Yield Prediction Model for Ingot Samples Based on Machine Learning and Data Augmentation

Abstract

The preparation of high-performance cadmium zinc telluride (CZT) radiation detector materials requires efficient ingot-level quality assessment before full downstream wafer testing. This study proposes a machine learning framework that predicts the product-level yield of test wafers from IV and double-sided spectral measurements of a limited number of standardized evaluation wafers from the same ingot. To address the small number of ingots and wafer-level variability, ingot-level aggregate, A/B-side consistency, threshold-ratio, and distributional features were combined with intra-ingot bootstrap augmentation. Among the evaluated regression models, Random Forest achieved the best held-out test performance under a leakage-safe protocol, with an MSE of 0.021, an MAE of 0.125, and a Pearson correlation coefficient of 0.646; XGBoost showed comparable performance, with an MSE of 0.023, an MAE of 0.128, and a Pearson correlation coefficient of 0.601. In a top-22% screening experiment, the average true yield of ingots selected by Random Forest and XGBoost reached 63.71% and 60.40%, respectively, exceeding the empirical Rule_IV_Abs baseline of 59.08%. These results indicate that the proposed framework can provide useful ranking and prioritization support for early CZT ingot screening, while remaining a decision-support tool rather than a replacement for wafer-level inspection.

1. Introduction

The manufacturing of radiation detector materials, such as CZT, involves cutting a single-crystal ingot into multiple wafers, each of which must satisfy stringent standards for electrical performance and detection efficiency [1,2]. Owing to the non-ideal Zn segregation coefficient and the non-uniform distribution of subgrain boundary networks during CZT crystal growth, wafers at different axial positions within the same ingot exhibit significant differences in the mobility-lifetime product ($\mu \tau$) and defect density [3]. These microscopic defects, encompassing point defects, dislocations, and grain boundaries, contribute to the intrinsic variability in wafer-level performance [4].

Traditional quality evaluation relies on manual multi-criterion judgment for individual wafers, including current–voltage (IV) characteristics, such as leakage current $\left|IV\right|\le 10\mathrm{n}\mathrm{A}$@500 V, and IV-curve shape classification, including linear, reverse-S, or forward-S types [5]. This process is not only time-consuming but also subject to subjective inconsistency. In particular, when ingot-level batches are processed, the impact of wafer-to-wafer variation on the overall yield is difficult to quantify. Veale et al. pointed out that, even when cross-IV tests are performed on both A and B sides, the current industry still lacks a systematic framework that links microscopic electrical measurements to macroscopic yield [5].

In recent years, material characterization techniques have enabled high-throughput acquisition of wafer-level data, including complete IV scans over a wide voltage range, double-sided spectral response parameters, such as energy resolution, photopeak count rate, and noise background, and spatial uniformity mapping [1,5]. However, because defect formation mechanisms in compound semiconductors are complex and characterization data are high-dimensional and sparse, the nonlinear relationship between measured results and final detector-level yield has not yet been adequately modeled. Amato et al. [6] emphasized that a key challenge in yield prediction for semiconductor manufacturing lies in handling imbalanced data and identifying the defect features that truly affect electrical failure [7,8].

Machine learning provides a systematic approach for multivariable relationship modeling and is particularly well suited to hierarchical data structures, such as the nested ingot–wafer relationship, and imbalanced-data problems [9,10]. Existing studies have demonstrated the effectiveness of machine learning in materials science, including performance prediction based on electrical characteristics [6], microscopic defect classification [9], and process-parameter optimization. Recent studies have further integrated XGBoost with SHAP (SHapley Additive exPlanations) to provide interpretable yield prediction and failure analysis in semiconductor manufacturing [11]. However, ingot-level yield prediction for radiation detector materials remains unexplored. Most existing studies focus on single-wafer defect identification and fail to exploit wafer-to-wafer variation information to infer the overall quality of an ingot. Consequently, early quality-control decisions still lack adequate data support.

Against this background, this study addresses the practical problem of early CZT ingot quality evaluation by developing an ingot-level prediction framework that maps standardized evaluation-wafer characterization data to the downstream product-level yield of test wafers. Unlike studies focused on single-wafer defect identification, the proposed framework aims to estimate the overall usability of wafers derived from the same ingot, thereby supporting early screening and processing-priority decisions. The main contributions are threefold: first, ingot-level descriptors are constructed from evaluation-wafer IV and double-sided spectral measurements to capture overall quality, wafer-to-wafer variability, and A/B-side consistency; second, intra-ingot bootstrap augmentation is introduced to improve model training under limited ingot-level sample size; and third, the framework is validated using a leakage-safe held-out ingot evaluation and compared with empirical production rules. The resulting models are intended for ranking and screening support rather than replacing downstream wafer-level inspection.

2. Data Description

The dataset contains characterization data from 94 cadmium zinc telluride ingots, comprising 33,832 downstream product/test wafers in total. For each ingot, two non-overlapping wafer subsets were distinguished. Product/test wafers were processed according to downstream product requirements and used only to calculate the ingot-level yield target, whereas standardized evaluation wafers were used exclusively to construct input features for prediction.

The raw files were organized in a dual-block row-wise spreadsheet format. The product/test-wafer block records wafer code, product specification, IV voltage, negative and positive IV current responses, IV-curve morphology, and final pass/fail judgement. The evaluation-wafer block records evaluation-wafer identifiers, standardized specifications, radiation source, IV measurements, preferred measurement side or orientation, and paired A/B-surface spectral descriptors, including FWHM, peak counts, valley counts, peak-to-valley ratio, count rate, peak-shape description, and noise level. The complete raw-file layout is provided in Appendix A.

