Abstractions of Graph Models
Building models, whether to explain or to predict observed data, is an exercise of describing how the values of observed variables depend on those of others. Black box models only describe relationships between observed variables, and they are evaluated by their ability to accurately describe the va...
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| Format: | Others |
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BYU ScholarsArchive
2020
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| Online Access: | https://scholarsarchive.byu.edu/etd/8455 https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=9455&context=etd |
| Summary: | Building models, whether to explain or to predict observed data, is an exercise of describing how the values of observed variables depend on those of others. Black box models only describe relationships between observed variables, and they are evaluated by their ability to accurately describe the values of observed variables in new situations not previously available to the model–such as the output response to a new set of inputs, for example. Black box models describe the observed behavior of the underlying system, but they may not correctly describe the way in which the system computes this behavior. White box models, on the other hand, describe the observed behavior and also incorporate hidden, intermediate variables that are used to describe the specific computation the underlying system uses to generate its observed behavior. In this sense, we say the white box model captures the structure of the system, in addition to its observed behavior. Since a given white box model may be accurately described by an infinite variety of black box models, all computing the same observed behavior but using different structures to do so, we say that any of these black box models is an abstraction of the white box model. This thesis constructs foundational pieces of a unifying theory of linear mathematical abstractions that are central to scientific modeling. It offers a precise description of the spectrums of grey box models linking any white and black box representation. There are various motivations for having this rich variety of representations of a given system. One key motivation is that of consilience, that is, to deepen our understanding of the modeling process by connecting various well developed theories under the umbrella of a broader theory. This work offers a precise relationship between Mason’s signal flow graphs [34] and Willem’s behavioral systems theory [66], in addition to linking the classical transfer function theory used by Nyquist [44], Bode [3], and Weiner [65] to the state space theory preferred by Kalman [27]. Another motivation comes from the application of identification or learning a model of the system from data. Learning problems trade off the number of a priori assumptions that one must make about a system, as well as the richness of available data, with the complexity of a model that one is able to confidently learn from measured observations. This work offers insight into these tradeoffs by characterizing them precisely over entire spectrums of grey box models of increasing complexity. A third motivation comes from the application of vulnerability analysis, which is the study of sensitivities of system behavior to structural perturbations in a grey-box model describing the attack surface, or representation of the system as visible to a potential attacker. The main results of this work, and its specific contributions, are as follows: 1. We define new graph-theoretic constructs and use them to create a unified framework for structural abstractions, 2. We demonstrate that there will always exist a complete, structure preserving, acyclic abstraction for every single-input, fully-connected system, 3. We define structural controllability of an abstraction of a system and argue why our definition is good, and 4. We show how complete abstractions preserve structural controllability. These results were accepted for publication in two papers at the 2020 International Federation of Automatic Control World Congress, each submitted to a different special invited session. These papers comprise Chapters 2 and 3, respectively, of the thesis presented here, and they are expected to appear in print July 2020. |
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