Error Propagation in Conditional Calculation Systems
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Solution Overview
Problem
Calculations that are based on alternative functions introduce errors due to measurement imprecision and rounding, and the uncertainty about which function to apply near the boundary between their domains can significantly impact the result, necessitating a method to account for these errors.
Innovation Solution
A system and method that calculate an error associated with a third calculation by considering the errors of the underlying functions and the probability of each function applying, using the formula √((μεa)^2 + ((1-μ)εb)^2 + μ(1-μ)(a-b)^2, where μ is the probability of applying the first function, εa and εb are the errors of the functions, and a and b are their respective results, to propagate this error to subsequent calculations.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Adaptability or versatility
If alternative calculations are used to handle different circumstances, then adaptability is improved, but measurement precision deteriorates due to uncertainty about which function applies
Solution Approach 1:
The calculation system is segmented into multiple alternative calculation functions (ƒ(x), g(x), etc.), each applicable to specific circumstances or domains. This segmentation allows the system to adapt to different conditions while maintaining measurement precision by calculating error propagation for each segment and combining them using the square root of sum of squares formula, which accounts for the uncertainty in selecting the appropriate function.
2Ease of operation
If conditional boundaries are used to switch between calculations, then ease of operation is improved, but manufacturing precision deteriorates due to boundary uncertainty
Solution Approach 1:
The system applies beforehand cushioning by pre-calculating the error propagation formula that includes boundary uncertainty. Instead of dealing with boundary issues when they arise, the error formula is designed in advance to accommodate the uncertainty of which function applies, using probability-weighted error components that cushion the impact of boundary ambiguity on final precision.
3Reliability
If probability-based function selection is used, then reliability is improved, but device complexity increases due to error propagation calculations
Solution Approach 1:
The system changes parameters by introducing probability weights (μ and 1-μ) as new parameters in the error propagation formula. These probability parameters quantify the reliability of each alternative calculation path and are integrated into the error calculation, allowing the system to maintain reliability through probabilistic reasoning while managing complexity through a standardized mathematical framework.
Data Source
AI summary
The result of a first calculation may be determined by one of two alternative calculations. Each of the alternative calculations has an error associated therewith (due to factors such as measurement imprecision, rounding, etc.), and these errors propagate to the first calculation. However, there may also be error in the first calculation due to uncertainty about which of the alternative calculations applies. Such error can be taken into account based on the respective probabilities that the first and second calculations will apply, as well as the difference in the results produced by the two calculations.