The predictive task is therefore not to reproduce the pass/fail decision of individual product wafers from their own measurements. Instead, the objective is to predict the downstream product-level yield of each ingot from standardized evaluation-wafer measurements, including cases where product/test specifications differ from the evaluation-wafer specification. To avoid information leakage, product/test-wafer IV measurements and pass/fail judgements were used only to compute the regression target and were not used as predictors.

Evaluation-wafer records were transformed into an ingot-level feature matrix using statistical summaries, A/B-side consistency descriptors, threshold-ratio features, and distributional variability measures. Missing values, which accounted for 3.2% of all values and mainly occurred in noise-related, low-count, failed-pressurization, over-range IV, or incomplete records, were imputed using parameters fitted on the training set only. The same preprocessing parameters were then applied to the validation and test sets.

3. Methods

3.1. Problem Definition

The objective of this study is to predict the product-level yield of test wafers in an ingot by using characterization data from a limited number of evaluation wafers from the same ingot. Let the g-th ingot contain $n_g^t$ test wafers. Then, its product-level yield is defined as

$$r_g={\displaystyle \frac{1}{n_g^t}}{\displaystyle \sum _{i=1}^{n_g^t}}\mathbb{I}(q_{g,i}=1),$$

(1)

where $\mathbb{I}\left(·\right)$ denotes the indicator function, and $q_{g,i}$ represents the pass/fail status of the i-th test wafer in the g-th ingot.

For product/test wafers, the recorded pass/fail status was obtained from the production quality-control dataset. This judgement was determined primarily from IV-related information, including the applied IV voltage, the current responses under negative and positive bias polarities, and the IV-curve morphology. Therefore, the product-level yield of the g-th ingot was calculated from the recorded pass/fail judgements of its product/test wafers:

$$q_{g,i}=\left\{\begin{array}{cc}1,\hfill & |I{V}_{g,i}^{-}|\le 1\times {10}^{-8}\mathrm{A}\mathrm{and}|I{V}_{g,i}^{+}|\le 1\times {10}^{-8}\mathrm{A},\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(2)

Therefore, the modeling task of this study can be formulated as follows. Ingot-level input features are constructed from the double-sided electrical performance and spectral information of evaluation wafers, and the product-level yield $r_g$ of test wafers is used as the regression target. In this way, a cross-level mapping is established from upstream evaluation wafers to the downstream overall quality of test wafers.

3.2. Machine Learning Algorithms

This study was conducted within a supervised learning framework, in which regression models were developed to predict the product-level yield of ingot materials. The essence of this task is to establish a complex mapping function from high-dimensional wafer characterization data to a continuous yield score. The main challenges include the following: (1) the data structure is hierarchical, with multiple wafers nested within each ingot [12,13]; (2) multicollinearity exists among features; and (3) the number of available ingot samples is limited ($N=94$). To address these challenges, multiple machine learning methods were systematically integrated, ranging from linear baselines to complex nonlinear models. Moreover, model robustness and generalization were comprehensively improved through careful feature engineering, data resampling, and hyperparameter optimization.

3.3. Regression Models

To comprehensively evaluate the performance of different modeling paradigms, six regression algorithms, together with a mean-value baseline, were implemented in this study:

(1) Multiple Linear Regression (MLR) [14]: This model was used as a performance baseline to preliminarily explore the linear relationship between features and the target variable. Its model form is $\widehat{y}={\mathbf{w}}^T\mathbf{x}+b$. Because no regularization was introduced, this model is highly prone to overfitting in the presence of high-dimensional features. However, its results provide a benchmark for comparison with subsequent complex models.

(2) Regularized linear models, namely Ridge and Lasso regression [15,16]: To address multicollinearity and overfitting, regularization terms were introduced. Ridge regression, using L2 regularization, constrains the magnitude of the weights by minimizing ${\left|\mathbf{w}\right|}_2^2$, thereby improving model stability. Lasso regression, using L1 regularization, produces sparse solutions by minimizing ${\left|\mathbf{w}\right|}_1$ and performs automatic feature selection. Its model form is ${min\mathbf{w}|\mathbf{y}-\mathbf{X}\mathbf{w}|}_2^2+\lambda {\left|\mathbf{w}\right|}_1$.

(3) Random Forest (RF) [17]: As a Bagging-based ensemble method, RF effectively reduces variance by constructing a large number of uncorrelated decision trees and averaging their predictions. Its mathematical form is $\widehat{y}={\displaystyle \frac{1}{B}}{\sum }_{b=1}^B{T}_b(\mathbf{x};{\Theta }_b)$, where $T_b$ denotes a single decision tree. RF can naturally handle nonlinearities and interaction effects and also provides feature-importance measures.

(4) XGBoost (Extreme Gradient Boosting) [18]: As an advanced gradient boosting framework, XGBoost optimizes the objective function $\mathcal{L}={\sum }_{i}l(y_i,{\widehat{y}}_i)+{\sum }_k\Omega \left(f_k\right)$, where $\Omega$ is the regularization term, by iteratively fitting the negative gradient, namely the residual, of the current model in a forward stage-wise additive manner. Its efficient handling of structured data, built-in regularization, and ability to process missing values made it particularly effective for this task [11].

