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Functional Autoregressive Modeling with Exogenous Variables for Day-Ahead Electricity Price Forecasting in the Croatian Electricity Market

This paper demonstrates that functional autoregressive models with exogenous variables (FARX(p)) significantly outperform traditional statistical and machine learning methods in forecasting day-ahead electricity prices in the Croatian market by effectively capturing high-frequency volatility, temporal dependencies, and demand-driven effects.

Original authors: Laila A. AL-Essa, Faheem Jan, Mehwish Tahir, Muhammad Wisal Khan

Published 2026-07-03

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CC BY 4.0

Original authors: Laila A. AL-Essa, Faheem Jan, Mehwish Tahir, Muhammad Wisal Khan

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict the price of electricity for tomorrow. In the past, people treated electricity prices like a string of separate beads on a necklace: they looked at the price at 1:00 AM, then the price at 2:00 AM, then 3:00 AM, as if each hour had no connection to the one before it.

This paper argues that this "bead-by-bead" approach misses the bigger picture. Instead, the authors suggest looking at the entire day's price as a single, smooth, flowing river. They call this Functional Data Analysis (FDA).

Here is a simple breakdown of what they did, how they did it, and what they found, using everyday analogies.

The Problem: The "Bead" vs. The "River"

Electricity prices are wild. They jump up and down, they have patterns that repeat every day (like a rush hour), and they are influenced by outside factors like how much power people are using.

The Old Way (Traditional Models): Imagine trying to predict the shape of a river by measuring the water level at just a few random spots. You might miss the curve of the riverbank or the speed of the current. Traditional models (like ARIMA or simple AI) treat every hour as an isolated point. They struggle to see the smooth flow of the day.
The New Way (Functional Models): The authors treat the 24 hours of electricity prices as one continuous, smooth curve—a "river." This allows them to see the whole shape of the day, not just isolated drops of water.

The Solution: Two Special Tools

The researchers built two new tools to predict these "price rivers":

FAR (The "Memory" Tool): This tool looks at the shape of yesterday's price river and uses that shape to guess what today's river will look like. It understands that if the river was high and steep yesterday morning, it's likely to be high and steep this morning too.
FARX (The "Memory + Weather" Tool): This is the upgraded version. It doesn't just look at yesterday's price river; it also looks at electricity demand (how much power people are using).
- The Analogy: Imagine predicting traffic. A "Memory" tool just looks at how fast cars moved yesterday. The "Memory + Weather" tool also looks at whether it's raining or if there's a big game on TV. The authors found that knowing the "traffic" (demand) helps predict the "speed" (price) much better.

The Experiment: The Croatian Test Drive

To test these tools, the authors used data from the Croatian electricity market from 2020 to 2024. They split the data into two piles:

The Practice Pile (2020–2023): They taught the models using this data.
The Test Pile (2024): They asked the models to predict the future prices for this year and checked how close they were to the actual numbers.

They compared their new "River" tools against the old "Bead" tools (standard statistics and common AI models like Neural Networks).

The Results: The Winner is Clear

The results were like a race where the new tools won by a landslide.

The Scoreboard: The authors used three scorecards to measure accuracy:
- MAE: How far off was the guess, on average?
- MAPE: What was the percentage error?
- RMSE: How bad were the biggest mistakes?
The Outcome: The FARX model (the one that used both the price shape and the demand) had the lowest errors across the board. It was the most accurate whether they were predicting the next hour, the next week, or the next month.
Why it won: The old models got confused by the complex, wiggly nature of electricity prices. The new models, by seeing the price as a smooth curve and adding the "demand" factor, could navigate the twists and turns much better.

The Takeaway

The paper concludes that if you want to predict electricity prices, stop looking at them as a list of separate numbers. Instead, look at them as a continuous, flowing shape.

For the Market: This helps companies and traders make better bets on prices, manage their risks, and plan their power usage more efficiently.
The Limitation: The authors admit they only tested this in Croatia. They also only used "demand" as the extra factor. They suggest that future versions could add other factors like weather or fuel prices to make the "river" even clearer.

In short: By treating electricity prices as a smooth, continuous story rather than a list of disconnected facts, and by listening to the "demand" chapter of that story, the authors created a crystal ball that sees the future of electricity prices more clearly than anyone else has before.

