Addressing Missing Data in Slope Displacement Monitoring: Comparative Analysis of Advanced Imputation Methods

Abstract

Slope displacement monitoring is essential for assessing slope stability and preventing catastrophic failures, particularly in geotechnically sensitive areas. However, continuous data collection is often disrupted by environmental factors, sensor malfunctions, and communication issues, leading to missing data that can compromise analysis accuracy and reliability. This study addresses these challenges by evaluating advanced machine learning models—SAITS, ImputeFormer, and BRITS (Bidirectional Recurrent Imputation for Time Series)—for missing data imputation in slope displacement datasets. Sensors installed at two field locations, Yangyang and Omi, provided high-resolution displacement data, with approximately 34,000 data points per sensor. We simulated missing data scenarios at rates of 1%, 3%, 5%, and 10%, reflecting both random and block missing patterns to mimic realistic conditions. The imputation performance of each model was evaluated using Mean Absolute Error, Mean Squared Error, and Root Mean Square Error to assess accuracy and robustness across varying levels of data loss. Results demonstrate that each model has distinct advantages under specific missingness patterns, with the ImputeFormer model showing strong performance in capturing long-term dependencies. These findings underscore the potential of machine learning-based imputation methods to maintain data integrity in slope displacement monitoring, supporting reliable slope stability assessments even in the presence of significant data gaps. This research offers insights into the optimal selection and application of imputation models for enhancing the quality and continuity of geotechnical monitoring data.

1. Introduction

The stability of natural and artificially constructed slopes is a subject of immense importance across various sectors, including civil engineering, environmental protection, and public safety [1,2,3]. Numerous studies have investigated the complex interplay of geological, hydrological, and mechanical factors influencing slope stability, employing advanced analytical and modeling approaches to understand landslide susceptibility and deformation behaviors [4,5,6]. In geotechnical engineering, slope displacement monitoring serves as a critical tool for understanding the stability and behavior of slopes, helping to identify early signs of failure and prevent potential catastrophes [7]. Given the high risks associated with slope instability, monitoring systems have become increasingly sophisticated, utilizing advanced displacement sensors to gather high-resolution time-series data across diverse conditions [8]. These data offer valuable insights into the progressive deformation behavior of slopes, which is essential for training accurate predictive models and implementing timely preventive actions. However, a persistent challenge within slope monitoring is the occurrence of missing data, an issue arising from several unavoidable sources, including extreme weather conditions, sensor malfunctions, communication lapses, and maintenance interruption [9]. Missing data, if not appropriately addressed, can lead to inappropriate analyses, lower model accuracy, and ultimately undermine the reliability of the monitoring system [10].

Traditional approaches to missing data imputation, such as mean, median, and linear interpolation, often fall short when applied to time-series data characterized by complex temporal dependencies and variability, as seen in slope displacement [11,12]. These conventional techniques assume a simple data structure and cannot capture the intricate patterns and temporal correlations inherent in high-resolution slope displacement data. Consequently, reliance on simplistic imputation methods can introduce significant bias, leading to erroneous conclusions and reduced sensitivity to early signs of slope instability. To improve the handling of missing data in slope monitoring, recent research has increasingly focused on leveraging machine learning models that excel in learning temporal dependencies and patterns. In particular, advanced imputation models such as SAITS (Self-Attention-based Imputation for Time Series), ImputeFormer, and BRITS (Bidirectional Recurrent Imputation for Time Series) have demonstrated considerable potential in reconstructing missing data in complex time-series datasets [13,14,15]. These models, grounded in deep learning architectures, utilize attention mechanisms and other sophisticated feature extraction techniques to capture long-range dependencies, and effectively impute missing gaps in data with minimal distortion of temporal trends. This capability is especially pertinent in slope monitoring, where the missing data may be both intermittent and block-based, reflecting real-world scenarios where data loss does not always follow a predictable pattern. Despite the promise of these advanced imputation models, few studies have systematically compared their performance specifically in the context of slope displacement monitoring [16,17,18,19]. This reveals a critical gap; although advanced imputation frameworks are increasingly recognized as valuable tools for reconstructing complex time-series data, few studies have systematically assessed their effectiveness for slope displacement monitoring. For instance, linear interpolation has been used to handle missing data in slope instabilities [20], while cubic spline interpolation has been applied to estimate time to slope failure [21]. In addition, a spatio-temporal kriging interpolation algorithm was introduced to improve slope data reconstruction [22], and a basic LSTM-based approach was explored for landslide detection [23]. Although these studies highlight the importance of addressing missing data, they do not fully evaluate the broader range of advanced deep learning architectures now available, underscoring the need for more comprehensive model comparisons. Given the high stakes associated with slope failures, understanding the relative performance of these models in accurately recovering missing displacement measurements is paramount.