(5) Feedforward neural network (FNN) [19,20]: An FNN architecture with two fully connected hidden layers was adopted. By stacking nonlinear activation functions, such as ReLU, the network learns deep representations of high-dimensional features: ${\mathbf{h}}^{\left(l\right)}=\sigma ({\mathbf{W}}^{\left(l\right)}{\mathbf{h}}^{(l-1)}+{\mathbf{b}}^{\left(l\right)})$. Combined with dropout and weight-decay regularization, this model was used to explore potentially complex nonlinear patterns in the data [21,22].

3.4. Feature Engineering

Wafer-level evaluation measurements were transformed into ingot-level descriptors to summarize overall electrical and spectral quality, intra-ingot variability, A/B-side consistency, and threshold-based process compliance. All engineered predictors were computed exclusively from evaluation wafers, while the product/test-wafer yield was used only as the regression target.

For an ingot g containing $n_g$ evaluation wafers, let $x_{ij}$ denote the value of the j-th wafer-level measurement for the i-th evaluation wafer. Four main categories of ingot-level features were constructed.

Statistical aggregate features. For each numerical measurement feature $x_j$, distributional statistics were calculated across the evaluation wafers within the same ingot. The basic aggregate descriptors included the mean ${\mu }_j$, standard deviation ${\sigma }_j$, coefficient of variation

$${\mathrm{CV}}_j={\displaystyle \frac{{\sigma }_j}{{\mu }_j}},{\mu }_j\ne 0,$$

(3)

median, minimum, maximum, and range

$$R_j=\underset{i}{max}\left(x_{ij}\right)-\underset{i}{min}\left(x_{ij}\right).$$

(4)

These features describe the overall performance level and wafer-to-wafer uniformity of an ingot. In addition, robust and distributional descriptors, including the 10th, 25th, 75th, and 90th percentiles, interquartile range, 10–90 percentile range, skewness, kurtosis, and median absolute deviation, were extracted to reduce sensitivity to outliers and to better characterize intra-ingot heterogeneity.

A/B-side difference features. Because several measurements were characterized separately on the A and B surfaces, inter-surface consistency was explicitly encoded. For the j-th paired measurement of the i-th evaluation wafer, the absolute A/B-side difference was defined as

$$d_{ij}=\left|x_{ij}^{\left(A\right)}-x_{ij}^{\left(B\right)}\right|.$$

(5)

The corresponding ingot-level difference descriptor was obtained by aggregating $d_{ij}$ across the evaluation wafers, for example,

$${\overline{d}}_j={\displaystyle \frac{1}{n_g}}{\displaystyle \sum _{i=1}^{n_g}}{d}_{ij}.$$

(6)

Additional statistics of the A/B-side difference distribution, such as the maximum, median, and percentile values, were also considered. These features complement the surface-specific descriptors by quantifying surface-to-surface asymmetry, which may reflect non-uniform contact behavior, local material inhomogeneity, or asymmetric detector response.

Threshold-based features. Domain-specific quality thresholds were used to convert discrete process rules into continuous ingot-level descriptors. For example, the leakage-current criterion

$$\left|IV\right|<1\times {10}^{-8}\mathrm{A}$$

(7)

corresponds to a 10 nA current threshold under the specified bias condition. For a given threshold ${\tau }_k$, the acceptable set was defined as

$$A_k=\{x:|x|<{\tau }_k\}.$$

(8)

The proportion of evaluation wafers satisfying the criterion within an ingot was then calculated as

$$p_k={\displaystyle \frac{1}{n_g}}{\displaystyle \sum _{i=1}^{n_g}}\mathbb{I}(x_{ik}\in A_k),$$

(9)

where $\mathbb{I}\left(·\right)$ denotes the indicator function, which takes the value 1 when the condition is true and 0 otherwise. Such threshold-ratio features provide a continuous representation of process compliance and are related to empirical production rules. The corresponding sample-size information was retained when applicable to indicate the amount of evaluation-wafer evidence supporting each threshold-based estimate.

It should be emphasized that all threshold-ratio or pass-rate-like descriptors used as input features were computed exclusively from evaluation-wafer measurements. They are therefore distinct from the product/test-wafer yield target, which was calculated from the recorded pass/fail judgements of downstream product/test wafers.

Complementary derived features and feature validity. Additional derived features were constructed only from variables available in the evaluation-wafer records and only when the transformation had a clear process or physical interpretation, ensuring consistency between the implemented pipeline and information available before yield evaluation.

The complete feature-engineering pipeline generated ingot-level descriptors from the original 52 wafer-level measurements, including 26 measurements from each of the A and B surfaces. In addition to the statistical, threshold-based, A/B-side difference, and interaction features described above, missing-pattern summaries and sample-size-related descriptors were included to account for variation in the number of valid evaluation records. To improve figure readability, abbreviated feature labels are used in the main text and figures, whereas the complete mapping between abbreviated labels, original variable names, definitions, units, and aggregation rules is provided in Appendix A.

To assess the redundancy structure among the most influential engineered descriptors, we calculated the Pearson correlation matrix between yield and the top 15 features ranked by XGBoost gain-based feature importance. The importance ranking was derived from XGBoost models fitted on the training data only after feature engineering and preprocessing. As shown in Appendix B.2, several high-importance descriptors exhibited moderate-to-strong pairwise correlations, particularly among count-rate-, peak-count-, valley-count-, and A/B-side-related features. This result indicates that some redundancy remained in the engineered feature space, while also confirming that the final feature set captured complementary information beyond simple marginal correlations with yield. The heatmap was used only as an exploratory visualization and not as an additional feature-selection, hyperparameter-tuning, or model-evaluation step.

