Sequence-based undersampling: an algorithm for managing imbalanced datasets

Imbalanced classification: classical and recent approaches

The issue of imbalanced data has been extensively studied in classification tasks (Johnson & Khoshgoftaar, 2019; Zhang et al., 2023), where the target variable is discrete and the class distribution is highly skewed. Foundational studies such as Chawla, Japkowicz & Kotcz (2004), Chawla (2005), He & Garcia (2009) provided systematic analyses of how data imbalance negatively affects learning algorithms, particularly in the detection of minority classes, and surveyed the early landscape of solutions. In response, various strategies have been proposed to improve minority class representation during training, broadly divided into: data-level methods, algorithm-level methods, and hybrid or integrated approaches.

The challenge of imbalance in regression

Imbalanced regression refers to predictive tasks in which the distribution of the continuous target variable is skewed, resulting in under-representation of certain value regions, typically extremes or areas of high importance. Unlike classification, where imbalance is defined over discrete classes, regression requires additional mechanisms to define what constitutes a “rare” or “relevant” value (Krawczyk, 2016).

To address this challenge, the most widely adopted approach relies on relevance functions—continuous mappings that assign an importance score to each target value based on domain knowledge or statistical heuristics (Branco, Torgo & Ribeiro, 2017). These functions $\varphi (y)$ are typically normalized to the [0, 1] range and allow the identification of critical target regions without discretizing the continuous space. Higher $\varphi (y)$ values correspond to rare or high-importance regions, while lower values are associated with common, low-impact areas. Relevance functions are used not only to guide resampling decisions but also to evaluate models through relevance-aware performance metrics (Avelino, Cavalcanti & Cruz, 2024), enabling a nuanced handling of imbalance in continuous prediction tasks.

Data-level approaches for imbalanced regression

Resampling methods that operate directly at the data level are commonly used to mitigate imbalance in regression tasks. These techniques aim to adjust the training distribution by adding or removing instances, without modifying the learning algorithm itself.

Oversampling methods include Random Oversampling (He & Garcia, 2009), Gaussian Noise Injection (Menardi & Torelli, 2014), SMOTE for Regression (Torgo et al., 2013), Synthetic Minority Over-Sampling Technique for Regression with Gaussian Noise (SMOGN) (Branco, Torgo & Ribeiro, 2017), and k-Nearest Neighbors Oversampling for Regression (KNNOR-Reg) (Belhaouari et al., 2025), synthesize new samples in regions deemed relevant, based on interpolation, noise perturbation, or neighborhood-based weighting. A recent focused review by Belhaouari et al. (2024) discusses several oversampling strategies for imbalanced regression and highlights the broad algorithmic space available for the generation of synthetic instances.

In contrast, undersampling methods focus on removing instances from overrepresented regions. Their design space is inherently narrower, as the task reduces to identifying and eliminating redundant or low-relevance values. Most techniques follow one of a few paradigms: random elimination, neighborhood disagreement, or rule-based filtering. Classical examples include Random Undersampling (He & Garcia, 2009), ENN, Tomek Links, and CNN, while more recent efforts like Selective Under-Sampling (SUS) (Aleksic & García-Remesal, 2025) represent attempts to formalize instance selection heuristics. Compared to oversampling, there have been fewer methodological developments in pure undersampling for regression, largely due to the conceptual simplicity of the problem. In addition to these strategies, hybrid approaches that combine both oversampling and undersampling have also been proposed. One notable example is the Weighted Relevance-based Combination Strategy (WERCS) (Branco, Torgo & Ribeiro, 2019), which adjusts the instance distribution by probabilistically duplicating highly relevant observations while removing less relevant ones, based on continuous relevance scores.

Table 1 provides a summary of representative data-level resampling techniques for regression, categorized by type and approach.

Sequence-aware resampling strategies

Most resampling techniques, whether for classification or regression, assume that instances are independent and identically distributed, regardless of the order in which they were recorded. This limits their applicability in datasets where structural or sequential patterns are intrinsic to the data distribution.

An important precedent is the work by Moniz, Branco & Torgo (2017), which explores adaptations of oversampling strategies for time series forecasting. Their methods incorporate temporal constraints to avoid disrupting the autocorrelation structure of sequential data. However, their focus lies in predictive modeling of time-dependent targets, using synthetic data generation, and does not address the problem of instance elimination within structurally redundant patterns.

In contrast, this work addresses a different scenario commonly found in environmental data, such as precipitation estimates: the overrepresentation of a single value (zero) occurring in long, homogeneous sequences. While the dataset includes timestamp information, the structure exploited by the proposed method does not rely on temporal features or patterns, but rather on the simple order in which the values were recorded. Any consistent indexing that preserves the original measurement order is sufficient. These repeated sequences often contain limited informational gain when observed exhaustively, making them suitable candidates for structured reduction.

