System Theory is a powerful paradigm to deal with abstract models of real processes in such a way to be accurate enough to capture the salient underlying dynamics while keeping the mathematical tools easy enough to be manageable. Its typical approach is to describe reality via a reduced subset of ordinary differential equations (ODE) linking the variables. A classical application is the circuits theory, linking the intensive (voltage) and extensive (current) variables across and through each simplified element by means of equilibrium laws at nodes and around elementary circuits. When such relationships are linear (like in ideal capacitors, resistances, and inductors, just to stay in the circuit field), a full battery of theorems does help in understanding the general properties of the ODE system. Positive systems, quite often used in compartmental processes like reservoirs in nature and pharmacologic concentration in medical therapy, enjoy most of the properties of the linear systems, with the nonlinear constraint of non negativity. More general nonlinear systems are less easily treatable unless a simple form of nonlinearity is taken into account like the ideal characteristic of a diode in circuit theory. When the physics of the process is quite known, like in the mentioned examples, it is quite easy to identify a small number of variables whose set would fully describe the dynamics of the process, once their interrelations are properly modeled: this is the classical way to approach such a problem.
Key Terms in this Chapter
Principal Component Analysis: Rearrangement of the data matrix in new orthogonal transformed variables ordered in decreasing order of variance.
Hybrid Systems: Hybrid systems’ evolution in time is composed by both smooth dynamics and sudden jumps.
K-Means: Iterative clustering technique subdividing the data in such a way to maximize the distance among centroids of different clusters, while minimizing the distance among data within each cluster. It is sensitive to initialization.
Model Identification: Definition of the structure and computation of its parameters best suited to mathematically describe the process underlying the data.
Hamming Clustering: A fast binary rule generator and variable selector able to build understandable logical expressions by analyzing the Hamming distance between samples.
Salient Variables: The real players among the many apparently involved in the true core of a complex business.
Unsupervised Clustering: Automatic classification of a dataset in two or more subsets on the basis of the intrinsic properties of the data without taking into account further contextual information.
Singular Value Decomposition: Algorithm able to compute the eigenvalues and eigenvectors of a matrix, also used to make principal components analysis.
Rule Inference: The extraction from the data of the embedded synthetic logical description of their relationships.