To reduce dimensionality and mitigate overfitting under the limited-sample condition, feature selection was performed using the training data only. Starting from 304 engineered features, near-zero-variance filtering was first applied during preprocessing, removing 37 low-variance features and retaining 267 features. Features with an absolute Pearson correlation coefficient with the target lower than 0.01 were then removed, leaving 243 features. Next, highly correlated redundant features were eliminated using a pairwise Pearson correlation threshold of 0.95. For each highly correlated feature pair, the feature with the lower absolute correlation with the target in the training set was discarded, resulting in 194 retained features. Finally, XGBoost-based feature importance was used for model-based selection, and the top 60 features were retained for final model training.

Model performance under different numbers of retained features was further examined, as shown in Figure 1. The final number of retained features was capped at 60 to balance information preservation and overfitting control. Importantly, the same fitted preprocessing and feature-selection pipeline was applied to the validation and held-out test sets without refitting, thereby preventing information leakage from the test ingots into the feature-selection process.

3.5. Intra-Ingot Bootstrap Augmentation

To mitigate the high variance caused by the limited number of independent ingots $(N=94)$, intra-ingot bootstrap resampling was used for training-set augmentation. All data splitting was performed at the original-ingot level. Each bootstrap replicate was indexed by its source ingot, and all replicates from the same ingot were assigned to the same training or held-out test partition. Grouped cross-validation within the training set was further performed using the original ingot identity as the grouping variable. Thus, no bootstrap-derived sample from a held-out test ingot was used during model fitting, feature selection, hyperparameter tuning, or model-fusion weight selection, and replicates from the same original ingot were not split across different cross-validation folds. This strategy prevents information leakage across data partitions [23,24].

For the g-th ingot, let ${\mathcal{E}}_g=\{{\mathbf{x}}_{g,1},\dots ,{\mathbf{x}}_{g,n_g}\}$ denote the evaluation-wafer records used for feature construction, and let

$${\mathcal{P}}_g=\{q_{g,1},\dots ,q_{g,n_g^t}\},q_{g,i}\in \{0,1\},$$

denote the product/test-wafer pass/fail records used to calculate the downstream yield. For each training ingot, the b-th bootstrap replicate was generated by independently resampling the two wafer sets with replacement:

$${\mathcal{E}}_g^{*\left(b\right)}=SampleWithReplacement({\mathcal{E}}_g,n_g),{\mathcal{P}}_g^{*\left(b\right)}=SampleWithReplacement({\mathcal{P}}_g,n_g^t),$$

where $b=1,\dots ,B$. The input features and the corresponding bootstrap yield target were then computed as

$${\mathbf{z}}_g^{*\left(b\right)}=\varphi \left({\mathcal{E}}_g^{*\left(b\right)}\right),r_g^{*\left(b\right)}=\psi \left({\mathcal{P}}_g^{*\left(b\right)}\right)={\displaystyle \frac{1}{n_g^t}}{\displaystyle \sum _{i=1}^{n_g^t}}{q}_{g,i}^{*\left(b\right)},$$

where $\varphi \left(·\right)$ denotes the feature-engineering function and $\psi \left(·\right)$ denotes the yield-calculation function.

The evaluation-wafer and product/test-wafer sets were resampled separately because they represent different wafer subsets from the same ingot and are not one-to-one paired observations. Accordingly, $r_g^{*\left(b\right)}$ reflects the sampling variability of the empirical downstream yield estimated from a finite number of product/test wafers. The bootstrap-derived replicates were used only as dependent pseudo-samples to improve robustness to within-ingot sampling fluctuations; they were not treated as new independent ingots. Final model performance was evaluated exclusively on original, non-augmented held-out test ingots.

3.6. Hyperparameter Tuning

Hyperparameters were optimized using Bayesian optimization [25,26] within five-fold cross-validation on the training set [27]. The search spaces covered model-specific parameters, including regularization strength for linear models, tree number and depth for ensemble models, learning rate and subsampling ratio for XGBoost, and hidden-layer configuration, dropout, and weight decay for the neural network. The selected configurations were then fixed before evaluation on the held-out test ingots.

To fully exploit the predictive potential of each model and avoid overfitting, systematic hyperparameter optimization was performed. Because the hyperparameter spaces of the models considered in this study, especially RF, XGBoost, and FNN, are high-dimensional and contain both continuous and discrete parameters, traditional grid search is computationally expensive. Therefore, Bayesian random optimization was adopted [25,26]. Based on previous evaluation results as prior information, this framework actively selects parameter combinations that are most likely to improve performance by constructing a surrogate model, such as a tree-structured Parzen estimator. Sampling is then conducted under the expected-improvement criterion for minimizing validation error. The optimization process was tightly integrated with five-fold cross-validation [27]. As a result, the optimal configurations of all models, such as the learning rate, tree depth, and subsampling ratio for XGBoost, and the number of layers, number of neurons, and Dropout rate for FNN, were determined while substantially reducing computational cost without compromising optimization quality.

4. Practical Validation

4.1. Data Collection and Preparation

As described in Section 2, each ingot contained downstream product/test-wafer records and standardized evaluation-wafer records. In the validation experiments, only the evaluation-wafer block was used to construct input features, while the product/test-wafer block was used solely to compute the ingot-level yield target from recorded production pass/fail judgements. This design ensured that the models predicted downstream product yield from early evaluation measurements rather than from information directly used in product-wafer acceptance decisions.