Existing undersampling methods remove overrepresented values based on statistical or geometric heuristics, treating each instance in isolation. The Sequence-Based Undersampling (SBU) algorithm introduces a different perspective: it considers the structure of consecutive identical values and selectively reduces redundancy by eliminating subsets of these repeated sequences. No previous method in the literature has been identified that explicitly addresses this type of structure-aware undersampling in regression tasks.

Alternative strategies: algorithmic and hybrid approaches

Aside from data-level resampling, several methods address imbalance through algorithmic adaptation or hybrid integration. These include Label Distribution Smoothing (LDS) and Feature Distribution Smoothing (FDS) (Yang et al., 2021), which regularize prediction distributions, as well as Balanced Mean Squared Error (MSE) (Ren et al., 2022) and DenseLoss (Steininger et al., 2021), which reweight the loss to emphasize rare target regions.

Hybrid strategies combine resampling with ensemble learning or loss adjustment. Examples include relevance-guided ensembles like REsampled Bagging (REBAGG) (Branco, Torgo & Ribeiro, 2018) and weighted oversampling with bagging, as in WSMOTE-Ensemble (Abedin et al., 2023). While effective, these methods operate under different assumptions and typically require model-specific adaptation. In contrast, the present work emphasizes a simple, interpretable data-level approach designed for temporally structured imbalance.

Dataset description

Passive microwave (PMW) sensors onboard satellites can measure the natural microwave radiation emitted by the Earth’s surface and atmospheric particles (Ulaby, Moore & Fung, 1982). PMW data are crucial for quantitative precipitation estimation (QPE) because electromagnetic radiation at microwave frequencies can penetrate clouds, allowing for more direct observation of precipitation that is possible using infrared frequencies. A more direct use of microwave data is through radar, as the radar reflectivity of hydrometeors is proportional to rainrate. The Core Observatory (CO) of NASA’s Global Precipitation Measurement (GPM) mission combines PMW data from the GPM Microwave Imager (GMI) sensor with radar data from the Dual-frequency Precipitation Radar (DPR) instrument, aiming to provide a global precipitation monitoring of the planet (Skofronick-Jackson et al., 2017) and thus advance climate research. A number of radiometers onboard other satellites complement the CO.

Data for this study come from the GPM constellation. It consists of six variables: five representing the radiometer measurements in five microwave channels and one corresponding to the radar estimate of precipitation from the US Multi-Radar/Multi-Sensor (MRMS) surface radar data, which is deemed as the truth. The data is publicly accessible through NASA’s Precipitation Processing System (PPS). The subset used to test the method contains approximately 7 million records spanning from January 2018 to December 2019 (Troncoso, 2024).

The PMW retrievals are floating-point measurements ranging from 100 to 300 K, representing the brightness temperatures at microwave frequencies. The target variable, instantaneous precipitation, ranges from 0 to 80 mm. The dataset includes the geographic location and the observations time, providing both spatial and temporal information to the measurements.

Figure 1 shows the histograms of the microwave radiances (input variables). The five channels approximate a normal distribution. In contrast, the precipitation distribution for the target variable, shown in Fig. 2, indicates that the majority of values are 0.0, representing no precipitation. This feature is well-known in precipitation science, where most algorithms that fuse PMW and radar data first screen for rainy pixels before estimating the actual precipitation quantity. However, even after removing the zeroes, the distribution remains imbalanced, with most cases corresponding to low rain rates. High rain rates are not scarce but are sparsely distributed. Moderate rates, which are crucial for agricultural and environmental studies, fall within the slope of a gentle leptokurtic distribution.

A long-standing goal of Quantitative Precipitation Estimation has been to simulate radar data using radiometer data, with the Core Observatory providing coincident observations as a privileged source of radiometer-radar n-tuples. The aim is for the radiometers in the constellation to deliver radar-like precipitation estimates, significantly improving the temporal and spatial resolution of the constellation. Achieving this goal presents several challenges. Firstly, there are many contradictory cases in the datasets, where the same set of radiances yields different precipitation values. Secondly, the distribution is highly skewed, meaning traditional training methods would poorly represent high rainfall rates in the modeling. This results in outputs where high rainfall rates are treated as outliers, and the distribution is artificially driven towards a normalized probability density function (pdf)—an unrealistic representation that misrepresents the true nature of precipitation.

Regression algorithms

Regression is a fundamental technique in supervised machine learning used to predict continuous numerical values of a target variable. Unlike classification, which deals with categorical outputs, regression models aim to learn a mapping function from input variables (X) to continuous output variables (Y). This approach is widely applied across various domains to forecast trends, estimate unknown quantities, and understand relationships between variables.

