2.1. Traditional Time Series Models

Traditional time series models have been extensively employed for the analysis and prediction of diverse time series data. These models rely on statistical methodologies and mathematical algorithms to capture patterns and trends within the data. Notable examples encompass autoregressive integrated moving average (ARIMA) models, which are widely used for forecasting and analyzing stationary time series data [16]. ARIMA models operate on the premise that future values in a time series can be foretold based on historical values and the model’s errors. Another traditional method involves the use of autoregressive (AR) models, which assume a linear dependence of future time series values on past values [17]. Furthermore, traditional models like exponential smoothing models, state space models, and seasonal and trend decomposition using LOESS (STL) models have been employed for time series analysis and prediction [18–20]. Combining traditional models can effectively capture the complexities of nonlinear and high-variance data. Reference [21] investigates the effectiveness of hybrid combinations of linear and nonlinear models for predicting Brent crude oil prices, demonstrating how decomposing the data into trend and stochastic components allows for more accurate day-ahead forecasts by addressing underlying patterns separately. Ensemble techniques are also highly effective in managing such intricate data patterns. Reference [22] introduces an ensemble approach tailored for electricity demand forecasting in the Peruvian market, where multiple models work together to refine predictions after preprocessing for seasonality, stationarity, and variance. Similarly, reference [23] employs multiple decomposition techniques for forecasting electricity consumption in Pakistan, breaking down time series into trend, seasonal, and stochastic components, each forecasted with optimized models. Building on these decomposition methods, recent studies have demonstrated the effectiveness of functional data analysis (FDA) in forecasting by isolating key data components for individual modeling. This method, which has been applied in various fields, allows each component—whether based on trends or stochastic processes—to be addressed with models tailored to their specific characteristics. Applications have demonstrated this approach in forecasting ozone concentration [24], wind speed [25], and electricity prices [26].

These models serve as a robust framework for comprehending and forecasting time series data and have found utility across various domains, including economics, finance, meteorology, and engineering. However, traditional models often struggle with long-term forecasting and fail to capture the complex, nonlinear dynamics inherent in transformer oil temperature data. Consequently, more advanced approaches, such as neural forecasting, have arisen to tackle these challenges and enhance the accuracy of time series forecasting.

2.2. Neural Forecasting Applications

Neural forecasting encompasses the utilization of neural network models to make predictions and forecasts across diverse domains. This approach has gained popularity due to its ability to capture complex patterns and relationships in data, particularly in tasks related to time series forecasting. The applications of neural forecasting are spanning various fields. In the domain of economics and finance, neural networks have been used to forecast stock prices and earnings per share, GDP, and foreign exchange rates [27–29]. In the energy sector, neural networks find applications in predicting electricity load, wind power generation, and solar wind activity [30–32]. In this context, a study utilized a classification predictive model to assess maintenance requirements for distribution transformers, thereby improving their reliability and cutting costs through predictive maintenance [33]. Another research employed artificial neural networks (ANNs) to predict wind and solar energy production, attaining higher accuracy than traditional models, which helps in balancing energy supply and demand [34]. Moreover, neural forecasting has been adopted in hydrology for the prediction of runoff and water demand [35]. In the transportation sector, neural networks have proven valuable in forecasting traffic patterns and emergency calls [36]. Additionally, neural forecasting has made inroads into the domains of marketing, tourism, and even geomagnetic storm prediction [32, 37, 38]. Although these methods have been successful, they are less suited for long-term forecasting tasks, which NHITS is designed to address more effectively.

2.3. Long-Sequence Time Series Forecasting

Long-sequence time series forecasting refers to predicting extended sequences of time series data with a focus on capturing intricate long-range dependencies. This forecasting challenge arises from the need to efficiently grasp precise long-range relationships between input and output variables [14]. Conventional time series forecasting methods, such as ARIMA models, can encounter difficulties when dealing with lengthy sequences [39]. To address this challenge, researchers have explored the utilization of deep learning models, including recurrent neural networks (RNNs) and transformers, for long-sequence time series forecasting. RNNs, which encompass long short-term memory (LSTM) networks, have demonstrated effectiveness in capturing long-sequence patterns and making multistep forecasts while considering multiple variables [40]. In this context, a bidirectional LSTM neural network was implemented to enhance the prediction of electrical loads. This approach significantly improves accuracy compared to previous models and aids in more effective energy planning and market management [41]. Transformers, which have gained popularity in natural language processing tasks, have been adapted for time series forecasting. Notably, the Informer model extends the transformer architecture to efficiently capture long-range dependencies between input and output variables [14]. DeepAR is another model that leverages AR recurrent networks for probabilistic forecasting, showcasing enhanced accuracy when compared to state-of-the-art methods [42]. In a separate study, a transformer model with attention-based encoder–decoder networks was proposed. This model incorporates attention mechanisms to focus on critical attributes and capture dependencies in multivariate time series prediction [43].

