Random Forest Implementation in Java
This section gives a brief overview of random forests and some comments about the features of the method.
We assume that the user knows about the construction of single classification trees. Random Forests grows many classification trees. To classify a new object from an input vector, put the input vector down each of the trees in the forest. Each tree gives a classification, and we say the tree "votes" for that class. The forest chooses the classification having the most votes (over all the trees in the forest).
Each tree is grown as follows:
If the number of cases in the training set is N, sample N cases at random - but with replacement, from the original data. This sample will be the training set for growing the tree. If there are M input variables, a number m<<M is specified such that at each node, m variables are selected at random out of the M and the best split on these m is used to split the node. The value of m is held constant during the forest growing. Each tree is grown to the largest extent possible. There is no pruning. In the original paper on random forests, it was shown that the forest error rate depends on two things:
The correlation between any two trees in the forest. Increasing the correlation increases the forest error rate. The strength of each individual tree in the forest. A tree with a low error rate is a strong classifier. Increasing the strength of the individual trees decreases the forest error rate. Reducing m reduces both the correlation and the strength. Increasing it increases both. Somewhere in between is an "optimal" range of m - usually quite wide. Using the oob error rate (see below) a value of m in the range can quickly be found. This is the only adjustable parameter to which random forests is somewhat sensitive.
It is unexcelled in accuracy among current algorithms. It runs efficiently on large data bases. It can handle thousands of input variables without variable deletion. It gives estimates of what variables are important in the classification. It generates an internal unbiased estimate of the generalization error as the forest building progresses. It has an effective method for estimating missing data and maintains accuracy when a large proportion of the data are missing. It has methods for balancing error in class population unbalanced data sets. Generated forests can be saved for future use on other data. Prototypes are computed that give information about the relation between the variables and the classification. It computes proximities between pairs of cases that can be used in clustering, locating outliers, or (by scaling) give interesting views of the data. The capabilities of the above can be extended to unlabeled data, leading to unsupervised clustering, data views and outlier detection. It offers an experimental method for detecting variable interactions. Remarks Random forests does not overfit. You can run as many trees as you want. It is fast. Running on a data set with 50,000 cases and 100 variables, it produced 100 trees in 11 minutes on a 800Mhz machine. For large data sets the major memory requirement is the storage of the data itself, and three integer arrays with the same dimensions as the data. If proximities are calculated, storage requirements grow as the number of cases times the number of trees.
To understand and use the various options, further information about how they are computed is useful. Most of the options depend on two data objects generated by random forests.
When the training set for the current tree is drawn by sampling with replacement, about one-third of the cases are left out of the sample. This oob (out-of-bag) data is used to get a running unbiased estimate of the classification error as trees are added to the forest. It is also used to get estimates of variable importance.
After each tree is built, all of the data are run down the tree, and proximities are computed for each pair of cases. If two cases occupy the same terminal node, their proximity is increased by one. At the end of the run, the proximities are normalized by dividing by the number of trees. Proximities are used in replacing missing data, locating outliers, and producing illuminating low-dimensional views of the data.
In random forests, there is no need for cross-validation or a separate test set to get an unbiased estimate of the test set error. It is estimated internally, during the run, as follows:
Each tree is constructed using a different bootstrap sample from the original data. About one-third of the cases are left out of the bootstrap sample and not used in the construction of the kth tree.
Put each case left out in the construction of the kth tree down the kth tree to get a classification. In this way, a test set classification is obtained for each case in about one-third of the trees. At the end of the run, take j to be the class that got most of the votes every time case n was oob. The proportion of times that j is not equal to the true class of n averaged over all cases is the oob error estimate. This has proven to be unbiased in many tests.
In every tree grown in the forest, put down the oob cases and count the number of votes cast for the correct class. Now randomly permute the values of variable m in the oob cases and put these cases down the tree. Subtract the number of votes for the correct class in the variable-m-permuted oob data from the number of votes for the correct class in the untouched oob data. The average of this number over all trees in the forest is the raw importance score for variable m.
