Operational Dst forecasting across solar cycle using physics-motivated gradient boosting

Abstract

A physics-motivated machine learning framework is developed for forecasting the hourly Dst index at lead times of 4–12 h. An expanded feature space (e.g., convolutional filters, rolling statistics, energy accumulation) is constructed from solar wind and interplanetary magnetic field measurements, going beyond usual lagged values. An XGBoost gradient boosting model is trained on solar cycles (SC) 20–24 and evaluated on the entirely independent SC 25 (August 2017 – November 2025, including the minimum phase between SC24 and SC25), comprising 73,056 hourly values. At a forecast horizon of 4 h, the model achieves MAE nT, RMSE nT, and . A threshold-based classification evaluation is also introduced as a complement to standard regression metrics, directly assessing storm detection performance at operationally relevant Dst thresholds. F1-scores of 0.75, 0.71, and 0.77 are obtained for moderate ( nT), intense ( nT), and severe ( nT) storm thresholds respectively. Interpretability analysis through SHAP values and Individual Conditional Expectation (ICE) plots reveal a context-dependent shift in dominant predictors: recent Dst history governs predictions under quiet conditions, while the solar wind electric field and convolution-derived IMF coherence features become dominant during active storm periods. The trained model is able to switch between relevant features, depending on the forecast horizon. This regime-dependent behavior is consistent with theoretical work on sustained solar wind energy transfer and emerges without explicit storm-phase conditioning, indicating that physics-motivated feature engineering guides the model towards physically meaningful predictions.

1 Introduction

Space weather describes the variable conditions driven by solar activity that affect the near-Earth environment. The consequences range from satellite orbital decay and communication disruption to power grid failures, as documented during the October 2003 storm that caused blackouts across the Swedish high-voltage transmission network (Pulkkinen et al., 2005) and during the February 2022 event that destroyed a fleet of Starlink satellites (Dang et al., 2022). At the other extreme, the same processes produce mid-latitude auroral displays, such as those widely observed following the May 2024 super-storm (Piersanti et al., 2025). Accurate forecasting of geomagnetic activity is therefore of direct operational relevance.

The Dst index (Disturbance Storm Time, Sugiura, 1964) is among the most widely used measures of geomagnetic storm intensity. It is derived from the horizontal magnetic component recorded at four low-latitude observatories, after removing the quiet-day variation, and is directly proportional to the kinetic energy of the ring current (Stern, 1984). Sustained southward interplanetary magnetic field (IMF) drives dayside magnetic reconnection (Dungey, 1961), injecting energy into the magnetosphere and intensifying the ring current; the resulting Dst depression defines the storm main phase (Gonzalez et al., 1994). Forecasting the Dst index is therefore a critical metric for evaluating the geoeffectiveness of solar transients and their subsequent risk to terrestrial infrastructure.

Machine learning methods have been increasingly applied to this problem over the past 2 decades. Early neural network approaches demonstrated the feasibility of multi-hour-ahead Dst prediction from solar wind inputs (Lazzús et al., 2017; Temerin and Li, 2002), and subsequent work explored ensemble methods (Xu et al., 2020), recurrent architectures (Gruet et al., 2018; Wintoft and Wik, 2021), and hybrid deep learning frameworks (Weng et al., 2024; Hu et al., 2023). More recently, gradient boosting methods have been applied with interpretability analysis (Ye et al., 2025). A summary of representative results across these approaches is provided in Table 1. This work will employ extreme gradient boosting (XGBoost, Chen and Guestrin, 2016) as the main learning algorithm. The selected method has already demonstrated strong performance even in other use cases, such as, for example, in predicting solar flare classes (Bringewald and Parisot, 2025), CME time arrival prediction (Shirke and Parihar, 2025), or as a benchmark model for solar wind predictions (Upendran et al., 2020).

Direct comparison across studies is complicated by significant methodological divergences. Test periods differ in duration, solar cycle phase, and relationship to the training data. Xu et al. (2020) report competitive metrics at –6 h, but their evaluation covers only 17 months and draws samples from storm intervals via random selection rather than temporal ordering, introducing a form of data leakage. Sierra-Porta et al. (2024) report over five solar cycles, but their test set is a randomly drawn 30% subset without enforcing temporal separation from training data. Zhang et al. (2023) achieve RMSE nT at hour on 7 days of data. These differences make direct numerical comparison unreliable, and motivate evaluating the present model under conditions that more closely reflect operational deployment.

This work develops a physics-motivated machine learning framework for forecasting the hourly Dst index at lead times of 4–12 h. An expanded feature space is constructed from solar wind and interplanetary magnetic field measurements, going beyond lagged values to incorporate convolutional filters, rolling statistics, and energy accumulation operators derived from established physical relationships. An XGBoost gradient boosting model is trained on solar cycles 20–24 and evaluated on the entirely independent SC 25 (August 2017 – November 2025), comprising 73,056 hourly values. A threshold-based classification evaluation complements standard regression metrics, directly assessing storm detection performance at operationally relevant Dst thresholds, a dimension largely absent from existing Dst forecasting literature.

