# Experimental Comparative Study on Self-Imputation Methods and Their Quality Assessment for Monthly River Flow Data with Gaps: Case Study to Mures River

Zsolt Magyari-Sáska, Ionel Haidu, Attila Magyari-Sáska

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Zsolt Magyari-Sáska, Ionel Haidu, Attila Magyari-Sáska. Experimental Comparative Study on Self-Imputation Methods and Their Quality Assessment for Monthly River Flow Data with Gaps: Case Study to Mures River. Applied Sciences, 2025, 15 (3), pp.1242. 10.3390/app15031242. hal-05033161

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Article

# Experimental Comparative Study on Self-Imputation Methods and Their Quality Assessment for Monthly River Flow Data with Gaps: Case Study to Mures River

Zsolt Magyari-Sáska <fnref n="1" />,* ①, Ionel Haidu <fnref n="2" />① and Attila Magyari-Sáska <fnref n="3" />

**Abstract:** Incomplete environmental datasets pose significant challenges in developing accurate predictive models, particularly in hydrological research. This study addresses data missingness by investigating gap imputation methodologies for datasets with 5–20% data absence, focusing on the Mureş River in Romania. Utilizing a novel approach, we applied various imputation techniques, including the ratio method, Kalman filtering, and machine learning algorithms (XGBoost, Gradient Boosting, Random Forest and CatBoost), while developing an innovative self-assessment metric for evaluating imputation performance without relying on external reference data. Through systematic analysis of hydrological station data from four monitoring points, we artificially introduced data gaps to rigorously test method applicability. The research demonstrates the feasibility of constructing a robust self-evaluation framework for selecting optimal imputation techniques, potentially enhancing data reliability and analytical precision in environmental and geospatial research. Our findings contribute a structured methodology for addressing data incompleteness, offering researchers a quantitative approach to improving dataset integrity and predictive modeling in complex environmental systems.

**Keywords:** missing data imputation; river discharge data; method evaluation; model performance; self-assessment; historical data

## 1. Introduction

Data integrity represents a critical foundation for scientific research, operational decision-making, and predictive modelling. Nonetheless, real-world datasets—especially those requiring continuous monitoring—frequently exhibit incompleteness. Missing data not only reduce statistical power but potentially introduce significant analytical bias if not meticulously addressed, thereby compromising research reliability and subsequent interpretative conclusions.

Imputation strategies for hydrological and climatological datasets demonstrate considerable variability in their treatment of correlation, autocorrelation, and periodicity [1,2]. The original ratio method emerges as a versatile technique [3], leveraging proportional relationships and has been used especially in environmental sciences [4] but also in sociology [5] and economics [6]. In hydrological contexts, this method proves especially valuable for reconstructing missing rainfall [7,8] or river flow data [9] by exploiting relationships between nearby monitoring stations or historical trends. However, its primary constraint lies in its dependence on external data sources, which may not always be available.

<figure><img src="image_2.png" /></figure>

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Academic Editor: Nir Krakauer

Received: 16 December 2024

Revised: 21 January 2025

Accepted: 24 January 2025

Published: 25 January 2025

**Citation:** Magyari-Sáska, Z.; Haidu, I.; Magyari-Sáska, A. Experimental Comparative Study on Self-Imputation Methods and Their Quality Assessment for Monthly River Flow Data with Gaps: Case Study to Mures River. *Appl. Sci.* **2025**, *15*, 1242. https://doi.org/10.3390/app15031242

**Copyright:** © 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

<footnote n="1">¹ Gheorgheni Extension, Faculty of Geography, Babes-Bolyai University, 535500 Gheorgheni, Romania</footnote>

<footnote n="2">² Laboratory LOTERR-EA7304, University of Lorraine, 57045 Metz, France; ionel.haidu@univ-lorraine.fr</footnote>

<footnote n="3">³ Faculty of Mathematics and Computer Science, Babes-Bolyai University, 400084 Cluj-Napoca, Romania; attila.magyari@stud.ubbcluj.ro</footnote>

<footnote>* Correspondence: zsolt.magyari@ubbcluj.ro</footnote>
In climatology and hydrology, widely utilized gap-filling methods often rely on data from nearby stations. To differentiate these approaches from the methods explored in the current research, the term “self-imputation” was introduced, signifying that the imputation process and its evaluation are conducted exclusively using the incomplete dataset itself, without incorporating external reference data.

To circumvent the methodological limitations in the absence of reference data, our research introduces a novel self-imputation approach. This innovative method capitalizes on the intrinsic periodic and autocorrelative properties of hydroclimatic systems, enabling data reconstruction without relying on external predictors. The approach presents a robust alternative for addressing data challenges in sparse observational networks, potentially enhancing the ability to manage incomplete environmental datasets.

ARIMA (AutoRegressive Integrated Moving Average) models represent a widely used imputation methodology for stationary time series analysis [10,11]. Despite their widespread application, these models present significant computational challenges when confronted with incomplete datasets. The presence of data gaps fundamentally disrupts the calculation of critical lagged terms and error correction mechanisms [12], thereby limiting their direct applicability in scenarios with substantial missing information. Complementary approaches, such as Kalman Filter techniques and their sophisticated extensions, including the Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF), have demonstrated remarkable versatility across diverse domains [13]. These methodological frameworks have proven particularly effective in dynamic data imputation, with applications spanning robotics [14], econometrics [15], and climate science [16].

