| We are surrounded by data, and understanding it can give a better view of the world. There are many ways to understand data. Some researchers propose new measures in order to gain specific insights. However, understanding these measures, themselves, is a crucial step in applying them. What do the measures represent? Do they present different aspects of the data, or do their meanings overlap? How are the measures related to each other? These are the types of questions that are addressed in this dissertation.;Part of my research in graduate school involved measuring the connectivity between different regions of the brain. Conditional Granger causality (cGC) was used as a measure to study the connections and the direction of information flow in the brain. Similarly, Magnitude-Squared Coherence (MSC) was used to measure the non-directional statistical dependence between regions. This study raised the following questions: Are MSC and cGC related to each other? Are alternative methods used to calculate cGC equal to cGC or do they actually represent different measures? Can MSC be generalized for application to vector-valued signals? If such generalizations exist, how are they related to each other?;It will be proven that a widely known method used to calculate the cGC, is not actually calculating the cGC, but a related feature, we call partitioned Granger causality (pGC). Researchers have defined many variations of generalized magnitude-squared coherence (GMSC) (such as those by Koopmans, Ramirez, Pascual-Marqui, etc.) and they can be classified into two types. In one category, the measures compare signals presented as rows of a single vector, and in the second category, the measures compare a pair of vector-valued signals. In each of the categories, new measures will be proposed to help in the goal of understanding the data and the measures. Many theorems and propositions will demonstrate the properties of each of the measures and their relationships with each other. |
