Hidden Markov Model Identification via Isomorphic Reduction
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Solution Overview
Problem
Existing methods for generating hidden Markov models from complex data sets are often empirical and struggle to distinguish between numerous possible models, particularly in systems with hidden states and transitions, leading to ambiguity and inefficiency in identifying the most likely graphical model.
Innovation Solution
A method that iteratively applies a series of rules to data sets to identify the most likely hidden Markov model by optimizing transition rates and eliminating isomorphic models, allowing the data to guide the construction and modification of the model without prior assumptions, using software tools like ModelGrower.py and NAUTY to handle graph partitions and eliminate duplicates.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If empirical methods are used to generate hidden Markov models from complex data sets, then the process can be completed with existing techniques, but the ability to distinguish between numerous possible models is poor and ambiguity remains high
Solution Approach 1:
The invention transforms the model identification problem from an empirical search process into a deterministic mathematical optimization problem by changing the parameter space to include transition rates and dwell times. This allows precise distinction between models through quantitative comparison of these parameters rather than relying on empirical fitting methods.
Solution Approach 2:
The invention replaces the empirical, trial-and-error mechanical process of model generation with a mathematical optimization system. By substituting the empirical approach with a formal optimization framework that maximizes likelihood functions, the system achieves precise model identification without the ambiguity of empirical methods.
2Reliability
If all possible graphical models are considered to identify the best hidden Markov model, then the most likely model can be found, but the computational complexity and time required increase combinatorially
Solution Approach 1:
The invention changes the problem parameters from considering all possible graphical structures to optimizing transition rates within a given graph structure. This parameter transformation reduces the combinatorial explosion by focusing optimization on continuous rate parameters rather than discrete structural choices, significantly reducing computational time while maintaining reliability.
Solution Approach 2:
The invention performs preliminary actions by fixing the graphical structure before optimization. By pre-determining the state transitions and connectivity, the system eliminates the need to evaluate all possible graphical models, thereby reducing computational complexity while preserving the ability to identify the most likely model through rate optimization.
3Adaptability or versatility
If hidden states and transitions are allowed in the model, then the model can represent complex systems more accurately, but the number of indistinguishable isomorphic models increases
Solution Approach 1:
The invention applies asymmetry by using directed transitions with specific rate parameters to distinguish between isomorphic models. By assigning asymmetric rate values to transitions between hidden states, the system can differentiate between models that would otherwise be structurally identical, enabling accurate representation of complex systems while maintaining model distinguishability.
Data Source
AI summary
The invention uses the ModelGrower program to generate possible candidates from an original or aggregated model. An isomorphic reduction program operates on the candidates to identify and exclude isomorphic models. A Markov model evaluation and optimization program operates on the remaining non-isomorphic candidates. The candidates are optimized and the ones that most closely conform to the data are kept. The best optimized candidate of one stage becomes the starting candidate for the next stage where ModelGrower and the other programs operate on the optimized candidate to generate a new optimized candidate. The invention repeats the steps of growing, excluding isomorphs, evaluating and optimizing until such repetitions yield no significantly better results.