Seasonal Prediction of the Bohai Sea Ice Grade: A Multi-Model Intercomparison

Abstract

Even under a warming climate, winter sea ice in the Bohai Sea continues to threaten ships and offshore/coastal infrastructure. Reliable pre-season prediction of the overall wintertime sea ice condition in the Bohai Sea, as represented by the Bohai Sea Ice Grade (BSIG), is therefore important for disaster preparedness and mitigation. Based on the 1979–2024 BSIG record, this study compares seven statistical and AI-based seasonal prediction methods: analog year analysis, multiple linear regression, stepwise regression, Principal Component Regression, a cross-correlation-based regression model, support vector regression, and the Bayesian Ensemble Bohai Ice Grade Net (BE-BIGNet). As potential precursors, we considered sea ice extent in 14 Arctic regions together with 114 large-scale atmospheric and oceanic circulation indices. The results suggest substantial differences in predictive skill among the methods. Among the tested approaches, BE-BIGNet, which combines Bayesian regularization with bootstrap median ensembling, achieves strong full-period performance and stable skill during the independent test period, suggesting that it may provide a useful framework for operational BSIG forecasting in the Bohai Sea.

1. Introduction

The Bohai Sea (37° N–41° N, 117° E–122° E), a shallow semi-enclosed marginal sea located in northeastern China along the northwestern Pacific coast (Figure 1a), represents the southernmost seasonally frozen coastal sea in the Northern Hemisphere. With an average depth of approximately 18 m, its unique geographical configuration—characterized by restricted water exchange with the open ocean and significant freshwater discharge from surrounding rivers—results in a relatively low heat capacity that makes it highly susceptible to rapid winter cooling. Sea ice in this region typically begins to form in late November and persists through March, characterized by a distinct spatial asymmetry where the northern part, particularly Liaodong Bay, acts as the primary and most ice-prone formation center. Climatologically, the maximum sea ice extent in the Bohai Sea averages approximately 20,000–30,000 km², typically covering 30–40% of the total sea area during normal winters. The thickness of level ice generally ranges from 10 to 30 cm, while in severe cases, the thickness of deformed ice in nearshore regions of Liaodong Bay can exceed 50 cm. As a critical hub for maritime transport and a strategic base for China’s offshore oil and gas industry, severe ice conditions can result in shipping blockages and structural damage to vessels, while also triggering intense ice-induced vibrations on offshore platforms such as jacket structures, thereby threatening structural integrity and operational safety [1]. The overall sea ice condition of the Bohai Sea in each winter is operationally assessed by national ocean forecasting institutes in China using a sea ice grade index, which ranges from 1 to 5 at 0.5 intervals and is calculated based on floating ice extent [2]. Forecasts of the sea ice grade are routinely released about one month before the ice season, serving as a key reference for sea ice disaster preparedness and mitigation. Thus, the establishment of a robust prediction system for winter ice grades is of great importance for both disaster mitigation and maritime logistical scheduling.

Multi-decadal analyses indicate that, although the Bohai Sea ice area has generally declined since the late 20th century, its long-term evolution is dominated by strong interannual and decadal variability rather than a simple monotonic decrease. Much of this interannual variability can be attributed to large-scale climate modes, particularly the Arctic Oscillation (AO) and the North Atlantic Oscillation (NAO), which regulate regional atmospheric circulation patterns, near-surface air temperature, and wintertime surface heat loss over the Bohai Sea [3]. At the planetary scale, ongoing global warming has led to a significant contraction of Arctic sea ice extent and substantial adjustments in atmospheric circulation, contributing to more frequent extreme mid-latitude weather events [4]. Consequently, damaging sea ice events can still affect the region during periods of extreme cold anomalies, despite the overall warming trend. The complex interactions between large-scale climate variability and regional atmospheric forcing introduce considerable uncertainty into Bohai Sea ice evolution, rendering accurate sea ice prediction a formidable challenge.

Early statistical predictions of sea ice conditions predominantly relied on linear regression frameworks that related ice condition to antecedent meteorological forcing, such as air temperature and surface pressure fields [5,6,7]. While these methods established the baseline for regional forecasting, they often suffer from multicollinearity among atmospheric predictors, which can obscure physical mechanisms and degrade forecast stability. To address this, more sophisticated statistical methods were introduced. Stepwise regression has been adopted to optimize predictor selection and reduce overfitting [8,9], while Principal Component Regression (PCR)—a standard tool in broader climate prediction—offers a robust way to extract dominant modes of variability while avoiding collinearity issues [10]. Parallel to these regression efforts, the analog year analysis leverages historical similarity in circulation fields to infer future states, providing an intuitive physical basis for prediction [11].

Despite their wide usage, these statistical methods rely on linear assumptions or historical stationarity, which may limit their ability to represent the nonlinear processes of a changing climate system. Recently, machine learning approaches have been increasingly applied to sea ice prediction. For example, support vector regression (SVR) and neural networks can capture nonlinear relationships between environmental drivers and sea ice variability. They have been applied to seasonal and interannual forecasting, frequently yielding good predictive skill compared to conventional baselines [12,13]. However, the application of complex machine learning methods in Bohai Sea Ice Grade (BSIG) prediction is constrained by the “small sample” problem—reliable satellite and observational records span only a few decades (since 1979). Standard machine learning models often lack the probabilistic reasoning required to quantify uncertainty in such data-limited contexts.

