A decision tree is a binary tree (tree where each non-leaf node has two child nodes). It can be used either for classification or for regression. For classification, each tree leaf is marked with a class label; multiple leaves may have the same label. For regression, a constant is also assigned to each tree leaf, so the approximation function is piecewise constant.
**Ordered variables.** The variable value is compared with a threshold that is also stored in the node. If the value is less than the threshold, the procedure goes to the left. Otherwise, it goes to the right. For example, if the weight is less than 1 kilogram, the procedure goes to the left, else to the right.
**Categorical variables.** A discrete variable value is tested to see whether it belongs to a certain subset of values (also stored in the node) from a limited set of values the variable could take. If it does, the procedure goes to the left. Otherwise, it goes to the right. For example, if the color is green or red, go to the left, else to the right.
So, in each node, a pair of entities (``variable_index`` , ``decision_rule
(threshold/subset)`` ) is used. This pair is called a *split* (split on
the variable ``variable_index`` ). Once a leaf node is reached, the value
assigned to this node is used as the output of the prediction procedure.
Sometimes, certain features of the input vector are missed (for example, in the darkness it is difficult to determine the object color), and the prediction procedure may get stuck in the certain node (in the mentioned example, if the node is split by color). To avoid such situations, decision trees use so-called *surrogate splits*. That is, in addition to the best "primary" split, every tree node may also be split to one or more other variables with nearly the same results.
The tree is built recursively, starting from the root node. All training data (feature vectors and responses) is used to split the root node. In each node the optimum decision rule (the best "primary" split) is found based on some criteria. In machine learning, ``gini`` "purity" criteria are used for classification, and sum of squared errors is used for regression. Then, if necessary, the surrogate splits are found. They resemble the results of the primary split on the training data. All the data is divided using the primary and the surrogate splits (like it is done in the prediction procedure) between the left and the right child node. Then, the procedure recursively splits both left and right nodes. At each node the recursive procedure may stop (that is, stop splitting the node further) in one of the following cases:
When the tree is built, it may be pruned using a cross-validation procedure, if necessary. That is, some branches of the tree that may lead to the model overfitting are cut off. Normally, this procedure is only applied to standalone decision trees. Usually tree ensembles build trees that are small enough and use their own protection schemes against overfitting.
Besides the prediction that is an obvious use of decision trees, the tree can be also used for various data analyses. One of the key properties of the constructed decision tree algorithms is an ability to compute the importance (relative decisive power) of each variable. For example, in a spam filter that uses a set of words occurred in the message as a feature vector, the variable importance rating can be used to determine the most "spam-indicating" words and thus help keep the dictionary size reasonable.
Importance of each variable is computed over all the splits on this variable in the tree, primary and surrogate ones. Thus, to compute variable importance correctly, the surrogate splits must be enabled in the training parameters, even if there is no missing data.
The split quality, a positive number. It is used to choose the best primary split, then to choose and sort the surrogate splits. After the tree is constructed, it is also used to compute variable importance.
Tree index in a ordered sequence of pruned trees. The indices are used during and after the pruning procedure. The root node has the maximum value ``Tn`` of the whole tree, child nodes have ``Tn`` less than or equal to the parent's ``Tn``, and nodes with :math:`Tn \leq CvDTree::pruned\_tree\_idx` are not used at prediction stage (the corresponding branches are considered as cut-off), even if they have not been physically deleted from the tree at the pruning stage.
The number of samples that fall into the node at the training stage. It is used to resolve the difficult cases - when the variable for the primary split is missing and all the variables for other surrogate splits are missing too. In this case the sample is directed to the left if ``left->sample_count > right->sample_count`` and to the right otherwise.
The structure contains all the decision tree training parameters. You can initialize it by default constructor and then override any parameters directly before training, or the structure may be fully initialized using the advanced variant of the constructor.
:param max_depth:The maximum possible depth of the tree. That is the training algorithms attempts to split a node while its depth is less than ``max_depth``. The actual depth may be smaller if the other termination criteria are met (see the outline of the training procedure in the beginning of the section), and/or if the tree is pruned.
:param regression_accuracy:Termination criteria for regression trees. If all absolute differences between an estimated value in a node and values of train samples in this node are less than this parameter then the node will not be split.
:param use_surrogates:If true then surrogate splits will be built. These splits allow to work with missing data and compute variable importance correctly.