Addressing this gap, the present study undertakes a comprehensive comparative analysis of SAITS, ImputeFormer, and BRITS models for missing data imputation in slope displacement data. Our objectives are to evaluate these models to assess their applicability in ensuring data reliability for slope monitoring. By implementing these models in a controlled experimental framework, we aim to provide a holistic understanding of their strengths and limitations in reconstructing missing data for high-stakes geotechnical applications. This research not only advances the understanding of imputation techniques for slope displacement monitoring but also highlights the broader implications of robust missing data handling for the reliability of slope stability assessments. Ultimately, our findings will contribute to the development of more resilient monitoring systems that can maintain high-quality data continuity even in the presence of significant data loss, thereby enhancing the efficacy of predictive analytics in slope stability and similar time-series applications in geotechnical engineering.

2. Materials & Method

This study employs an advanced, IoT-enabled displacement monitoring system designed to address the challenges of slope stability assessment through accurate, continuous data acquisition. The system integrates strain gauge-based sensors, a resilient infrastructure for data transmission, and a self-sustained gateway, ensuring stable data flow and high sensitivity to displacement changes.

2.1. Sensor, IoT, and Gateway Setup

For accurate slope displacement monitoring, this study utilizes a high-sensitivity displacement sensor integrated with an IoT-based communication system. The sensor system is designed to capture even minor shifts in slope stability, ensuring early detection of potential failures. This setup includes a strain gauge-based displacement sensor, an IoT-enabled sensor node, and a solar-powered gateway that together form a robust monitoring network capable of operating in remote and challenging environments.

2.1.1. Displacement Sensor Design

The displacement sensor uses a strain gauge as the core detection mechanism, which operates based on variations in electrical resistance. The gauge detects voltage changes in response to strain, allowing precise measurements of displacement. To enhance data quality, drift signals and noise components are minimized, yielding clean digital outputs from analog signals. The displacement detection capability of the sensor is verified to 0.01 mm, establishing its suitability for high-sensitivity applications in geotechnical monitoring. The sensor is designed specifically for underground installation, ensuring accurate measurements in environments where traditional sensors may fail. It features high tensile strength—approximately three times greater than conventional rebar—as well as excellent electrical insulation and corrosion resistance due to its construction from Glass Fiber Reinforced Polymer (GFRP). Figure 1 shows the integrated IoT sensor node mounted on a GFRP rod, highlighting the compact, durable design that ensures stable measurements even in harsh environmental conditions. The GFRP pole measures 3000 mm in length and 30 mm in diameter, with five units installed to provide comprehensive coverage. Each GFRP rod includes two strain gauges (5 × 1.9 mm, 12 mm diameter) to facilitate simultaneous axial and lateral displacement measurements, achieved through a dual-circuit design.

2.1.2. IoT Sensor Node and Gateway

The sensor node is optimized for Internet of Things (IoT) applications, incorporating LoRa (Long Range) technology for low-power, long-range communication. LoRa modulation enables reliable displacement data transmission, achieving both low power consumption and high sensitivity, which enhances the signal-to-noise ratio [24]. This setup ensures that accurate data can be consistently transmitted despite low power demands, making it ideal for slope monitoring where real-time data continuity is critical. The IoT sensor node is designed to operate in a self-contained power setup, supported by solar charging and a low-power sleep mode for energy efficiency. The node collects and transmits data at intervals ranging from 10 milliseconds to 1 s, allowing real-time monitoring and flexibility in data resolution based on environmental needs (

Table 1. Specifications of each item used in this study.

Item	Specifications
Power Source	Solar-powered with low-power sleep mode
Data Transmission	LoRa-based communication, optimized for low power and high signal-to-noise ratio
Measurement Frequency	Adjustable from 10 ms to 1 s

).

2.1.3. Environmental and Durability Considerations

In designing this IoT-enabled monitoring system, specific environmental conditions were accounted for, ensuring that all components are resilient to extreme weather. The system is fully operational between −35 °C and 70 °C and has an IP65 rating, providing reliable performance under harsh conditions. However, in environments where surface temperatures may exceed 70 °C, the system’s performance could be compromised, necessitating additional protective measures or alternative sensor configurations capable of withstanding more extreme temperature ranges. Additionally, compliance with KC standards for electromagnetic compatibility certifies the system’s stability and suitability for deployment in geotechnical monitoring environments. In summary, this high-sensitivity displacement monitoring system, combining robust sensor technology with advanced IoT communication protocols, ensures reliable and continuous data acquisition in slope monitoring applications. The combination of precise displacement sensors, energy-efficient IoT nodes, and an autonomous gateway establishes a comprehensive framework for capturing critical slope displacement data, which is fundamental to addressing the challenge of missing data in geotechnical monitoring.

2.2. Machine Learning Algorithms for Imputation

In slope displacement monitoring, missing data are a prevalent issue that can hinder accurate analysis and early warning capabilities. This study employs advanced machine learning models specifically designed for time-series imputation to address data gaps in slope displacement measurements. The following models—SAITS, ImputeFormer, and BRITS—are utilized due to their ability to capture complex temporal dependencies and nonlinear patterns, which are critical in the context of displacement monitoring.