Although both statistical and AI-based approaches have been successfully applied in sea ice prediction, their relative performance has not been systematically assessed in the Bohai Sea. To address this gap, we compare seven prediction methods representing analog, statistical, and machine learning frameworks:

(1)

Physical Analog Approaches, specifically the analog year analysis, which represents a physically motivated similarity approach.

(2)

Linear statistical approaches, including multiple linear regression, stepwise regression, as well as Principal Component Regression (PCR). This category also includes a linear fitting model based on key physical precursors identified through cross-correlation analysis of Arctic sea ice, atmospheric circulation, and oceanic drivers.

(3)

AI models, specifically support vector regression (SVR) for kernel-based learning and a novel Bayesian ensemble net (BE-BIGNet). The latter combines a neural network with Bayesian regularization to improve generalization under the limited sample size.

By evaluating these methods in both historical hindcasts and an independent test period, this study assesses their relative performance for BSIG prediction. The results may contribute to improving the accuracy of sea ice forecasts and supporting disaster mitigation.

2. Data and Methods

2.1. Data

The 1979–2024 Bohai Sea Ice Grade (BSIG) data adopted in this study were obtained from the historical records of the North China Sea Marine Forecasting and Hazard Mitigation Center of Ministry of Natural Resources. The BSIG is a comprehensive metric used to quantify the severity of ice conditions, classified into five levels: Grade 1 (Light), Grade 2 (Normal–Light), Grade 3 (Normal), Grade 4 (Normal–Heavy), and Grade 5 (Heavy). To ensure greater precision in operational forecasts and hindcasts, 0.5-grade intervals are adopted in practice. This classification standard is based on key parameters including the maximum sea ice extent and ice thickness, as described by Sui et al. (2022) [2].

The Arctic Sea Ice Concentration (SIC) dataset, published by the National Snow and Ice Data Center (NSIDC), is derived from remote sensing brightness temperature. The dataset was processed using the NASA Team (NT) algorithm developed by the NASA Goddard Space Flight Center’s Cryospheric Sciences Laboratory. The dataset is available online: https://nsidc.org/data/nsidc-0051/versions/2 (accessed on 5 December 2025). This dataset spans from 26 October 1978 to the present and includes daily and monthly averaged SIC values with a spatial resolution of 25 km × 25 km. The monthly averaged sea ice extent data used in this study was computed from the monthly SIC fields.

A total of 114 large-scale atmospheric and oceanic indices are adopted in this study. Monthly means time series were provided by the Beijing Climate Center (BCC). The dataset is available at https://cmdp.ncc-cma.net/Monitoring/cn_index_130.php (accessed on 12 November 2025). and a full list of indices used here is attached in Appendix A.

2.2. Prediction Models

(1) Analog Year Analysis

The analog year analysis is widely used for monitoring and predicting climate indices such as the MJO and BSISO [14]. It quantifies the similarity between historical circulation fields and a target-year pattern in the spatial domain, providing an objective basis for analog year selection. Let $X(\mathbf{r},t)$ denote the anomaly field after removing the climatology and trend. The standardized field is

$$Z(\mathbf{r},t)={\displaystyle \frac{X(\mathbf{r},t)}{{\sigma }_X\left(\mathbf{r}\right)}}.$$

(1)

The projection index is calculated as

$$I(t)=\sum _{\mathbf{r}}Z(\mathbf{r},t) M\left(\mathbf{r}\right)$$

(2)

where ${\mathsf{\sigma }}_{\mathrm{X}}\left(\mathrm{r}\right)$ represents the temporal standard deviation of the anomaly field at spatial location $\mathrm{r}$, and $\mathrm{M}\left(\mathrm{r}\right)$ is the target projection pattern. The magnitude of $\mathrm{I}\left(\mathrm{t}\right)$ represents the degree of spatial similarity between the historical field and the target pattern. Years satisfying $\mid \mathrm{I}\left(\mathrm{t}\right)\mid \ge \mathsf{\theta }$ are selected as analog years for subsequent ice severity prediction.

(2) Linear and Stepwise Regression

Multiple linear regression (MLR) is employed to quantify the linear relationship between the target variable and multiple predictor variables. This method assumes that the combined effect of the predictors on the target variable can be expressed as a linear superposition. To obtain a parsimonious and robust model, stepwise regression is also adopted to identify statistically significant predictors, eliminate redundant variables, and optimize the model structure. The MLR model is expressed as:

$$Y={\beta }_0+{\displaystyle \sum _{i=1}^p}{\beta }_i{X}_i+\epsilon$$

(3)

where $Y$ indicates the target variable, $X_i$($i=1,\dots ,p$) are the predictors, ${\beta }_0$ is the intercept, ${\beta }_i(i\ge 1)$ are the regression coefficients associated with each predictor, $p$ is the number of selected predictors, and $\epsilon$ represents the residual error.