Machine Learning (ML) has emerged as a transformative methodology in addressing data imputation challenges within hydrology and climatology, demonstrating unprecedented capabilities in extracting intricate patterns from incomplete environmental datasets [17–19]. Researchers have increasingly developed hybrid methodological frameworks that ingeniously integrate deterministic seasonal trend analysis with stochastic fine-tuning techniques, thereby enhancing predictive accuracy and computational resilience [20].

Ensemble methodological approaches, integrating deterministic models with machine learning-based stochastic techniques, represent a promising strategy for substantially enhancing imputation robustness [21]. This sophisticated approach enables more precise reconstruction of datasets characterized by complex periodicity and inherent measurement noise. Furthermore, machine learning’s remarkable capacity for cross-dataset generalization introduces innovative pathways for predictive analytics and anomaly detection within hydrological and climatological research domains.

In the scientific literature, much research critically examines and comparatively evaluates diverse data imputation methodologies [22–25]. For hydrological time series, the majority of studies predominantly rely on data from nearby stations [26–29] or on external referential datasets [30] for executing gap-filling operations, rather than exploring the intrinsic relational dynamics and inherent structural dependencies within the original dataset itself.

The emergent challenge, therefore, lies in developing more sophisticated analytical frameworks that valorises the endogenous informational architecture of datasets, achieving a self-imputation for missing data reconstruction.

The present study undertakes a comprehensive investigation of self-imputation techniques specifically designed to address missing data challenges in monthly environmental datasets. Our research objectives include the following: (1) systematically evaluating multiple self-imputation methodologies; (2) comparing the relative effectiveness of these techniques; (3) developing an evaluation metric for assessing the self-imputation performance without the use of external data.
The outcomes of this research are expected to significantly advance self-reliant impu-
tation frameworks, particularly for regions with limited observational infrastructure. By
improving data integrity and analytical precision, these findings have potential implications
for enhancing environmental monitoring capabilities in data-sparse contexts.

**2. Materials and Methods**

## 2.1. *Study Site, Physical Framework and Data*

Our research utilized average monthly discharge measurements from four monitor-
ing stations located on the Romanian section of the Mureș River (Figure 1), spanning a
comprehensive period from 1977 to 2014.

<figure><img src="image_3.png" /><figcaption>**Figure 1.** Course of Mureş River in Romania and the considered four hydrological stations.</figcaption></figure>

In time series datasets, such as river flow records, missing values emerge from both
random occurrences and systematic variations. Addressing these data gaps necessitates
sophisticated imputation techniques, as conventional methods like mean substitution
or linear interpolation frequently prove inadequate in capturing complex temporal in-
terdependencies [31,32]. Advancing imputation accuracy demands methodological ap-
proaches that can effectively leverage the intrinsic temporal and stochastic characteristics
of environmental data.

Interpolation under conditions of limited monthly observations requires a robust
physical framework that facilitates generating contextually appropriate values for data-
sparse periods. This framework essentially involves identifying and exploiting natural
relationships among sparsely observed environmental measurements.

Hydrological and climatological datasets exhibit distinctive periodicities and auto-
correlational patterns, fundamentally driven by Earth's heliogeophysical characteristics—including its spherical morphology, orbital dynamics, and axial inclination. These fundamental planetary attributes generate annual solar radiation and surface heating cycles, manifesting as a predictable 12-month periodicity in hydroclimatic elements. Such cyclical patterns reveal intrinsic similarities between corresponding monthly observations within and across years (Figure 2), and their structured arrangement (Figure 3) further highlights the serial correlation and autocorrelation inherent in such time series.
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Cyclical patterns in time series influenced by Earth's heliogeophysical factors.</figcaption></figure>

<table><caption>**Figure 3.** Structured arrangement of time series data.</caption><thead><tr><th></th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th><th>10</th><th>11</th><th>12</th></tr></thead><tbody><tr><th>Year 1</th><td>-0.11</td><td>-0.06</td><td>0.04</td><td>0.01</td><td>-0.07</td><td>0.03</td><td>0.11</td><td>0.10</td><td>0.01</td><td>-0.14</td><td>-0.28</td><td>-0.16</td></tr><tr><th>Year 2</th><td>-0.20</td><td>-0.12</td><td>-0.02</td><td>-0.07</td><td>-0.16</td><td>0.01</td><td>0.13</td><td>0.12</td><td>0.04</td><td>-0.14</td><td>-0.31</td><td>-0.11</td></tr><tr><th>Year 3</th><td>-0.09</td><td>-0.06</td><td>0.10</td><td>0.00</td><td>0.04</td><td>0.07</td><td>0.17</td><td>0.16</td><td>0.07</td><td>-0.07</td><td>-0.18</td><td>-0.06</td></tr><tr><th>Year 4</th><td>-0.04</td><td>0.00</td><td>0.06</td><td>-0.02</td><td>-0.02</td><td>0.15</td><td>0.19</td><td>0.23</td><td>0.08</td><td>-0.02</td><td>-0.14</td><td>-0.03</td></tr><tr><th>Year 5</th><td>-0.03</td><td>-0.04</td><td>0.14</td><td>0.12</td><td>0.09</td><td>0.14</td><td>0.21</td><td>0.23</td><td>0.08</td><td>-0.04</td><td>-0.21</td><td>-0.11</td></tr><tr><th>Year 6</th><td>-0.11</td><td>-0.20</td><td>0.01</td><td>-0.03</td><td>-0.01</td><td>0.10</td><td>0.22</td><td>0.18</td><td>0.05</td><td>-0.08</td><td>-0.21</td><td>-0.10</td></tr><tr><th>Year 7</th><td>-0.06</td><td>-0.11</td><td>0.02</td><td>0.02</td><td>0.01</td><td>0.16</td><td>0.29</td><td>0.23</td><td>0.12</td><td>-0.02</td><td>-0.18</td><td>-0.03</td></tr><tr><th>Year 8</th><td>-0.02</td><td>-0.05</td><td>0.07</td><td>0.05</td><td>0.05</td><td>0.21</td><td>0.30</td><td>0.27</td><td>0.12</td><td>-0.03</td><td>-0.17</td><td>-0.05</td></tr><tr><th>Year 9</th><td>-0.04</td><td>-0.09</td><td>0.07</td><td>0.03</td><td>0.04</td><td>0.21</td><td>0.29</td><td>0.29</td><td>0.13</td><td>-0.03</td><td>-0.17</td><td>-0.08</td></tr><tr><th>Year 10</th><td>-0.08</td><td>-0.16</td><td>-0.04</td><td>-0.09</td><td>-0.05</td><td>0.12</td><td>0.23</td><td>0.25</td><td>0.01</td><td>-0.12</td><td>-0.27</td><td>-0.20</td></tr><tr><th>Year 11</th><td>-0.14</td><td>-0.20</td><td>-0.04</td><td>-0.08</td><td>-0.03</td><td>0.08</td><td>0.22</td><td>0.23</td><td>0.03</td><td>-0.11</td><td>-0.24</td><td>-0.14</td></tr><tr><th>Year 12</th><td>-0.12</td><td>-0.19</td><td>-0.13</td><td>-0.05</td><td>-0.03</td><td>0.08</td><td>0.22</td><td>0.19</td><td>0.00</td><td>-0.11</td><td>-0.28</td><td>-0.19</td></tr></tbody></table>