4.2. Model Development and Evaluation

The 94 ingots were partitioned at the ingot level into training and held-out test sets using a fixed random seed, and the same split was used for all models. Hyperparameter tuning and feature selection were performed using the training data only through five-fold cross-validation. Bootstrap augmentation was applied exclusively to the training ingots after the split, whereas the held-out test set contained 19 original, non-augmented ingots.

Models were implemented using scikit-learn and XGBoost in Python 3.11.9 [28]. Performance was evaluated on the held-out test ingots using MSE, MAE, and Pearson/Spearman correlation coefficients. Different intra-ingot bootstrap multiples were examined to assess the effect of training-set augmentation.

4.3. Experimental Results

Table 1. Predictive performance of models without and with intra-ingot bootstrap augmentation.

Setting	Model	MSE (95% CI)	MAE (95% CI)	Pearson’s r (95% CI)
Baseline (Boot = 0)	Mean	0.036 (0.016–0.060)	0.144 (0.090–0.203)	NA
	MLR	0.270 (0.128–0.473)	0.414 (0.287–0.571)	0.145 (−0.115–0.383)
	Lasso	0.036 (0.016–0.060)	0.144 (0.090–0.203)	NA
	Ridge	0.045 (0.019–0.075)	0.166 (0.106–0.228)	0.067 (−0.417–0.517)
	RF	0.034 (0.017–0.054)	0.147 (0.094–0.200)	0.251 (−0.188–0.591)
	XGBoost	0.059 (0.033–0.096)	0.207 (0.151–0.271)	0.244 (−0.134–0.571)
	NN	0.048 (0.022–0.074)	0.178 (0.118–0.233)	0.221 (−0.202–0.609)
Augmentation (Boot = 6)	Mean	0.036 (0.016–0.059)	0.144 (0.090–0.203)	NA
	MLR	0.041 (0.018–0.071)	0.155 (0.092–0.219)	0.296 (−0.160–0.616)
	Lasso	0.034 (0.016–0.055)	0.142 (0.089–0.197)	0.204 (−0.312–0.652)
	Ridge	0.044 (0.021–0.073)	0.164 (0.106–0.231)	0.194 (−0.278–0.583)
	RF	0.021 (0.011–0.032)	0.125 (0.090–0.163)	0.646 (0.346–0.839)
	XGBoost	0.023 (0.012–0.039)	0.128 (0.089–0.168)	0.601 (0.227–0.845)
	NN	0.031 (0.018–0.047)	0.149 (0.111–0.187)	0.611 (0.272–0.824)

Note. Baseline denotes the original training condition without intra-ingot bootstrap augmentation. Augmentation denotes the setting in which six intra-ingot bootstrap resampling iterations were applied to each training ingot. The held-out test set consisted of original, non-augmented ingots. Values are reported as point estimates with 95% bootstrap confidence intervals in parentheses. MSE and MAE are dimensionless because yield is expressed as a fraction between 0 and 1. For MSE and MAE, lower values indicate better predictive performance; for Pearson’s r, higher values indicate stronger linear association. Pearson’s r is undefined for constant-prediction baselines and is therefore denoted as NA. The best performance for each metric is highlighted in bold.

compares model performance under two representative training conditions: no intra-ingot bootstrap augmentation and augmentation with Boot = 6. In both cases, evaluation was conducted on the same held-out set of 19 original, non-augmented ingots. To reflect the limited test-set size, 95% confidence intervals for MSE, MAE, and Pearson’s correlation coefficient were estimated by bootstrap resampling of the held-out test ingots.

Without augmentation, most models showed limited improvement over the mean baseline, indicating that the small number of independent training ingots constrained model fitting. After applying moderate intra-ingot bootstrap augmentation, nonlinear models improved markedly, particularly the tree-based ensembles. Random Forest achieved the best overall performance, with an MSE of 0.021, an MAE of 0.125, and a Pearson correlation coefficient of 0.646 (p = 0.0014). XGBoost showed comparable performance, with an MSE of 0.023, an MAE of 0.128, and a Pearson correlation coefficient of 0.601 (p = 0.0033). The neural network also benefited from augmentation in terms of correlation, although its error metrics remained less competitive.

Linear models showed weaker and less consistent predictive ability, suggesting that nonlinear relationships and feature interactions are important for mapping evaluation-wafer characteristics to ingot-level yield. Nevertheless, because the held-out test set contained only 19 ingots, the results should be interpreted primarily as evidence of ranking and screening utility rather than precise absolute yield estimation.

The complete results across all bootstrap augmentation settings are provided in Appendix C 

Table A3. Predictive performance of models across bootstrap settings.