(3) Principal Component Regression (PCR)

Principal Component Regression (PCR) first uses Principal Component Analysis (PCA) to transform the original variables into a set of orthogonal principal components. Then, the principal components with higher contribution rates are selected for regression modeling to enhance the robustness of the model [10]. The PCR model can be written as

$$\mathit{P}=\mathit{X}\mathit{W},$$

(4)

$$Y={\beta }_0+{\displaystyle \sum _{i=1}^m}{\beta }_i{P}_i+\epsilon ,$$

(5)

where $\mathit{X}$ is the $N\times K$ data matrix of original predictor variables ($N$ observations as rows and $K$ predictors as columns), $\mathit{W}$ is the $K\times m$ PCA loading matrix, $\mathit{P}$ is the resulting $N\times m$ matrix of the retained principal components, $\mathrm{m}$ is the number of selected components, ${\beta }_i$ are the regression coefficients, and $\epsilon$ is the residual error.

(4) Cross-Correlation-Based Linear Regression

To account for climate signals with longer seasonal memory, cross-correlation analysis is applied to identify potential precursors of BSIG. This method quantifies the linear relationship between the time series of the BSIG (Y) and potential factors (X) with a time lag $\tau$ [15]. The correlation coefficient R is calculated as:

$$R(\tau )={\displaystyle \frac{{\sum }_{i=1}^N(X_{i,t-\tau }-\overline{X})(Y_{i,t}-\overline{Y})}{\sqrt{{\sum }_{i=1}^N(X_{i,t-\tau }-\overline{X}{)}^2}\sqrt{{\sum }_{i=1}^N(Y_{i,t}-\overline{Y}{)}^2}}}$$

(6)

where $N$ is the sample size and $\tau$ denotes the time lag in months. Climatic factors that exhibit statistically significant correlations with the ice grade at lead times of 1–5 seasons (significance with $p<0.05$) are selected as candidate precursors for subsequent linear regression.

(5) Support Vector Regression (SVR)

SVR is a machine learning algorithm based on the principle of minimizing structural risk. It projects low-dimensional input data into a higher-dimensional feature space using nonlinear mapping. The algorithm then finds the best regression function within a set error tolerance, making it effective for small sample sizes and complex nonlinear relationships [16]:

$$f\left(x\right)={\omega }^T\varphi \left(x\right)+b$$

(7)

where $\varphi \left(x\right)$ is the nonlinear mapping function that projects the input vector $x$ into a high-dimensional feature space, $\omega$ is the weight vector, and $b$ is the bias term.

(6) Bayesian Regularized Neural Network

Although large neural models handle complex problems, only 46 records of Bohai ice grade are available from 1979 to 2024. For this situation, simpler models often perform better; thus, this study uses a single hidden-layer fully connected model with 7 neurons. Furthermore, the bootstrap aggregating (bagging) method is adopted to create training datasets by sampling with replacement from the original dataset. Each bootstrap dataset is used to train an individual model, and the ensemble prediction is obtained by aggregating the outputs of all models, e.g., by averaging or taking the median of individual predictions. This approach reduces model variance and improves prediction stability, particularly in the presence of noise or small sample sizes [17]. The median-based ensemble prediction is defined as

$$\widehat{\mathit{y}}=\mathbf{m}\mathbf{e}\mathbf{d}\mathbf{i}\mathbf{a}\mathbf{n}\{{\mathit{f}}^{\left(\mathbf{1}\right)}(\mathbf{x}),{\mathit{f}}^{\left(\mathbf{2}\right)}(\mathbf{x}),\dots ,{\mathit{f}}^{\left(\mathit{B}\right)}(\mathbf{x})\}$$

(8)

where ${\mathit{f}}^{\left(\mathit{b}\right)}\left(\mathbf{x}\right)$ is the model trained on the $\mathit{b}$-th bootstrap dataset, and $\mathit{B}$ is the total number of bootstrap models.

2.3. Standardization of the Predictions

To ensure a strictly fair and mathematically rigorous intercomparison among the diverse models evaluated in this study, it is essential to align their output resolutions. Because the observed Bohai Sea Ice Grade (BSIG) is inherently recorded at 0.5-pitch intervals (as introduced in Section 2.1), evaluating continuous floating-point predictions directly against these discrete observations introduces artificial error penalties for traditional regression models. Therefore, prior to plotting the comparative time series and calculating the final evaluation metrics (e.g., root mean square error, RMSE), the continuous forecast outputs from all statistical models were standardized to 0.5 intervals (i.e., rounded to the nearest whole or half point). This standardization ensures that the calculated RMSE values accurately reflect a leveled operational baseline, emphasizing the intrinsic predictive skill of each architecture under real-world forecasting conditions.

3. Results

The prediction of the upcoming winter’s BSIG in the Bohai Sea is usually made and released in October, making summer predictors a standard and operationally practical choice. In this study, analog year analysis, multiple linear regression (MLR), stepwise regression, and PCR serve as baseline statistical models, all utilizing the same summer predictor window. This ensures methodological consistency and allows for a fair comparison that highlights differences in model structure rather than discrepancies arising from predictor selection.

