ${\displaystyle m_{n}(\mathbf {x} ,\mathbf {\Theta } _{j})}$ the predicted value at point ${\displaystyle \mathbf {x} }$ by the ${\displaystyle j}$-th tree, where ${\displaystyle \mathbf {\Theta } _{1},\ldots ,\mathbf {\Theta } _{M}}$ are independent random variables, distributed as a generic random variable ${\displaystyle \mathbf {\Theta } }$, independent of the sample ${\displaystyle {\mathcal {D}}_{n}}$. This random variable can be used to describe the randomness induced by node splitting and the sampling procedure for tree construction. The trees are combined to form the finite forest estimate ${\displaystyle m_{M,n}(\mathbf {x} ,\Theta _{1},\ldots ,\Theta _{M})={\frac {1}{M}}\sum _{j=1}^{M}m_{n}(\mathbf {x} ,\Theta _{j})}$. For regression trees, we have ${\displaystyle m_{n}=\sum _{i=1}^{n}{\frac {Y_{i}\mathbf {1} _{\mathbf {X} _{i}\in A_{n}(\mathbf {x} ,\Theta _{j})}}{N_{n}(\mathbf {x} ,\Theta _{j})}}}$, where ${\displaystyle A_{n}(\mathbf {x} ,\Theta _{j})}$ is the cell containing ${\displaystyle \mathbf {x} }$, designed with randomness ${\displaystyle \Theta _{j}}$ and dataset ${\displaystyle {\mathcal {D}}_{n}}$, and ${\displaystyle N_{n}(\mathbf {x} ,\Theta _{j})=\sum _{i=1}^{n}\mathbf {1} _{\mathbf {X} _{i}\in A_{n}(\mathbf {x} ,\Theta _{j})}}$.

Thus random forest estimates satisfy, for all ${\displaystyle \mathbf {x} \in [0,1]^{d}}$, ${\displaystyle m_{M,n}(\mathbf {x} ,\Theta _{1},\ldots ,\Theta _{M})={\frac {1}{M}}\sum _{j=1}^{M}\left(\sum _{i=1}^{n}{\frac {Y_{i}\mathbf {1} _{\mathbf {X} _{i}\in A_{n}(\mathbf {x} ,\Theta _{j})}}{N_{n}(\mathbf {x} ,\Theta _{j})}}\right)}$. Random regression forest has two levels of averaging, first over the samples in the target cell of a tree, then over all trees. Thus the contributions of observations that are in cells with a high density of data points are smaller than that of observations which belong to less populated cells. In order to improve the random forest methods and compensate the misestimation, Scornet[40] defined KeRF by

${\displaystyle {\tilde {m}}_{M,n}(\mathbf {x} ,\Theta _{1},\ldots ,\Theta _{M})={\frac {1}{\sum _{j=1}^{M}N_{n}(\mathbf {x} ,\Theta _{j})}}\sum _{j=1}^{M}\sum _{i=1}^{n}Y_{i}\mathbf {1} _{\mathbf {X} _{i}\in A_{n}(\mathbf {x} ,\Theta _{j})},}$

which is equal to the mean of the ${\displaystyle Y_{i}}$'s falling in the cells containing ${\displaystyle \mathbf {x} }$ in the forest. If we define the connection function of the ${\displaystyle M}$ finite forest as ${\displaystyle K_{M,n}(\mathbf {x} ,\mathbf {z} )={\frac {1}{M}}\sum _{j=1}^{M}\mathbf {1} _{\mathbf {z} \in A_{n}(\mathbf {x} ,\Theta _{j})}}$, i.e. the proportion of cells shared between ${\displaystyle \mathbf {x} }$ and ${\displaystyle \mathbf {z} }$, then almost surely we have ${\displaystyle {\tilde {m}}_{M,n}(\mathbf {x} ,\Theta _{1},\ldots ,\Theta _{M})={\frac {\sum _{i=1}^{n}Y_{i}K_{M,n}(\mathbf {x} ,\mathbf {x} _{i})}{\sum _{\ell =1}^{n}K_{M,n}(\mathbf {x} ,\mathbf {x} _{\ell })}}}$, which defines the KeRF.

