Quantifier Elimination for Model Parameter Accuracy Verification
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Solution Overview
Problem
Conventional methods for fitting model parameters in differential equations struggle to accurately distinguish between experimental and numerical errors, leading to uncertainties in the accuracy of obtained parameters and initial values, with no effective method to control or clarify computation accuracy.
Innovation Solution
A program utilizing the quantifier elimination method is employed to read and compute differential equations with numeric time-series values, determining the ranges of model parameters and verifying the accuracy of computations by adjusting significant figures and numerical solving methods.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If conventional parameter fitting methods are used to estimate model parameters by minimizing sum of squared residuals, then model parameters can be obtained, but the accuracy of the obtained parameters cannot be correctly discussed because it cannot be distinguished whether standard deviation is generated by experimental error or numerical calculation
Solution Approach 1:
The patent segments the error analysis by separating experimental errors from numerical calculation errors. It introduces a quantifier elimination method that processes differential equations to identify and distinguish between errors arising from experimental data and errors from numerical computations, thereby resolving the ambiguity in accuracy assessment
Solution Approach 2:
The patent introduces a quantifier elimination method as an intermediary tool between the differential equation model and the parameter estimation process. This intermediary systematically analyzes the mathematical structure to identify and separate different error sources, enabling accurate discussion of parameter accuracy without confusion between experimental and numerical error contributions
2Measurement precision
If conventional fitting computation is performed repeatedly until sum of squared residuals becomes minimum, then model parameters are obtained, but there is no method of easily clarifying the factors of computation accuracy or controlling the accuracy
Solution Approach 1:
The patent applies preliminary action by using the quantifier elimination method to pre-analyze the differential equations and identify accuracy factors before the parameter fitting process begins. This preliminary analysis clarifies the computational accuracy factors and enables control mechanisms to be established in advance, rather than attempting to control accuracy during or after the fitting process
3Productivity
If the influence of difference between accuracy of time-series computations and fitting computation is present, then parameter estimation can be performed, but the accuracy of obtained values of parameters cannot be correctly discussed
Solution Approach 1:
The patent extracts the accuracy analysis function from the conventional fitting computation process. By applying the quantifier elimination method separately to analyze the differential equations, it extracts and isolates the accuracy factors, enabling accurate discussion of parameter accuracy independent of the parameter estimation process itself
Data Source
AI summary
A program causes a computer to perform a step 1 of reading a model from a storage apparatus; a step 2 of computing difference equations by replacing variables with numeric values of a time-series that are stored in the storage apparatus; a step 3 of determining ranges of parameters of the model using a quantifier elimination method and a step 4 of verifying numbers of significant figures that show an accuracy of a computation by the determined ranges of parameters of the model.