Genomic Signal Processing and Regulatory Networks: Representation, Dynamics and Control

Genomic Signal Processing (GSP) is a discipline to study the processing of genomic signal. GSP studies a large collection of genomic sequence instead of individual gene. The aim of GSP is to integrate the theory and methods of signal processing with the global understanding of genomics. In GSP, one would find a proper representation of the genomic information and employ various signal processing methodologies such as detection, prediction, classification, control, etc. In this thesis, we propose the concept of mapping equivalence theory for the numerical representation of symbolic data. We propose a framework for the analysis of different numerical mappings undergoing transformation by an analytic operator using Taylor's expansion. Moreover, we emphasize the investigation of first- and second-order operators including the correlation function and Fourier transform. We also provide an analysis of the correlation between different numerical mappings of a symbolic sequence. In particular, we derive conditions for strong equivalence captured by perfect correlation among distinct mappings. We explore a relaxed similarity measure between distinct numerical mappings. Specifically, we provide conditions for weak equivalence which is characterized by preservation of the local extrema of the representation. We also introduce an abstract mapping model and extend the concept of equivalence to the generalized mapping model. We extend the mapping equivalence theory for iterated operator. We provide a method for analyzing the consistency between different mappings under iterations of operator. We define different concepts of mapping equivalence. We show the necessary and sufficient condition for consistency under iteration of affine operator. We present a few theoretical results for the equivalent mappings on the concept of Fatou and Julia Set. We give the definition of stability under iteration of operator and show the stability issue can be viewed as a special case of mapping equivalence. We also establish the connection of stability to Fatou and Julia set. We propose the Non-cooperative stochastic game (NCSG) model for control of the genetic regulatory networks and formulate the intervention problem into solving the Nash equilibrium (NE). We show that the Markov decision process (MDP) is a special case of NCSG and the solving methods are provided. The definition of NE in this context has been proposed and the existences for both infinite and finite horizon cases have been proven. We provide a constructive method for solving the approximate NE.