2.2.1. Self-Attention-Based Imputation for Time Series (SAITS)

SAITS, or Self-Attention-based Imputation for Time Series, leverages the self-attention mechanism to focus on specific time points within a sequence, allowing it to detect and reconstruct temporal dependencies even in the presence of missing values. Unlike traditional imputation techniques, SAITS does not assume a fixed temporal structure; instead, it dynamically assigns importance to relevant parts of the sequence, an approach particularly beneficial for the variable nature of slope displacement data. The SAITS model architecture includes multiple layers of self-attention units, where each layer captures temporal dependencies of varying lengths, enabling the model to recognize both local and global patterns. The core operation of each self-attention unit is the scaled dot-product self-attention, shown in Figure 2, where each input vector is projected into query (Q), key (K), and value (V) vectors. These vectors undergo dot-product operations to calculate attention scores, which are then scaled, softmaxed, and applied to the values (V) to generate context-aware representations. This structure allows SAITS to dynamically weigh information across the sequence, focusing on points that hold the most relevant information for accurate imputation [25].

Each layer consists of several attention heads, which simultaneously process different portions of the sequence. This multi-head setup allows SAITS to capture complex temporal relationships by viewing the data from multiple perspectives, enabling it to reconstruct missing values more accurately [26,27]. For instance, in a case where sensor data are missing intermittently, SAITS can adaptively focus on related segments of the sequence to infer plausible values. Additionally, the use of positional encodings in SAITS provides a mechanism to track the order of time steps, ensuring that the model maintains an accurate temporal sequence while performing imputation. To further enhance its ability to handle real-world data gaps, SAITS includes residual connections that reduce the risk of vanishing gradients during backpropagation, enabling deeper networks and allowing the model to handle large, complex datasets. SAITS is trained using a masked data approach, where segments of the sequence are deliberately removed and subsequently predicted by the model. This training paradigm encourages SAITS to learn efficient reconstruction strategies, making it highly suitable for time-series data with both randomly distributed and block-based missing patterns, as is often the case in slope displacement monitoring.

2.2.2. ImputeFormer

ImputeFormer is a Transformer-based model designed for spatio-temporal data imputation that integrates low-rank inductive biases to enhance both performance and generalizability (Figure 3). Unlike conventional Transformer architectures, which can inadvertently capture high-rank, noisy correlations in time-series data, ImputeFormer applies a low-rankness constraint, balancing the preservation of informative signals with the minimization of noise. This approach enables ImputeFormer to generalize effectively across diverse spatio-temporal imputation tasks, including datasets with heterogeneous missing patterns [28].

ImputeFormer incorporates a projected attention mechanism to enforce low-rank structures within temporal interactions. By projecting temporal data into a lower-dimensional space, ImputeFormer captures essential temporal patterns while simultaneously reducing the rank of the attention map, enhancing computational efficiency and mitigating the risk of overfitting to noise. This mechanism operates in two stages: first, a compact representation of the temporal data is formed using a learnable projection vector; this representation is then expanded to cover the entire sequence. This process achieves linear complexity, making it particularly efficient for longer sequences. To capture spatial dependencies without requiring a predefined graph structure, ImputeFormer employs embedded attention based on node embeddings that represent spatial relationships. Each node embedding serves as a dense abstraction of individual time series, enabling ImputeFormer to construct a correlation map through pairwise interactions. This map is subsequently used to impute missing values across spatially connected nodes, enhancing imputation quality without incurring the computational cost of conventional full spatial attention.

ImputeFormer also integrates a Fourier sparsity loss, which regularizes the spectral properties of the imputed data [29,30]. This loss guides the model to produce imputations that align with the sparse frequency spectrum characteristic of spatio-temporal data, minimizing overfitting by constraining high-frequency noise. The Fourier sparsity loss operates by applying the Fast Fourier Transform (FFT) on the imputed values and calculating the L1 norm of the transformed values, thereby enforcing a low-rank solution within the frequency domain [31]. Through these integrated mechanisms, ImputeFormer achieves a balance between model expressivity and generalizability, making it well-suited for complex imputation tasks across diverse spatio-temporal datasets, including traffic flow, air quality, and solar energy production data.

2.2.3. BRITS

BRITS is an imputation method specifically designed to handle missing values in time-series data by utilizing bidirectional recurrent neural networks (RNNs). Unlike traditional imputation methods that often rely on assumptions about the underlying data generation process (e.g., linearity or smoothness), BRITS directly learns the missing values within a recurrent dynamical system without making restrictive assumptions about the data structure. The key innovation in BRITS is its use of bidirectional RNNs, which allow the model to impute missing values by incorporating both past (forward direction) and future (backward direction) information in the sequence. This bidirectional approach provides a more robust imputation framework, especially for time series with nonlinear dynamics, where forward-only models might struggle. BRITS treats the missing values as variables within the RNN’s computational graph, allowing them to be updated iteratively through backpropagation. This mechanism provides delayed gradients in both the forward and backward passes, enhancing the accuracy of the imputed values. Additionally, BRITS employs a consistency constraint that enforces agreement between forward and backward predictions, further improving imputation quality. Experimental results have shown that BRITS outperforms many existing methods, particularly in settings with multivariate and irregularly spaced time-series data, making it suitable for applications such as health care and environmental monitoring.