The paper is organized as follows: Section 2 describes the dataset; Section 3 describes the modeling approach; Section 4 details the feature engineering; Section 5 presents the computational framework and evaluation methodology; Section 6 reports forecasting results; Section 7 presents the physical interpretation through feature importance analysis; Section 8 summarizes the conclusions.

2 Data sourcing

The dataset used in this study consists of hourly-resolved solar wind and magnetospheric measurements from NASA’s OMNIWEB interface (https://omniweb.gsfc.nasa.gov/form/dx1.html), covering November 1976 to November 2025 across SCs 21–25. OMNIWEB provides standardized observations from multiple L1 spacecraft missions (IMP-8, Wind, ACE, DSCOVR) located approximately 1.5 million kilometers upstream of Earth. Measurements include interplanetary magnetic field components (, , in GSM coordinates), magnetic field magnitude (scalar and field average), plasma speed, proton density, temperature, and derived parameters such as flow pressure, electric field, plasma beta and magnetosonic Mach. Magnetospheric response is quantified through the hourly Dst index, which, as already mentioned, measures equatorial ring current strength. Dst (Sugiura, 1964) serves as the primary geomagnetic storm indicator used in this study.

Hourly temporal resolution was chosen as the best time resolution available in the OMNI database. Specifically, substorm dynamics lasts between 2–4 h (Kepko et al., 2015), ring current development 6–12 h (Zhang et al., 2007), and circulation changes 12–24 h. Beyond prediction accuracy, hourly resolution preserves the physical granularity required to investigate how individual parameters and engineered features drive magnetospheric response. This is a prerequisite for the interpretability analysis presented in Section 7. The parameters are mapped into meaningful features as detailed in Section 4.

3 Methods

The present study aims to leverage a physics-motivated machine learning approach that integrates domain expertise with predictive modeling to forecast hourly Dst index values with a time lead of hours from the moment of measurement . The framework combines (1) physics-based feature engineering (Raissi et al., 2019), incorporating magnetospheric variables, solar wind parameters, and temporal dynamics; (2) gradient boosting models (Chen and Guestrin, 2016) to capture nonlinear relationships; and (3) interpretability analysis using SHapley Additive exPlanations (SHAP) (Lundberg and Lee, 2017) and Individual Conditional Expectation (ICE) plots (Goldstein et al., 2014) to identify feature importance and interaction patterns. This approach serves two purposes: confirming that the model recovers known magnetospheric relationships, and providing a data-driven method to identify feature interactions whose physical interpretation warrants further investigation. Furthermore, it also addresses interpretability concerns regarding black box models, offering a path for experts to validate the methodology (Rudin, 2019).

The problem is formally defined as follows. Let represent the collection of raw observational parameters at time , including solar wind plasma properties, interplanetary magnetic field components, and derived quantities such as dynamic pressure and electric field. The objective is to construct a physics-guided feature space where , incorporating variables ranging from simple time-lagged measurements to complex energy-based integrals representing magnetospheric state dynamics. This expanded feature representation enables systematic investigation of geomagnetic storm phenomena through controlled experimentation, where hypotheses can be tested by adding or removing physics-motivated predictors from a baseline model.

Let the target variable represent the Dst index value at the prediction horizon :

The modeling objective is to determine a nonlinear transformation dependent of parameters, that minimizes the prediction error between model estimates and observed Dst values under the norm, where

This optimization problem can be formulated as:where represents the number of training samples. The norm is employed rather than the conventional norm to ensure robust treatment of extreme storm events (Bektas and Sisman, 2010). While loss assumes Gaussian error distributions and heavily penalizes large deviations, loss corresponds to Laplacian error distributions and treats all prediction errors uniformly. This choice prevents the model from overfitting to rare but intense geomagnetic storms, promoting generalization across different phases of the solar cycle where storm intensity distributions vary significantly.

In terms of statistical modelling, Extreme Gradient Boosting (XGBoost) algorithm will be used as the underlying learning algorithm, leveraging its well-established performance on structured data. This is a gradient-based ensemble method (Chen and Guestrin, 2016) that constructs predictions through a sequential ensemble of decision trees (Quinlan, 1986).

While Random Forests aggregate independent decision trees built in parallel, boosting algorithms such as XGBoost construct trees sequentially, where each new model is trained to correct the residual errors of the current ensemble. This iterative refinement progressively reduces the bias of weak learners, which is the defining principle behind the gradient boosting framework. Each successive tree is trained to minimize the residual errors of the current ensemble, with the final prediction given by:where represents the total number of trees in the ensemble, is the learning rate, and is the -th tree’s contribution to the ensemble. The learning procedure is controlled by several key hyperparameters: tree depth determines the complexity of individual learners, regularization parameters ( for L2 and for L1 penalties) prevent overfitting, and subsampling ratios for features and observations control variance. The algorithm supports interaction constraints between predictors, enabling explicit grouping of physical domains (e.g., magnetic field, plasma properties, electromagnetic coupling) to prevent spurious correlations. Beyond native feature importance metrics such as gain, the tree-based ensemble structure enables efficient computation of SHAP values for feature importance analysis and Individual Conditional Expectation (ICE) plots for investigating feature interactions.