3.1. Historical Data Collection

This study leverages the ETT dataset1, an accessible multivariate time series dataset consisting of four subdatasets containing electric load data and oil temperature records from two different electrical stations located in separate regions [14]. Variations between regions may reflect different operational environments, which could influence the oil temperature trends. The ETT dataset is accessible in two resolutions, either every 15 min or hourly intervals. Table 2 describes the available datasets and their specific attributes.

3.2. Data Preprocessing

3.3. Feature Selection

The ETT dataset contains different features, including date points, electrical transformer oil temperature, and six distinct categories of external power load features. Features such as electrical load are important to the model as they are directly linked to transformer operation and potential temperature changes. The prediction of electrical transformer oil temperature serves a vital role as an early warning for a predictive maintenance system. As oil temperature rises, it signifies increased demand and stress on the transformer. By forecasting temperature fluctuations, proactive management of maintenance and load distribution becomes possible, averting overheating and optimizing electricity consumption. This approach ensures the reliability and effectiveness of the system [5]. Table 3 outlines the various attributes present within the ETT dataset. This study compares the influence of employing multivariate and univariate time series for predicting the oil temperature of transformers using the ETT datasets. In the multivariate scenario, all dataset features are utilized, whereas in the univariate scenario, only the OT variable serves as the sole input to the model. The aim is to analyze the performance of the model under each scenario.

3.4. Model Selection

The NHITS model was chosen to predict the oil temperature of transformers in the ETT dataset. The model incorporates two key techniques: multirate signal sampling and hierarchical interpolation. The multirate signal sampling of the input signal is achieved by the MaxPool layer, with the sampling rate determined by the kernel size. A larger kernel size tends to highlight the long-term characteristics of the time series, while a smaller kernel size does the opposite. This allows the model to decrease memory consumption and computational demands, thereby improving its performance in long-term forecasting. Additionally, the NHITS model effectively addresses seasonality by isolating seasonal trends through its multirate signal sampling, which captures recurring patterns within different time scales of the transformer oil temperature data. This hierarchical approach ensures that seasonal fluctuations are recognized and interpreted as part of the core patterns rather than as anomalies. The hierarchical interpolation is then utilized to merge the predictions of all stacks, with each stack focusing on a different time scale. This lowers the computational demands by reducing the cardinality of predictions, thereby preventing overfitting and enhancing forecast accuracy. By using this hierarchical approach, the model is able to tackle both short-term and long-term forecasting tasks more effectively.

3.5. Model Training and Hyperparameter Optimization

The data are divided into training, validation, and testing subsets, apportioned at a ratio of 60%–20%–20%, respectively. The training set is utilized for training the model parameters, the validation set facilitates hyperparameter tuning, and the test set is employed for model evaluation. Figure 3 illustrates the seasonal trends in the transformer oil temperature, which our model must capture accurately to provide reliable long-term forecasts. The figure presents the time series of the OT variable with the utilized data partitions within the ETTm₁ dataset, along with an enlarged section highlighting the variable’s seasonality. The model structure comprises 3 stacks, with a maximum of 2 blocks in each stack. Every block consists of 2 MLP layers, with 512 hidden units. The complete set of hyperparameters used for the NHITS configuration is detailed in Table 4.

3.6. Model Evaluation

In this context, y_τ and $$ represent the actual and observed values at time τ, with t indicating the initial time index and H denoting the forecast horizon or time steps over which the metric is calculated.

4.1. Forecasting Accuracy

The NHITS model proposed in this study was evaluated against the Informer model [14]. The Informer model utilizes a ProbSparse Self-Attention Mechanism to lower time complexity and memory consumption by concentrating on essential parts of the sequence. Additionally, it employs Self-Attention Distilling to handle long input sequences more effectively by highlighting important attention areas and reducing cascading layer inputs. This makes the Informer model an efficient transformer-based solution for long-sequence time series forecasting. In this evaluation, ARIMA serves as a baseline for the comparison. The results for both the Informer model and ARIMA are derived from the same study [14].

