Binary decision trees

To keep the description of the regions simple, binary decision trees with a root, nodes, and leaves, the R, s, are favored. Each region is split into two further regions over and over again termed a recursive binary partitioning. The decision for the binary split is based on a constant, s, as shown in Figure 5.34b. 

Li, Q., Bermak, A., A Low-Power Hardware-Friendly Binary Decision Tree Classifier for Gas Identification. Journal of Low Power Electronics and Applications, 1(1), (2011)45-58. 

We explain the basic principles of the BDD representation on an example. Fig. 1 presents the binary decision tree for the 2-bit comparison x y h X2 V2- The binary tree representation of a formula is exponential in the number of free boolean propositions in the formula. 

Figure 6.1 A binary decision tree to aid the selection of electrolytic cell design for a new process. Reproduced from Ref [12] with permission from lUPAC 2001.

Cabrera et al. [50] modeled a set of 163 drugs using TOPS-MODE descriptors with a linear discriminant model to predict p-glycoprotein efflux. Model accuracy was 81% for the training set and 77.5% for a validation set of 40 molecules. A "combinatorial QSAR" approach was used by de Lima et al. [51] to test multiple model types (kNN, decision tree, binary QSAR, SVM) with multiple descriptor sets from various software packages (MolconnZ, Atom Pair, VoSurf, MOE) for the prediction of p-glycoprotein substrates for a dataset of 192 molecules. Best overall performance on a test set of 51 molecules was achieved with an SVM and AP or VolSurf descriptors (81% accuracy each). 

The main limitation of classification trees is their instability. Small changes in the data can result in a completely different tree. This is due to the hierarchical structure of the binary decisions, where a slightly different split on top can cause completely different splits subsequently. A procedure called bagging can reduce this instability by averaging many trees (Hastie et al. 2001). 

Decision Trees are also a well-known technique in the field [151]. They arrange a subset of the descriptor components in a hierarchical fashion (a binary tree) such that on a particular node in the tree a classification on a single descriptor component decides whether the left or the right branch underneath is followed. The leaves of the tree determine the overall classification label. Decision trees have been found useful, especially on large-scale descriptors like binary pharmacophore descriptors [152]. 

The orthogonality of a set of molecular descriptors is a very desirable property. Classification methodologies such as CART (11) (or other decision-tree methods) are not invariant to rotations of the chemistry space. Such methods may encounter difficulties with correlated descriptors (e.g., production of larger decision trees). Often, correlated descriptors necessitate the use of principal components transforms that require a set of reference data for their estimation (at worst, the transforms depend only on the data at hand and, at best, they are trained once from some larger collection of compounds). In probabilistic methodologies, such as Binary QSAR (12), approximation of statistical independence is simplified when uncorrelated descriptors are used. In addition,... 

Decision trees [135] can be used to identify and segment spectra when discriminating rules are known or desired (Fig. 8.8). A binary tree consists of nodes in which a single parameter is used as a discriminant. After a series of nodes are traversed, leaf nodes of the tree are encountered in which all the objects are labeled as belonging to a particular class. Decision trees can be axis parallel or oblique. Axis-parallel trees are called so because they correspond to... 

The mathematical representation of this operational procedure has a main decision to take either to represent the tanks in an aggregated manner or include each tank as a model instance and obtain additionally as model result the alternation schemes for all products. For the later, such detail level at the mathematical representation results in higher model size due to a new set, new variables (either continuous and binary, expanding the decision tree size) and higher number of equations. Three key aspects have to be considered when modeling individual tanks i) the allocation of tanks to products, ii) the tanks operational cycle and hi) the initialization data. 

Another routine develops a decision tree of binary choices which, taken as a whole, can classify the members of the training set. The decision tree generated implements a piecewise linear discriminant function. The decision tree is developed by splitting the data set into two parts in an optimal way at each node. A node is considered to be terminal when no advantageous split can be made a terminal node is labelled to associate with the pattern class most represented among the patterns present at the node. 

Sinnamon, R., Andreas, J., January 1996. Fault tree analysis and binary decision diagrams. In Proceedings of the Reliability and Maintainability Symposium. 

Outline In Sections 2 to 5, we overview the foundations of symbolic model checking Kripke structures, the class of models considered binary decision diagrams, an efficient data structure to represent such structures elements of fixpoint theory in lattices syntax and semantics of computation tree logic. In Section 6, we present the main result of the paper a sufficient condition for a given property to be an invariant of a given model. We also show how to incorporate the computation of this sufficient condition in CTL model checking. 

One of the first pattern recognition applications in mass spectrometry was the attempt to determine the molecular formula by a decision tree C120, 128, 1293. The decision tree contained several binary classifiers. Each of the classifiers decided whether a compound contains more atoms than a given number- A run through the decision tree yields the molecular formula of an unknown whose low resolution mass spectrum is known. A tree with 26 classifiers was necessary for a set of 346 compounds of formulas --i 6 0-3 0-2 spectra with an artifi-... 

A decision tree is a binary rooted tree, i.e. there is one initial node v ,. the root, and each node (other than a leaf) has exactly two successors. The leaves are also called terminal nodes, all remaining nodes are internal nodes. Internal nodes beeu decision rules of the form Xj < a, terminal nodes bear function values %. The node numbering is such that an internal node has successors V2fc+i and V2fc+2> see Figure 6.4. 

In constructing a decision tree, the learning set is successively partitioned into two disjoint subsets. Partitioning is done according to a binary decision rule Xj < a. Let be the index set of the observations represented by V. Further,... 

J. D. Andrews, S.J. Dunnett, Event Tree Analysis Using Binary Decision Diagrams, Lough-... 

Andrews, J.D. Dunnett, S.J. 2000. Event-tree analysis using binary decision diagrams. IEEE Transactions on Reliability 49(2) 230-238. 

The various evaluation and selection criteria are either of a binary nature, i.e., they allow yes-no decisions leading to immediate acceptance or rejection of a reaction, or they allow the assignment of a numerical value to a reaction which reflects its quality. In combination with upper or lower limits this numerical value may also serve to accept or reject a reaction. These quality factors have to be carried over several levels of the synthesis tree (see Chapter 6). 

The hierarchic principle underlying the approach to a construction of a chemical name (parent, substituent, substituent-on-substituent, etc.) was the decisive factor in the design of the data format for effective name generation analysis. The data structure finally implemented has been built around the concept of an ordered binary tree. ... 

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