In order to aid the gradient descent procedure training data is usually transformed to achieve specific distributional properties: approximate normality, bounded variance (homoscedasticity), symmetry, consistent scale across features, and absence of extreme outliers that can cause gradient instability. To this end, component-wise transformations such as Signed-Logarithmic and Sinh-Arcsinh, are independently applied to to achieve these desirable statistical properties (LeCun et al., 2012).

Signed Logarithmic Transform is applied in cases where raw data analysis can be misleading when parameter characteristics are not apparent at the original scale. For example, SW Plasma Temperature values span four orders of magnitude ( to ), while magnetic field parameters contain both positive and negative measurements with physical significance. The signed logarithmic transformation addresses both challenges by preserving sign information while revealing patterns in log-space. It is defined as follows:

Sinh-Arcsinh transformation (Jones and Pewsey, 2009) provides independent control over skewness and kurtosis, compared to Yeo-Johnson transformation (Yeo and Johnson, 2000) that optimizes only one parameter for both properties. For example, plasma temperature after signed logarithmic transformation may achieve near-zero skewness but retain heavy tails, requiring separate tail weight adjustment without affecting the achieved symmetry (as seen in Figure 1). Sinh-Arcsinh transformation is defined as follows:where controls tail weight (kurtosis) independently of , which adjusts asymmetry (skewness).

The full transformation pipeline applied to is as follows. , , scalar , and SW proton density receive only signed logarithmic transformation. SW plasma speed and electric field are transformed with Sinh-Arcsinh alone. Alpha/proton ratio, SW plasma temperature, plasma , and flow pressure each require both steps: signed log followed by Sinh-Arcsinh, with plasma temperature using . Parameters not listed are retained in their original form.

4 Feature engineering

The construction of the feature space addresses a fundamental challenge in applying machine learning to magnetospheric physics: while non-linear models like XGBoost can theoretically learn complex functional relationships from raw data, the learning process may follow statistically convenient but physically meaningless paths. Without proper guidance, gradient boosting algorithms may exploit spurious correlations, ignore physically relevant predictors, or learn unstable patterns that fail to generalize across different solar cycle conditions.

To constrain the learning process toward physically meaningful solutions, is formulated using domain expertise, encoding established magnetospheric theory directly into the model inputs. This approach provides three relevant advantages: (1) it offers physically meaningful starting points that guide the optimization toward interpretable solutions; (2) it incorporates established physical relationships as explicit features rather than requiring the model to rediscover fundamental physics; and (3) it enables investigation of complex coupling mechanisms through controlled feature experimentation.

The following subsections will detail how initial measurements are mapped into an expanded representation through techniques such as temporal embedding operators, rolling statistics, energy accumulation operators or state characterizations.

4.1 Lagged measurements

Lagged measurements were used in order to incorporate additional past context, regarding solar wind and magnetospheric processes. This would allow the model to observe changes in different features across a predefined window of time (for instance the change in solar wind speed in a 12 h window). Past Dst index values are also incorporated as lagged features from 0 to 4 h. This is distinct from an autoregressive model as the objective is not to extrapolate Dst index through its own past trajectory, but to use recent index values as a compact proxy for the current magnetospheric state, complementing the solar wind and derived features rather than replacing them. The lag transformation creates delayed versions of each parameter:where represents the lag time in hours, selected based on known physical response timescales. Magnetic field parameters utilize shorter lags to capture rapid field reconfiguration, while plasma parameters employ longer lags reflecting slower thermal and density evolution processes.

4.2 Rolling statistical operators

Rolling statistical operators quantify temporal evolution patterns across input features . At each time step , a rolling window of size collects the most recent observations:where indexes time within the window. Three quantities characterize the empirical distribution of each window.

Standard deviation measures dispersion, where elevated variance tends to precede storm onset while quiet intervals exhibit low variability:where denotes the sample mean over the window.

To avoid redundancy, time windows can be computed with a step size larger than 1, leading to more abrupt changes. The median tracks shifts in central tendency across stepped windows of size and step , where is the overlap fraction:where denotes the empirical probability over window .

The Wasserstein distance measures the distributional difference between consecutive stepped windows, representing the minimum transport cost to transform one distribution into the other:where is the set of all joint distributions with marginals and corresponding to consecutive windows.

Standard deviation is computed on rolling windows to ensure a value at every hour with a 12 h window size for magnetic measurements and a 24 h window for plasma. Median and Wasserstein distance are computed on stepped windows, which require sufficient temporal separation to detect meaningful distributional shifts. Since stepped windows produce one value per step rather than per hour, linear interpolation restores hourly resolution:where indexes consecutive window endpoints and denotes either or .

4.3 Takens’ embedding operators

Simple statistical summaries of a time series do not capture its dynamical structure: two signals with identical mean and variance can exhibit qualitatively different trajectories. Takens’ embedding theorem (Takens, 1981) addresses this by showing that the phase space of an unknown dynamical system can be reconstructed from a scalar time series alone. For a parameter , the -dimensional embedding with time delay constructs the phase space vector:where is the embedding dimension and is the time delay in hours. According to the theorem, this reconstruction preserves the topological properties of the underlying dynamical system, provided is sufficiently large.

