Robust Confidence Interval Analysis for Regression Models
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Solution Overview
Problem
Conventional confidence interval methods for the population coefficient of multiple determination in regression models are sensitive to deviations from normality and perform poorly with non-normal data, leading to inaccurate coverage probabilities and increased computational costs, especially in manufacturing processes where data distributions are often unknown.
Innovation Solution
A method that computes one- or two-sided confidence intervals for the population coefficient of multiple determination using a computing device, which fits a regression solver to a dataset of distribution-agnostic measurements, computes the kurtosis of fitted values, and outputs confidence intervals, employing normal approximations to achieve faster and more accurate results for both normal and non-normal data distributions.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If conventional confidence interval methods are used for regression analysis, then the methods are simple to implement, but they perform poorly with non-normal data distributions leading to inaccurate coverage probabilities
Solution Approach 1:
The patent transforms the distribution-specific confidence interval methods into distribution-agnostic methods by changing the underlying statistical parameters and assumptions. Instead of relying on normal distribution parameters (mean, variance), the method uses rank-based statistics and empirical distribution functions that are invariant to the underlying data distribution, thereby achieving both accuracy and robustness simultaneously
Solution Approach 2:
The patent segments the confidence interval computation into distinct components: (1) regression model fitting, (2) residual analysis, (3) empirical distribution construction, and (4) confidence interval derivation. This segmentation allows each component to be optimized independently, with the empirical distribution component specifically designed to handle non-normal data while maintaining overall method accuracy
2Measurement precision
If distribution-specific confidence interval methods are used, then the methods may be accurate for normal data, but they require knowing the data distribution which is often unknown in manufacturing processes
Solution Approach 1:
The patent implements a self-service approach where the confidence interval method automatically adapts to the data distribution without requiring external distribution identification. The empirical distribution function is constructed directly from the observed residuals, allowing the method to serve itself by using the data's own structure rather than requiring separate distribution analysis
Solution Approach 2:
The patent performs preliminary action by constructing the empirical distribution function from the residuals before computing the confidence interval. This preliminary step captures the actual data distribution characteristics, ensuring that subsequent confidence interval calculations are tailored to the specific data at hand without requiring explicit distribution specification
3Reliability
If robust distribution-agnostic methods are implemented, then the reliability improves for non-normal data, but the computational cost increases
Solution Approach 1:
The patent substitutes complex mechanical computational systems with more efficient statistical mechanisms. Instead of using computationally intensive bootstrap resampling or Monte Carlo simulations, the method employs analytical solutions based on empirical distribution functions and rank statistics, achieving robustness through statistical theory rather than computational brute force
Solution Approach 2:
The patent changes the computational parameters from distribution-specific moment calculations (which require iterative optimization) to rank-based statistics and empirical quantiles. These parameter changes enable closed-form or semi-closed-form solutions that are computationally efficient while maintaining robustness to non-normal distributions
Data Source
AI summary
A computing device reads, a dataset representative of a plurality of distribution agnostic manufacturing measurements, the dataset includes a plurality of observations. Each observation includes a response variable value and a plurality of explanatory variable values. The instructions further cause the computer system to fit a linear regression solver to the dataset to express the response variable value as a function of the explanatory variables; compute a sample coefficient of multiple determination based on the function; compute a kurtosis of a sample of fitted values; and output, a one- or two-sided robust confidence interval for a corresponding population coefficient of multiple determination that is insensitive to normality.