If the values of this score from tree to tree are independent, then the standard error can be computed by a standard computation. The correlations of these scores between trees have been computed for a number of data sets and proved to be quite low, therefore we compute standard errors in the classical way, divide the raw score by its standard error to get a z-score, ands assign a significance level to the z-score assuming normality.
If the number of variables is very large, forests can be run once with all the variables, then run again using only the most important variables from the first run.
For each case, consider all the trees for which it is oob. Subtract the percentage of votes for the correct class in the variable-m-permuted oob data from the percentage of votes for the correct class in the untouched oob data. This is the local importance score for variable m for this case, and is used in the graphics program RAFT.
Every time a split of a node is made on variable m the gini impurity criterion for the two descendent nodes is less than the parent node. Adding up the gini decreases for each individual variable over all trees in the forest gives a fast variable importance that is often very consistent with the permutation importance measure.
The operating definition of interaction used is that variables m and k interact if a split on one variable, say m, in a tree makes a split on k either systematically less possible or more possible. The implementation used is based on the gini values g(m) for each tree in the forest. These are ranked for each tree and for each two variables, the absolute difference of their ranks are averaged over all trees.
This number is also computed under the hypothesis that the two variables are independent of each other and the latter subtracted from the former. A large positive number implies that a split on one variable inhibits a split on the other and conversely. This is an experimental procedure whose conclusions need to be regarded with caution. It has been tested on only a few data sets.
These are one of the most useful tools in random forests. The proximities originally formed a NxN matrix. After a tree is grown, put all of the data, both training and oob, down the tree. If cases k and n are in the same terminal node increase their proximity by one. At the end, normalize the proximities by dividing by the number of trees.
Users noted that with large data sets, they could not fit an NxN matrix into fast memory. A modification reduced the required memory size to NxT where T is the number of trees in the forest. To speed up the computation-intensive scaling and iterative missing value replacement, the user is given the option of retaining only the nrnn largest proximities to each case.
When a test set is present, the proximities of each case in the test set with each case in the training set can also be computed. The amount of additional computing is moderate.
The proximities between cases n and k form a matrix {prox(n,k)}. From their definition, it is easy to show that this matrix is symmetric, positive definite and bounded above by 1, with the diagonal elements equal to 1. It follows that the values 1-prox(n,k) are squared distances in a Euclidean space of dimension not greater than the number of cases. For more background on scaling see "Multidimensional Scaling" by T.F. Cox and M.A. Cox.
Let prox(-,k) be the average of prox(n,k) over the 1st coordinate, prox(n,-) be the average of prox(n,k) over the 2nd coordinate, and prox(-,-) the average over both coordinates. Then the matrix
cv(n,k)=.5*(prox(n,k)-prox(n,-)-prox(-,k)+prox(-,-)) is the matrix of inner products of the distances and is also positive definite symmetric. Let the eigenvalues of cv be l(j) and the eigenvectors nj(n). Then the vectors
x(n) = (Öl(1) n1(n) , Öl(2) n2(n) , ...,)
have squared distances between them equal to 1-prox(n,k). The values of Öl(j) nj(n) are referred to as the jth scaling coordinate.
In metric scaling, the idea is to approximate the vectors x(n) by the first few scaling coordinates. This is done in random forests by extracting the largest few eigenvalues of the cv matrix, and their corresponding eigenvectors . The two dimensional plot of the ith scaling coordinate vs. the jth often gives useful information about the data. The most useful is usually the graph of the 2nd vs. the 1st.
Since the eigenfunctions are the top few of an NxN matrix, the computational burden may be time consuming. We advise taking nrnn considerably smaller than the sample size to make this computation faster.
There are more accurate ways of projecting distances down into low dimensions, for instance the Roweis and Saul algorithm. But the nice performance, so far, of metric scaling has kept us from implementing more accurate projection algorithms. Another consideration is speed. Metric scaling is the fastest current algorithm for projecting down.
Generally three or four scaling coordinates are sufficient to give good pictures of the data. Plotting the second scaling coordinate versus the first usually gives the most illuminating view.