To extract geometric features from the reconstructed trajectory, consecutive coordinate pairs , are formed from each embedding vector:where denotes the -th component of . The sequence traces a trajectory in the 2D coordinate plane. Storm periods and quiet periods produce qualitatively different trajectories: the former tend to follow smooth, directed paths while the latter exhibit irregular, space-filling patterns. Figure 2 illustrates this contrast for solar wind speed.

FIGURE 1

FIGURE 2

Three geometric invariants summarize each trajectory. Jump magnitude , measures the largest single-step displacement:

Path efficiency , compares the straight-line displacement between trajectory endpoints to the total path length , with low values indicating chaotic or looping evolution:

Turning angle variance , measures trajectory smoothness through the spread of angular deviations between consecutive segments :

These invariants are scale-independent: they capture dynamical transitions encoded in trajectory geometry rather than in absolute parameter values. For SW Plasma Temperature and Plasma there are 6 Embedding Dimension with time delay of 3 h and 2 h, respectively. SW Proton Density, Flow Pressure, Alpha/Proton Ratio have six embedding dimensions and 2 h time delay. For SW Plasma Speed there are five embedding dimensions with 1 h time delay. , B scalar and the electric field have 4 embedding dimensions and 1 h time delay.

4.4 Convolution operators

Convolution operations are fundamental in signal processing and Fourier analysis, traditionally employed for noise reduction, impulse response characterization, frequency domain filtering, and PDE solutions. In the context of time series analysis, convolution enables pattern detection through template matching, where specific kernel functions emphasize temporal signatures of interest while suppressing irrelevant variations (Turin, 1960). For continuous one dimensional signals, convolution is defined as the integral operation:where represents the input signal and the convolution kernel.

In discrete time series applications, this operation reduces to the finite summation:where denotes the discrete time series, represents the kernel of length , and indicates the time index.

In image processing, convolution can be used to detect edges, sharpen or blur an image. Space weather observations constitute discrete temporal measurements that can be interpreted as sampled signals containing embedded physical signatures. Analogous to image processing applications, appropriate kernel selection can reveal specific patterns within the observed signal evolution. Rather than learning kernel shapes from data, a set of kernels is proposed whose profiles correspond to known physical signatures: rapid southward IMF transitions associated with magnetic reconnection, compression events driven by CME sheaths or stream interaction regions, and gradual parameter enhancements characteristic of co-rotating interaction region development (Gonzalez et al., 1994; Borovsky and Steinberg, 2006).

Each kernel encodes domain knowledge as an explicit signal template, reducing the burden on the gradient boosting model to rediscover temporal patterns from scratch. Figure 3 illustrates the storm onset kernel applied to scaled measurements. The convolved signal responds strongly to the sustained southward turning near hour 20, preceding the storm interval (shaded) by approximately 20 h. Within the storm interval itself, the response is suppressed, consistent with the kernel’s sensitivity to transitions rather than sustained negative values. The complete set of kernels are listed in the following equations, while the physical meaning is described in: Table 2.

FIGURE 3

4.5 Energy loading state

Magnetospheric energy balance is influenced by both solar wind energy input and ring current dissipation processes. The Akasofu epsilon parameter quantifies the solar wind power available for magnetospheric coupling (Akasofu, 1981), while ring current decay follows exponential timescales of 7–10 h (Dessler and Parker, 1959; Hamilton et al., 1988). Wang et al. (2014) has computed an improved Akasofu (1981) formula using 3D magnetohydrodynamics simulations. The net energy balance provides a direct measure of magnetospheric “stress” that determines the system’s readiness to develop storms under continued driving. Thus, the energy injection rate is (Wang et al., 2014):where is the solar wind number density ] is the solar wind velocity [km/s], , is the IMF clock angle, , , and being IMF components [nT], and the scaling factor converts to appropriate units.

The energy loading state is computed as an exponentially-weighted accumulation of the injection rate:where with hour controls the decay rate. This formulation represents a first-order approximation to the continuous energy balance: recent injection contributes strongly while older energy contributions fade exponentially.

4.6 Magnetotail loading state

Magnetotail magnetic flux accumulation controls substorm onset timing and intensity through the loading-unloading cycle (McPherron, 1970; Baker et al., 1996). Dayside magnetic reconnection transfers magnetic flux to the magnetotail lobes, where it accumulates until substorm onset releases the stored energy. The rate of flux transfer depends on the IMF orientation and solar wind mass flux, with efficient loading occurring during sustained southward or weakly northward IMF conditions.

The magnetotail flux loading rate captures the combined effects of IMF orientation and mass transport:where is the reconnection-favorable IMF component (extended to weakly northward conditions up to +3 nT), is the solar wind velocity [km/s], is the proton density [N/cm³], and the scaling factor converts to appropriate units. Multiple reconnection thresholds (0, +3, and +5 nT) and integration timescales (2 and 4 h) were tested to capture varying substorm loading conditions.

The magnetotail loading state accumulates flux transfer with exponential memory:where with hours controls the accumulation rate for substorm-scale loading, and hour is the temporal resolution. This formulation provides exponential weighting of flux accumulation over the typical substorm growth phase duration, with the magnetotail loading state providing a solar wind-based proxy for processes reflected in the AE auroral electrojet index.