2.3. Imputation Performance Evaluation

Evaluating the accuracy and reliability of imputation models is essential to determine their effectiveness in reconstructing missing slope displacement data. To ensure that the selected machine learning models—SAITS, ImputeFormer, and BRITS—are adequately suited for this task, we use a various quantitative performance metrics and experimental validation.

2.3.1. Experimental Setup

To simulate real-world missing data conditions, we introduce missing values into the slope displacement dataset according to several patterns that reflect common issues encountered in field monitoring. The missing data rate is varied to examine each model’s robustness across different levels of data loss. This multi-pattern approach ensures a comprehensive evaluation of the models under conditions that resemble actual monitoring challenges.

2.3.2. Performance Evaluation Metrics

The imputation models are evaluated using three key performance evaluation metrics that provide a comprehensive view of their accuracy and reliability in handling missing data:

• Mean Absolute Error (MAE): MAE measures the average magnitude of errors between the original ($y_i)$ and imputed values $\left({\widehat{y}}_i\right)$, providing insight into the absolute accuracy of each model. Lower MAE values indicate a closer approximation of missing values to the true data points, making MAE a primary metric for assessing each model’s baseline performance.

  $$MAE={\displaystyle \frac{1}{n}}{\displaystyle \sum }_{i=1}^n\left|y_i-{\widehat{y}}_i\right|$$
• Mean Squared Error (MSE): MSE measures the average squared difference between the original values ($y_i$) and the imputed values (${\widehat{y}}_i$). By squaring the errors, MSE places a higher emphasis on larger deviations, making it particularly useful for identifying significant discrepancies between actual and imputed data points. Lower MSE values indicate that the imputation model is effective in minimizing large errors, making it a key metric for assessing the robustness of imputation performance.
• Root Mean Square Error (RMSE): RMSE, which emphasizes larger errors due to its squared term, is useful for assessing each model’s ability to handle high deviations in imputation. In slope displacement data, large deviations are often critical indicators of displacement trends; hence, a lower RMSE score signifies the models’ capacity to accurately reconstruct missing values even in cases with high variability.

$$RMSE=\sqrt{{\displaystyle \frac{{\sum }_{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}{n}}}$$

Each model’s performance is assessed by averaging these metrics across multiple runs, ensuring that results are not influenced by random variations in missing data patterns.

2.3.3. Model Comparison and Analysis

By conducting this comprehensive analysis across multiple patterns and metrics, we gain insights into each model’s suitability for imputation in slope displacement monitoring. The results will guide the selection of the most appropriate imputation strategy, contributing to enhanced data quality and continuity in displacement monitoring applications.

3. Experiment

The experimental framework, as illustrated in Figure 4, provides a systematic approach for evaluating missing data imputation methods in slope displacement monitoring. This framework involves five primary stages: data collection and preprocessing, simulation of missing data, model input and imputation, imputed output generation, and performance evaluation.

3.1. Data Description

This study collected slope displacement data from sensors installed at two distinct field sites in South Korea: Yangyang and Omi. These sites were selected in consultation with local government authorities, who identified steep slopes which are considered high-risk due to their potential for large-scale damage in the event of slope failure. The Yangyang site, selected for its diverse slope conditions, features six sensors strategically installed at different locations to capture a comprehensive view of displacement behaviors across varying terrain. The Omi site, while smaller, includes two sensors strategically positioned to monitor critical slope areas. Figure 5 presents one such installation, illustrating how the solar-powered IoT sensor node is deployed directly on a rocky slope, providing insights into the local terrain and environmental factors influencing data integrity. The multi-location setup provides a valuable range of displacement data that enhances this study’s capacity to evaluate imputation models across different environmental conditions. Although these two locations provide a valuable range of displacement data suitable for testing the robustness of imputation models, we acknowledge that they may not fully represent the diversity of all potential geotechnical environments. As such, the present results should be viewed as a proof-of-concept demonstration rather than a definitive conclusion. In future work, we plan to expand our data collection efforts to multiple sites with varying geological and climatic conditions to enhance the generalizability and applicability of our findings.

Each sensor recorded approximately 34,000 data points, offering high-resolution temporal data suitable for detailed time-series analysis. These data points represent measurements taken over an extended monitoring period, providing insights into both short-term fluctuations and longer-term trends in slope stability. The dataset is well-suited for evaluating imputation models, as it allows for testing the models’ robustness under realistic conditions of sensor-distributed data with varying potential for data loss. Additionally, photos documenting the sensor installations at both sites are available, providing visual context for the physical setup and aiding in understanding the environmental challenges each sensor may encounter. This documentation supports this study by offering insights into factors that may influence data integrity, such as terrain variability, weather exposure, and installation angles. The dataset thus presents a unique, real-world basis for assessing missing data imputation methods, contributing to a thorough evaluation of each model’s performance in realistic slope monitoring applications.