Model	MSE (CI)	MAE (CI)	Pearson’s r (CI)	Spearman’s $\mathit{\rho }$ (CI)
Boot = 0
Mean	0.036 (0.016–0.060)	0.144 (0.090–0.203)	0.000 (0.000–0.000)	0.000 (0.000–0.000)
MLR	0.270 (0.128–0.473)	0.414 (0.287–0.571)	0.145 (−0.115–0.383)	0.183 (−0.264–0.497)
Lasso	0.036 (0.016–0.060)	0.144 (0.090–0.203)	0.000 (0.000–0.000)	0.000 (0.000–0.000)
Ridge	0.045 (0.019–0.075)	0.166 (0.106–0.228)	0.067 (−0.417–0.517)	0.061 (−0.420–0.568)
RF	0.034 (0.017–0.054)	0.147 (0.094–0.200)	0.251 (−0.188–0.591)	0.282 (−0.230–0.651)
XGBoost	0.059 (0.033–0.096)	0.207 (0.151–0.271)	0.244 (−0.134–0.571)	0.316 (−0.108–0.647)
NN	0.048 (0.022–0.074)	0.178 (0.118–0.233)	0.221 (−0.202–0.609)	0.261 (−0.181–0.670)
Boot = 2
Mean	0.036 (0.016–0.060)	0.144 (0.090–0.203)	0.000 (0.000–0.000)	0.000 (0.000–0.000)
MLR	0.057 (0.036–0.082)	0.209 (0.158–0.261)	0.284 (0.004–0.592)	0.251 (−0.134–0.667)
Lasso	0.036 (0.016–0.059)	0.144 (0.090–0.202)	0.222 (−0.356–0.690)	0.247 (−0.422–0.751)
Ridge	0.046 (0.025–0.069)	0.183 (0.136–0.234)	0.174 (−0.214–0.555)	0.141 (−0.294–0.566)
RF	0.036 (0.020–0.054)	0.160 (0.113–0.207)	0.243 (−0.201–0.592)	0.131 (−0.402–0.552)
XGBoost	0.031 (0.015–0.049)	0.136 (0.086–0.189)	0.390 (−0.032–0.696)	0.323 (−0.183–0.730)
NN	0.048 (0.027–0.069)	0.185 (0.133–0.238)	0.363 (−0.003–0.633)	0.375 (−0.065–0.729)
Boot = 4
Mean	0.036 (0.016–0.060)	0.144 (0.090–0.203)	0.000 (0.000–0.000)	0.000 (0.000–0.000)
MLR	0.046 (0.019–0.075)	0.159 (0.105–0.220)	0.295 (−0.026–0.601)	0.401 (−0.087–0.742)
Lasso	0.036 (0.016–0.059)	0.144 (0.090–0.203)	0.204 (−0.312–0.652)	0.189 (−0.399–0.707)
Ridge	0.046 (0.020–0.080)	0.170 (0.116–0.234)	0.121 (−0.368–0.535)	0.183 (−0.349–0.626)
RF	0.033 (0.017–0.052)	0.147 (0.099–0.194)	0.326 (0.026–0.654)	0.356 (−0.057–0.784)
XGBoost	0.036 (0.019–0.053)	0.155 (0.107–0.202)	0.330 (0.056–0.618)	0.328 (−0.029–0.670)
NN	0.057 (0.021–0.101)	0.173 (0.105–0.249)	-0.226 (−0.700–0.299)	-0.129 (−0.662–0.504)
Boot = 6
Mean	0.036 (0.016–0.059)	0.144 (0.090–0.203)	0.000 (0.000–0.000)	0.000 (0.000–0.000)
MLR	0.041 (0.018–0.071)	0.155 (0.092–0.219)	0.296 (−0.160–0.616)	0.335 (−0.131–0.700)
Lasso	0.034 (0.016–0.055)	0.142 (0.089–0.197)	0.204 (−0.312–0.652)	0.189 (−0.399–0.707)
Ridge	0.044 (0.021–0.073)	0.164 (0.106–0.231)	0.194 (−0.278–0.583)	0.152 (−0.374–0.563)
RF	0.021 (0.011–0.032)	0.125 (0.090–0.163)	0.646 (0.346–0.839)	0.595 (0.262–0.817)
XGBoost	0.023 (0.012–0.039)	0.128 (0.089–0.168)	0.601 (0.227–0.845)	0.490 (0.011–0.796)
NN	0.031 (0.018–0.047)	0.149 (0.111–0.187)	0.611 (0.272–0.824)	0.643 (0.209–0.892)
Boot = 8
Mean	0.036 (0.016–0.060)	0.144 (0.090–0.203)	0.000 (0.000–0.000)	0.000 (0.000–0.000)
MLR	0.046 (0.024–0.073)	0.176 (0.127–0.234)	0.201 (−0.240–0.587)	0.183 (−0.276–0.643)
Lasso	0.035 (0.016–0.057)	0.143 (0.090–0.200)	0.204 (−0.312–0.652)	0.189 (−0.399–0.707)
Ridge	0.045 (0.022–0.077)	0.179 (0.131–0.233)	0.218 (−0.296–0.652)	0.222 (−0.327–0.687)
RF	0.028 (0.014–0.044)	0.139 (0.094–0.186)	0.481 (0.024–0.754)	0.352 (−0.140–0.750)
XGBoost	0.031 (0.015–0.048)	0.141 (0.095–0.193)	0.406 (0.070–0.712)	0.304 (−0.225–0.739)
NN	0.036 (0.016–0.060)	0.144 (0.090–0.203)	0.000 (0.000–0.000)	0.000 (0.000–0.000)

Note. Boot denotes the number of intra-ingot bootstrap resampling iterations applied to each training ingot. The best performance for each metric is highlighted in bold.

.

4.4. Visualization Analysis

Visualization analyses were conducted to examine three aspects of the proposed framework: the effect of retained feature number, the influence of intra-ingot bootstrap augmentation, and the interpretability of the strongest nonlinear models. These analyses support model selection and help identify the engineered feature categories most associated with ingot-level yield.