4.7 Coupling efficiency state

Magnetospheric response to solar wind driving varies with the magnetosphere’s recent activity history (Newell et al., 2007; Borovsky, 2013). A recently-disturbed magnetosphere may exhibit altered response to additional solar wind forcing compared to a quiet magnetosphere under similar driving conditions, representing saturation or preconditioning effects. The coupling efficiency quantifies the ratio of past magnetospheric activity to current solar wind energy input:where is the AE auroral electrojet index, is the solar wind energy input rate (Equation 9), and provides numerical stability. Temporal lags are applied as defined in Table 3. This ratio quantifies whether past magnetospheric activity was large relative to current driving, indicating potential saturation or memory effects that modify the magnetosphere’s response capacity.

4.8 Recovery state

A recently disturbed magnetosphere exhibits reduced capacity to respond to new perturbations compared to a well-recovered system (Gonzalez et al., 1994). Recovery state approximates this capacity based on elapsed time since the last storm occurred. Using signal processing techniques (Du et al., 2006), storm periods are identified through peak detection on inverted Dst time series. The algorithm requires a peak threshold (, , or nT depending on storm intensity), minimum prominence (10–20 nT), and minimum duration (1–2 h) to distinguish sustained disturbances from transient threshold crossings. The peak detection returns storm end times and corresponding storm durations for each identified event.

The recovery state takes into account a time factor , and recovered Dst :

The temporal component keeps track of how much time passed since last identified event:where is the time when the most recent storm ended, is the lookback period and the 24-h normalization represents the characteristic ring current decay timescale (Dessler and Parker, 1959; Hamilton et al., 1988).

The recovery rate quantifies active Dst restoration over the most recent 2-h period:where is the Dst index in nT and time steps are 1 h. Only positive changes contribute, filtering out continuing disturbances. The additive constant in Equation 11 preserves time-based recovery during stable Dst periods.

This formulation produces low recovery values immediately after storm periods, with the elapsed time naturally accounting for storm duration through the identified storm end time. Multiple timescales capture different recovery processes. The short-scale variant uses an 8-h lookback with a nT threshold, targeting substorm recovery. The medium-scale uses 24 h and nT, aligned with ring current recovery. The long-scale extends to 48 h at nT, capturing global circulation recovery.

Individual components appear in isolation across the literature: lagged solar wind inputs are standard, and coupling functions such as the Akasofu epsilon parameter are well established. The systematic construction of an expanded, physics-motivated feature space from first principles, designed to support both forecasting and interpretability analysis, represents a contribution of this work.

5 Computational framework

Building on the feature definitions outlined above, this section describes the computational methodology (which is publicly available at https://github.com/ionutcatalinsandu/operational-dst-forecasting). The main objective is to achieve accurate statistical predictions while preserving physical consistency throughout the learning pipeline. Rather than treating Dst forecasting as a purely data-driven regression task, the experimental design prioritizes operational realism and physical understanding. The model incorporates domain-informed features and is evaluated on hourly measurements partitioned by solar cycles. However, average error metrics alone do not capture operational utility. For instance, a model may achieve low RMSE while failing to detect an approaching storm. To address this, storm event detection is introduced as a complementary evaluation perspective, assessing the model’s ability to correctly identify operationally significant thresholds, such as Dst nT.

Experiments are conducted using 1 hour resolution data spanning 49 years of solar wind and magnetospheric observations. Available data was split based on solar cycles (SC), training set containing SCs 20, 21, 23 and 24. For validation SC 22 was selected, and finally testing was conducted on the on-going cycle 25. In this work, solar cycle is considered to start at the same time as the minimum phase between the two consecutive cycles, not the SC start listed in the SILSO database (https://www.sidc.be/SILSO/home). This distribution ensures exposure to diverse solar activity levels while maintaining strict temporal separation between data splits. The training set spans November 1976 through July 2017, yielding 268,056 hourly samples. Validation covers July 1985 through August 1995, corresponding to SC 22 and contributing 89,136 samples. The test set begins in August 2017 and extends through November 2025, providing 73,056 samples drawn entirely from SC 25.

The assignment of solar cycles to each split was deliberate. Splitting by complete solar cycles rather than arbitrary time windows ensures that each subset would capture the full dynamical character, including the distinct phase activity during a cycle’s progression. The training set was composed of as many and as varied cycles as possible to expose the model to the broadest range of solar conditions. SC 22 was selected for validation on account of its activity level being broadly comparable to SC 21, while SC 25 was reserved for testing given its relevance to near-future geomagnetic prediction, which was the primary operational motivation of this work.

The current split distinguishes the present study from the majority of works surveyed in Table 1, which typically employ random or ratio-based temporal splits. The closest precedent is Sierra-Porta et al. (2024), whose dataset spans the same solar cycles. However, their train, test division follows a fixed 70:30 ratio rather than cycle boundaries.

Source data from older solar cycles frequently contain missing values. Missing observations are addressed through regression-based imputation, where each missing value is estimated from the remaining parameters present in that row. Imputation is applied only when at least 40% of measurements are available for a given sample, ensuring the regression has sufficient information to produce reliable estimates. Features are subsequently computed on this imputed dataset. Samples where more than 50% of values remain missing after imputation are excluded from the analysis. The estimator used for imputation is XGBoost (Ye et al., 2025), as it is regarded as one of the best models for structured data. For this purpose, the default configuration is used.