The 12-month periodicity in hydroclimatic datasets provides insights into seasonal behaviors but simultaneously introduces non-stationarity, demanding sophisticated pre-processing for effective modeling. Historically, both deterministic seasonal models [33] and stochastic approaches [34–36] have been employed by researchers to address such challenges, recognizing the complex interplay between periodicity and autocorrelation in environmental data. Contemporary imputation methodologies continue to evolve, increasingly refining techniques to address non-stationary series characteristics.

The data preprocessing approach demonstrated methodological flexibility, with two data arrangement strategies: (a) traditional time series approach by maintaining chronological sequence; (b) tabular configuration by organizing monthly data vertically by year.

By strategically leveraging the inherent 12-month periodicity characteristic of hydroclimatic systems, ML models can be systematically trained using innovative tabular data configurations, organizing each year as a discrete row and individual months as columnar variables. This sophisticated architectural approach enables advanced ML algorithms to comprehensively explore and exploit periodic interdependencies, facilitating robust predictive capabilities for reconstructing missing data points.

To rigorously validate our imputation methodologies, we implemented a systematic approach to creating artificial data gaps with 4 data removal scenarios: 5%, 10%, 15%, and 20% data point elimination using uniform distribution as a randomization method. In this study, we introduced artificial data gaps using a uniform distribution to provide an unbiased assessment of the method's foundational capabilities, while acknowledging that in real-world applications, certain data points may carry more importance and thus warrant a more tailored gap-generation approach. To ensure the robustness of validation we have generated 1000 distinct random gap masks for each scenario. The frequency of gaps between months for different scenarios is presented in Figure 4. This comprehensive experimental design ensures statistically robust analysis, with each monitoring station subjected to 4000 distinct imputation test scenarios (1000 masks × 4 missingness levels), used in all imputation models.
<figure><img src="image_5.png" /><figcaption>Figure 4. Frequency of gaps between years and months for different scenarios using uniform distribution.</figcaption></figure>

The stationarity of all four original dataset was evaluated using the Augmented Dickey-Fuller (ADF) test, and the results confirmed that each dataset was stationary.

The algorithm for gap creation, as well as the original and masked data, is available on GitHub https://github.com/zsmagyari/SelfImpute.

## 2.2. Imputation Methods

### 2.2.1. Ratio Method (R)

The ratio method as proposed by Paulhus and Kohler [37] relies on the assumption that the relationship between two variables can be expressed as a proportion. If two variables, X and Y, are linearly related, with $Y = k \times X$, where k is a constant, the ratio method estimates the missing value of Y by using the known value of X and the constant. The value of k can be derived from the observed data where both X and Y are present (Equation (1))

$$k = \frac{\sum_{i=1}^{n} y_i}{\sum_{i=1}^{n} x_i} \qquad (1)$$

where

$n$—period length;

$x_i, y_i$—paired values

The ratio method offers several compelling advantages that make it a valuable approach to data imputation. One of its primary strengths is its ability to preserve the proportional relationships between variables, ensuring that the imputed values are consistent with the underlying data structure. This feature is particularly crucial in datasets where maintaining variable relationships is essential for accurate analysis. Additionally, the method's simplicity and computational efficiency make it accessible and practical for researchers, especially in resource-constrained environments or when dealing with large datasets. Unlike more complex techniques, the ratio method requires minimal computational resources while still delivering reliable results.