4.4.1. Effects of Feature Number and Bootstrap Augmentation

Figure 1 shows that model performance depended on both the number of retained features and the degree of bootstrap augmentation. For feature selection, tree-based ensemble models generally outperformed linear models and the neural network. Performance improved when the retained feature set increased from a small subset to an intermediate size, indicating that multiple engineered descriptors contributed complementary information. However, further increasing the number of features did not yield consistent gains and could introduce redundant or noisy variables under the limited ingot-level sample size. Therefore, the final feature number was capped at 60 to balance information retention and overfitting control.

The bootstrap analysis showed a similar trade-off. Moderate intra-ingot bootstrap augmentation improved the performance of Random Forest and XGBoost, with the best overall results observed around Boot = 6. However, performance did not increase monotonically with larger bootstrap multiples, suggesting that excessive resampling may amplify ingot-specific fluctuations rather than provide additional independent information. These results support moderate training-set augmentation rather than aggressive oversampling.

4.4.2. Feature Importance Analysis

The feature-importance results in Figure 2 indicate that the strongest predictors were mainly associated with spectral signal strength, signal-to-background quality, energy resolution, count-rate variability, A/B-side consistency, and the amount of valid evaluation-wafer evidence. In both XGBoost and Random Forest, peak-count descriptors such as B_Peak_Cnt_max and A_Peak_Cnt_max ranked highly, suggesting that spectral response strength was strongly associated with downstream yield. Other important variables included valley-count, PVR, FWHM, count-rate dispersion, and A/B-side difference descriptors.

Although the exact rankings differed between XGBoost and Random Forest, both models emphasized physically meaningful feature categories rather than relying on a single raw measurement or threshold rule. This pattern is consistent with the expected behavior of CZT detector materials, where poor spectral response, unstable count rates, surface-to-surface inconsistency, and wafer-to-wafer non-uniformity may reflect defect heterogeneity or contact-related instability that reduces downstream wafer usability.

4.4.3. SHAP-Based Model Interpretation

The SHAP summary plots in Figure 3 further show that the contribution of individual features was nonlinear and context dependent. Important descriptors could have either positive or negative SHAP values depending on their magnitude and interaction with other features, indicating that no single measurement alone determined the predicted yield. For example, high peak-count descriptors may indicate stronger spectral response, but their contribution depends on accompanying FWHM, count-rate stability, and A/B-side consistency.

The SHAP results also confirm that both models used information from multiple engineered feature groups, including threshold-derived descriptors, spectral-response statistics, distributional variability measures, and A/B-side difference features. These explanations should be interpreted as statistical associations rather than causal evidence. Nevertheless, their consistency with feature-importance rankings and detector-physics intuition supports the plausibility of the learned prediction patterns.

In summary, the visualization analysis confirms three major findings. First, an intermediate number of carefully selected features provides a better balance between information richness and overfitting control than either too few or too many retained variables. Second, moderate intra-ingot bootstrap augmentation improves the generalization of tree-based ensemble models, whereas excessive augmentation does not guarantee further performance gains. Third, the most influential predictors are associated with spectral response strength, energy resolution, signal-to-noise behavior, count-rate variability, A/B-side consistency, and threshold-derived pass-rate information. These observations explain why Random Forest and XGBoost achieved stronger test performance and practical screening capability than linear models and empirical rules alone.

4.5. Evaluation Experiment of Model Prediction and Empirical Rules

To assess practical screening value, each method was used to rank the held-out test ingots, and the average true product/test-wafer yield of the top 22% selected ingots was compared with that of the remaining ingots. The 22% selection ratio was chosen to match the approximate proportion of ingots satisfying the predefined evaluation-wafer-based production rule. Model-based methods ranked ingots by predicted yield, whereas empirical rules ranked ingots by scores computed from evaluation-wafer measurements.

For the model-based screening methods, the final models were trained under the selected configuration with six bootstrap resampling iterations and 60 retained features. This configuration was determined using the training data and cross-validation procedure and was fixed before evaluating the held-out test set. No product/test-wafer IV measurements, IV-curve morphology, or pass/fail judgements were used as model input features.

Two evaluation-wafer-based empirical screening rules were considered. The first, Rule_IV_Abs, was based on the proportion of evaluation wafers satisfying the absolute IV-current threshold:

$$S_g^{\mathrm{abs}}={\displaystyle \frac{1}{M_g}}{\displaystyle \sum _{j=1}^{M_g}}\mathbb{I}\left(min\left(|I{V}_{g,j}^{-}|,|I{V}_{g,j}^{+}|\right)\le 1\times {10}^{-8}\mathrm{A}\right),$$

with the corresponding binary rule defined as $Rul{e}_g^{\mathrm{abs}}=\mathbb{I}\left(S_g^{\mathrm{abs}}\ge 0.70\right)$.

The second, Rule_IV_Shape, was based on the proportion of evaluation wafers with acceptable IV-curve morphology: ${\mathcal{C}}_{\mathrm{acc}}=\{\mathrm{reverse}\mathrm{S}1,\mathrm{linear},\mathrm{positive}\mathrm{S}1\}$. Its binary output was defined as

$$S_g^{\mathrm{shape}}={\displaystyle \frac{1}{M_g}}{\displaystyle \sum _{j=1}^{M_g}}\mathbb{I}\left(C_{g,j}\in {\mathcal{C}}_{\mathrm{acc}}\right),$$

where $C_{g,j}$ denotes the IV-curve morphology of the j-th evaluation wafer. The binary output of this rule was then defined as $Rul{e}_g^{\mathrm{shape}}=\mathbb{I}\left(S_g^{\mathrm{shape}}\ge 0.70\right)$.