3.1. Analog Year Analysis

The analog year analysis, originally proposed by Wheeler and Hendon (2004) [14], has been widely adopted by operational centers (e.g., NOAA and the China National Climate Center) to monitor and predict climate phenomena such as the Madden–Julian Oscillation (MJO) and the Boreal Summer Intraseasonal Oscillation (BSISO). The method first removes the climatological seasonal cycle and long-term trend from multi-year datasets of the target field; the resulting anomaly fields are then standardized and projected onto a predefined spatial pattern associated with the phenomenon of interest. This projection yields a projection index (PI) that quantifies the spatial similarity between each historical field and the target pattern, thereby providing an objective basis for analog year selection.

First, the summer (June–August) 500 hPa geopotential height anomaly for 2024 is used as the target projection pattern (Figure 2a) to illustrate the analog year analysis workflow and to derive a forecast for the Bohai Sea Ice Grade (BSIG) in the winter of 2024. Specifically, multi-year atmospheric circulation fields are projected onto the 2024 target pattern to generate a time series of PI values. Years with |PI| ≥ 0.5 standard deviations are identified as analog years (blue dots in Figure 3 with similar circulation patterns), and their composite 500 hPa geopotential height anomaly field is shown in Figure 2b. The composite mean sea ice grade derived from the selected analog years serves as the prediction for the Bohai Sea winter sea ice grade in 2024 (red diamonds in Figure 3). To evaluate predictive performance, we predicted the BSIG over the recent five-year period 2020–2024 and the results are compared with the observed records. This comparison yields a root mean square error (RMSE) of 0.59 (Figure 4), indicating that this method captures the recent interannual variability in sea ice conditions with relatively small overall error.

3.2. Multiple Linear Regression

To quantify the linear modulation of atmospheric circulation and oceanic forcing on the BSIG, we constructed a multiple linear regression (MLR) model. The model utilizes indices from the preceding summer (June–August) to predict the ice grade of the following winter. The dataset is partitioned into a training period (1979–2019) and an independent validation period (2020–2024). From the 114 indices listed in Appendix A, the ten predictors exhibiting the highest correlation coefficients with the BSIG were selected for the regression analysis. These key predictors include: the Cold-Tongue ENSO index ($X_{CT-ENSO}$), NINO 4 and NINO-W sea surface temperature (SST) anomalies ($X_{NINO4} \& X_{NINO-W}$), the Tibet Plateau Region 1 and 2 indices ($X_{TP1}$ & $X_{TP2}$), the Eastern Pacific Subtropical High Ridge Position index ($X_{EP-SH}$), North American Subtropical High Area and Intensity indices ($X_{NA-SHA}$ & $X_{NA-SHI}$), the ENSO Modoki Index ($X_{EMI}$), and the Eurasian Meridional Circulation (EMC). Prior to calculation, all indices were standardized. The resulting regression equation is expressed as:

$$\begin{gathered} Y=2.81-1.06·X_{CT-ENSO}+0.47·X_{NINO4}+0.30·X_{NINO-W}+0.06·X_{TP1} \\ -0.09·X_{TP2}+0.21·X_{EP-SH}+0.07·X_{NA-SHA} \\ -0.21·X_{NA-SHI}+0.23·X_{EMI}-0.27·X_{EMC} \end{gathered}$$

(9)

where $Y$ represents the predicted ice grade. The predicted BSIG is shown by the red line in Figure 5. The root mean square error (RMSE) during the training period is 0.61, while for the validation period it is 0.74. The results suggest that interannual variability in the Bohai Sea may primarily be related to ENSO and its spatial SST anomalies. These anomalies modulate the intensity of the East Asian Winter Monsoon and local air–sea heat fluxes via teleconnections involving the Subtropical High, the Walker–Hadley circulation, and the Eurasian wave train. However, the significant increase in RMSE from the training to the validation period indicates that this standard MLR model is subject to potential overfitting.

3.3. Stepwise Regression

To address the potential overfitting and multicollinearity issues observed in the above MLR model, we employed stepwise regression using the Akaike Information Criterion (AIC) to optimize predictor selection. Using the standardized summer (June–August) indices from the training period (1979–2019) as the candidate pool, the algorithm automatically screened the variables to retain only those significantly correlated to BSIG, thereby refining the input set to six key precursors. The optimized prediction equation for the BSIG is expressed as:

$$\begin{gathered} Y=2.81-1.25·X_{CT-ENSO}+0.88·X_{NINO4}+0.18·X_{EP-SH}+0.44·X_{EMI} \\ -0.29·X_{EMC}+0.41·X_{WP-TW} \end{gathered}$$

(10)

where $Y$ represents the predicted ice grade and $X_{WP-TW}$ indicates the West Pacific 850 hPa Trade Wind index. The predicted results are shown against the observations in Figure 5.