3.2. Data Missingness Simulation

While the actual rate of missing data in the collected slope displacement dataset is less than 1%, factors such as extreme weather conditions, equipment issues, or the presence of significant outliers may necessitate the removal of additional data points, potentially increasing the overall missing rate. To evaluate the robustness of the imputation models under varying levels of data loss, this study simulates different rates of missing data, setting them at 1%, 3%, 5%, and 10%. The missing data are introduced using two main patterns to reflect realistic conditions encountered in field monitoring:

• Random Missing Data: Individual data points are randomly removed throughout the dataset, representing cases of intermittent data loss due to temporary disruptions such as brief communication issues or minor sensor malfunctions. This pattern is intended to test the models’ ability to interpolate based on surrounding context and capture short-term trends despite sporadic data gaps.
• Block Missing Data: Consecutive sequences of data points are removed to simulate prolonged outages, which can occur due to sensor power failures, communication breakdowns during adverse weather, or maintenance needs. This pattern presents a greater challenge for imputation, as the models must rely on extended temporal dependencies to fill in these larger gaps effectively.

Each simulated missing rate (1%, 3%, 5%, and 10%) is applied across both patterns, providing a comprehensive framework for assessing the imputation models under diverse data loss scenarios as illustrated in Figure 6. By experimenting with increasing missing data rates, this setup allows us to examine how each model’s performance scales with the extent of data loss. This approach helps in identifying models that can effectively handle more extensive missing data, which is crucial for ensuring reliable slope monitoring under adverse field conditions. The simulated missingness, combined with real-world patterns, provides a rigorous basis for evaluating the robustness and accuracy of the imputation models, supporting this study’s goal of establishing reliable data handling methods for slope displacement monitoring.

3.3. Data Preprocessing

Prior to model training, a series of preprocessing steps were applied to ensure data consistency and enhance the models’ ability to handle missing values effectively. First, the continuous slope displacement measurements were segmented into fixed-length time windows using a sliding window approach. This method preserves temporal information and facilitates the capture of both short-term variations and longer-term trends, while also providing consistent input shapes for the imputation models.

Next, we applied min-max normalization to the segmented time-series data. This scaling method transforms the displacement values into a common range (e.g., [0, 1]), reducing the impact of outliers and ensuring that all variables contribute comparably to the models’ training process. By normalizing the data, we address the risk of certain features dominating others due to their larger numeric ranges, thereby improving the stability and convergence of the machine learning models.

3.4. Machine Learning Model Development

The implementation of SAITS, ImputeFormer, and BRITS models for missing data imputation was conducted in Python version 3.11.7 on a Linux system with an x86_64 architecture. The experiments utilized PyTorch version 2.2.2+cu121, NumPy version 1.26.3, and Pandas version 2.1.4. All computations were performed on a dedicated workstation with sufficient CPU and GPU resources to handle large-scale time-series data. To ensure reproducibility, a fixed random seed (42) was applied to all data splitting, model initialization, and training procedures. Missing values were simulated in the data at rates of 1%, 3%, 5%, and 10% using random and block patterns, as outlined in Section 3.2, to ensure each model’s robustness against different types of missingness. The key hyperparameters for each model were optimized based on performance across several metrics: MAE, MSE, RMSE, and the R² metric. For SAITS, the number of attention heads was set to 4, 6, or 8, allowing the self-attention mechanism to weigh temporal relationships across different parts of the sequence. The number of self-attention layers was adjusted between 2 and 6 to balance model complexity and temporal learning depth. The hidden layer size was tuned with values of 64, 128, and 256, enabling the model to capture intricate temporal patterns and nonlinearities in the data.

Dropout rates between 0.1 and 0.5 were explored to mitigate overfitting, with lower dropout rates used for simpler configurations and higher rates for deeper networks. For ImputeFormer, the number of fully connected layers was adjusted between 2 and 5, providing the model flexibility in learning relationships among features within the dataset. Batch size was tuned in the range of 16 to 64 to balance training stability and computational efficiency, with larger batches helping to smooth gradients during training. The learning rate was examined in the range of $1\times {10}^{-3}$ to $1\times {10}^{-4}$, with smaller learning rates favoring training stability. Additionally, L2 regularization was applied to prevent overfitting, with lambda values tested between 0.01 and 0.001 to balance model accuracy and generalization. For BRITS, the number of LSTM layers was configured to 1–3, achieving effective temporal dependency capture without overcomplicating the model architecture. The hidden state dimension was tuned in the range of 64 to 256, where higher dimensions allowed the model to retain more information from past data points. Dropout rates were set between 0.2 and 0.5, optimizing regularization while preserving model accuracy. Learning rates from $1\times {10}^{-3}$ to $1\times {10}^{-4}$, were tested to identify the rate that achieved the best convergence while minimizing error.

