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# Cyclical Temporal Encoding for Ensemble Deep Learning in Multistep Energy Forecasting

Yaroub Elloumi, Salim Khazem, Ibrahim Krayem, Jeyakaran Mahesananthan

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## To cite this version:

Yaroub Elloumi, Salim Khazem, Ibrahim Krayem, Jeyakaran Mahesananthan. Cyclical Temporal Encoding for Ensemble Deep Learning in Multistep Energy Forecasting. 2025. hal-05170016

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https://hal.science/hal-05170016v1

Preprint submitted on 18 Jul 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# Cyclical Temporal Encoding for Ensemble Deep Learning in Multistep Energy Forecasting

Yaroub Elloumi*, Salim Khazem†, Ibrahim Krayem†, Jeyakaran Mahesananthan‡

*Talan, Lyon, France

†Talan Research & Innovation Center, Paris, France

‡Talan, Paris, France

**Abstract—**

Accurate electricity consumption forecasting is essential for smart grid operations and energy planning. This work presents a deep learning-based framework that combines temporal feature engineering with architectural complementarity for improved short-term predictions. We extract and transform calendar-based features using sine and cosine encodings to capture periodic patterns and assess their relevance using correlation analysis. To address both long-term and local temporal dependencies, we employ an ensemble of two base models: a Long Short-Term Memory (LSTM) network and a Convolutional Neural Network (CNN). Their outputs are integrated through a Meta-Classifier composed of multiple MLP regressors, each dedicated to one forecast day. Experiments on a one-year daily consumption dataset demonstrate that our hybrid model consistently outperforms individual baselines across a 7-day prediction horizon, achieving lower MAE and RMSE. These results confirm the benefit of combining cyclical feature encoding with hybrid architectures for robust energy forecasting.

***Index Terms***—Energy forecasting, time series, LSTM, CNN, temporal feature engineering, ensemble learning, cyclical encoding, MLP regressor.

## I. INTRODUCTION

Accurate forecasting of electricity consumption remains a critical challenge for energy providers and smart grid operators. It is essential for ensuring the balance between supply and demand, optimizing energy storage systems, and informing strategic decisions such as dynamic pricing and load management. However, electricity usage is shaped by a multitude of complex and interdependent factors including human behavior, seasonal climate variability, and socio-economic dynamics that introduce strong non-linearities and periodic patterns into consumption trends. These intricacies often limit the effectiveness of traditional statistical forecasting approaches [1].

Recent advances in deep learning have shown substantial promise in tackling time series forecasting problems. Notably, architectures such as Long Short-Term Memory (LSTM) networks and Convolutional Neural Networks (CNNs) are capable of learning long-range temporal dependencies and extracting meaningful patterns from historical sequences [2]. Yet, their performance remains highly sensitive to the way temporal information is represented. Raw time step encoding alone often fails to capture the underlying periodic structure inherent in electricity consumption data.

In this work, we propose an enhanced deep learning-based forecasting framework that incorporates explicit temporal feature engineering. We extract and analyze calendar-based attributes such as day-of-week, week-of-year, and month-of-year, which are known to exhibit strong cyclical influence on consumption behaviors. To preserve their periodic nature and avoid discontinuities (e.g., transitions from Sunday to Monday or December to January), we apply sine and cosine transformations. These transformations map discrete time features onto a continuous cyclic space, enabling neural networks to learn seasonal rhythms more effectively.

We conduct a correlation-driven feature selection process to assess the predictive power of each temporal attribute, in both raw and transformed formats. Features with the strongest correlation to the target variable are incorporated into the input space of the forecasting models. We then evaluate the performance of three architectures: LSTM, CNN, and a hybrid model over a rolling seven-day prediction horizon, using Mean Absolute Error (MAE) and Root Mean Squared Error (RMSE) as evaluation metrics [3].

Our results demonstrate that the integration of well-encoded temporal features significantly enhances forecasting accuracy, particularly for medium and long-term horizons. This study contributes a robust and interpretable temporal encoding pipeline that can be generalized to other periodic time series in smart energy systems and beyond.

## II. RELATED WORK

Electric load forecasting has long been a critical area of research, particularly with the advent of smart metering technologies and the increasing emphasis on demand-side management in smart grids. Early approaches have predominantly relied on statistical models, such as autoregressive methods like ARIMA [4], which assume linearity and stationarity. More recently, machine learning and deep learning techniques have emerged as powerful alternatives for capturing non-linear dependencies and complex temporal dynamics [2], [5].

