Enhancing Limited-Sample Probability of Detection Estimation Using Models and Advanced Regression Techniques

Abstract

The probability of detection (POD) is a fundamental metric for evaluating the performance of nondestructive evaluation (NDE) techniques. However, traditional empirical approaches to POD estimation often require extensive measurements, making them costly in terms of time, budget, and resources. In scenarios with limited data, conventional estimation methods frequently fail to capture the underlying relationship between signal responses and flaw sizes, as well as the variability introduced by testing conditions, influencing factors, and inherent uncertainties. Moreover, standard linear regression models, commonly used in POD analysis, rely on assumptions that are often violated when sample sizes are small, resulting in biased or imprecise estimates. To overcome these challenges, this study investigates advanced regression techniques and their integration with physics-based models for limited-sample POD (LS-POD) estimation. LS-POD here is defined as POD estimation when the sample size is below the threshold typically required by conventional methods. We explore a range of information-augmentation approaches, including physics-informed regression and Bayesian methods, which incorporate prior knowledge to improve the characterization of the signal-flaw relationship and the variability of NDE procedures. Additionally, we adapt advanced statistical techniques, such as Box-Cox transformation, robust regression, weighted linear regression, and bootstrapping, to mitigate the impact of assumption violations commonly encountered in small-sample contexts. These methods are further integrated to simultaneously leverage existing knowledge and address statistical assumption violations. We conduct comprehensive simulation studies using both synthetic and empirical datasets to evaluate the performance of these approaches under a variety of LS-POD scenarios. The results are benchmarked against conventional POD estimates derived from large-sample data. Our findings indicate that incorporating prior knowledge and employing assumption-resilient regression techniques can significantly enhance the accuracy and precision of LS-POD estimation. The combined use of information-augmentation and assumption-correction strategies yields further improvements. These results provide practical insights for NDE practitioners, facilitating the selection and application of appropriate LS-POD methods tailored to specific data conditions and application needs.