This hyperparameter tuning process enabled each model to maximize performance for accurate and reliable imputation. Hyperparameter tuning was performed using grid search and cross-validation techniques across multiple validation subsets extracted from the complete dataset. Cross-validation was essential to generalize model performance, ensuring that hyperparameters were not biased towards specific data segments or missing data patterns. Models were trained with an early stopping criterion based on validation MAE and RMSE to prevent overfitting and reduce training time. Each model’s hyperparameters were selected based on their performance across the simulated missing rates and patterns, targeting configurations that minimized MAE, MSE, and RMSE. This process allowed for an objective assessment of each model’s capability to handle varied missingness rates (1%, 3%, 5%, 10%) and patterns (random, block) in slope displacement data.

4. Results and Discussion

This section presents the empirical results of the SAITS, ImputeFormer, and BRITS models across varying rates of missing data (1%, 3%, 5%, and 10%) and different missingness patterns (random and block). The performance of each model was evaluated using MAE, MSE, and RMSE, allowing a comprehensive assessment of each model’s imputation capabilities.

4.1. Performance Evaluation Result

The performance evaluation includes a comparative analysis of both traditional statistical imputation methods (Mean and Linear Imputation) and advanced machine learning models (SAITS, ImputeFormer, and BRITS), examining their effectiveness across different missing data patterns and rates. Mean Imputation replaces missing values with the mean of the observed data for each feature. Linear Imputation estimates missing values by linearly interpolating between adjacent observed points. Each method was assessed for its ability to accurately reconstruct missing values under various scenarios, including random and block missing data patterns, with missing rates of 1%, 3%, 5%, and 10%.

The results demonstrate that conventional statistical approaches, while straightforward to implement, often fall short in handling the complexities of time-series data. Mean imputation, for example, exhibited higher errors across most scenarios due to its inability to account for temporal dependencies. Particularly in random missing data patterns, the Mean Imputation method yielded significantly higher MAE and RMSE values, such as an MAE of 28.2 and an RMSE of 39.0 at a 1% missing rate. Similarly, Linear Imputation showed variable performance, performing well at low missing rates (e.g., MAE of 9.7 and RMSE of 20.9 for 1% random missing data), however struggling with block missing patterns or higher missing rates, where RMSE values reached as high as 18,447.5 for a 1% block missing rate.

In contrast, the advanced machine learning models consistently outperformed traditional methods across all scenarios. In evaluating the results across models, clear trends emerged. BRITS showed resilience and accuracy particularly in random missing data patterns, demonstrating its strength in handling shorter, more isolated gaps due to its bidirectional recurrent structure, which can effectively capture local temporal dependencies. This capability allowed BRITS to consistently maintain low error values across low and moderate random missing rates, proving to be a robust choice for cases where data gaps occur intermittently. ImputeFormer, with its Transformer-based architecture, displayed versatility, managing to achieve relatively stable performance across both random and block missing data. It was particularly effective at moderate missing rates, demonstrating a balanced approach that made it adaptable to mixed missing data conditions. However, as missing rates increased, ImputeFormer’s accuracy gradually decreased, though it continued to handle moderate and structured gaps better than other models.

SAITS, on the other hand, excelled in handling block missing data at lower missing rates, capitalizing on its self-attention mechanism to capture long-range dependencies across consecutive missing values. This made it an advantageous choice in settings where data loss is clustered in continuous sequences. However, SAITS struggled to interpolate well under random missing patterns and higher missing rates, indicating that it may be less suited for settings with frequent, sporadic gaps. These findings suggest that while each model performs well within specific contexts, there is no single model that consistently outperforms across all missing data conditions.

In practical terms, this highlights the value of selecting imputation strategies based on the anticipated missing data pattern. For instance, BRITS may be favored in environments with intermittent missing data, while SAITS might be more suitable in applications where missing data occurs in blocks. ImputeFormer offers a balanced option for datasets with moderate rates of missing data across different patterns. This evaluation provides a comprehensive understanding of each model’s capabilities, supporting the selection of optimal imputation strategies in slope displacement monitoring. The following Results section presents the specific metrics achieved by each model, further illustrating these observations with detailed data (

Table 2. Imputation Performance of SAITS, ImputeFormer, and BRITS across Different Missing Rates and Patterns.