Tree-based methods such as XGBoost (Chen and Guestrin, 2016) utilize an extended parameter set beyond the model weights which are optimized during training. These additional parameters, commonly referred to as hyperparameters, are fixed during optimization, but govern model characteristics such as tree complexity and learning rate, significantly influencing generalization. For instance, an excessive learning rate may overshoot optimal solutions, while an insufficient one risks convergence to local minima. Selecting appropriate hyperparameters typically requires numerous training runs, and while experienced practitioners may develop intuition for good candidates, the search can be formalized as a mathematical optimization problem.

Hyperparameter optimization is conducted using Optuna (Akiba et al., 2019), which employs the Tree-structured Parzen Estimator (TPE) algorithm. TPE separately models the probability distributions of hyperparameter configurations associated with good and poor performance using kernel density estimation, selecting new candidates by maximizing expected improvement. The training procedure follows gradient boosting with second-order gradients, and the complete configuration is detailed in Table 4. An extra hyperparameter has been added to alleviate class imbalance effects, aiming to increase the weight of interesting samples while decreasing the weight of common positive samples.

6 Results

The primary objective of this method is Dst index forecasting, producing hourly predictions at a lead time of hours. While the model was initially trained for hours, results remain robust up to hours, as reflected in Table 1. Throughout the paper, hours will denote a model trained to predict at that given horizon. Performance is quantified using standard regression metrics: Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), Coefficient of Determination , and Pearson Correlation Coefficient . The model is evaluated on Solar Cycle 25, covering August 2017 to November 2025, a period entirely excluded from training. Testing on an Full Solar Cycle provides a direct assessment of generalization to unseen geomagnetic activity regimes.

Across all forecast horizons, the proposed method yields MAE below 8 nT and above 0.50, even at hours. At hours, RMSE reaches 8.36 nT and , competitive with Laperre et al. (2020) (RMSE = 8.17 nT, ) and Lazzús et al. (2017) (RMSE = 8.82 nT, ), while being evaluated on a substantially larger and temporally independent test set (73,056 samples from Solar Cycle 25, versus 33,120 and 58,440 samples, respectively). Direct numerical comparison with all listed approaches is complicated by significant methodological heterogeneity: studies differ in test period, temporal resolution, sample size, and forecast horizon.

Several entries are further discussed. Xu et al. (2020) report competitive metrics at –6 h (RMSE = 7.03–8.09 nT), however their evaluation covers only 17 months and is restricted exclusively to storm intervals, with samples drawn via random selection rather than temporal (see Table 1). This challenges temporal causality and introduces a form of data leakage, likely influencing reported performance. Sierra-Porta et al. (2024) report unusually high and values, though the evaluation is conducted on a randomly drawn 30% subset spanning over 5 decades, without enforcing temporal separation between training and test data. Zhang et al. (2023) achieve RMSE = 3.20 nT at hour, evaluated on only 7 days.

Performance of the proposed method remains stable across data splits, with values of 0.84, 0.80, and 0.79 for training, validation, and test sets respectively (Table 5), suggesting consistent generalization across solar cycles with distinct activity profiles. The full test period forecast is displayed in Figure 4, with a focused 6-month window in Figure 5.

FIGURE 4

FIGURE 5

To complement standard regression metrics, a linear fit between predicted and observed Dst values provides a more direct characterization of estimation bias. The slope and intercept of this fit quantify the degree to which the model under or overestimates across the dynamic range of the index. Figure 6 shows the results for both the full test (Figure 6a) period and a focused (Figure 6b) 6-month window.

FIGURE 6

The fitted slope of approximately 0.80 over the full test period indicates a underestimation of extreme storm intensities, becoming more pronounced below nT. This pattern is consistent with the relative scarcity of intense storm events in the training data, a known challenge in geomagnetic storm prediction. The focused period yields a similar slope of 0.77, confirming that the bias is not an artifact of the evaluation window but a structural characteristic of the model. For typical magnetospheric activity, predictions remain well-calibrated, with the regression line closely following the perfect prediction diagonal in the to 0 nT range.

6.1 Threshold-based classification

From an operational standpoint, predicting the onset and severity of geomagnetic storms is of paramount importance. Standard regression metrics evaluate prediction accuracy across the full dynamic range of the Dst index, where quiet-time samples ( nT) constitute the majority of observations. A model that accurately reproduces quiet-time values while failing to detect approaching storms can still achieve low RMSE. This is a limitation well documented in forecasting evaluation literature (Wilks, 2011), but not addressed in Dst prediction studies. To assess operational utility directly, a threshold-crossing evaluation is introduced: Dst nT for moderate storms, Dst nT for intense storms, and Dst nT for severe storms. For a given threshold , the problem is cast as binary classification: any hour where Dst is labeled as an active event (1), while hours above are labeled as quiet (0). This formulation allows the proposed regression model to function as an event detection mechanism, though it does not resolve storm phases such as onset, main phase, or recovery (a distinction that could be considered in future). Applying this labeling to the test set, transforms the regression outputs and inputs into sequences of binary values. This allows evaluation via standard classification metrics. Classification performance is quantified using the confusion matrix framework (Wilks, 2011), built from True Positive (TP), True Negative (TN), False Positive (FP), and False Negative (FN) counts. Three derived metrics are reported. Precision measures the fraction of predicted storm events that correspond to actual storms:

Recall measures the fraction of actual storm events that are successfully detected:

The two metrics capture complementary aspects of model behavior: precision penalizes false alarms, while recall penalizes missed events. The F1-score combines both as their harmonic mean, ranging from 0 to 1:

These values are computed for the test period (Aug. 2017 – Nov. 2025) and are displayed in Table 6.