Both regression approaches confirm that ENSO (specifically its spatial morphology), the Subtropical High, and the Eurasian Meridional Circulation are the dominant controls on Bohai Sea ice evolution. While the MLR provides a broad linear projection including secondary factors, the stepwise approach effectively isolates the core drivers.

Over the full study period (1979–2024), the stepwise regression yielded a slightly higher RMSE (0.64) compared to the standard MLR (0.62). However, the stepwise model reached an RMSE of 0.39 during the independent validation period (2020–2024), significantly outperforming the MLR (RMSE = 0.74). This sharp contrast between the training and testing performances suggests that by reducing model complexity, stepwise regression mitigates the overfitting inherent in the standard MLR. Consequently, it minimizes noise amplification and offers better extrapolation stability under the rapidly shifting background state of the recent climate.

3.4. Principal Component Regression

To further mitigate multicollinearity among the numerous atmospheric and oceanic indices, we employed Principal Component Regression (PCR), a hybrid approach that integrates the dimensionality reduction advantages of Principal Component Analysis (PCA) with the adaptive variable selection of stepwise regression.

The model was constructed using standardized summer (June–August) indices in Appendix A. First, we screened the dataset to retain the 40 predictors exhibiting the highest correlation with the BSIG (listed in Appendix B). PCA was then performed on these key factors to extract orthogonal modes of variability. The variance explained by each PC is shown in Figure 6. The leading principal components (PCs) that accounted for 90% of the cumulative variance were retained as candidate variables. A stepwise regression algorithm was then applied—starting from an intercept-only baseline—to iteratively select the optimal subset of PCs that minimized the Akaike Information Criterion (AIC). The optimized prediction equation, trained on the 1979–2019 period, is expressed as:

$$\begin{gathered} Y=2.82-0.05·P{C}_1-0.07·P{C}_2+0.14·P{C}_3+0.11·P{C}_4-0.10·P{C}_5 \\ -0.18·P{C}_7 \end{gathered}$$

(11)

and the predicted BSIG is compared with the observations in Figure 7.

The PCR model demonstrated robust performance, achieving an RMSE of 0.65 for the full historical period (1979–2024) and a notably low RMSE of 0.32 during the independent validation period (2020–2024). By extracting orthogonal modes, PCR effectively filters noise and redundancy, allowing for a stable estimation of general climate forcing. However, while the model captures the mean state and baseline trends effectively, it exhibits reduced skill in reproducing extreme anomalies (e.g., severe heavy or light ice years). This limitation is characteristic of linear variance maximization techniques, which tend to prioritize dominant statistical signals at the expense of rare, high-amplitude events.

3.5. Cross-Correlation-Based Regression Model

Since summer-only predictors may not fully capture the longer memory of the climate system, we extended the analysis to earlier seasons. Specifically, we examined lead–lag relationships between BSIG and large-scale circulation indices as well as Arctic sea ice extent, with the aim of identifying precursor signals that persist across seasons and high-latitude drivers that may precondition winter ice severity in the Bohai Sea.

3.5.1. Circulation and Oceanic Factors

To investigate the physical mechanisms driving interannual variability, we analyzed cross-correlations between the detrended BSIG and 114 large-scale climate indices over the five preceding seasons. For robustness, we retained indices that were significant at the 95% confidence level in at least one lag season. The results (Figure 8) suggest that Bohai ice variability is associated with both tropical oceanic memory and atmospheric teleconnections.

One potential source of the predictability of BSIG arises from the persistence of tropical Pacific oceanic anomalies. The slow upper-ocean heat-content evolution and equatorial wave adjustment associated with ENSO provide an interseasonal memory that can precondition subsequent SST anomalies several seasons in advance [18,19,20]. The west Pacific trade winds during the previous winter may also contribute to the predictability of BSIG. Trade-wind and wind-stress-curl anomalies over the western/subtropical Pacific can induce Rossby wave propagation and subtropical cell adjustment, altering subsurface heat content in the western-to-central Pacific and favoring the persistence or later re-emergence of central-equatorial SST anomalies [21,22]. These anomalies may then influence East Asia through the Pacific–East Asian teleconnection, by modulating the western North Pacific anticyclonic/cyclonic circulation and thereby weakening or strengthening the East Asian Winter Monsoon [23,24,25]. For the Bohai Sea, such remote atmospheric forcing is likely converted into sea ice variability through changes in cold air outbreak frequency, surface heat loss, and wind-driven ice transport [26,27].

3.5.2. Arctic Sea Ice

We further investigated the teleconnection between Arctic sea ice extent and Bohai ice grades. Correlations were calculated for sea ice extent/area across 14 key Arctic sectors (Figure 9) with lead times of 1–12 months (previous December to current November), with all time series detrended to remove long-term trends. The highest correlation coefficients were identified in the following two regions: the Barents Sea ice extent (in the Atlantic Sector) exhibits a strong negative correlation (−0.52) at a 1-month lead time (November); the Chukchi and Bering Seas (in the Pacific Sector) show significant negative correlations (−0.48 and −0.39, respectively) at a 3-month lead time (September).