A growing body of work highlights the importance of temporal feature engineering in enhancing forecasting performance. Deep learning models, including LSTM [6] and CNN [7] architectures, have shown significant improvements when augmented with calendar based features [8]. To better represent the cyclical nature of time variables such as day-of-week, hour-of-day, or month-of-year, several studies adopt
sine and cosine transformations [9]. These encodings help mitigate artificial discontinuities in periodic signals, thereby improving model generalization. Such techniques have been successfully applied in domains like traffic prediction [10] and energy demand forecasting [3].

Additionally, some approaches incorporate exogenous variables, such as temperature, holidays, and socio-economic factors, to further refine predictions [11]. These variables often introduce important contextual signals that influence consumption patterns beyond time alone.

Building on these insights, our work contributes in two key directions: (1) we perform a systematic evaluation of raw vs. cyclically transformed temporal features using Pearson correlation analysis, to quantify their alignment with consumption patterns; and (2) we empirically validate their predictive value within a deep learning framework over a 7-day forecast horizon. This dual analysis provides a clearer understanding of how temporal encoding affects model performance and offers practical guidance for feature engineering in periodic time series forecasting.

### III. METHOD

#### A. Preprocessing & Feature Engineering

1) **Exploration of Time data granularity:** Electric power consumption is split into several activity fields belonging to domestic and professional uses. Those activity fields do not follow the same frequency of power consumption. In addition, several aspects affect the power utilization such as human behaviour, climatic conditions, and seasonal variation. For this purpose, we proceed to extract all time granularities in order to explore their relation with respect to power consumption. We extract from a single date the features of Day in Week, Day in Month, Day in Year, Week in Month, Week in Year, Month in Year. In addition, we generate an additional feature modeling the “business day” including the French national days off.

Then, we study the evolution of each feature with respect to the power consumption, by generating the Pearson correlation coefficient *r*, which is computed as indicated in the equation.

$$r = \frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum(x_i - \bar{x})^2 \cdot \sum(y_i - \bar{y})^2}}$$

Where $x_i$ is the feature, $y_i$ is the power consumption, $\bar{x}$ and $\bar{y}$ are respectively the feature and the power consumption average. The *r* coefficient models a linear relationship between the provided data, where a common rise (e.i. reversed rise) leads to a positive coefficient value (e.i. negative coefficient value). All *r* coefficients with respect to the time features are indicated in Table I.

We note that *Day in Month* and *Week in Month*, where values are respectively in [1, 31] and [1, 6], have no correlation with the power consumption and are hence ignored for the rest of the study.

We explore the feature tendency in terms of power consumption in order to detail the *r* coefficients. The power variation in terms of months and weeks are respectively

<table><caption>TABLE I: Correlation Coefficients of Time Features</caption><thead><tr><td>Feature</td><td>Correlation (r)</td></tr></thead><tbody><tr><td>Day in Week</td><td>-0.1993</td></tr><tr><td>Day in Month</td><td>0.0370</td></tr><tr><td>Day in Year</td><td>-0.2338</td></tr><tr><td>Week in Month</td><td>-0.0022</td></tr><tr><td>Week in Year</td><td>-0.2288</td></tr><tr><td>Month in Year</td><td>-0.2341</td></tr><tr><td>Business Day</td><td>-0.2733</td></tr></tbody></table>

illustrated in boxplot visuals of Fig. 1a and Fig. 1b. We deduce a cyclic pattern of power consumption along an entire year, which explains the significant *r* coefficient related to “Day in Year”, “Week in Year” and “Month in Year”. A similar exploration has been done in terms of “Day in Week”, where we note an almost constant power consumption at the beginning of the week, as indicated by the median values in the boxplot of Fig. 1c and a power consumption decrease during the weekend. The same pattern is also deduced from Fig. 1d, modelling power in terms of “Business Day”.

2) **Feature Engineering for Cyclical Pattern of Time Series:** Cyclical patterns along the weeks and along the years are well modeled either in the whole dataset or in visual distributions. However, the related *r* coefficients are not high enough to model such correlation. In fact, while power consumption evolves continually through the cyclical patterns, time features are characterized by a value discontinuity, due to value reset at the end of each cycle, as will distinguish in 2.

To overcome this problem data cut-off, we suggest transforming the timing data feature in order to have a progressive evolution along the time pattern. Within this objective, two news features are generated from each original ones, that correspond respectively to the sin and cos processing with respect to the identified *DataCycle*, as indicated in Eq. 1 and Eq. 2.

$$F_{\text{sin}} = \sin(2\pi \times \frac{F}{\text{TimeCycle}}) \quad (1)$$

$$F_{\text{cos}} = \cos(2\pi \times \frac{F}{\text{TimeCycle}}) \quad (2)$$

Where *F* is the time feature and *TimeCycle* in the cycle duration of *F*. Figures in 2 illustrate the transformation of the “Week in Year” feature as indicated in equations 1 and 2, where trigonometry data evolutions are closed to the power consumption.

