in some casessuch as the one just discussedit may be useful to use splitting rule that involves several variablesas opposed to single one the decision regarding the split type clearly depends on the problem domain for examplefor logical (binaryvariables our domain knowledge may indicate that different behavior is expected when both xi and ( jare true in this casewe will naturally introduce decision rule of the forms( {xi true and truecategorical variables when an explanatory variable is categorical with labels (levelssay { }the splitting rule is generally defined via partition of the label set { kinto two subsets specificallylet and be partition of { kthenthe splitting rule is defined via ( { lfor the general supervised learning casefinding the optimal partition in the sense of minimal loss requires one to consider subsets of { kconsequentlyfinding good splitting rule for categorical variables can be challenging when the number of labels is large missing values missing data is present in many real-life problems generallywhen working with incomplete feature vectorswhere one or more values are missingit is typical to either completely delete the feature vector from the data (which may distort the dataor to impute (guessits missing values from the available datasee [ tree methodshoweverallow an elegant approach for handling missing data specificallyin the general casethe missing data problem can be handled via surrogate splitting rules [ when dealing with categorical (factorfeatureswe can introduce an additional category "missingfor the absent data the main idea of surrogate rules is as follows firstwe construct decision (regression or classificationtree via algorithm during this construction processthe solution of the optimization problem ( is calculated only over the observations that are not missing particular variable suppose that tree node has splitting rule ( { xfor some and threshold xfor the node we can introduce set of alternative splitting rules that resemble the original splitting rulesometimes called the primary splitting ruleusing different variables and thresholds namelywe look for binary splitting rule ( jx) jsuch that the data split introduced by will be similar to the original data split from sthe similarity is generally measured via binary misclassification losswhere the true classes of observations are determined by the primary splitting rule and the surrogate splitting rules serve as classifiers considerfor examplethe data in table and suppose that the primary splitting rule at node is {age that isthe five data points are split such that the left and the right child of contains two and three data pointsrespectively nextthe following surrogate splitting rules can be considered