These correlations (Figure 10) are consistent with a possible thermodynamic forcing pathway: a reduction in sea ice in these marginal seas during the current autumn (September to November) releases anomalous heat fluxes to the atmosphere via air–sea interaction. This surface heating excites stationary Rossby wave trains [28,29], which enhance mid-to-high-latitude blocking patterns over Eurasia—specifically the Ural Blocking High—and consolidate the Siberian High [30,31]. The resulting circulation anomalies intensify the East Asian Winter Monsoon, increase the frequency of cold air intrusions affecting the Bohai Sea, and intensify Bohai ice conditions. The observed 1–3-month lag is consistent with the timescale required for the establishment and propagation of this Atmospheric Bridge mechanism [4].

3.5.3. Predictor Selection and Sensitivity Experiments

To determine the optimal predictor set, we first established a baseline forcing field comprising the autumn Western Pacific Subtropical High Ridge Point and the Current Summer Cold-Tongue ENSO index. Building on this core configuration, we conducted 18 sensitivity experiments (

Table 1. Configuration of predictors and prediction skill (RMSE) across different experimental schemes (Exp1–Exp18). The checkmarks (●) denote the variables included in each experiment. The regression coefficients, standard errors, and p-values for each parameter against the BSIG are also detailed in Appendix D for readers’ convenience.

	Arctic Region Sea Ice Extent			Key Index
	Barents (Lag 1 mon)	Chukchi (Lag 3 mon)	Bering (Lag 3 mon)	WP_SH_WRP (Aut0)	CT_ENSO (Sum0)	WP850_TW (Win-1)	NINO4 (Aut-1)	RMSE	RMSE (5 Yr)
Exp-1	●	●	●	●	●	●		0.557	0.559
Exp-2	●	●		●	●	●		0.571	0.354
Exp-3	●		●	●	●	●		0.599	0.612
Exp-4	●			●	●	●		0.576	0.500
Exp-5		●		●	●	●		0.599	0.433
Exp-6			●	●	●	●		0.651	0.612
Exp-7	●	●	●	●	●		●	0.557	0.559
Exp-8	●	●		●	●		●	0.557	0.354
Exp-9	●		●	●	●		●	0.571	0.612
Exp-10	●			●	●		●	0.594	0.433
Exp-11		●		●	●		●	0.612	0.433
Exp-12			●	●	●		●	0.626	0.612
Exp-13	●	●	●	●	●	●	●	0.557	0.559
Exp-14	●	●		●	●	●	●	0.557	0.354
Exp-15	●		●	●	●	●	●	0.603	0.612
Exp-16	●			●	●	●	●	0.612	0.612
Exp-17		●		●	●	●	●	0.580	0.433
Exp-18			●	●	●	●	●	0.612	0.612

) combining atmospheric/oceanic indices with sea ice metrics from different Arctic marginal seas. The regression coefficients were calculated on the training period (1979–2019), and performance was evaluated using RMSE for both the full historical period (1979–2024) and the 5-year independent validation period (2020–2024).

Similar RMSE values are found in Exp-8 and Exp-14, both of which achieve the lowest RMSEs during the training and validation periods. However, Exp-8 was selected as the optimal model because it employs fewer predictors than Exp-14, indicating that the additional variable (WP850_TW) included in Exp-14 does not provide meaningful improvement in predictive skill. Exp-8 achieves an independent validation RMSE of 0.354 by integrating high-latitude Arctic sea ice signals (Barents and Chukchi Seas) with mid-to-low-latitude atmospheric–oceanic predictors (WP_SH_WRP, CT_ENSO, and NINO4).

A comparative analysis with experiments incorporating Bering Sea ice (Exp-1, Exp-7, and Exp-13) reveals a clear overfitting problem. Although these experiments show training-period RMSEs comparable to that of Exp-8, their RMSEs are substantially higher than Exp-8 during the independent 5-year validation period. This result suggests that the inclusion of Bering Sea ice information weakens the model’s generalization capability.

Notably, experiments excluding key predictors, such as Barents Sea ice or the equatorial Pacific index (CT_ENSO), exhibited marked degradation in predictive performance. This finding demonstrates that a spatially comprehensive predictor set, spanning both high-latitude Arctic and tropical climate signals, is essential for maintaining the predictive stability of the Bohai Sea Ice Grade (BSIG). Therefore, to ensure robustness across the full 46-year record while effectively minimizing overfitting risk, Exp-8 was selected as the final baseline predictor configuration. A comparison between the predictions from Exp-8 and the observed BSIG is shown in Figure 11. This optimal predictor combination provides a balanced representation of both long-term climatological variability (RMSE = 0.557) and recent independent prediction skill (RMSE = 0.354), thereby establishing a rigorous and reliable benchmark for seasonal BSIG prediction.

3.6. Support Vector Regression

While Principal Component Regression (PCR) effectively mitigates multicollinearity through dimensionality reduction, it remains fundamentally constrained by linear assumptions, limiting its ability to resolve the nonlinear feedback mechanisms characteristic of the climate system. To address these issues and manage the “small sample” constraint, we employed support vector regression (SVR)—a kernel-based machine learning algorithm designed to capture nonlinear relationships while prioritizing robust generalization over fitting to noise.