The classification results reflect patterns observed in the regression analysis. Precision values exceed 0.76 across all thresholds, indicating that predicted storm events correspond to actual storms with high reliability. Recall values range from 0.65 to 0.74, meaning a non-negligible fraction of actual storms remain undetected. This conservative behavior (favoring fewer false alarms) is consistent with the underestimation of extreme Dst values identified in the scatter analysis (Figure 6). Notably, the severe storm threshold (Dst nT) achieves the highest precision (0.87) and F1-score (0.77) despite having only 100 events, suggesting that the most intense storms produce sufficiently distinct solar wind signatures to be reliably identified. The moderate threshold (Dst nT) shows the most balanced precision-recall trade-off, with an F1-score of 0.75 across 2,074 events.

7 Physical interpretation through feature importance

To gain insight into the predictive behavior of the trained XGBoost model, an interpretability analysis is conducted. While XGBoost achieves strong predictive performance, its ensemble nature make it difficult to directly interpret feature importance. To address this, two complementary techniques are employed: SHAP and ICE plots. Together, these methods offer both a global and a local perspective on model behavior.

7.1 SHapley additive exPlanations (SHAP)

For tree-based models, SHapley Additive exPlanations (SHAP) values (Lundberg and Lee, 2017) provide a principled method for decomposing individual predictions into feature contributions. For any prediction , SHAP values satisfy the additive decomposition:where is the expected model output over the training data and quantifies how feature shifts the prediction away from this baseline. For tree ensembles, SHAP values are computed exactly using the TreeSHAP algorithm (Lundberg et al., 2020), which exploits the decision tree structure to efficiently track feature contributions along each decision path. Global feature importance is then summarised by the mean absolute SHAP value across all samples:where is the SHAP value for feature in sample . This ranking reflects the average magnitude of each feature’s contribution regardless of direction.

For this study, the default model with prediction horizon of h will be used. Figure 7a displays the top 15 features ranked by this criterion. The SHAP analysis reveals a clear dominance of recent Dst history, with the current and lagged values (up to 4 h) occupying the top positions in the evaluation set. This is physically consistent: the Dst index evolves continuously, and its recent trajectory encodes the accumulated magnetospheric energy state more compactly than any single solar wind parameter. Among solar wind inputs, the electric field (E) and the component in GSM coordinates emerge as the most influential drivers, confirming the established role of southward IMF in ring current energization (Gonzalez et al., 1994).

FIGURE 7

A notable shift occurs when restricting the analysis to storm-time samples (Dst nT, Figure 7b). The electric field (E) rises above several Dst lag features, and conv_storm_onset_short_BZ_nT_GSM (subsection 4.4; Equation 3) enters the top rankings, a feature which was absent from the full-test list. This suggests that during active storm conditions, the temporal coherence of southward IMF driving becomes more predictive than the instantaneous magnetospheric state alone, supporting theoretical work on efficient energy transfer during sustained southward IMF intervals (Borovsky and Funsten, 2003). The magnetotail loading state and flow pressure features appear consistently across both subsets, reflecting their role in characterizing the background magnetospheric preconditioning that modulates coupling efficiency (Borovsky, 2021). This context-dependent feature ranking indicates that the model implicitly distinguishes between quiet and storm-time regimes without explicit conditioning.

The interaction matrix in Figure 8 quantifies pairwise feature dependencies as mean absolute SHAP interaction values. The strongest interaction occurs between the current and 1-h lagged Dst (0.765), consistent with the high temporal autocorrelation of the index. Interactions between Dst current and its successive lags decrease monotonically from 0.496 at 2 h to 0.282 at 4 h, reflecting the gradual decay of magnetospheric memory over the prediction window. The E electric field shows its strongest off-diagonal interaction with in GSM coordinates (0.339) and a secondary interaction with Scalar B (0.201): since the dawn-to-dusk electric field is proportional to the product of solar wind velocity and southward IMF, the model independently rediscovers this physical coupling without it being explicitly encoded as a single feature (Gonzalez et al., 1994). The convolution-based compression feature interacts most strongly with Dst current (0.209) among solar wind-derived inputs, suggesting that magnetospheric compression events modulate the current state rather than acting independently. Flow pressure exhibits uniformly weak interactions across all features (all ), indicating its contribution to Dst prediction is largely additive.

FIGURE 8

7.2 Individual Conditional Expectation analysis

While SHAP values quantify the average contribution of each feature across the dataset, they do not reveal how individual predictions respond to feature variation. Individual Conditional Expectation (ICE) plots (Goldstein et al., 2014) address this by tracing the predicted output for a single sample as one feature varies across its observed range, holding all other features fixed. This sample-level resolution is particularly relevant for regime-dependent phenomena such as magnetospheric dynamics, where the same solar wind parameter may produce qualitatively different responses depending on the preconditioning state.