:param max_categories:Cluster possible values of a categorical variable into ``K`` :math:`\leq` ``max_categories`` clusters to find a suboptimal split. If a discrete variable, on which the training procedure tries to make a split, takes more than ``max_categories`` values, the precise best subset estimation may take a very long time because the algorithm is exponential. Instead, many decision trees engines (including ML) try to find sub-optimal split in this case by clustering all the samples into ``max_categories`` clusters that is some categories are merged together. The clustering is applied only in ``n``>2-class classification problems for categorical variables with ``N > max_categories`` possible values. In case of regression and 2-class classification the optimal split can be found efficiently without employing clustering, thus the parameter is not used in these cases.
:param use_1se_rule:If true then a pruning will be harsher. This will make a tree more compact and more resistant to the training data noise but a bit less accurate.
:param truncate_pruned_tree:If true then pruned branches are physically removed from the tree. Otherwise they are retained and it is possible to get results from the original unpruned (or pruned less aggressively) tree by decreasing ``CvDTree::pruned_tree_idx`` parameter.
:param priors:The array of a priori class probabilities, sorted by the class label value. The parameter can be used to tune the decision tree preferences toward a certain class. For example, if you want to detect some rare anomaly occurrence, the training base will likely contain much more normal cases than anomalies, so a very good classification performance will be achieved just by considering every case as normal. To avoid this, the priors can be specified, where the anomaly probability is artificially increased (up to 0.5 or even greater), so the weight of the misclassified anomalies becomes much bigger, and the tree is adjusted properly. You can also think about this parameter as weights of prediction categories which determine relative weights that you give to misclassification. That is, if the weight of the first category is 1 and the weight of the second category is 10, then each mistake in predicting the second category is equivalent to making 10 mistakes in predicting the first category.
Decision tree training data and shared data for tree ensembles. The structure is mostly used internally for storing both standalone trees and tree ensembles efficiently. Basically, it contains the following types of information:
#. Training data preprocessed to find the best splits more efficiently. For tree ensembles, this preprocessed data is reused by all trees. Additionally, the training data characteristics shared by all trees in the ensemble are stored here: variable types, the number of classes, a class label compression map, and so on.
There are two ways of using this structure. In simple cases (for example, a standalone tree or the ready-to-use "black box" tree ensemble from machine learning, like
:ref:`Boosting` ), there is no need to care or even to know about the structure. You just construct the needed statistical model, train it, and use it. The ``CvDTreeTrainData`` structure is constructed and used internally. However, for custom tree algorithms or another sophisticated cases, the structure may be constructed and used explicitly. The scheme is the following:
The structure is initialized using the default constructor, followed by ``set_data``, or it is built using the full form of constructor. The parameter ``_shared`` must be set to ``true``.
* The **first two** methods follow the generic :ocv:func:`CvStatModel::train` conventions. It is the most complete form. Both data layouts (``tflag=CV_ROW_SAMPLE`` and ``tflag=CV_COL_SAMPLE``) are supported, as well as sample and variable subsets, missing measurements, arbitrary combinations of input and output variable types, and so on. The last parameter contains all of the necessary training parameters (see the :ocv:class:`CvDTreeParams` description).
* The **last** method ``train`` is mostly used for building tree ensembles. It takes the pre-constructed :ocv:class:`CvDTreeTrainData` instance and an optional subset of the training set. The indices in ``subsampleIdx`` are counted relatively to the ``_sample_idx`` , passed to the ``CvDTreeTrainData`` constructor. For example, if ``_sample_idx=[1, 5, 7, 100]`` , then ``subsampleIdx=[0,3]`` means that the samples ``[1, 100]`` of the original training set are used.
:param preprocessedInput:This parameter is normally set to ``false``, implying a regular input. If it is ``true``, the method assumes that all the values of the discrete input variables have been already normalized to :math:`0` to :math:`num\_of\_categories_i-1` ranges since the decision tree uses such normalized representation internally. It is useful for faster prediction with tree ensembles. For ordered input variables, the flag is not used.
The method traverses the decision tree and returns the reached leaf node as output. The prediction result, either the class label or the estimated function value, may be retrieved as the ``value`` field of the :ocv:class:`CvDTreeNode` structure, for example: ``dtree->predict(sample,mask)->value``.
:param resp:If it is not null then size of this vector will be set to the number of samples and each element will be set to result of prediction on the corresponding sample.
The method calculates error of the decision tree. In case of classification it is the percentage of incorrectly classified samples and in case of regression it is the mean of squared errors on samples.
