Generalised Probability of Detection

In traditional non-destructive evaluation (NDE), the standard \(\hat{a}\) vs \(a\) approach relies on several restrictive assumptions: linearity, homoscedasticity (constant variance), and normality of residuals. Interaction with complex physics—such as crack roughness—often breaks these assumptions.

To address this, digiqual implements a generalized framework based on the work of Malkiel et al. (2025). This article showcases the specific functions used to relax these assumptions.

1. Automated Model Selection: `fit_robust_mean_model()`

Instead of forcing a linear relationship, digiqual treats the expectation model as a selection problem.

Functionality: This function evaluates a pool of candidate models, including polynomials (up to degree 10) and Gaussian Process (Kriging) models.
The Relaxation: It uses 10-fold Cross-Validation (CV) to calculate the Mean Squared Error (MSE) for every candidate.
Selection Logic: By identifying the model that minimizes the CV error, the system automatically balances bias and variance without manual tuning.

# From pod.py: Selecting the best fit among Poly 1-10 and Kriging
best_model_info = min(cv_scores, key=cv_scores.get)
best_type, best_params = best_model_info

2. Modeling Heteroscedasticity: `predict_local_std()`

In many simulations, signal scatter increases with flaw size. digiqual replaces the “constant variance” assumption with a localized estimation.

Functionality: It implements a Nadaraya-Watson Gaussian kernel average smoother.
The Relaxation: The function calculates the variance at a specific point by taking a weighted average of the surrounding squared residuals.
Bandwidth Optimisation: The optimize_bandwidth function uses Leave-One-Out Cross-Validation (LOO-CV) to determine the optimal “window” (sigma) for this smoothing.

# From pod.py: Calculating weights for the local variance
weights = stats.norm.pdf(diff, loc=0, scale=bandwidth)
local_std = np.sqrt(weights @ sq_residuals)

3. Inferring Error Distributions: `infer_best_distribution()`

The assumption that noise is always Gaussian (Normal) is often inaccurate for safety-critical inspections.

Functionality: This function standardizes the residuals (calculating Z-scores) using the local standard deviation estimated above.
The Relaxation: It fits the Z-scores against a suite of distributions: Normal, Gumbel (left/right skewed), Logistic, Laplace, and t-Student.
Selection Logic: The best-fitting distribution is selected using the Akaike Information Criterion (AIC), which rewards likelihood while penalizing overfitting.

# pod.py identifies the best analytical distribution to use for PoD
for dist_name in candidates:
    params = dist_obj.fit(z_scores)
    aic = 2 * k - 2 * log_likelihood  # Lower is better

4. Analytical PoD and Bootstrapping

Once the models for the mean, variance, and distribution are established, the PoD is calculated.

Curve Generation: `compute_pod_curve()`

This function analytically derives the PoD at every point by calculating the probability that the signal exceeds the threshold, based on the inferred distribution and local variance.

Confidence Bounds: `bootstrap_pod_ci()`

Because the model is now non-linear and non-normal, traditional analytical confidence bounds are not tractable. digiqual uses Bootstrap resampling (typically 1000 iterations) to generate 95% confidence intervals by refitting the models to resampled datasets.

Example Summary

By calling study.pod(), these functions work in sequence:

fit_robust_mean_model finds the non-linear trend.
fit_variance_model and predict_local_std capture the growing noise.
infer_best_distribution selects the true shape of the error.
bootstrap_pod_ci quantifies the reliability of the resulting curve.

1. Automated Model Selection: fit_robust_mean_model()

2. Modeling Heteroscedasticity: predict_local_std()

3. Inferring Error Distributions: infer_best_distribution()