To evaluate and compare the performance of different regression models, the following metrics were used:

• R-squared (R²): measures the proportion of variance in the dependent variable explained by the model. Ranges from 0 to 1, with values closer to 1 indicating a better fit.

• Root Mean Squared Error (RMSE): represents the average magnitude of the prediction errors. Lower values indicate better model performance, with greater sensitivity to larger errors.

• Pearson Correlation Coefficient (PR): assesses the linear relationship between predicted and actual values, ranging from −1 (perfect negative) to 1 (perfect positive). Values closer to −1 or 1 represent a stronger linear association, where −1 indicates an inverse correlation and 1 indicates a direct correlation. A value near 0 suggests little to no linear relationship between the variables.

In the initial phase of this work, a pre-configured Python library was employed to automate the construction and evaluation of multiple regression models quickly (Pandala, 2022). This approach provided a broad overview of the performance of various algorithms without requiring manual implementation or fine-tuning. This initial exploration phase helped to identify the most promising models for the given dataset, focusing on baseline performance metrics while not yet considering the dataset’s inherent imbalance.

The results presented in Table 2 illustrate the performance of various pre-configured regression models applied to the dataset for the training period of April and the testing period of May. Upon examination of the top 10 models, it is observed that the RMSE values maintain relative stability, lacking significant enhancements. Consequently, the analysis will focus on R² and PR metrics; the latter exhibits the greatest variance among the models. The Multi-layer Perceptron Regressor (MLPRegressor) achieved the highest R² value of 0.63, indicating its effectiveness in capturing the underlying data patterns. This model’s superior performance can be attributed to its capability to learn complex nonlinear relationships among variables, which is essential when the data exhibits such complexities.

In contrast, the Extra Trees Regressor demonstrated competitive R² performance with a value of 0.61 while executing significantly faster than the MLP, with a time of only 82.66 s compared to the MLP’s 460.49 s but with a significant lower PR of 0.783 compared to 0.795 for the MLP. The Random Forest Regressor, although yielding a high R² of 0.62, took even longer to execute at 462.15 s and also has a worse PR. This highlights a trade-off between execution time and predictive performance, as models like LGBM and XGB also yielded reasonable R² scores of 0.58 while maintaining low execution times of 1.17 s and 1.61 s, respectively.

While this preliminary tool primarily facilitates the quick comparison of different models, it must be recognized that the default settings may not produce optimal performance. Therefore, the next phase focused on refining the 10 best performing models identified in Table 2. Using the scikit-learn library (scikit-learn developers, 2024), a systematic hyperparameter tuning process was implemented to improve the predictive accuracy of these selected models.

The results of the hyperparameter tuning are summarized in Table 3, which presents the performance metrics of the top 5 regression models after tuning.

The results presented in Table 3 indicate the improvements achieved through hyperparameter tuning for the top regression models. Notably, the Multilayer Perceptron Regressor (MLPRegressor) maintained its leading position, achieving a R² value of 0.64, which demonstrates its ability to effectively model the underlying complexities of the data. Similarly, both the Extra Trees Regressor and Bagging Regressor showed a slight enhancement, confirming their effectiveness as ensemble methods that enhance predictive performance by reducing variance through the aggregation of predictions from multiple trees.

The KNeighborsRegressor, which was initially in the top 7 (see Table 2), also experienced an improvement in predictive accuracy, with its R² value increasing from 0.58 to 0.62, claiming the top 3 spot, while also boasting the shortest execution time among the top models. This model’s performance can be attributed to its simplicity and the local nature of its predictions, which allows it to capture local patterns efficiently.

The five models that showed the best performances are detailed below:

• MLPRegressor: a neural network-based approach that can capture complex non-linear relationships (Hornik, Stinchcombe & White, 1989).

• KNeighborsRegressor: a method that predicts the value of a new observation by averaging the values of its nearest neighbors in the feature space (Altman, 1992).

• ExtraTreesRegressor: an ensemble method that aggregates the predictions of multiple decision trees to improve accuracy and control overfitting (Geurts, Ernst & Wehenkel, 2006).

• BaggingRegressor: ensemble technique that fits base regressors on random subsets of the original dataset and then aggregates their predictions (Breiman, 1996).

• RandomForestRegressor: another ensemble learning method that constructs multiple decision trees during training and outputs the average prediction of individual trees for regression tasks. This technique enhances predictive accuracy and mitigates overfitting by leveraging the diversity of the ensemble (Breiman, 2001; Liaw & Wiener, 2002).

Among these, Multilayer Perceptron (MLPRegressor) was chosen as the reference model for further development, as it demonstrated slightly superior performance across the various evaluation metrics. This neural network model was preferred due to its flexibility in modeling complex patterns in data, making it particularly suitable for this regression task. Following the selection of the reference model, additional efforts were directed towards addressing the issue of data imbalance to further enhance its performance.