Despite its advantages, the ratio method faces several limitations and challenges that can impact its effectiveness in certain contexts. One major limitation is its reliance on the assumption of a stable proportional relationship between variables, which may not hold true in all datasets, especially those with complex or nonlinear dynamics. This dependence on proportionality restricts its applicability to scenarios where such relationships are evi-
dent. Additionally, the method is highly sensitive to outliers, as extreme values can distort the calculated ratio and lead to inaccurate imputations. The ratio method is unsuitable for datasets where no strong relationships exist between the variables of interest, limiting its scope. Although traditionally used for gap imputation in hydrological data, its inability to account for noise makes it a suboptimal model for imputation purposes.

In this study, we employ a self-imputation approach, raising the question of which data series should serve as the predictor. To address this, we created an adapted version of the ratio method, wherein pairwise correlations were calculated for each row and column (A,B) based on non-missing values. These correlation values were weighted by the proportion of usable value pairs (nA,B) relative to the total length of the row or column (N), giving priority to scenarios with longer, gap-free data pairs (Equation (2)).

$$w_{A,B} = \frac{n_{A,B}}{N} \qquad wCor(A, B) = w_{A,B} \cdot Cor(A, B) \qquad (2)$$

By considering the correlations between different column pairs, we tested for variable lag values measurable across months, while correlations between different row pairs allowed for the testing of lag values measurable across years. Thus, for imputing missing values in any given row or column, the predictor role is assigned to the row or column with the highest weighted correlation value (r* or l*), determined by the amount of usable (non-missing) data (Equations (3)–(5)).

$$r^* = \arg \max_k wCor(x_{i,\cdot}, x_{k,\cdot}) l^* = \arg \max_l wCor(x_{\cdot,j}, x_{\cdot,l}) \qquad (3)$$

$$k = \frac{\sum x_{i,j}}{\sum x_{r^*,j}} \quad \text{OR} \quad k = \frac{\sum x_{i,j}}{\sum x_{i,l^*}} \qquad (4)$$

$$X_{i,j} = k \cdot x_{r^*,j} \quad \text{OR} \quad X_{i,j} = k \cdot x_{i,l^*} \qquad (5)$$

where

wCor(A,B)—the weighted correlation between datasets A and B

x_{i,j}—the value of the dataset at row i and column j

x_{i,\cdot}—row i of the dataset

x_{\cdot,j}—column j of the dataset

X_{i,j}—the imputed value for x_{i,j}

### 2.2.2. ARIMA-Based Kalman Filter (K)

AutoRegressive Integrated Moving Average (ARIMA) models have been a cornerstone in time series forecasting since their formalization by Box and Jenkins in 1970 [38]. The ARIMA process models time series data as a function of its past values (autoregression), past forecast errors (moving average), and differencing operations to achieve stationarity. ARIMA models are denoted by ARIMA(p,d,q), where p represents the order of the autoregressive part, d the degree of differencing and q the order of the moving average.

The Kalman filter, introduced by Rudolf E. Kálmán in 1960 [39], is a recursive algorithm used to estimate the state of a linear dynamic system. It operates by predicting the state of the system and then updating this prediction using noisy observations. The algorithm comprises two phases a prediction step generating estimates of the current state using the system's model and update step which refines these estimates based on observed data. Mathematically, the Kalman filter works by minimizing the mean squared error of the state estimates. Its robustness to noise and ability to provide real-time updates make it a valuable tool for time series imputation.

The integration of ARIMA models and Kalman filters has been explored as a solution to the weaknesses of each individual method. While ARIMA models excel at capturing
temporal dependencies and forecasting, the Kalman Filter handles real-time state esti-
mation and noise reduction [40]. By representing ARIMA models in a state-space form,
researchers have developed hybrid approaches that offer enhanced performance in imputa-
tion tasks [41].

In our current research, we utilized an ARIMA model as the state space model within
the Kalman filter framework. The model employs an autoregressive parameter of 12, reflect-
ing the monthly resolution of the data, a differencing degree of 0, and a moving average
parameter of 1. By analyzing the monthly flow data, we observed only a pronounced
monthly seasonality and no significant yearly trend requiring differencing or additional
seasonal terms. This situation is characteristic for hydrological data series. Consequently,
the ARIMA(12,0,1) model—capturing a year’s worth of monthly lags and a single moving
average term—is sufficient to address the temporal structure without overlapping yearly
trends. This method adheres to the principles of ARIMA-based Kalman filtering. The
ARIMA state space model captures the underlying temporal dynamics of the data and
the Kalman filter recursively updates state estimates, ensuring optimal imputation for
missing values. The described imputation method was coded in R [42] using the imputeTS
package [43], which is designed to support researchers by providing a range of gap-filling
methods. Among these, the *na_kalman* function facilitates Kalman filter-based imputa-
tion. The *na_kalman* function requires two inputs: the time series with missing values and
the specified time series model. For this research, the time series model was defined as
ARIMA(12,0,1).

2.2.3. Machine Learning (ML) Methods

Machine Learning (ML), a branch of Artificial Intelligence (AI), has rapidly evolved
into a transformative technology. ML enables systems to learn and improve from expe-
rience without explicit programming, making it a key driver of automation in various
industries. This automation often involves analyzing vast amounts of data and applying
ML algorithms to uncover patterns and insights critical to intelligent decision-making and
were successfully used also in data imputation [44]. Common categories of ML algorithms
include supervised, unsupervised, semi-supervised, and reinforcement learning, each
catering to specific types of data and applications [45].

The effectiveness of ML solutions depends significantly on the quality of data and the
choice of algorithms, which can be mostly classified as classification, regression, clustering
and deep learning. In the realm of data imputation, ML offers innovative solutions for man-
aging incomplete datasets, a common issue in real-world scenarios [46,47]. By leveraging
the principles of ML, researchers can enhance the accuracy of predictive models and the
reliability of data-driven decisions [48]. In our research we focused on the evaluation of
four ML regressor techniques.