The construction of Centered KeRF of level ${\displaystyle k}$ is the same as for centered forest, except that predictions are made by ${\displaystyle {\tilde {m}}_{M,n}(\mathbf {x} ,\Theta _{1},\ldots ,\Theta _{M})}$, the corresponding kernel function, or connection function is

${\displaystyle {\begin{aligned}K_{k}^{cc}(\mathbf {x} ,\mathbf {z} )=\sum _{k_{1},\ldots ,k_{d},\sum _{j=1}^{d}k_{j}=k}&{\frac {k!}{k_{1}!\cdots k_{d}!}}\left({\frac {1}{d}}\right)^{k}\prod _{j=1}^{d}\mathbf {1} _{\lceil 2^{k_{j}}x_{j}\rceil =\lceil 2^{k_{j}}z_{j}\rceil },\\&{\text{ for all }}\mathbf {x} ,\mathbf {z} \in [0,1]^{d}.\end{aligned}}}$

Uniform KeRF is built in the same way as uniform forest, except that predictions are made by ${\displaystyle {\tilde {m}}_{M,n}(\mathbf {x} ,\Theta _{1},\ldots ,\Theta _{M})}$, the corresponding kernel function, or connection function is

${\displaystyle K_{k}^{uf}(\mathbf {0} ,\mathbf {x} )=\sum _{k_{1},\ldots ,k_{d},\sum _{j=1}^{d}k_{j}=k}{\frac {k!}{k_{1}!\ldots k_{d}!}}\left({\frac {1}{d}}\right)^{k}\prod _{m=1}^{d}\left(1-|x_{m}|\sum _{j=0}^{k_{m}-1}{\frac {(-\ln |x_{m}|)^{j}}{j!}}\right){\text{ for all }}\mathbf {x} \in [0,1]^{d}.}$

Predictions given by KeRF and random forests are close if the number of points in each cell is controlled:

Assume that there exist sequences ${\displaystyle (a_{n}),(b_{n})}$ such that, almost surely,

${\displaystyle a_{n}\leq N_{n}(\mathbf {x} ,\Theta )\leq b_{n}{\text{ and }}a_{n}\leq {\frac {1}{M}}\sum _{m=1}^{M}N_{n}{\mathbf {x} ,\Theta _{m}}\leq b_{n}.}$

Then almost surely,

${\displaystyle |m_{M,n}(\mathbf {x} )-{\tilde {m}}_{M,n}(\mathbf {x} )|\leq {\frac {b_{n}-a_{n}}{a_{n}}}{\tilde {m}}_{M,n}(\mathbf {x} ).}$

When the number of trees ${\displaystyle M}$ goes to infinity, then we have infinite random forest and infinite KeRF. Their estimates are close if the number of observations in each cell is bounded:

Assume that there exist sequences ${\displaystyle (\varepsilon _{n}),(a_{n}),(b_{n})}$ such that, almost surely

• ${\displaystyle \operatorname {E} [N_{n}(\mathbf {x} ,\Theta )]\geq 1,}$
• ${\displaystyle \operatorname {P} [a_{n}\leq N_{n}(\mathbf {x} ,\Theta )\leq b_{n}\mid {\mathcal {D}}_{n}]\geq 1-\varepsilon _{n}/2,}$
• ${\displaystyle \operatorname {P} [a_{n}\leq \operatorname {E} _{\Theta }[N_{n}(\mathbf {x} ,\Theta )]\leq b_{n}\mid {\mathcal {D}}_{n}]\geq 1-\varepsilon _{n}/2,}$

Then almost surely,

${\displaystyle |m_{\infty ,n}(\mathbf {x} )-{\tilde {m}}_{\infty ,n}(\mathbf {x} )|\leq {\frac {b_{n}-a_{n}}{a_{n}}}{\tilde {m}}_{\infty ,n}(\mathbf {x} )+n\varepsilon _{n}\left(\max _{1\leq i\leq n}Y_{i}\right).}$

Assume that ${\displaystyle Y=m(\mathbf {X} )+\varepsilon }$, where ${\displaystyle \varepsilon }$ is a centered Gaussian noise, independent of ${\displaystyle \mathbf {X} }$, with finite variance ${\displaystyle \sigma ^{2}<\infty }$. Moreover, ${\displaystyle \mathbf {X} }$ is uniformly distributed on ${\displaystyle [0,1]^{d}}$ and ${\displaystyle m}$ is Lipschitz. Scornet[40] proved upper bounds on the rates of consistency for centered KeRF and uniform KeRF.

Providing ${\displaystyle k\rightarrow \infty }$ and ${\displaystyle n/2^{k}\rightarrow \infty }$, there exists a constant ${\displaystyle C_{1}>0}$ such that, for all ${\displaystyle n}$, ${\displaystyle \mathbb {E} [{\tilde {m}}_{n}^{cc}(\mathbf {X} )-m(\mathbf {X} )]^{2}\leq C_{1}n^{-1/(3+d\log 2)}(\log n)^{2}}$.

Providing ${\displaystyle k\rightarrow \infty }$ and ${\displaystyle n/2^{k}\rightarrow \infty }$, there exists a constant ${\displaystyle C>0}$ such that, ${\displaystyle \mathbb {E} [{\tilde {m}}_{n}^{uf}(\mathbf {X} )-m(\mathbf {X} )]^{2}\leq Cn^{-2/(6+3d\log 2)}(\log n)^{2}}$.