The stability and reliability of steel products are paramount for modern industries, from appliance manufacturing to aerospace. Among various quality issues, the slag inclusion defect is a critical surface defect originating during the continuous casting process. These inclusions, typically embedded 2 to 10 mm beneath the surface, not only mar the final product’s appearance but also significantly degrade its mechanical properties. It is widely acknowledged within metallurgical research that fluctuations in the mould level are a primary contributor to the formation of slag inclusion defect. However, most existing studies rely on fluid dynamics models or vortex simulations. There is a notable scarcity of data-driven analyses that delve deeply into the relationship between specific mould level fluctuation characteristics and the occurrence of slag inclusion defect. This study aims to bridge this gap by mining and statistically validating key features from mould level time-series data that correlate with slab quality records, thereby providing a foundational methodology for defect root cause analysis and prediction.
Our research is grounded in a comprehensive dataset from a leading steel plant, comprising mould level sensor readings and corresponding hot-rolling defect logs over a two-month production period. The core objective is to systematically extract, evaluate, and interpret features from the mould level fluctuation signal that have a statistically significant correlation with the presence of a slag inclusion defect. The overarching workflow encompasses signal representation, extensive feature extraction, rigorous correlation hypothesis testing, and effective feature selection, culminating in a model-based validation of the mined features’ efficacy.
1. Signal Representation and Preprocessing
The raw mould level fluctuation signal, defined as the deviation of the actual level from its setpoint, is inherently noisy and may contain transient anomalies. Direct analysis on the raw signal can be misleading. Therefore, we employ two complementary preprocessing techniques to obtain cleaner signal representations that preserve underlying patterns related to the slag inclusion defect.
Piecewise Aggregate Approximation (PAA): This method operates in the time domain, segmenting a time series of length \(N\) into \(n\) contiguous frames (where \(n < N\)) and representing each frame by its mean value. PAA effectively smooths local spikes and suppresses short-duration anomalies, providing a more robust representation. For a time series \(X = x_1, x_2, …, x_N\), the PAA representation \(\bar{X} = \bar{x}_1, \bar{x}_2, …, \bar{x}_n\) is calculated as:
$$
\bar{x}_i = \frac{n}{N} \sum_{j=\frac{N}{n}(i-1)+1}^{\frac{N}{n}i} x_j
$$
We determined an optimal segment size by evaluating the Dynamic Time Warping (DTW) distance between the original and PAA-smoothed signals. DTW is adept at measuring similarity between two temporal sequences which may vary in speed. The average DTW distance was minimized and found acceptable at a window size of 5.
Low-Pass Filtering (LPF): This technique operates in the frequency domain, attenuating signal components with frequencies higher than a specified cutoff frequency \(f_c\). It removes high-frequency noise while retaining the lower-frequency fluctuations which are often of metallurgical interest. We applied a digital low-pass filter with a cutoff frequency ratio of 0.1 (relative to the Nyquist frequency). The choice was validated by observing that filtering at this frequency removed noise while revealing clearer periodic patterns and maintaining a low average DTW distance from the original signal.
We treat the Original Signal, the PAA-smoothed Signal, and the Low-Pass Filtered Signal as three distinct but related representations. The rationale is that features correlating with the slag inclusion defect should be inherent and discoverable across these representations, with each representation potentially emphasizing different aspects (e.g., global trend vs. denoised detail). Processing the signal with both PAA and LPF sequentially was avoided as it could over-smooth and distort the signal excessively.
2. Comprehensive Feature Extraction
From each of the three signal representations, we extracted a comprehensive set of 853 features designed to capture the multifaceted nature of mould level fluctuations. These features span both time and frequency domains, combining general time-series descriptors with domain-specific insights related to casting process stability.
| Feature Category | Description & Examples | Count |
|---|---|---|
| Global Statistical Features | Basic descriptors of the entire signal: mean, variance, standard deviation, root mean square (RMS), skewness, kurtosis, complexity measures, stationarity tests. | ~50 |
| Segmented Statistical Features | Statistics computed over different quantiles or fixed segments of the signal (e.g., mean of values between the 20th and 60th percentile). | ~120 |
| Model-based Coefficients | Parameters from fitting simple models to the signal, such as coefficients from an Autoregressive (AR) model of order 10. These capture temporal dependencies. $$x_t = c + \sum_{i=1}^{10} \phi_i x_{t-i} + \epsilon_t$$ | ~30 |
| Shape and Pattern Features | Descriptors of signal morphology: number of peaks, length of longest consecutive subsequence above/below a threshold, maximum/minimum derivative, counts of crossings at specific levels (e.g., crossing zero). | ~150 |
| Special Value & Interval Features | Domain-informed counts, such as the number of data points where the fluctuation falls within specific intervals like \([-2, -1)\) or \([1, 2)\) mm, which are critical for process control. | ~100 |
| Frequency Domain Features | Magnitudes of Fast Fourier Transform (FFT) coefficients, capturing the power of specific frequency components. Also includes coefficients from Continuous Wavelet Transform (CWT). | ~400 |
| Boolean Features | Features derived from thresholds, e.g., “Is the variance greater than 1?” or “Is the standard deviation larger than 0.15 times the range?”. | ~3 |
This exhaustive feature set ensures that various potential manifestations of unstable mould level behavior linked to the slag inclusion defect are quantified and made available for analysis.