Imputation Method	Pattern	Missing Rate (%)	MAE	MSE	RMSE
Mean imputation	Random	1	28.2	1521.1	39.0
Mean imputation	Random	3	34.7	90,931.5	301.5
Mean imputation	Random	5	39.4	609,442.0	780.7
Mean imputation	Random	10	60.3	21,896,192.4	4679.3
Mean imputation	Block	1	29.5	3772.7	61.4
Mean imputation	Block	3	44.4	986,135.8	993.0
Mean imputation	Block	5	29.3	5829.9	76.4
Mean imputation	Block	10	65.9	23,356,052.1	4832.8
Linear imputation	Random	1	9.7	435.1	20.9
Linear imputation	Random	3	77.9	18,700,375.6	4324.4
Linear imputation	Random	5	54.5	11,706,671.5	3421.5
Linear imputation	Random	10	43.0	21,945,710.4	4684.6
Linear imputation	Block	1	841.5	340,310,434.1	18,447.5
Linear imputation	Block	3	31.9	1,218,551.8	1103.9
Linear imputation	Block	5	13.6	5403.4	73.5
Linear imputation	Block	10	53.0	23,479,679.3	4845.6
SAITS	Random	1	1.43	11.34	3.37
SAITS	Random	3	2.41	9,541.94	97.68
SAITS	Random	5	7.35	437,946.43	661.78
SAITS	Random	10	8.09	439,525.97	662.97
SAITS	Block	1	0.94	2.93	1.71
SAITS	Block	3	12.18	737,333.76	858.68
SAITS	Block	5	1.72	3444.00	58.69
SAITS	Block	10	4.94	232,504.61	482.19
ImputeFormer	Random	1	0.78	5.08	2.25
ImputeFormer	Random	3	0.81	7.28	2.70
ImputeFormer	Random	5	0.95	248.94	15.78
ImputeFormer	Random	10	4.57	220,725.42	469.81
ImputeFormer	Block	1	0.84	8.27	2.88
ImputeFormer	Block	3	1.29	52.24	7.23
ImputeFormer	Block	5	1.43	3169.29	56.30
ImputeFormer	Block	10	5.15	234,489.16	484.24
BRITS	Random	1	0.76	4.84	2.20
BRITS	Random	3	0.73	2.96	1.72
BRITS	Random	5	2.48	9537.28	97.66
BRITS	Random	10	1.27	2942.83	54.25
BRITS	Block	1	0.99	118.42	10.88
BRITS	Block	3	2.03	9716.57	98.57
BRITS	Block	5	1.54	3267.75	57.16
BRITS	Block	10	1.35	3203.19	56.60

).

4.2. Discussion

Figure 7 presents imputation results for three advanced models—(a) SAITS, (b) ImputeFormer, and (c) BRITS—each illustrating how missing data points are reconstructed to form a continuous and plausible time-series. In these plots, the blue line represents the original data, the orange line indicates the data with artificially introduced missing values, and the green line shows the imputed data. By comparing how closely the green line aligns with the blue, readers can assess each model’s ability to handle different missingness patterns and rates. This visual representation highlights which models maintain stable performance, how they deal with isolated versus prolonged data gaps, and the extent to which their imputed values preserve underlying trends and temporal dependencies.

Under random missing patterns, BRITS and ImputeFormer exhibited strong performance, especially at lower missing rates. At a 1% random missing rate, BRITS achieved the lowest MAE of 0.76 and RMSE of 2.20, closely followed by ImputeFormer with an MAE of 0.78 and RMSE of 2.25. These results indicate that both BRITS and ImputeFormer were effective at handling minimal data loss, likely due to their architectures, which are optimized to capture short-term dependencies essential for interpolating isolated missing points in random patterns. As the missing rate increased to 3%, BRITS continued to demonstrate stability, with a slight improvement in MAE to 0.73 and RMSE to 1.72. ImputeFormer maintained comparable accuracy with an MAE of 0.81 and RMSE of 2.70. At this rate, SAITS showed higher error values than both BRITS and ImputeFormer, with an MAE of 2.41 and RMSE of 97.68. This trend continued at a 5% missing rate, where BRITS and ImputeFormer demonstrated resilience, achieving MAE values of 2.48 and 0.95, respectively. However, SAITS recorded substantially higher error metrics at this rate, with an MAE of 7.35 and RMSE of 661.78, suggesting that its performance may be less suited for scenarios with frequent random gaps in data. At the highest random missing rate of 10%, BRITS maintained relatively low error metrics (MAE = 1.27, RMSE = 54.25), while ImputeFormer’s performance slightly deteriorated, yielding an MAE of 4.57 and RMSE of 469.81. SAITS displayed the highest error metrics among the models under this condition, with an MAE of 8.09 and RMSE of 662.97, indicating that it struggles to interpolate effectively in cases of extensive random missing data. Overall, these results highlight BRITS’s robustness in handling random missing data, particularly at low to moderate missing rates, and ImputeFormer’s reliability for moderate levels of random missingness.

For block missing patterns, where missing values occur in consecutive sequences, SAITS demonstrated improved performance relative to its results for random missing data, particularly at lower missing rates. At a 1% block missing rate, SAITS achieved an MAE of 0.94 and RMSE of 1.71, outperforming both BRITS and ImputeFormer in terms of accuracy for low-rate block missingness. These results suggest that SAITS’s self-attention mechanism, which can capture long-range dependencies, is beneficial for handling continuous gaps in data. ImputeFormer and BRITS recorded slightly higher MAE and RMSE values under this condition, with ImputeFormer yielding an MAE of 0.84 and RMSE of 2.88, while BRITS achieved an MAE of 0.99 and RMSE of 10.88. However, an anomaly emerged at the 3% block missing rate, where both SAITS and BRITS exhibited unexpectedly large errors (e.g., SAITS: MAE = 12.18, RMSE = 858.68; BRITS: MAE = 2.03, RMSE = 98.57). This spike may be attributed to sensor-specific noise, localized data artifacts, or other complexities in the on-site detection data. Further investigation such as applying denoising techniques, collecting additional validation data, or refining the models’ preprocessing steps would be necessary to clarify this discrepancy.