Prototypes are a way of getting a picture of how the variables relate to the classification. For the jth class, we find the case that has the largest number of class j cases among its k nearest neighbors, determined using the proximities. Among these k cases we find the median, 25th percentile, and 75th percentile for each variable. The medians are the prototype for class j and the quartiles give an estimate of is stability. For the second prototype, we repeat the procedure but only consider cases that are not among the original k, and so on. When we ask for prototypes to be output to the screen or saved to a file, prototypes for continuous variables are standardized by subtractng the 5th percentile and dividing by the difference between the 95th and 5th percentiles. For categorical variables, the prototype is the most frequent value. When we ask for prototypes to be output to the screen or saved to a file, all frequencies are given for categorical variables.
Random forests has two ways of replacing missing values. The first way is fast. If the mth variable is not categorical, the method computes the median of all values of this variable in class j, then it uses this value to replace all missing values of the mth variable in class j. If the mth variable is categorical, the replacement is the most frequent non-missing value in class j. These replacement values are called fills. The second way of replacing missing values is computationally more expensive but has given better performance than the first, even with large amounts of missing data. It replaces missing values only in the training set. It begins by doing a rough and inaccurate filling in of the missing values. Then it does a forest run and computes proximities.
If x(m,n) is a missing continuous value, estimate its fill as an average over the non-missing values of the mth variables weighted by the proximities between the nth case and the non-missing value case. If it is a missing categorical variable, replace it by the most frequent non-missing value where frequency is weighted by proximity.
Now iterate-construct a forest again using these newly filled in values, find new fills and iterate again. Our experience is that 4-6 iterations are enough.
When there is a test set, there are two different methods of replacement depending on whether labels exist for the test set.
If they do, then the fills derived from the training set are used as replacements. If labels no not exist, then each case in the test set is replicated nclass times (nclass= number of classes). The first replicate of a case is assumed to be class 1 and the class one fills used to replace missing values. The 2nd replicate is assumed class 2 and the class 2 fills used on it.
This augmented test set is run down the tree. In each set of replicates, the one receiving the most votes determines the class of the original case.
The training sets are often formed by using human judgment to assign labels. In some areas this leads to a high frequency of mislabeling. Many of the mislabeled cases can be detected using the outlier measure. An example is given in the DNA case study.
Outliers are generally defined as cases that are removed from the main body of the data. Translate this as: outliers are cases whose proximities to all other cases in the data are generally small. A useful revision is to define outliers relative to their class. Thus, an outlier in class j is a case whose proximities to all other class j cases are small.
Define the average proximity from case n in class j to the rest of the training data class j as:
The raw outlier measure for case n is defined as
This will be large if the average proximity is small. Within each class find the median of these raw measures, and their absolute deviation from the median. Subtract the median from each raw measure, and divide by the absolute deviation to arrive at the final outlier measure.
In unsupervised learning the data consist of a set of x -vectors of the same dimension with no class labels or response variables. There is no figure of merit to optimize, leaving the field open to ambiguous conclusions. The usual goal is to cluster the data - to see if it falls into different piles, each of which can be assigned some meaning.
The approach in random forests is to consider the original data as class 1 and to create a synthetic second class of the same size that will be labeled as class 2. The synthetic second class is created by sampling at random from the univariate distributions of the original data. Here is how a single member of class two is created - the first coordinate is sampled from the N values {x(1,n)}. The second coordinate is sampled independently from the N values {x(2,n)}, and so forth.
Thus, class two has the distribution of independent random variables, each one having the same univariate distribution as the corresponding variable in the original data. Class 2 thus destroys the dependency structure in the original data. But now, there are two classes and this artificial two-class problem can be run through random forests. This allows all of the random forests options to be applied to the original unlabeled data set.
If the oob misclassification rate in the two-class problem is, say, 40% or more, it implies that the x -variables look too much like independent variables to random forests. The dependencies do not have a large role and not much discrimination is taking place. If the misclassification rate is lower, then the dependencies are playing an important role.
Formulating it as a two class problem has a number of payoffs. Missing values can be replaced effectively. Outliers can be found. Variable importance can be measured. Scaling can be performed (in this case, if the original data had labels, the unsupervised scaling often retains the structure of the original scaling). But the most important payoff is the possibility of clustering.