3. Correlation Analysis and Feature Selection
To identify which of the 853 extracted features are significantly associated with the slag inclusion defect, we performed rigorous statistical hypothesis testing. For each feature \(X_\varphi\), we test:
$$
H_{0\varphi}: \text{\(X_\varphi\) is not correlated with the slag inclusion defect label \(Y\).}
$$
$$
H_{1\varphi}: \text{\(X_\varphi\) is correlated with the slag inclusion defect label \(Y\).}
$$
The p-value resulting from this test quantifies the probability of observing the data if the null hypothesis \(H_{0\varphi}\) were true.
Testing Methods:
- For real-valued features, we employed the non-parametric two-sample Kolmogorov-Smirnov (KS) test. The KS statistic \(D_{n,m}\) measures the maximum distance between the empirical distribution functions \(F_{1,n}(x)\) and \(F_{2,m}(x)\) of the feature values for the normal slabs (\(n\) samples) and defective slabs (\(m\) samples), respectively:
$$
D_{n,m} = \sup_x | F_{1,n}(x) – F_{2,m}(x) |
$$
A small p-value indicates the distributions for normal and defective slabs are significantly different. - For Boolean features, we used Fisher’s exact test. The data is arranged in a 2×2 contingency table, and the test calculates the exact probability of the observed distribution under the null hypothesis of independence.
Multiple Testing Correction: With hundreds of simultaneous tests, the chance of false discoveries (Type I errors) increases dramatically. We controlled the False Discovery Rate (FDR) using the Benjamini-Yekutieli procedure. For a desired FDR level \(q\), the procedure sorts all p-values \(p_{(1)} \leq p_{(2)} \leq … \leq p_{(n_\varphi)}\) and finds the largest index \(k\) such that:
$$
p_{(k)} \leq \frac{k \cdot q}{n_\varphi \cdot \sum_{i=1}^{n_\varphi} \frac{1}{i}}
$$
All features with p-values less than or equal to \(p_{(k)}\) are deemed significantly correlated with the slag inclusion defect. This method provides a robust guard against identifying spurious correlations.
4. Validation via Weighted Random Forest Modeling
To validate the practical utility of the mined features, we constructed predictive models. A common alternative is to use deep learning models like 1D Convolutional Neural Networks (1D-CNN) directly on raw signals. We compared this approach against using our engineered features within a classical machine learning model.
Baseline: 1D-CNN on Raw Signals. We built a 1D-CNN with multiple convolutional layers followed by average pooling and a linear classifier. To handle class imbalance, a weighted version of the cross-entropy loss was also tested. The results were poor for the mould level data, with test Area Under the Curve (AUC) around 0.53, indicating the model failed to generalize, likely due to the high noise and subtle patterns related to the slag inclusion defect.
Our Approach: Feature-Based Weighted Random Forest (WRF). We trained a Random Forest classifier on the selected feature sets. To address the severe class imbalance (12,463 normal vs. 240 defective slabs), we implemented instance weighting where the weight for a class is inversely proportional to its frequency. The model hyperparameters included 100 trees and a maximum depth of 10. We experimented with feature sets selected from different signal representations:
| Feature Selection Source | # Features Selected | Test AUC | Interpretation |
|---|---|---|---|
| All features (Original Signal) | 817 | 0.6564 | Baseline with all features. |
| Selected (Original Signal) | 45 | 0.7156 | Improvement from selection. |
| Selected (PAA Signal) | 45 | 0.6476 | Features from smoothed view. |
| Selected (LPF Signal) | 244 | 0.6598 | Features from denoised view. |
| Intersection (Orig. + PAA) | 17 | 0.6212 | Very stable core features. |
| Intersection (Orig. + LPF) | 21 | 0.7213 | Best performing set. |
| Intersection (PAA + LPF) | 34 | 0.7024 | Good alternative set. |
| Intersection (All three) | 12 | 0.6600 | Most robust core features. |
The results are conclusive: The model built on the 21 features derived from the intersection of the Original and Low-Pass Filtered signal selections achieved the highest test AUC (0.7213), substantially outperforming the 1D-CNN baseline (0.5273). This demonstrates that our feature mining methodology successfully extracts discriminative information that a raw signal model misses. Furthermore, the features selected from the intersection of all three signal representations (12 features) represent the most stable and reliable characteristics associated with the slag inclusion defect, ideal for root cause analysis.
5. Interpretation of Key Mould Level Features Related to Slag Inclusion
The final and most crucial step is interpreting the mined features to derive actionable process insights. We focus on the 12 features common to all three signal representations. Analyzing their distribution plots for normal versus defective slabs reveals clear patterns.