As missing rates increased to 5% and 10% in block patterns, ImputeFormer displayed stable error metrics, while BRITS performed comparably. At the highest block missing rate, ImputeFormer and BRITS maintained moderate performance levels, whereas SAITS exhibited a notable decline, suggesting it may be less reliable for extensive block missingness.

While we employed MAE, MSE, and RMSE to evaluate imputation performance, these metrics each have inherent limitations. MAE provides an average measure of the absolute differences between predicted and actual values, treating all errors equally. MSE emphasizes larger errors more heavily due to the squaring of residuals, making it sensitive to outliers but potentially overshadowing smaller discrepancies. RMSE, as the square root of MSE, is often more interpretable in terms of the original data units but still retains the sensitivity to large deviations. Given these considerations, using multiple metrics allows for a more comprehensive assessment, yet we acknowledge that each metric captures only certain aspects of model performance. In future work, we may consider incorporating additional measures, such as R² or domain-specific performance metrics, to provide further insights and highlight differences in imputation performance.

In summary, the empirical results reveal that BRITS performs best in scenarios with low to moderate random missing data, where it achieves consistently low MAE and RMSE values. ImputeFormer showed versatility across both random and block missing data patterns, particularly at moderate missing rates, while maintaining relatively stable error metrics. SAITS was effective at low-rate block missingness but struggled under higher missing rates and random patterns, where shorter dependencies are critical for accurate imputation. The anomaly observed at a 3% block missing rate underscores the importance of considering data quality and potential artifacts when interpreting model performance. These findings provide a comprehensive basis for selecting the appropriate model for different missing data patterns in slope displacement monitoring applications and suggest avenues for future work to address unexpected performance variations.

5. Conclusions

This study set out to address the critical challenge of missing data in slope displacement monitoring by evaluating three advanced machine learning under controlled experimental conditions. Our primary research questions focused on determining how effectively these models handle varying rates and patterns of missing data. By simulating missingness at 1%, 3%, 5%, and 10% using both random and block patterns, we obtained a comprehensive understanding of each model’s performance and their respective strengths and limitations.

The results show that model performance varies significantly based on the pattern and rate of missing data. BRITS, with its bidirectional recurrent architecture, achieved consistently high accuracy in scenarios with random missing data, particularly at lower missing rates. This outcome suggests that BRITS is well-suited for cases of intermittent data loss, where short-term dependencies are critical for accurate interpolation.

Conversely, SAITS demonstrated relatively stronger performance at low block missing rates, capitalizing on its self-attention mechanism to effectively capture long-range dependencies. This advantage suggests that SAITS may be particularly suitable in environments where prolonged outages or equipment malfunctions create extended gaps in the data, but remain at lower levels of missingness. However, as the block missing rate increases, SAITS’s performance notably declines. The Transformer-based ImputeFormer model showed versatility at moderate missing rates, but its performance declined under higher rates and more complex missing patterns, indicating that it may be better suited to applications with low to moderate data loss. Collectively, these findings emphasize the importance of aligning imputation model selection with the specific missing data pattern, as no single model performed optimally across all conditions. In practical terms, our study suggests that adopting a hybrid strategy, using BRITS for random gaps and SAITS for block gaps, could improve data quality in slope displacement monitoring.

Beyond model selection, successful real-world implementation depends on practical considerations. Computational resources, including processing power and memory, influence the feasibility of near-real-time imputation. While these models typically run efficiently on modern hardware, field deployment may require edge computing solutions or cloud-based infrastructures to handle continuous data streams. Training time must also be considered, as models may need periodic retraining when environmental conditions or sensor configurations change. Integrating these models with existing monitoring systems involves establishing reliable data transfer protocols, ensuring compatibility with data storage formats, and developing user-friendly interfaces that facilitate interpretation by engineers and decision-makers.

In the future, exploring additional missing data patterns (e.g., seasonal, periodic, or trend-based missingness) could better reflect real-world scenarios and enhance the generalizability of these methods. Furthermore, exploring developing ensemble models that combine the strengths of multiple imputation methods to handle a broader range of missing data scenarios in real time. Additionally, expanding this evaluation to include other types of geotechnical monitoring data would further validate the applicability of these models across various engineering contexts. In conclusion, machine learning-based imputation methods show significant potential for improving data reliability in slope displacement monitoring. By enhancing the continuity of time-series data, these methods support more accurate and timely assessments of slope stability, ultimately contributing to safer and more effective geotechnical monitoring systems.