Formally, for a trained model , feature , and a grid sampled from the empirical distribution of , the ICE function for sample is:

The complete ICE curve for sample is the collection , and repeating this across all samples exposes heterogeneity in feature-response relationships that averaged metrics such as SHAP would suppress. Thus, ICE curves provide a complementary view of how individual features influence predictions. For this analysis, ICE curves are computed on scaled features (Equations 1, 2), restricted to storm-period samples (Dst nT) to focus on magnetospheric responses under active conditions. Three features are examined: the electric field (E), the magnetotail loading state (Equation 10), and Scalar .

The electric field (Figure 9a) shows a qualitatively different response. The mean ICE is approximately flat for strongly negative values (scaled ), suggesting that under sustained strong driving the current magnetospheric state, already captured by Dst lags, absorbs most of the predictive signal. The mean steepens by roughly 15 nT as the field approaches zero, where the transition from active injection to recovery produces a detectable change in prediction slope. The narrowing of the IQR toward positive values is consistent with fewer storm-time samples reaching this range.

FIGURE 9

The magnetotail loading state (Figure 9b) exhibits a near-linear mean decline of approximately 10 nT across its full scaled range. The IQR spans roughly 120 nT throughout and remains nearly parallel across the full range, indicating that this feature contributes a small, consistent negative offset to predictions while the variance is dominated by other features encoding the current magnetospheric state.

The most pronounced nonlinearity appears in Scalar (Figure 9c). The mean response is flat across the bulk of the scaled range (0.52–0.64), then drops abruptly by approximately 30 nT beyond a threshold near 0.66. Individual ICE curves fan out sharply in this region, with some trajectories reaching below nT, indicating that extreme total field values amplify storm intensity in a strongly state-dependent manner. This is consistent with CME-driven magnetic field enhancements that precede or accompany the most intense geomagnetic storms (Gonzalez et al., 1994).

Across all three features, the wide spread of individual ICE curves, which do not collapse onto the mean, reflects genuine sample heterogeneity: identical feature values produce substantially different predictions depending on the broader magnetospheric state. Storm intensity emerges from the joint configuration of multiple drivers, rather than from any single parameter in isolation.

8 Conclusion

A physics-motivated XGBoost framework for hourly Dst forecasting has been developed and evaluated across an independent solar cycle. The main methodological contribution of this work is the construction of an expanded feature space that encodes magnetospheric domain knowledge directly into model inputs: temporal lags of the Dst index as a compact proxy for the current ring current state, energy accumulation integrals derived from the Wang et al. (2014) coupling function, matched-filter convolution kernels targeting specific solar wind signatures, and preconditioning indices capturing magnetotail loading and recovery history.

The proposed model was trained on Solar Cycles 20, 21, 23, and 24, comprising 268,056 samples, achieving MAE nT and at hours (see Table 5). Validation is performed on Solar Cycle 22, an independent cycle entirely withheld from training, yielding MAE nT and across 89,136 samples. The modest degradation in performance between training and validation is indicative of a well-calibrated model, with no significant signs of overfitting.

The valuation was conducted on August 2017 to November 2025 (see Table 1), a period entirely excluded from training, the model achieves MAE nT and at hours. These results are competitive with existing approaches while being obtained under substantially more demanding evaluation conditions: 73,056 hourly values from an independent solar cycle, without temporal overlap between training and test data. The forecast capabilities remain reliable even at longer time horizons (4, 8 or 12 h), revealing the ability of the model to rely on different features for predictions.

A threshold-based classification framework (see Table 6) is introduced alongside standard regression metrics to directly measure storm detection capability, a dimension of operational utility not addressed in most existing Dst forecasting studies. F1-scores of 0.75, 0.71, and 0.77 are obtained for moderate, intense, and severe storm thresholds respectively, with precision exceeding 0.76 across all levels. The severe storm class ( nT) achieves the highest precision (0.87) despite containing only 100 events, suggesting that the most intense storms produce sufficiently distinct solar wind signatures to be reliably identified.

Interpretability analysis using SHAP values and ICE plots confirms that the model recovers established magnetospheric coupling relationships. Recent Dst history dominates predictions under quiet conditions, reflecting the inertial character of the ring current. During storm periods, the solar wind electric field and convolution-derived IMF coherence features rise in importance, consistent with theoretical work on efficient energy transfer during sustained southward IMF intervals (Borovsky and Funsten, 2003). This context-dependent feature weighting emerges without explicit storm-phase conditioning, indicating that the physics-motivated feature space provides sufficient structure for the model to implicitly distinguish between magnetospheric regimes.

Several directions remain open for future work. The underestimation of predicted Dst values reflects a fundamental challenge in rare-event forecasting: the most consequential storms are the least represented in any training set. The present framework identifies storm-time intervals through threshold crossing, but does not resolve storm phases. Distinguishing onset, main phase, and recovery phase constitutes a natural extension towards a fully operational storm characterization. More broadly, the physics-motivated feature space developed here provides a controlled experimental environment: individual features can be added, removed, or reparameterized to test hypotheses about magnetospheric coupling directly within the model.

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