XGBoost (XGB) is a high-performance machine learning system known for its effective-
ness and scalability. It is widely used for achieving top results in tasks such as classification,
regression, and ranking. XGBoost's success lies in its innovative features, including a
sparsity-aware algorithm for handling sparse data [49], a weighted quantile sketch for
scalable learning, and support for parallel and distributed computing.

Gradient Boosting (GB) is a machine learning technique that combines multiple “weak”
models, to build a strong predictive model. It works iteratively, adding new models to
correct errors made by the ensemble, focusing on refining the model with each step, making
it highly adaptable to tasks like regression, classification, and ranking. Its flexibility allows
for customization with various loss functions, enabling it to handle diverse applications
effectively, capturing complex data patterns and improving accuracy [50].
Random Forest (RF) is a robust machine learning technique that combines multiple decision trees to create a strong predictive model. By introducing randomness during tree construction, random forests enhance model diversity, which improves overall performance [51]. This approach makes them highly effective for handling high-dimensional data and mitigating the risk of overfitting. Widely used across various domains, random forests are valued for their simplicity, versatility, and ability to provide reliable predictions with minimal tuning [52].

CatBoost (CB) is a gradient boosting algorithm specifically designed to address challenges like target leakage and categorical feature handling. It introduces two innovative techniques: ordered boosting, which prevents prediction bias by avoiding target leakage, and an advanced method for processing categorical features effectively. By overcoming statistical issues present in other gradient boosting methods, CatBoost achieves superior performance on diverse datasets [53]. Its design makes it particularly robust for tasks with heterogeneous features and complex data dependencies, outperforming tools like XGBoost and LightGBM in many benchmarks [54].

In our research considering the machine learning-based approach, gaps within a column are predicted using the values from other columns. To ensure the predictor columns are free of missing values, any initial gaps in these columns has been imputed using median imputation. The considered ML approach leverages the inherent periodicity of the data through our dual data arrangement strategy. In the tabular configuration, each row represents a year with months as columns, allowing the ML algorithms to naturally capture seasonal patterns. The chronological sequence arrangement preserves temporal ordering and enables the capture of autoregressive component. The ML models implicitly learn these temporal relationships through the feature patterns.

In all imputation scenarios, once a column is predicted and imputed, its values are incorporated as predictors for subsequent imputations. To optimize the use of available information, the gap-filling process prioritizes columns with the highest number of missing values, imputing these first. This approach forms the baseline method, referred to as Progressive (P). Beyond this, three additional prediction strategies were evaluated: (1) Stabilized (PS)—this method involves iteratively imputing the full dataset until a stable state is reached, defined either by the median of the changes between consecutive imputations falling below a predefined threshold or by reaching a maximum number of iterations; (2) Progressive Duplicate (PD)—an extension of the Progressive method, where imputed columns are also used as predictors during subsequent imputations; and (3) Progressive Duplicate Stabilized (PDS)—a combination of the progressive duplicate approach with the stabilization procedure applied to ensure convergence.

The presented and evaluated imputation methods were applied for prediction of missing data values and through this for reconstruction of hydrological data series, making them suitable for subsequent calculations. There are many rivers around the world that, in the early years of hydrometric measurements (1930s–1960s), did not have systematic measurements, and in some months, there were no monthly averages (especially in less developed countries). In addition, there were no measurements of other hydrometeorological elements (e.g., precipitation or evaporation) in neighbouring basins, so methods that rely on reference data for reconstruction could not be applied. There are many rivers across the world that, in the early years of hydrometric measurements (1930s–1960s), did not have systematic measurements, and in some months there were no monthly averages (especially in less developed countries). In addition, there were no measurements of other hydrometeorological elements (e.g., precipitation or evaporation) in neighboring basins, so methods that rely on reference data for reconstruction could not be applied.
## 2.3. *Evaluation Methods*

The evaluation of the imputation methods is conducted by comparing their performance to the original data. The performance assessment involves both the use of static indicators (mean absolute error, standard deviation, and the number of extreme values in errors) for locations with gaps, as well as the preservation of temporal correlations based on the Autocorrelation Function (ACF) between the original and the imputed series. In situations where real data necessary for comparison and evaluation are unavailable, innovative solutions must be developed. In a recent study by us, focusing on interpolating the surface areas of Moroccan lakes using satellite imagery, the ratio of existing to missing data was approximately 50–50%. In that case, we re-imputed the original data based on the imputations and used their comparison to assess the accuracy of the gap-filling process [55]. However, this approach could not be applied in the present study, as data gaps in hydrological stations typically range from 5–20%, making it impractical to re-impute 80–95% of the dataset. Consequently, we employed a novel approach and we developed an innovative evaluation metric.

The rationale for this approach is grounded in the following concept: starting with the original dataset that contains missing values, we selected an equivalent set of adjacent time points surrounding these gaps. We hypothesized that the magnitude of errors generated when applying the same imputation method to these new gaps would closely mirror the errors observed in the original imputation process. This approach allows us to assess the performance of imputation techniques without relying on complete datasets (Figure 5).