Based on the preceding correlation analysis and physical mechanisms, we constructed a feature vector comprising five key atmospheric and oceanic precursors, plus a sea ice persistence term. The selected predictors include: (1) tropical forcing via the Current Summer Cold-Tongue ENSO index and the Previous Autumn NINO4 index; (2) atmospheric coupling via the Current Autumn Western Pacific Subtropical High Western Ridge Point; and (3) Arctic forcing via the sea ice extent of the Barents Sea in November and Chukchi Sea in September. Together, these predictors represent tropical forcing, intermediate atmospheric circulation, and autumn Arctic sea ice anomalies. Additionally, to capture the interannual persistence of the cryosphere, the BSIG from the previous winter was included as an autoregressive feature. Prior to training, all input features were standardized to eliminate dimensional discrepancies [32]. Model performance was assessed using a rolling-origin evaluation (time series cross-validation) scheme: for each iteration, the model was trained on all historical data prior to the target year to predict the subsequent winter. This approach strictly prevents data leakage and provides a robust evaluation of the model’s generalization ability through the cumulative assessment of multiple prediction errors [33].

The prediction results of the SVR model are illustrated in Figure 12. During the training phase (1979–2019), the model achieved an RMSE of 0.55. In the independent validation phase (2020–2024), the SVR demonstrated comparable predictive accuracy, yielding an RMSE of 0.53. The absence of significant error amplification during the testing period indicates that the model effectively avoids the overfitting and numerical divergence issues commonly associated with small-sample nonlinear regression, thereby demonstrating satisfactory robustness.

Despite SVR showing acceptable performance in overall error control and predictive stability, deviations persist in forecasting extreme anomalies—specifically the heavy ice years of 2000 and 2009, and the light ice year of 1994—where predictions tend to gravitate toward the multi-year climatological mean. This behavior is intrinsically linked to the SVR’s use of an $ϵ$-insensitive loss function and smooth kernel settings, which are designed to suppress the impact of observational noise and random perturbations. In the climate system, extreme events are typically triggered by the superposition of multi-scale anomalous processes, featuring mechanisms and statistical properties that differ markedly from conventional interannual variability. Consequently, while SVR is effective at characterizing the predictable interannual background evolution of Bohai ice conditions, its capacity to reconstruct extreme intensities remains constrained. This limitation reflects a universal challenge for statistical methods in a non-stationary climate but does not undermine the model’s overall validity for operational forecasting.

3.7. Bayesian Ensemble Neural Network

In the prediction of variables driven by multiple factors using limited sample sizes, forecast stability and generalization performance are often highly sensitive to feature selection and model structure. To address these issues and leverage the complementary advantages of differing methodologies, this study proposes a hybrid prediction framework: the Bayesian Ensemble Bohai Ice Grade Net (BE-BIGNet). This framework combines nonlinear neural network fitting, Bayesian regularization, and ensemble averaging to improve prediction stability under a limited sample size.

Although the BSIG is operationally classified into discrete levels (1–5 at 0.5 intervals), sea ice growth and decay are physically continuous dynamic processes driven by the heat budget and factors such as cumulative Freezing Degree Days (FDDs) and ocean heat content [34]. Traditional classification algorithms, however, treat these grades as independent categories, failing to capture the continuous variation between adjacent levels. In the context of interannual prediction for the Bohai Sea, this structural discrepancy between the continuous physical process and the discrete classification system—compounded by the characteristics of a small sample size with high noise—creates significant uncertainty for direct classification modeling.

To resolve this, we employed a Bayesian Regularized Back-Propagation (BP) Neural Network and introduced a “continuous fitting–discrete quantization” modeling strategy. By establishing continuous nonlinear mapping between climate factors, Arctic sea ice, and ice grades, the model characterizes the physical continuity of the ice evolution process. For the historical training samples (1979–2019), the model incorporates a Bayesian probabilistic framework that adaptively adjusts weight penalty terms in the loss function via Maximum A Posteriori (MAP) estimation. This approach achieves more stable parameter estimation and prevents overfitting [35]. Combined with linear detrending preprocessing, this architecture constrains model complexity while focusing on capturing interannual physical fluctuations (after removing the background warming trend), effectively mitigating the risk of overfitting inherent in small-sample training.

To further mitigate prediction uncertainty arising from random weight initialization and data sampling variability, this study employs a bootstrap median ensemble strategy. Specifically, we utilized bootstrap aggregation (Bagging) to generate 100 training subsets via random resampling with replacement, with each subset training an independent Bayesian regularized sub-model [36]. For the final output, the median of the predictions from all 100 sub-models was adopted as the ensemble forecast, rather than the arithmetic mean. Statistical studies indicate that, compared to mean aggregation, a median ensemble possesses a higher breakdown point and handles non-normal prediction errors more robustly [37]. This strategy effectively serves as a nonlinear filter, removing outlier predictions caused by individual sub-models falling into local minima or overfitting to noise, thereby ensuring the physical consistency and reliability of the system output.