In the context of the ETT dataset, Table 5 showcases the outcomes of benchmarked models for both univariate and multivariate time series, specifically in the case of short forecasting horizons. On the other hand, Table 6 details the results for long forecasting horizons. The top-performing outcomes are indicated in bold. Lower MSE and MAE values signify superior model performance. The reported metrics represent the averages derived from three separate runs.

Tables 5 and 6 demonstrate that the NHITS model consistently outperforms the Informer and ARIMA models across various datasets and forecasting horizons. The exception is the ETTh₂ dataset for Horizon ∈ {24, 336}, where the Informer model performs marginally better than NHITS. Table 7 highlights the relative improvement in MSE and MAE of the NHITS model over the Informer model. On average, NHITS shows a 51.37% improvement in MSE and a 37.83% improvement in MAE. This significant enhancement underscores NHITS’s superior capability in capturing both short-term and long-term dependencies in the data, facilitated by its hierarchical interpolation mechanism and multirate signal sampling technique.

4.2. Univariate and Multivariate Forecasting

Univariate forecasting and multivariate forecasting represent two distinct methodological approaches applied in the field of predictive modeling and time series analysis. Univariate forecasting entails making predictions based on a single variable, whereas multivariate forecasting involves the simultaneous consideration of multiple variables to inform predictive outcomes [45].

Univariate forecasting typically offers relative simplicity and computational efficiency compared to its multivariate counterpart. It is particularly advantageous when relationships between variables are ill-defined or when there is limited availability of data. In contrast, multivariate forecasting methods address predictive challenges by comprehensively considering the interactions among multiple variables. These techniques excel in capturing dependencies and correlations between variables, leading to more precise forecasts in specific scenarios [46].

In the univariate benchmark for the ETT dataset, the target variable is the OT variable, with all other fields, presented in Table 2, excluded from consideration. Conversely, in the multivariate benchmark, the models generate predictions for every feature within the dataset, and the evaluation metrics are computed as averages across all dataset features.

Figure 4 presents the average MSE and MAE for the NHITS model over various forecasting horizons for both univariate and multivariate time series on the ETT datasets. The figure demonstrates that univariate forecasts maintain relatively low and stable MSE and MAE values across different horizons, indicating high accuracy. In contrast, multivariate forecasts start with higher MSE and MAE values than univariate forecasts, and these metrics increase significantly as the forecast horizon extends.

4.3. Forecast Horizon

Long-horizon forecasting and short-horizon forecasting represent distinct temporal prediction approaches that focus on different forecasting time frames. Long-horizon forecasting involves predicting events in the distant future, while short-horizon forecasting focuses on predicting events shortly. Long-horizon forecasting, which involves making predictions for the distant future, is particularly challenging due to the inherent unpredictability of forecasts, increasing uncertainties, and the compounding of errors over time [47].

Tables 5 and 6 display the outcomes of forecasting over short and long-term horizons, respectively, as applied to the ETT dataset. In the case of short-horizon forecasting, we considered horizons of 24 and 48 data points, encompassing 1 day and 2 days for univariate and multivariate time series within the ETTh_1,2 datasets, and 6 h and 12 h for the ETTm_1,2 datasets. For long-horizon forecasting, we selected horizons of 336 and 720 data points for ETTh_1,2, and for ETTm_1,2 we selected horizons of 288 and 672. That is equivalent to 14 days and 30 days for ETTh_1,2, and 72 h and 168 h for ETTm_1,2. This configuration was applied for both the univariate and the multivariate time series.

Figure 4 shows that both MSE and MAE for univariate forecasts remain relatively low and stable across various horizons, indicating high accuracy, especially for shorter horizons (24 and 48 units). As the forecasting horizon extends to 288, 336, and 672 units, both metrics show a slight increase but tend to stabilize beyond 288 units. In contrast, multivariate forecasts display higher MSE and MAE values even at shorter horizons, reflecting the added complexity and potential errors from considering multiple variables simultaneously. For longer horizons (336 and 720 units), there is a significant increase in both MSE and MAE, particularly at the 336-unit mark and a noticeable spike at 720 units, indicating that multivariate forecasts become significantly less accurate over very extended periods.

4.4. Temporal Resolution

ETTm_1,2 and ETTh_1,2 represent two versions of the same dataset, with different temporal resolutions. Both datasets capture information related to electric load and oil temperature. However, ETTm_1,2 maintains a high temporal resolution of 15 min, providing fine-grained insights into fluctuations and patterns over shorter time intervals. In contrast, ETTh_1,2 opts for a lower temporal resolution, with data points collected at hourly intervals, offering a broader perspective with less granularity but spanning longer periods.