Technical Summary: Functional Autoregressive Modeling with Exogenous Variables for Day-Ahead Electricity Price Forecasting in the Croatian Electricity Market

Problem Statement
Electricity price forecasting (EPF) remains a complex challenge due to the high volatility, non-linear dependencies, multiple seasonalities, and the increasing influence of external market factors inherent in modern energy markets. Traditional statistical methods (e.g., ARIMA, VAR) and machine learning approaches (e.g., ANN, LSTM) often struggle to model the day-to-day dynamics of high-frequency electricity price data effectively. These methods typically treat hourly prices as discrete observations, potentially losing the continuous temporal structure and smooth evolution of price curves. Furthermore, while exogenous factors like electricity demand significantly influence prices, few studies have integrated such information into functional forecasting frameworks, particularly within emerging European markets like Croatia, which is undergoing rapid transformation in renewable energy integration and market interconnectivity.

Methodology
The study proposes a two-component forecasting framework that decomposes the electricity price series ( $X_i$ ) into a deterministic part ( $D_i$ ) and a stochastic part ( $Y_i$ ):
$X_i = D_i + Y_i$

Deterministic Component ( $D_i$ ): Modeled using a Generalized Additive Model (GAM) to capture systematic phenomena such as seasonal trends, day-of-the-week effects, holidays, and long-term trends without strong parametric assumptions.
Stochastic Component ( $Y_i$ ): Modeled using Functional Data Analysis (FDA) techniques. The hourly price data is transformed into continuous curves using Fourier basis expansion.
- Dimensionality Reduction: Functional Principal Component Analysis (FPCA) is applied to extract the leading eigenfunctions and scores, reducing the dimensionality of the data while preserving temporal dependence.
- Core Models:
  - FAR(p): A Functional Autoregressive model of order $p$ is used to model the lagged functional response of the stochastic component.
  - FARX(p): An extension of FAR(p) that incorporates functional exogenous variables. In this study, electricity demand is treated as a functional covariate to improve forecast accuracy.
- Forecasting Process: The framework estimates FPC scores, fits a Vector Autoregressive (VAR) model to these scores (incorporating exogenous FPC scores for FARX), and reconstructs the final curve using the Karhunen-Loève expansion.
Benchmarks: The proposed models are compared against classical statistical models (ARIMA, VAR) and machine learning models (ANN, ARNN).
Data: The study utilizes hourly electricity price and demand data from the Croatian market (Zagreb) from January 2020 to December 2024. The data is split into a training/validation set (2020–2023) and an out-of-sample testing set (2024).

Key Contributions

Development of Functional Models: The paper develops and tests Functional Autoregressive (FAR(p)) and Functional Autoregressive with Exogenous Variables (FARX(p)) models specifically for day-ahead EPF using hourly price curves.
Integration of Exogenous Information: It uniquely integrates electricity demand as a functional exogenous covariate within the FARX framework, demonstrating its impact on forecast accuracy.
Comprehensive Evaluation: The study provides an extensive empirical comparison across multiple time horizons (hourly, daily, weekly, and monthly) using Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), and Root Mean Square Error (RMSE).
Market-Specific Analysis: It offers novel empirical findings on the viability of FDA approaches in the Croatian electricity market, a region previously under-studied in the context of functional time series forecasting.

Results
Empirical results indicate that the functional forecasting models, particularly FARX(p), outperform all benchmark models (ARIMA, VAR, ANN, ARNN, and FAR(p)) across all evaluation metrics and time horizons.

Overall Performance: FARX(p) achieved the lowest errors: MAE (11.94), MAPE (8.34%), and RMSE (9.49), compared to the next best model (FAR(p)) and significantly better than classical methods like ARIMA (MAE 20.73).
Temporal Stability: The superiority of FARX(p) was consistent across all 24 hourly periods, all days of the week, and all months of the year.
Exogenous Impact: The inclusion of electricity demand as a functional variable consistently reduced forecasting errors, confirming that external market factors provide significant predictive value beyond historical price trends.
Data Representation: The transformation of discrete hourly data into smooth functional curves via Fourier basis functions successfully captured intraday patterns and reduced noise, with the first three functional principal components explaining the vast majority of variability (the first alone accounted for ~97%).

Significance and Claims
The paper claims that Functional Data Analysis offers a robust and accurate alternative to traditional and deep learning methods for electricity price forecasting. By treating price data as continuous curves rather than discrete points, the proposed methodology better captures the intrinsic temporal dependence and complex dynamics of the market.

The authors assert that the FARX(p) model is a significant advancement because it successfully integrates exogenous information (electricity demand) into the functional framework, leading to superior predictive accuracy and robustness. This approach provides market participants with a reliable tool for:

Formulating more informed bidding strategies.
Enhancing risk management.
Improving operational planning and generation scheduling.

The study concludes that while the results are promising, they are specific to the Croatian market and the variables tested. The authors note limitations regarding the linear functional relationship assumption and the exclusion of other potential exogenous variables (e.g., renewable generation, weather, fuel prices), suggesting future research should expand the multivariate FARX structure and explore non-linear or regime-switching functional models.