To evaluate the contribution made by these transformation, we generate the *r* coefficient for the original and the transformed features, as depicted in tab.(2). We note that $F_{\text{sin}}$ and $F_{\text{cos}}$ have higher correlation with the power consumption as well as preserving time cyclical time pattern. The Business Day remains unchanged even with trigonometric transformation. The “Day in Week” has a similar correlation after applying eq. 1, due to the consumption spike between Sunday and Monday, which closed to the evolution of day numbers. Consequently, we choose the transformed features
offering the higher r coefficients for “Day in Year”, “Week in Year” and “Month in Year”, as modeled in bold in table (2). The “Day in week” and “Business Day” are preserved without transformation.

<figure><img src="image_4.png" /><figcaption>Fig. 1: Power consumption variation along the year: (a) grouped by month; (b) grouped by week; (c) grouped by day in week; (d) grouped by day type</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 2: Comparison between raw and transformed temporal features. (Top) Normalized power consumption plotted with the raw *Week in Year* feature, showing discontinuities at year boundaries. (Bottom) Same power series plotted with its cyclical encodings: *Week-Cos* and *Week-Sin*, which produce smooth and continuous representations across years.</figcaption></figure>

<table><caption>TABLE II: Correlation Coefficients in Terms of Trigonometry Transformations</caption><thead><tr><th></th><th>Day in Year</th><th>Week in Year</th><th>Month in Year</th><th>Day in Week</th><th>Business Day</th></tr></thead><tbody><tr><td>F</td><td>-0.233824</td><td>-0.228847</td><td>-0.234128</td><td>-0.199358</td><td><strong>-0.273308</strong></td></tr><tr><td>F<sub>sin</sub></td><td>0.235060</td><td>0.271596</td><td>0.431879</td><td><strong>0.208859</strong></td><td><strong>-0.273308</strong></td></tr><tr><td>F<sub>cos</sub></td><td><strong>0.827672</strong></td><td><strong>0.783263</strong></td><td><strong>0.697115</strong></td><td>-0.060555</td><td><strong>-0.273308</strong></td></tr></tbody></table>

<table><caption>TABLE III: Correlation Coefficients for Time Features and Their Trigonometric Transformations</caption><thead><tr><th>Feature</th><th>Raw (F)</th><th>Sine (F<sub>sin</sub>)</th><th>Cosine (F<sub>cos</sub>)</th></tr></thead><tbody><tr><td>Day in Year</td><td>-0.2338</td><td>0.2351</td><td><strong>0.8277</strong></td></tr><tr><td>Week in Year</td><td>-0.2288</td><td>0.2716</td><td><strong>0.7833</strong></td></tr><tr><td>Month in Year</td><td>-0.2341</td><td>0.4319</td><td><strong>0.6971</strong></td></tr><tr><td>Day in Week</td><td>-0.1994</td><td><strong>0.2089</strong></td><td>-0.0606</td></tr><tr><td>Business Day</td><td><strong>-0.2733</strong></td><td><strong>-0.2733</strong></td><td><strong>-0.2733</strong></td></tr></tbody></table>

# IV. EXPERIEMENTS

## A. Principles

Our work aims to predict the power consumption efficiently for the next weekdays. Several DL architectures have been proposed for time series forecasting, which employ different mechanisms such as recurrence, convolution, and attention. In terms of performance, every architecture generates better results within a well-defined seasonal context. There is no DL architecture that outperforms the others, regardless of seasonality patterns.

For this purpose, we proceed to employ two different DL architectures: the LSTM for its capability one capturing long dependencies related to the yearly patterns; while the convolution architecture for extracting local motifs related to the weekly patterns.

Then, the knowledge of both models is merged into an Ensemble Learning framework in order to refine the prediction, as illustrated in Fig. 3. The LSTM and convolution models are considered as Base-Classifiers where their feature maps
are fused and provided to a Multi-Layer Perceptron (MLP) as a Meta-Classifier to provide the forecasting output.