<figure><img src="image_6.png" /><figcaption>**Figure 5.** Imputation and performance assessment steps including forward and backward imputation phase, without relying on complete dataset.</figcaption></figure>

# 3. Results and Discussion

Our research implemented a rigorous methodology for assessing interpolation techniques using the Mureş River's monthly average flow dataset. The evaluation framework was designed to comprehensively test different imputation methods performance across various data missingness scenarios. Data preparation involved utilizing a complete monthly average flow dataset from the Mureş River into which we systematically introduced data
gaps with 5%, 10%, 15%, and 20% missingness levels, generating 1000 distinct samples for each gap percentage.

The gap preparation methodology ensures a controlled and reproducible comparison environment by applying a uniform missing data pattern across the dataset, effectively eliminating potential biases in gap placement. The selected gap rates were chosen to reflect real-world data characteristics, where environmental and scientific datasets frequently exhibit similar levels of missingness. This approach enhances the practical relevance of our research findings, bridging the gap between theoretical methodologies and actual observational challenges.

By implementing a large sample size of 1000 distinct datasets for each missingness level, we significantly enhanced the analytical robustness of our research. This expansive approach minimizes the influence of random variations and enables statistically rigorous conclusions. The comprehensive testing framework allows for an objective assessment of various imputation methods' suitability by measuring their performance against a known ground truth.

Visualization of an imputation result is exemplified in Figure 6, which provides a representative illustration showcasing the original data series, identified gap sections with their original values, and the imputed data for those gap sections. To ensure transparency and reproducibility, the complete research datasets are publicly available on GitHub at https://github.com/zsmagyari/SelfImpute.

There are various basic metrics for evaluating imputation errors, of which we chose the Mean Absolute Error (MAE), as it is less sensitive to outliers [56]. Alongside this, our performance evaluation utilized another two statistical metrics: the number of outliers in error values (OUTL) and the standard deviation of errors (SD). These metrics were calculated as average values across the 1000 sample datasets, providing a comprehensive and nuanced understanding of each imputation method's effectiveness. To ensure precise outlier identification, we employed Tukey's method [57], which relies on the interquartile range of the data. The R function collecting the basic statistical metrics of evaluation are publicly available on GitHub at https://github.com/zsmagyari/SelfImpute.

<figure><img src="image_7.png" /><figcaption>Figure 6. *Cont.*</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>**Figure 6.** Imputed river flow values with random forest, Kalman filter, ratio method and CatBooster for sample #512 with 20% gaps—Glodeni station, monthly data series.</figcaption></figure>

Figure 7 presents a comprehensive comparative analysis of three key performance metrics across different imputation methodologies, focusing on the P scenario for machine learning methods. The results reveal significant insights into the effectiveness of various interpolation techniques.

The Ratio method consistently demonstrated inferior performance compared to the ARIMA-based Kalman filter. Both methods exhibited an increasing (MAE) as the percentage of data gaps expanded, indicating reduced accuracy with greater data missingness.

Machine learning algorithms demonstrated superior performance due to their advanced pattern recognition capabilities. Unlike traditional methods that rely on single-column data, these algorithms leverage complex internal learning mechanisms to identify and reconstruct missing information across multiple dimensions. The CB method emerged as the most robust regression technique, maintaining remarkably consistent MAE values across different locations and scenarios.

An interesting anomaly was observed at the Arad monitoring station, where the Kalman filter unexpectedly outperformed machine learning methods, highlighting the nuanced nature of data imputation challenges.

The research emphasizes that MAE represents only one dimension of imputation quality. Extreme values in error series and their variability, as illustrated in Figure 8, play crucial roles in comprehensively evaluating interpolation effectiveness. The standard deviation of error values provides additional critical insights into the reliability and precision of different imputation approaches.

The analysis of extreme value characteristics revealed nuanced insights across different imputation methodologies. The RF and GB methods demonstrated remarkable consistency, exhibiting the lowest occurrence of extreme values across multiple sampling locations.

The Kalman filter presented an intriguing performance profile. While maintaining moderate MAE metrics, its effectiveness substantially diminished when confronting extreme value scenarios, positioning it in a comparatively weaker performance category.
<figure><img src="image_9.png" /><figcaption>**Figure 7.** Imputation error metrics for all stations in case of different methods by gap percentage.</figcaption></figure>

Contrary to expectations, the ratio method—ranking lowest in MAE assessments—displayed better performance when addressing extreme value distributions. A robust, positively correlated relationship exists between the percentage of missing data and the frequency of extreme value occurrences, as illustrated in Figure 7.

<figure><img src="image_10.png" /><figcaption>**Figure 8.** *Cont.*</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>**Figure 8.** Extreme monthly river flow values for RF 620 (high number of extreme values) and RF (788 no extreme) samples—Glodeni station.</figcaption></figure>

Error variance analysis revealed minimal ranking differentials compared to MAE positioning. The Arad station presented a notable exception, where the CB method distinguished itself by exhibiting the smallest variance, thereby occupying the optimal position.