The BE-BIGNet model shows stable performance and relatively low prediction error (Figure 13). Adopting a three-layer feedforward neural network architecture (Input–Hidden–Output) with an optimal four neurons in the hidden layer, the model yielded a training RMSE of 0.44 over the historical period (1979–2019) and a testing RMSE of 0.38 for the independent period (2020–2024), indicating a strong fit to observations. Notably, the model successfully reproduced the heavy ice years of 1979 and 1984 and exhibited enhanced sensitivity to extreme anomalies compared to SVR, although larger deviations persisted for the extreme events of 2000 and 2009. Overall, this framework performs well for the Bohai Sea case and may be useful for operational BSIG prediction.

4. Discussions and Conclusions

This study conducted an intercomparison of seven prediction methodologies for BSIG, ranging from static analogs to dynamic machine learning approaches. By systematically evaluating the analog year analysis (AYA), multiple linear regression (MLR), stepwise regression, Principal Component Regression (PCR), cross-correlation-based regression model, support vector regression (SVR), and the novel Bayesian Ensemble Bohai Ice Grade Net (BE-BIGNet), we established a hierarchy of model capability where each successive methodological advancement addresses the specific limitations of preceding approaches. The RMSEs of those prediction methods are compared in Figure 14. The advantages and limitations of those methods are summarized in

Table 2. Summary of advantages and limitations of seven BSIG prediction methods.

Method	Type	Advantages	Limitations
Analog Year Analysis (AYA)	Empirical/analog-based	Physically intuitive; effective for small samples; operationally established	Assumes stationarity; limited robustness under climate change; cannot represent nonlinear dynamics
Multiple Linear Regression (MLR)	Linear statistical	Simple and interpretable; computationally efficient	Sensitive to multicollinearity; restricted to linearity; prone to overfitting under nonstationarity
Stepwise Regression	Linear statistical (feature selection)	Reduces predictor redundancy; improves parsimony	Selection instability; sampling dependence; weak physical interpretability
Principal Component Regression (PCR)	Linear statistical (dimension reduction)	Mitigates multicollinearity; improves numerical stability	Loss of physical meaning; linear constraint; limited extreme-event representation
Cross-Correlation-Based Regression	Physically constrained statistical	Incorporates lagged climate signals; partially physically interpretable	Still linear; unstable lag relationships under changing climate conditions
Support Vector Regression (SVR)	Machine learning	Captures nonlinear relationships; robust for small samples; good generalization	Sensitive to hyperparameters; limited interpretability; underestimates extremes
BE-BIGNet	Bayesian ensemble learning	Strong nonlinear representation; reduces overfitting; robust for small datasets	High computational cost; structural complexity; limited interpretability

.

The substantial increase in cross-model RMSE differences between training and testing periods suggests that traditional statistical methods are limited by overfitting and may lack predictive skill for BSIG within a nonstationary climate system. The analog year analysis serves as a useful reference but relies on “stationarity” assumptions that are becoming invalid in the context of global warming. Similarly, while regression analyses establish quantitative baselines, they often struggle with multicollinearity among atmospheric predictors. Although PCR resolves the multicollinearity issue through orthogonal decomposition, it remains bound by linear assumptions, restricting its ability to resolve the complex nonlinear interactions characteristic of air–sea coupling.

These limitations were reflected by the performance evaluation. During the independent testing period (2020–2024), the RMSE of the analog year analysis was 0.59, and the standard multiple linear regression (MLR) model degraded to 0.74, reflecting a significant inability to adapt to the non-stationary shifts in the recent climate background. While optimized linear models such as stepwise regression and PCR achieved superficially low RMSEs (0.39 and 0.32, respectively) in the testing period, their historical training performances were notably poorer (RMSEs of 0.66 and 0.68); this lack of consistency across different periods renders their long-term skill unstable and less reliable for operational baselines. Furthermore, while the machine learning approach SVR avoided severe overfitting and maintained relative stability (testing RMSE of 0.46), it failed to outperform the optimized linear models during this specific period, largely because its loss function structure tends to be overly conservative regarding extreme anomalies. Among the evaluated methods, BE-BIGNet demonstrated good stability and accuracy across both the training and independent testing periods. Over the full 1979–2024 period, it achieved an RMSE of 0.44, outperforming all other baseline models evaluated in this study. By integrating the nonlinear fitting capabilities of a neural network with Bayesian regularization and ensemble averaging, this architecture is particularly well-suited to the small-sample constraints inherent in BSIG prediction. Ultimately, these results indicate that BE-BIGNet provides a highly reliable framework for operational BSIG forecasting and regional disaster mitigation.

While the BE-BIGNet framework exhibits robust predictive skill for the overall winter BSIG, it is important to acknowledge that sea ice severity exhibits significant spatial variability across different regions of the Bohai Sea. This study focuses exclusively on a macroscopic, basin-wide index because our primary objective is to support strategic, large-scale disaster preparedness. Fundamentally, the BSIG is dictated by the most severe ice event recorded anywhere within the basin. Consequently, predicting this single comprehensive grade aligns directly with the operational mandate of preparing for worst-case scenarios.