5.1. Overall Fluctuation Magnitude
Features like Variance, Standard Deviation, Root Mean Square (RMS), and Absolute Energy consistently show higher values for slabs with a slag inclusion defect. This confirms the fundamental principle that larger deviations from the setpoint are detrimental.
$$ \text{Var}(X)_{\text{defective}} > \text{Var}(X)_{\text{normal}}, \quad \text{RMS}(X)_{\text{defective}} > \text{RMS}(X)_{\text{normal}} $$
Process Implication: Tightening control loops (e.g., PID parameters for the stopper rod or slide gate) to minimize the overall magnitude of level fluctuations is paramount for reducing the slag inclusion defect rate. The goal should be to keep the variance below 1 mm².
5.2. Specific Fluctuation Intervals
The features count_interval_[-2,-1) and count_interval_[1,2) (number of data points where fluctuation is between -2 & -1 mm and 1 & 2 mm, respectively) are elevated for defective slabs.
Process Implication: Control strategies should specifically target reducing the occurrence and duration of fluctuations in these “danger zones.” This might involve adjusting the controller’s gain or response time when the level deviates beyond ±1 mm.
5.3. Temporal Structure and Autocorrelation
The second coefficient (\(\phi_2\)) from a 10th-order Autoregressive (AR) model is more negative for normal slabs. This coefficient relates the current level to its value two seconds prior (\(x_{t-2}\)). A more negative value indicates a stronger corrective tendency or negative correlation at that specific lag.
$$ x_t = … + \phi_2 x_{t-2} + … $$
Process Implication: The control system’s response should be tuned to introduce appropriate negative correlation at short time lags (around 2 seconds) to dampen oscillations, rather than allowing persistent deviations that characterize slabs prone to slag inclusion defect.
5.4. Spectral (Frequency) Components
The magnitudes of the 10th and 16th FFT coefficients are higher for defective slabs. Given the signal length and sampling frequency, these correspond to low-frequency oscillations in the ranges of approximately 0.015-0.0225 Hz and 0.025-0.0375 Hz.
Process Implication: These frequencies may be linked to process disturbances like casting speed variations, tundish level changes, or periodic nozzle clogging/unclogging. Identifying and mitigating the source of these low-frequency disturbances is crucial. This could involve stabilizing tundish flow, optimizing argon injection, or reviewing mold oscillator settings to avoid resonant frequencies.
5.5. Process Stability and Tail Behavior
The 90% mass quantile index is closer to 1 (the end of the time series) for defective slabs. This indicates that for slabs with a slag inclusion defect, 90% of the signal’s “energy” or variation is concentrated earlier, leaving a disproportionately volatile tail section.
Process Implication: The casting process becomes significantly less stable toward the end of a sequence, often during slowdowns or stops. Special attention must be paid to control strategies during sequence tails, ladle changes, and planned stops to prevent severe level fluctuations that lead to slag entrapment.
5.6. Boolean Rule-based Features
Features like “standard deviation > 0.15 * range” and “variance > standard deviation” show different prevalence between classes. These features act as simple, interpretable rules. For instance, “variance > standard deviation” is equivalent to “standard deviation > 1” (since Var = σ² > σ implies σ > 1). This rule triggers more often for defective slabs, reinforcing the finding that fluctuations with σ > 1 mm are problematic.
6. Conclusion and Process Recommendations
This study successfully demonstrates a data-driven methodology for mining interpretable features from mould level fluctuation signals that are strongly correlated with the occurrence of slag inclusion defect. By combining advanced signal processing, exhaustive feature engineering, rigorous statistical testing, and validation through robust modeling, we move beyond qualitative assessments to quantitative, feature-level insights.
The key technical findings are:
- Features mined from the intersection of original and low-pass filtered signals provide the best predictive power for identifying slabs with a slag inclusion defect, outperforming direct deep learning on raw data.
- A small set of 12 highly stable features, common across all signal representations, captures the core aberrant behavior leading to defects.
- These features offer direct interpretability, allowing metallurgists and process engineers to understand not just *that* the level was unstable, but *how* it was unstable.
Based on the distribution analysis of the mined features, we can formulate specific, actionable process control recommendations to mitigate the slag inclusion defect:
- Primary Objective: Reduce Overall Fluctuation. Prioritize control loop optimization to minimize variance and standard deviation. Aim for a process where the level fluctuation standard deviation consistently remains below 1 mm.
- Target Specific Deviation Ranges. Implement enhanced control actions or alarms when the level deviation enters the \([-2, -1)\) and \([1, 2)\) mm intervals.
- Suppress Low-Frequency Oscillations. Investigate and eliminate root causes of disturbances in the 0.015-0.04 Hz frequency band, which our analysis links to the slag inclusion defect.
- Stabilize Process Tails. Develop and implement specialized control protocols for the end of casting sequences, during ladle changes, and at other transitional phases to prevent the severe instability observed in the tail-end of signals from defective slabs.
- Monitor Derived Indicators. Use simple Boolean indicators like “standard deviation > 1 mm” as real-time performance metrics for mould level control.
This feature mining framework provides a powerful foundation for building real-time decision support systems. The validated features can serve as inputs for online prediction models or be directly monitored on a dashboard to alert operators to developing conditions conducive to the formation of a slag inclusion defect. Ultimately, this approach translates industrial big data into tangible process knowledge, driving improvements in product quality and operational consistency in continuous casting.