To comprehensively assess method performance, we developed a new metric, the Error Index (EI), by combining the three previously discussed indicators in their normalized forms. The weights, expressing their relative importance were calculated using pairwise correlation values, with each indicator's correlation values summed as absolute values and subsequently normalized as shown in Equations (6)–(8). Complementing the EI, a Performance Index (PI) was defined as the inverse value of the EI (Equation (9)).

$$ 
\begin{aligned}
w_{mae} &= |r_{mae-extr}| + |r_{mae-sd}| \quad w_{extr} = |r_{extr-mae}| + |r_{extr-sd}| \quad w_{sd} \\
&= |r_{sd-mae}| + |r_{sd-extr}|
\end{aligned} 
\qquad (6) $$

$$ 
\begin{aligned}
W_{MAE} &= \frac{w_{mae}}{w_{mae}+w_{extr}+w_{sd}} \quad W_{EXTR} = \frac{w_{extr}}{w_{mae}+w_{extr}+w_{sd}} \quad W_{SD} \\
&= \frac{w_{sd}}{w_{mae}+w_{extr}+w_{sd}}
\end{aligned} 
\qquad (7) $$

$$ EI = W_{MAE} \cdot MAE + W_{EXTR} \cdot EXTR + W_{SD} \cdot SD \qquad (8) $$

$$ PI = 1 - EI \qquad (9) $$

where

xyz—placeholder for: mae—mean absolute error; sd—standard deviation; extr—outliers number

$r_{xyz1-xyz2}$—Pearson correlation between xyz1 and xyz2 values across 1000 samples

$W_{XYZ}$—normalized weight for the xyz component

MAE, EXTR, SD—normalized values of mean absolute error, number of outliers and standard deviation

Table 1 summarizes the obtained weights across all stations having similar values. Generally, the errors' standard deviation holds the greatest importance, followed by the mean absolute error.

<table><caption>**Table 1.** Weight values of the three indicators in case of different stations.</caption><thead><tr><th></th><th>W<sub>MAE</sub></th><th>W<sub>EXTR</sub></th><th>W<sub>SD</sub></th></tr></thead><tbody><tr><td>Suseni</td><td>38.5%</td><td>15.8%</td><td>45.7%</td></tr><tr><td>Glodeni</td><td>35.4%</td><td>15.1%</td><td>49.5%</td></tr><tr><td>Alba Iulia</td><td>37.1%</td><td>14.2%</td><td>48.7%</td></tr><tr><td>Arad</td><td>37.7%</td><td>17.5%</td><td>44.8%</td></tr></tbody></table>

The developed performance assessment also allows for the comparison of the different scenarios for every ML regressor (Figure 9). Our visualization employed color coding
to highlight performance insights. The cyan-highlighted variant represents the most effectively performing scenario for a given regressor, while the overall optimal method is distinguished in green (Figure 9). A clear performance pattern emerged, demonstrating the CB method's superior capabilities across multiple scenarios.

<figure><img src="image_12.png" /><figcaption>**Figure 9.** Performance index of different imputation methods.</figcaption></figure>

The calculated imputation performance index revealed that the PS and PDS variants often outperformed the P method which underscores the importance of context-specific approach selection in data imputation.

For each sample, we calculated the autocorrelation and compared these values to those of the original series. This analysis allowed us to determine the percentage of samples in which the imputed series did not differ significantly (at a significance level of 0.05) from the temporal correlation of the original series, indicating that these characteristics were preserved.

The results obtained align with those indicated by the performance index, with the CB method ranking first in terms of the proportion of cases where the temporal correlation of the reconstructed samples does not differ significantly from that of the original series. The subsequent positions are occupied by other well-performing methods, GB and RF. These rankings remain consistent for both forward and backward interpolation (Table 2).

<table><caption>**Table 2.** Percentage of cases with preserved temporal correlation for different imputation methods.</caption><thead><tr><td>Method</td><td>Forward Imputation</td><td>Backward Imputation</td></tr></thead><tbody><tr><td>R</td><td>98.64%</td><td>92.82%</td></tr><tr><td>K</td><td>99.37%</td><td>93.91%</td></tr><tr><td>CB</td><td>99.75%</td><td>95.94%</td></tr><tr><td>GB</td><td>99.54%</td><td>95.22%</td></tr><tr><td>XGB</td><td>99.37%</td><td>94.21%</td></tr><tr><td>RF</td><td>99.48%</td><td>94.58%</td></tr></tbody></table>

To address the challenge of evaluating imputation methods in scenarios lacking reference data, we have developed an innovative assessment metric. The methodology involved a strategic approach: starting with the original dataset containing missing values, we identi-
fied an equivalent number of adjacent time points surrounding these gaps. The underlying hypothesis suggested that error magnitudes resulting from applying identical imputation methods to newly created gaps would closely approximate errors observed in the original imputation process.

The proposed methodology combines the results of forward imputation (applied to the original dataset with missing values) and backward imputation (applied to newly created adjacent gaps). An imputation method was considered effective if the errors from both approaches aligned with a linear function originating at zero and maintaining a 45-degree slope. This alignment indicates equal contributions of errors from both interpolations, reflecting the method's consistency regardless of the scenario. Symmetry along the 45-degree line minimizes biases, as deviations would suggest a method's overfitting or underfitting in specific cases. This criterion ensures robustness and generalizability, avoiding disproportionate sensitivities to initial conditions or subsequent adjustments. By aligning error points along the 45-degree line, the interpolation method demonstrates balanced performance, reliability, and accuracy, making it optimal for applications requiring consistent data estimation. We have performed backward imputation using the ratio method, ARIMA-based Kalman filter, and P variants of machine learning imputation methods, calculating for each case the slope and intercept of regression equations derived from original and backward-imputed value pairs. The R function collecting the basic statistical metrics of evaluation are publicly available on GitHub at https://github.com/zsmagyari/SelfImpute.

Our analysis provides robust substantiation for the initial research hypotheses. Across the four monitoring stations, the CB method demonstrated the highest performance, consistently exhibiting low deviation from the 45-degree reference line and presenting the lowest intercept values, as illustrated in Figure 10.

