Artificial Adaptive Systems Using Auto Contractive Maps

Auto-Contractive Maps (Auto-CM) is a newer approach to artificial adaptive systems (AASs). In turn, AASs encompass the subject of artificial neural networks (ANNs). This chapter is an introduction to AASs.

Artificial Adaptive Systems include Artificial Neural Networks (ANNs or simply neural networks as they are commonly known). The philosophy of neural networks is to extract from data the underlying model that relates this data as an input/output (domain/range) pair. This is quite different from the way most mathematical modeling processes operate. Most mathematical modeling processes normally impose on the given data a model from which the input to output relationship is obtained. For example, a linear model that is a “best fit” in some sense, that relates the input to the output is such a model. What is imposed on the data by artificial neural networks is an a priori architecture rather than an a priori model. From the architecture, a model is extracted. It is clear, from any process that seeks to relate input to output (domain to range), requires a representation of the relationships among data. The advantage of imposing an architecture rather than a data model, is that it allows for the model to adapt. Fundamentally, a neural network is represented by its architecture. Thus, we look at the architecture first followed by a brief introduction of the two types of approaches for implementing the architecture—supervised and unsupervised neural networks. Recall that Auto-CM, which we discuss in Chap. 3, is an unsupervised ANN while K-CM, discussed in Chap. 6, is a supervised version of Auto-CM. However, in this chapter, we show that, in fact, supervised and unsupervised neural networks can be viewed within one framework in the case of the linear perceptron. The chapter ends with a brief look at some theoretical considerations.

This chapter focuses on Auto-Contractive Maps, which is a particularly useful ANN. Moreover, the relationship between Auto-Contractive Map (Auto-CM), which is the main topic of this monograph, its relationship to other ANNs and some illustrative example applications are presented.

One of the most powerful aspects of our approach to neural networks is not only the development of the Auto-CM neural network but the visualization of its results. In this chapter we look at two visualization approaches—the Minimal Spanning Tree (MST) and the Maximal Regular Graph (MRG). The resultant from Auto-CM is a matrix of weights. This weight matrix naturally fits into a graph theoretic framework since the weights connecting the nodes will be viewed as edges and the weights as the weights on these edges.

We have looked at how to visualize the relationships among the elements of a dataset in Chap. 4. This chapter is devoted to the use of Auto-CM in the transformation of datasets for the purpose of extracting further relationships among data elements. The first transformation we call the FS-Transform, which looks beyond an all or nothing, binary relationship that is typical of most ANNs. The extraction of these perhaps more subtle relationships can be thought of as gradual relationships, zero denoting no relationship is present and one denoting a full/complete relationship that is absolutely present. It is thus, akin to a fuzzy set. The second transformation is one, which “morph” the delineation between records and variables within records that we call Hyper-Composition.

This section is devoted to a more advanced type of Auto-CM that is supervised.

We compare Auto-CM with various other methods that extract patterns from data. The way that we measure the results of comparisons uses MST.

We look at how to use Auto-CM in the context of datasets that are changing in time. We modify our approach while keeping the original philosophy of Auto-CM.