In addition to standalone machine learning models and empirical rules, we evaluated two hybrid screening strategies, denoted as XGBoost + Abs and RF + Abs. These hybrid methods combined the predicted product/test-wafer yield from XGBoost or Random Forest with the binary output of Rule_Eval_IV_Abs. Specifically, the final screening score was computed as a weighted average of the model-predicted yield and the 0/1 output of Rule_Eval_IV_Abs: $\alpha {\widehat{r}}_g+(1-\alpha )Rul{e}_g^{\mathrm{abs}}$, where ${\widehat{r}}_g$ denotes the model-predicted product/test-wafer yield of the g-th ingot and $\alpha$ is the combination weight. The combination weight $\alpha$ was determined exclusively on the training set using five-fold cross-validation, and the resulting fixed weight was then applied to the held-out test set. Therefore, all results reported in

Table 2. True product-wafer yield of the top 22% held-out test ingots selected by different evaluation methods. Top22%_Avg denotes the average true product-wafer yield of the selected ingots, Rest Avg denotes the average yield of the remaining ingots, and Improvement is defined as Top22%_Avg minus Rest Avg in percentage points. XGBoost + Abs and RF + Abs combine the model-predicted yield with the 0/1 output of Rule_IV_Abs using weights determined by five-fold cross-validation on the training set. p-values are given by two-sample t-tests on the held-out test set. The best performance for the key metric (i.e. Improvement) is highlighted in bold.

Method	Top22% Avg	Rest Avg	Improvement	p-Value
XGBoost	60.40	33.87	26.53	0.0083
RF	63.71	32.92	30.79	0.0026
Rule_IV_Abs	59.08	34.23	24.85	0.0307
Rule_IV_Shape	43.15	38.80	4.36	0.3352
XGBoost + Abs	61.94	33.43	28.51	0.0052
RF + Abs	63.71	32.92	30.79	0.0026

reflect held-out test-set performance.

As shown in Table 2, Random Forest achieved the highest average true yield among the selected top 22% ingots, reaching 63.71%, followed by XGBoost at 60.40%. Both exceeded the Rule_IV_Abs baseline of 59.08%, whereas Rule_IV_Shape provided only limited screening improvement. The hybrid XGBoost + Abs strategy slightly improved over standalone XGBoost, suggesting that the empirical absolute-current rule supplied complementary information. In contrast, RF + Abs produced the same selected set as standalone RF, indicating that Random Forest had already captured most of the useful information contained in this rule for the present split.

5. Discussion

The results demonstrate that standardized evaluation-wafer measurements contain useful information for estimating downstream product/test-wafer yield at the ingot level. This is practically important because the evaluation wafers and product/test wafers may follow different specifications: the former provide standardized early characterization, whereas the latter reflect downstream product requirements. The proposed framework therefore addresses a more challenging cross-specification setting than same-wafer or same-specification prediction, but one that is directly relevant to early production screening.

The comparison between machine learning models and empirical screening rules further indicates that data-driven models can complement production-inspired criteria. The empirical rules based on evaluation-wafer IV magnitude and IV-curve morphology provide transparent and easily interpretable references. However, such rules rely on fixed thresholds and may not fully capture multivariate interactions among IV response, spectral peak characteristics, surface-to-surface consistency, and noise-related descriptors. In contrast, models such as Random Forest and XGBoost can integrate multiple types of evaluation-wafer features and generate continuous yield estimates, which are useful for ranking ingots under a limited screening quota.

The hybrid screening results suggest that rule-based information can complement model predictions, although the benefit was model dependent. In particular, Rule_IV_Abs slightly improved XGBoost-based screening but did not further improve Random Forest, implying that RF had already captured much of the rule-relevant information in this split. These findings support a practical deployment strategy in which machine learning scores are used for ranking, while empirical rules remain transparent reference criteria. The interpretation should remain tied to the defined data roles: evaluation-wafer measurements serve as predictors, and product/test-wafer records serve only to define the downstream yield target.

6. Conclusions

This study developed a machine learning-assisted framework for early ingot-level yield screening of CZT detector materials. Standardized evaluation-wafer IV and double-sided spectral measurements were transformed into ingot-level descriptors capturing overall quality, distributional variability, A/B-side consistency, and threshold-based compliance. Product/test-wafer records were used only to define the downstream yield target, enabling leakage-safe prediction from early evaluation data.

Tree-based ensemble models showed the strongest performance. Under moderate intra-ingot bootstrap augmentation, Random Forest achieved an MSE of 0.021, an MAE of 0.125, and a Pearson correlation coefficient of 0.646 on the held-out test ingots, while XGBoost achieved an MSE of 0.023, an MAE of 0.128, and a Pearson correlation coefficient of 0.601. In the top-22% screening experiment, Random Forest and XGBoost selected ingots with average true yields of 63.71% and 60.40%, respectively, exceeding the empirical Rule_IV_Abs baseline of 59.08%.

These results indicate that data-driven models can provide useful ranking and prioritization support for early CZT ingot screening. However, given the limited number of independent ingots, the framework should be used as a decision-support tool rather than as a replacement for wafer-level inspection or experimental validation. Future work should focus on prospective validation with newly produced ingot batches, expansion of the dataset, incorporation of boule-level and spatial characterization information, and development of bounded or probabilistic yield-prediction models.