A predictable correlation emerged, revealing that higher percentage gap variations corresponded directly to more pronounced statistical divergences. The CB method's performance was particularly compelling, with the RF method and, in certain instances, the Kalman filter closely approximating its precision.

The proposed self-assessment efficiency methodology found strong empirical validation through this approach. A significantly high correlation coefficient of 0.95 was observed between the performance index and the linear regression slope's deviation from the 45-degree reference line obtained through backward interpolation.

This statistically significant correlation (Figure 11) fundamentally substantiates our methodological framework. The proximity to a perfect correlation suggests considerable reliability and reproducibility of our self-assessment technique across diverse data interpolation scenarios. These findings validate the research methodology and also provide a promising framework for future environmental and hydrological data reconstruction efforts.

A Dell Precision 3660 workstation—equipped with an i9-12900K, 16-core 3200 MHz processor and 32 GB of memory—was used for the modelling tasks. Substantial differences in computation speed emerged across the various imputation models. The ratio model proved to be the fastest, filling an average of 332 samples in one second with a 5% data gap. The Kalman filter followed in speed, roughly twice as slow by filling an average of 152 samples per second. By contrast, the ML models were markedly slower: XGB managed 8.5 samples on average, and the GB and RF models each filled 0.80–0.85 samples, while CB was the slowest, at just 0.17 samples per second.

The ML models were not sensitive to the proportion of missing data, whereas the ratio model and the Kalman filter displayed different sensitivities: the ratio model took increasingly more time as the gap size grew, whereas the Kalman filter required progressively less time. At a 20% gap, for example, the ratio method filled an average of 78 samples, whereas the Kalman filter filled 431 samples. The reason is that, as the number of gaps grows, the
ratio method becomes increasingly time-consuming due to the need for numerous iterative
ratio lookups for each missing data point, whereas the Kalman filter, by skipping its update
step when data is absent and thus performing fewer computationally intensive updates,
ultimately reduces its runtime as the proportion of missing values rises.

<figure><img src="image_13.png" /><figcaption>**Figure 10.** Deviation from ideal intercept and slope in case of different imputation methods considering all four gap percentage scenario.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Figure 11.** Correlation between performance index and slope value deviation to 45° in a case of backward interpolation.</figcaption></figure>
## 4. Conclusions

Our research investigated self-interpolation methodologies for monthly time series data, employing both classical and machine learning techniques. A notable discovery emerged: ML methods not originally designed for time series interpolation (ex. CatBoost or Gradient Boost) exhibited high performance. This important efficacy originates primarily from the inherent monthly periodicity, with predictor columns effectively capturing values across diverse monthly contexts.

A cornerstone of this research is the development of an innovative self-assessment methodology, enabling the evaluation of imputation methods without reliance on external reference datasets. This approach not only enhances the reliability of gap-filling techniques but also broadens their applicability to regions with limited observational infrastructure. By simulating diverse missingness scenarios and applying systematic validation metrics, the study highlights the critical interplay between accuracy, computational efficiency, and methodological simplicity in selecting optimal imputation strategies.

Our simulation analyses revealed the CatBooster method as the most performant approach. However, this superior performance comes with a significant trade-off in computational complexity, particularly when compared to the competitive random forest and Kalman filter methods. Traditional mean absolute error metrics proved insufficient for comprehensive method assessment. We identified critical additional performance indicators: variance of error value series and frequency of extreme values.

The research's second phase focused on developing a methodology for assessing interpolation success in scenarios lacking reference datasets. Our breakthrough approach involves the backward interpolation technique, systematic method comparison and context-specific interpolation technique identification

The research findings offer significant practical solutions for interpolation challenges in real-world scenarios, particularly in environments with missing reference stations, an absence of supplementary reconstructive data sources, and complex temporal data characteristics. The imputation methods described in the article show strong potential for generalization to other fields, particularly those dealing with time series data that exhibit periodic patterns or structural relationships, such as medical monitoring, industrial sensor data, or financial time series, in cases when the probability of a data point being missing is independent of any other observed or unobserved variable. However, successful generalization requires that the data exhibit sufficient internal structure and periodicity, with a low missing data percentage (5–20% as studied), which in most real-world applications is an acceptable amount to impute without severe bias. The novel performance assessment methodology using adjacent time points is especially valuable across disciplines since it addresses the common challenge of evaluating imputation accuracy without complete reference datasets. The article's Performance Index, which combine multiple evaluation metrics, offer a sophisticated framework that could be adapted for quality assessment in different domains. However, researchers would need to carefully evaluate whether their specific data exhibits enough internal structure to support imputation and potentially adjust the evaluation metrics to match their field's requirements. Our study demonstrates the potential of advanced self-interpolation techniques to address critical data reconstruction challenges, providing a robust framework for handling incomplete time series datasets across various environmental and scientific domains.

**Author Contributions:** Conceptualization, I.H. and Z.M.-S.; methodology, I.H. and Z.M.-S.; software, Z.M.-S. and A.M.-S.; validation, I.H. and Z.M.-S.; formal analysis, I.H., Z.M.-S. and A.M.-S.; investigation, I.H. and Z.M.-S.; resources, I.H.; writing—original draft preparation, I.H., Z.M.-S. and A.M.-S.; writing—review and editing, I.H., Z.M.-S. and A.M.-S.; visualization, Z.M.-S. and A.M.-S.; All authors have read and agreed to the published version of the manuscript.
**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** All generated data are available from GitHub https://github.com/zsmagyari/SelfImpute.

**Conflicts of Interest:** The authors declare no conflicts of interest.

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