LOGICAL CLASSIFICATION TREES IN RECOGNITION PROBLEMS

The paper is dedicated to algorithms for constructing a logical tree of classification. Nowadays, there exist many algorithms for constructing logical classification trees. However, all of them, as a rule, are reduced to the construction of a single classification tree based on the data of a fixed training sample. There are very few algorithms for constructing recognition trees that are designed for large data sets. It is obvious that such sets have objective factors associated with the peculiarities of the generation of such complex structures, methods of working with them and storage. In this paper, we focus on the description of the algorithm for constructing classification trees for a large training set and show the way to the possibility of a uniform description of a fixed class of recognition trees. A simple, effective, economical method of constructing a logical classification tree of the training sample allows you to provide the necessary speed, the level of complexity of the recognition scheme, which guarantees a simple and complete recognition of discrete objects.


Introduction
As of today, various algorithms for constructing logical classification trees are known [7]. However, all of them, as a rule, are reduced to the construction of a single classification tree from the data of a fixed training sample. Note that in the literature there are very few algorithms for constructing logical trees for training samples of large volume. It is clear that this is based on objective factors associated with the features of the generation of such complex structures, methods of working with them and storage [5]. Even using the tools of Java or C#, it is necessary to provide the implementation of special data structures for working with logical trees, and ready-made libraries (LightGBM, XGBoost), although close ideologically (logical tree scheme), do not allow to implement the concept of an algorithmic classification tree, which consists of a set of verticesdifferent types of Autonomous classification algorithms. However, the main drawback in the construction of logical trees is the lack of algorithms and methods that would allow uniformly describe different algorithms for pattern recognition in the form of tree structures.
The ability to represent the recognition function as a logical tree has great advantages over other representations of classification schemes [4]. It should be noted that the algorithms for generating classification trees according to the training sample complement the methodology of the branched feature selection approach and allow to build simple and effective rules for the classification of discrete objects [1].
In this paper, we will focus on the description of the algorithm for constructing logical training samples for a large volume and show the way to the possibility of a uniform description of a fixed class of logical trees.

Statement of the recognition problem
In fact, the central task of pattern recognition is to build such a system that for each object that will be presented to it, would give the number of the class to which the object belongs [3,4,5]. The general problem of pattern recognition (classification) can be formulated in the following simplified form. Let on some set M of objects w a given partition R into a finite number m of subsets (classes, images) H i (i = 0, ..., m): As a rule, in the problems of pattern recognition at the beginning is given some a priori information about R the nature of the partition and, depending on the nature of this information, it is customary to consider the following recognition tasks with training, without training, with self-learning. In this paper, the main attention will be paid to the problems of recognition with the previous training.
It should be noted that the fixed set of features that are characterized w, is always the same for all objects that are considered in solving this problem. Each feature can take values from different sets of valid feature values. For example, very often signs take values from a set the value of a sign can be a distribution function of some random variable. Objects that belong to one class are characterized by a certain commonality of their features, and objects from different classes do not have such a commonality, therefore the solution to the problem of recognition is to somehow highlight and describe this commonality or its absence.
When recognizing images, the most important, and sometimes the only given information about the partition is the training sample: It is on the basis of it in the problems with the previous training that the classification rule is built, and the solution is to determine the class to which the object that is being studied belongs. The main requirement that is imposed on the training sample is the most complete and adequate description of the nature of the breakdown H 0 , H 1 , ..., H m . In real tasks to fulfill this requirement is not only difficult, and sometimes impossible, because the nature of the partition may not even know the experimenter, and its full description will lead to a significant increase in the amount of training sample, and thereforeand the time spent on the learning process of the system. Therefore, the process of "error accumulation" begins at this stage, and even the most perfect recognition system in this case will be ineffective and powerless. Each for which the predicate () Pw will be true. This partition can be set using an arbitrary function of the form: As rudimentary signs there may be predicates, that is, deterministic characteristics of the host either 0 or 1. From them we will demand that they, in a sense, were the simplest (were a description of a fixed image). A sign that can be obtained in some way from the elementary features 12 , ,..., n    we call a generalized feature.

The scheme of construction of a logical recognition tree
The main purpose of the algorithms of recognition methods based on the logical tree, which will be presented below, is to maximize the value of () M Wf [2,5,7]. The latter means that the algorithms of the logical tree should be found for the training sample (2)  deterministic. Thus, we pose the problem of optimal approximation of the probability sample (2)  Wf [5] relative to the training sample (2)  The case (a) occurs when the sample (2) is the data of some experiment (for example, computer measurements), which are recorded in permanent memory. The learning algorithm in this case is a multiple sample processing (2). Note that the sample (2) can be very large. Therefore, the algorithms of processing of the sample should be such that it would be in their work sample (2) are not recorded in memory.
If there is no case (a) and stored data in permanent memory, then we have case (b). In this case, all the pairs that are processed in the step i d are not remembered, and therefore some other series of training pairs of the form (2) are supplied in the step Most of the methods presented below are structured so that they can be applied in both case (a) and case (b). For certainty, we further assume that there is a case (a), that is, at each step i At the heart of all recognition methods in the form of a logical tree is one schematic diagram, which is called the scheme of the tree method. In this scheme, initially selected some elementary feature 1 1  . This characteristic requires that the magnitude 1 1 () M W  is relatively (2) were possibly the greatest. Let us note once 1 1 () M W  that it is calculated according to the method [5,6].
The following steps of the method of the logical tree, it is convenient to interpret with the help of a tree (Fig. 1).
In each vertex of the tree (Fig. 1)   Suppose that only three steps of the tree method are carried out and 1 2   2  3  3  3  1  1  2  1  2  3 , , , , ,      all the signs obtained as a result of these steps. The logical tree that we get for these three steps will have the form shown in (Fig. 2). Each related pair (2) is the corresponding defined path of the tree (Fig. 2). This path is implemented as follows.