Transition Graph Properties of Target Class Classification

TL;DR

The structure of realistic transition graphs is studied, which makes it possible to find classification inconsistencies, helping to transfer it into the desired form and further clarifies the investigated framework.

Abstract

Target class classification is a mixed classification and transition model whose integrated goal is to assign objects to a certain, so called target or normal class. The classification process is iterative, and in each step an object in a certain class undergoes an action attached to that class, initiating the transition of the object to one of the classes. The sequence of transitions, which we call class transitions, must be designed to provide the final assignment of objects to the target class. The transition process can be described in the form of a directed graph, and the success of the final classification is mainly due to the properties of this graph. In our previous research we showed that the desirable structure of the transition graph is an oriented rooted tree with orientation towards the root vertex, which corresponds to the normal class. It is clear that the transition graph of an arbitrary algorithm (policy) may not have this property. In this paper we study the structure of realistic transition graphs, which makes it possible to find classification inconsistencies, helping to transfer it into the desired form. The medical interpretation of dynamic treatment regime considered in the article further clarifies the investigated framework.

Figures (4)

• Figure 1: An example TG in graphical form a) and in its algebraic form b). Rows in b) correspond to vertices of TG, and these rows sum to 1. A successful path through a transition graph is a series of edges forming a path beginning at some start state and ending at a final state. Concatenating the edges visited will yield the path string.
• Figure 2: Graph types that may appear as components of general $sdTCC$ (loops allowed). Graph of type $v_0$ is an in-branching tree to the vertex $v_0$ that corresponds to the target class. Graph “loop” is similar to case a) and their number in $sdTCC$ is equal to the number of “loop” vertices. The reminder components of $sdTCC$ have structure of simple cactus graphs with one oriented cycle with incoming tree-type fragments like one given in c). Number of vertices and structural particularities at components can be diverse.
• Figure 3: The core part of target class classification graph structure and its relation to the complementary part of the structure.
• Figure 4: $TCC$ graph structure split to components: $v_0$, isolated-loop and in-loop, and the part $SC(G-G_0-G_l)$.

Theorems & Definitions (4)

• theorem thmcountertheorem
• proof
• theorem thmcountertheorem
• proof