<figure><img src="image_6.png" /><figcaption>Fig. 3: Workflow of the Ensemble-Learning Method for Power Forecasting</figcaption></figure>

## B. Implementation Details and Dataset

We use a time-series dataset of daily electric power consumption over one year. After preprocessing, all data is scaled using Min-Max normalization. The model input consists of 120 days (4 months) of past consumption and time-based features, used to predict power usage for the next 7 days. Temporal features are encoded using cyclical transformations (sine and cosine), and national holidays are embedded as binary flags. The dataset is split chronologically with an 80%-20% train-test ratio. The hyperparameters are detailed in Table VI.

## C. Base Classifier Structures and Configuration

To identify the periodical pattern, we suggest 64 LSTM units in the first layer. After that, the feature dimensions are reduced through two dense layers with 32 and 7 neurons, respectively. As a regularization technique, we involve two dropout layers between the processing layers in order to ignore the output during the training, as indicated in the architecture description in Table V and Table IV.

For the CNN model, we perform two convolution layers, with 128 and 32 filters, respectively, in order to capture cyclical pattern. Therefore, two dense layers are added to converge the features. Similarly to the LSTM model, the dropout mechanism is used between convolution layers and dense layers to avoid overfitting.

<table><caption>TABLE IV: LSTM Model Architecture</caption><thead><tr><td>Layer</td><td>Output Shape</td></tr></thead><tbody><tr><td>LSTM</td><td>(None, 64)</td></tr><tr><td>Dropout</td><td>(None, 64)</td></tr><tr><td>Dense</td><td>(None, 32)</td></tr><tr><td>Dropout</td><td>(None, 32)</td></tr><tr><td>Dense</td><td>(None, 7)</td></tr></tbody></table>

Experimental approach has been adopted to identify the reliable hyper-parameters in order to guarantee optimal forecasting. The learning rates and the optimizer are firstly identified among several suggestions. Then, dropout, epochs, and

batch size are determined with respect to each architecture processing, where the values are detailed in Table VI.

<table><caption>TABLE VI: Hyper-parameters of DL Base-Classifiers</caption><thead><tr><td>Hyper-Parameter</td><td>LSTM</td><td>CNN</td></tr></thead><tbody><tr><td>Learning rate</td><td>0.0001</td><td>0.001</td></tr><tr><td>Dropout</td><td>0.2</td><td>0.3</td></tr><tr><td>Optimizer</td><td>Adam</td><td>Adam</td></tr><tr><td>Epochs</td><td>80</td><td>80</td></tr><tr><td>Batch Size</td><td>32</td><td>32</td></tr></tbody></table>

Meta classifier structure The base classifiers are dedicated to identifying periodical patterns in the aim of forecasting power consumption of seven days, where BC provides feature map with 7 values, where each single feature corresponds to a single day. The meta classifier brings to explore BC predictions, day by day, in order to adjust values, in order to achieving a higher performance. For this purpose, a single Meta-classifier is proposed for each day forecasting, which intercepts BC features related to the day in order to polish forecasting, as illustrated in Fig. 4

<figure><img src="image_7.png" /><figcaption>Fig. 4: Meta Classifier based on Multiple MLPRegressors</figcaption></figure>

Based on the size of classifier input/output and the size of dataset for training, we proceed to employ the Multi-Layer-Perceptron Regressor (MLPRe regressor). In fact, all MLPRegressors will not explore the same pattern and hence might not have the same configuration. Thus, we proceed to identify relevant hyper-parameters per MLP regressor. Based on the large number of hyperparameter combinations which must be repeated for each MC, we perform a GRIDS search approach for each parameter in terms of the MLP regressor where parameters and different values are detailed in Table VII. Therefore, specific hyper-parameters are identified for MLPRe regressor where forecasting values are exposed in Section IV.

## V. RESULTS

This section presents the experimental results of our forecasting models over a seven-day prediction horizon. We
<table><caption>TABLE VII: HYPER-PARAMETER VALUES OF MLPREGRESSOR</caption><thead><tr><th>Hyper-parameter</th><th>Values</th></tr></thead><tbody><tr><td>Hidden Layers</td><td>(10,8,6), (10,6,6), (10,6,4), (8,8,4), (8,8,6), (8,6,4), (8,6,6), (6,6,6), (6,6,4), (6,4,4)</td></tr><tr><td>Activation Function</td><td>relu, tanh</td></tr><tr><td>Solver</td><td>adam, SGD</td></tr><tr><td>Learning Rate</td><td>0.0001, 0.0002, 0.0003, 0.0004, 0.0005</td></tr><tr><td>LR Update</td><td>adaptive, constant</td></tr></tbody></table>

evaluate the effectiveness of transformed temporal features and compare the forecasting performance of three different approaches: LSTM, CNN and a Hybrid model (MC). Performance is reported in terms of Mean Absolute Error (MAE) and Root Mean Square Error (RMSE), computed for each forecasted day and averaged across the horizon.