Facing data imbalance

The imbalance problem is evident, with a ratio of values where precipitation is greater than 0 compared to non-precipitation values of 1:8 (across all subsets). This imbalance is particularly distinctive because, in general, literature and other datasets often exhibit an imbalance with a range of values containing the majority of observations, as seen in Fig. 3. However, in this case, the value of 0, representing the absence of precipitation, is the only specific value that is overrepresented.

To address this, the first approach was to use different data preprocessing techniques (see Table 4). These include feature scaling and normalization, which, although they do not directly address the imbalance itself, play a crucial role in stabilizing the training process and ensuring numerical consistency across features.

The results from Table 4 indicate that scaling and normalization applied to the input features significantly improve model performance, as demonstrated by the higher R² and Pearson correlation values compared to the reference with no preprocessing.

Furthermore, after testing different transformation functions, the log1p logarithmic transformation, when applied together with input scaling, produced the best model performance. However, this improvement was only slightly better compared to using the same logarithmic transformation in conjunction with input normalization, achieving an R² of 0.708 and a Pearson correlation of 0.849. This combination outperformed all other preprocessing techniques. The logarithmic transformation was particularly effective because it compresses larger values more than smaller ones, reducing the skewness and creating a distribution more conducive to learning. By mitigating extreme variations in the output variable, this transformation enables the model to better capture underlying patterns in the data.

These findings highlight the importance of selecting appropriate preprocessing techniques tailored to both the input and output variables. While input scaling alone provided a significant improvement, the introduction of a logarithmic transformation on the output further optimized the model’s predictive capacity. This was the optimal preprocessing configuration, and it will serve as the foundation for subsequent steps aimed at addressing data imbalance, including the application of under-sampling and over-sampling techniques.

Undersampling and oversampling

Based on the models described in Table 1, the following experiments implement and evaluate representative undersampling and oversampling techniques to address the specific imbalance characteristics of the dataset.

Initially, five classical and widely adopted oversampling techniques were evaluated: Random Oversampling (RO), SMOTE for Regression, Gaussian Noise Injection (GN), SMOGN, and KNNOR-Reg. These methods were selected for their relevance as foundational baselines in the literature (He & Garcia, 2009; Torgo et al., 2013; Menardi & Torelli, 2014; Branco, Torgo & Ribeiro, 2017; Belhaouari et al., 2025), as well as their availability in established open-source frameworks (Wu, Kunz & Branco, 2022). KNNOR-Reg, in particular, represents a more recent contribution that synthesizes samples through neighborhood-weighted aggregation and does not rely on a relevance function, in contrast to the other methods considered.

It is important to note that most of these methods rely on a relevance function to identify high-importance regions and guide the generation of synthetic samples. This dependency adds an extra layer of model complexity and tuning effort.

Figure 4 illustrates the density distribution after applying each method, and Table 5 summarizes the model performance. Contrary to expectations, all oversampling methods led to a consistent degradation in both R² and Pearson correlation metrics. This is particularly significant given that these techniques represent the standard practice in the literature for addressing regression imbalance (Belhaouari et al., 2024).

Table 5:

Oversampling comparison in the MLP model (April–May 2018).

Technique	R2	Pearson
Without oversampling	0.712	0.846
KNNOR-Reg	0.643	0.839
Random oversampling	0.623	0.817
SMOGN	0.564	0.812
SMOTE	0.587	0.809
Gaussian noise	0.501	0.800

DOI: 10.7717/peerj-cs.3078/table-5

These results suggest that oversampling techniques may not be suitable in this case. The generation of synthetic samples can introduce noise or artificial structures that distort the original distribution, particularly when high-density areas (e.g., zero-precipitation regions) dominate the dataset. Neural networks are especially sensitive to such distortions, which may alter the optimization landscape and exacerbate overfitting. The fact that even SMOGN—one of the most refined oversampling techniques—underperforms in this context reinforces the idea that oversampling is not a reliable solution for this type of structured imbalance.

On the other hand, undersampling is often seen as a natural complement to oversampling, particularly for mitigating the dominance of majority values. However, eliminating values without considering the structure of the dataset may break continuity or remove relevant transitional states. This is especially true in our dataset, where the dominant value (zero) appears in long, consecutive sequences.

While techniques such as ENN, Tomek Links, and CNN have been considered, they are typically designed to remove boundary noise or misclassified instances in classification tasks, and are not directly applicable when a single target value overwhelmingly accounts for the imbalance. In this case, regardless of the strategy used, the goal of any undersampling method becomes the selective reduction of the zero value—raising the need for structure-aware solutions.