Table 8 displays the performance metrics for the univariate time series using ETTm_1,2 and ETTh_1,2 across various periods. The periods under consideration are t ∈ {1, 3, 7} days, corresponding to the horizon H ∈ {76, 288, 672} for ETTm_1,2 and H ∈ {24, 72, 168} for ETTh_1,2. An analysis of these results reveals distinct patterns for each case. For the 1-day forecast period, ETTm₁ shows significantly higher MSE and MAE than ETTh₁, with ETTm₂ and ETTh₂ exhibiting similar patterns. This suggests that the higher temporal resolution of ETTm₁ may introduce more variability, leading to greater errors in short-term forecasts. Over a 3-day forecast horizon, the performance of ETTm datasets improves relative to ETTh datasets, with both ETTm₁ and ETTm₂ demonstrating lower MSE and MAE compared to their ETTh counterparts. This indicates that the higher granularity in ETTm datasets becomes beneficial over medium-term forecasts, possibly due to capturing more detailed fluctuations that aid in improving prediction accuracy. Finally, in the long-term 7-day forecasts, ETTm₁ maintains lower error metrics compared to ETTh₁, a trend that is also observed in ETTm₂ and ETTh₂. The reduced error in ETTm datasets for longer horizons suggests that the finer temporal resolution allows the model to maintain accuracy even as the forecast period extends, likely by better capturing underlying trends and periodic patterns.

5.1. Forecasting Accuracy

The NHITS model demonstrated a notable decrease in both MSE and MAE compared to the Informer model, highlighting its superior capability for long-sequence forecasting. NHITS efficiently captures both long-term and short-term characteristics of time series data using its multirate signal sampling technique. Additionally, its hierarchical interpolation mechanism integrates predictions across various time scales, enhancing forecast accuracy and providing interpretable forecast decompositions. This approach allows for more precise insights into trends and seasonal patterns within the data. The model’s architecture, which includes multiple stacks and blocks operating at different temporal resolutions, effectively manages the diverse patterns and trends present in the data, further distinguishing it from the Informer model.

5.2. Univariate and Multivariate Forecasting

There is a clear performance gap between univariate and multivariate forecasts, with univariate forecasts consistently showing lower MSE and MAE values, highlighting their superior accuracy and stability across different horizons. In contrast, multivariate forecasts exhibit higher variability and error rates, especially for long-term predictions. This trend is observed in the NHITS, Informer, and ARIMA models and can be attributed to the higher dimensionality and complexity of multivariate data, which demand more computational resources and sophisticated handling. In the ETT dataset, the multivariate time series’ dimensionality is seven times greater. The lower error rates in univariate forecasting suggest that focusing on single-variable predictions might be more effective for applications requiring high accuracy, while multivariate forecasts, though potentially more informative, require careful handling.

5.3. Impact of Forecast Horizon

Another key finding is the model’s performance across different forecast horizons. Short-horizon forecasts were consistently more accurate than long-horizon ones, a trend seen in other models as well due to increased uncertainty and error accumulation over longer periods. Although the NHITS model’s hierarchical structure and multirate sampling approach help mitigate these issues, long-term predictions remain challenging. Nonetheless, NHITS still outperformed the Informer model.

5.4. Impact of Temporal Resolution

The performance differences between ETTm and ETTh datasets across different forecast horizons highlight the impact of temporal resolution on the NHITS model’s forecasting accuracy. Higher temporal resolution (15-minute intervals) in ETTm datasets generally leads to better performance in medium to long-term forecasts. In contrast, lower temporal resolution (hourly intervals) datasets might be more suitable for very short-term predictions but tend to lag in accuracy over extended periods. This analysis underscores the importance of selecting appropriate temporal resolutions based on the specific forecasting horizon and application requirements. For applications needing high accuracy over longer forecast periods, utilizing higher temporal resolution data can be advantageous. Conversely, for short-term forecasts, lower temporal resolutions may suffice and potentially reduce computational complexity.

The NHITS model’s ability to efficiently handle long-term dependencies, coupled with its hierarchical interpolation, allows it to outperform traditional models that rely on simpler time series extrapolation techniques. While NHITS shows promising results, its performance in different contexts or with larger datasets warrants further investigation to ensure generalizability. These results highlight the importance of selecting forecasting methods that suit the specific needs of each application, striking a balance between accuracy and computational efficiency.