## A. Impact of Time-Based Feature Engineering

We first assess the correlation between each engineered time feature and power consumption using the Pearson correlation coefficient. Table I reports the correlation values for raw calendar-based features. As observed, most features, particularly *Day in Year*, *Week in Year*, *Month in Year*, and *Business Day* exhibit a moderate linear relationship with consumption.

To better capture cyclical patterns inherent in temporal data, we applied trigonometric transformations (sine and cosine) to each periodic feature. As shown in Table II, the transformed features exhibit substantially higher correlation with power consumption. For instance, the cosine-transformed *Day in Year* reaches a correlation coefficient of 0.83, compared to -0.23 in its raw form. This confirms the utility of cyclical encoding in modeling seasonality and improving feature interpretability.

## B. Forecasting Performance

We trained and evaluated each model using a sliding window approach: a 120-day input sequence was used to predict the next 7 days of power consumption. Input features included normalized power values along with selected time-based features, with cyclical transformations applied where appropriate.

Table VIII and Figures 5 summarize the MAE and RMSE for each model across the 7-day horizon. We observe that the hybrid MC model consistently outperforms LSTM and CNN baselines across all prediction days. Notably, while all models show a gradual increase in prediction error with longer horizons, the MC model maintains lower error variance and better stability.

# VI. DISCUSSION & CONCLUSION

The results indicate that time-aware feature engineering plays a pivotal role in improving forecast accuracy. The use of cyclical encodings enables the models to better internalize seasonality without the drawbacks of ordinal encodings. Moreover, the MC model's superior performance suggests that hybrid architectures, which integrate both sequential and

<table><caption>TABLE VIII: 7-Day Forecasting Results: MAE and RMSE for Each Model</caption><thead><tr><th rowspan="2">Day</th><th colspan="2">LSTM</th><th colspan="2">CNN</th><th colspan="2">MC</th></tr><tr><th>MAE</th><th>RMSE</th><th>MAE</th><th>RMSE</th><th>MAE</th><th>RMSE</th></tr></thead><tbody><tr><td>1</td><td>0.072</td><td>0.098</td><td>0.081</td><td>0.109</td><td><strong>0.065</strong></td><td><strong>0.088</strong></td></tr><tr><td>2</td><td>0.076</td><td>0.104</td><td>0.085</td><td>0.112</td><td><strong>0.068</strong></td><td><strong>0.092</strong></td></tr><tr><td>3</td><td>0.079</td><td>0.108</td><td>0.088</td><td>0.117</td><td><strong>0.070</strong></td><td><strong>0.096</strong></td></tr><tr><td>4</td><td>0.084</td><td>0.112</td><td>0.093</td><td>0.121</td><td><strong>0.072</strong></td><td><strong>0.098</strong></td></tr><tr><td>5</td><td>0.086</td><td>0.115</td><td>0.095</td><td>0.123</td><td><strong>0.073</strong></td><td><strong>0.100</strong></td></tr><tr><td>6</td><td>0.088</td><td>0.118</td><td>0.098</td><td>0.126</td><td><strong>0.075</strong></td><td><strong>0.103</strong></td></tr><tr><td>7</td><td>0.090</td><td>0.120</td><td>0.099</td><td>0.127</td><td><strong>0.076</strong></td><td><strong>0.104</strong></td></tr><tr><td>Avg.</td><td>0.082</td><td>0.111</td><td>0.091</td><td>0.119</td><td><strong>0.071</strong></td><td><strong>0.097</strong></td></tr></tbody></table>

<figure><img src="image_8.png" /><figcaption>(a) MAE per Day per Model</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>(b) RMSE per Day per Model</figcaption></figure>

<caption>Fig. 5: Error metrics across the 7-day forecast horizon: (a) Mean Absolute Error (MAE), (b) Root Mean Square Error (RMSE).</caption>

convolutional components, are more robust for multistep forecasting tasks. Future research could also explore integrating spatial and longitudinal feature propagation strategies, as demonstrated in [12], to enhance the temporal modeling of periodic signals. Moreover, leveraging compact and structured representations of temporal patterns, inspired by approaches like PolygoNet [13], may reduce computational complexity while preserving predictive accuracy. Such techniques could make energy forecasting models more interpretable and more suitable for deployment on resource-constrained platforms.

Overall, our approach achieves reliable day-ahead to week-ahead predictions and demonstrates the value of combining temporal domain knowledge with deep learning models.

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