RNN(p) for Power Consumption Forecasting

TL;DR

This work introduces RNN($p$), a simple nonlinear autoregressive model with $p$ Jordan feedbacks designed to capture multi-scale seasonality in time series, particularly for energy data. It provides closed-form gradient expressions for three learning algorithms—RTRL, BPTT, and AAD—and derives their leading-time and space-usage complexities, showing that BPTT suffers exponential growth for $p\ge 2$ while AAD achieves linear scaling in sequence length. The authors demonstrate strong forecasting performance on two power-demand datasets (New England load and London net load), with multi-lag configurations delivering notable accuracy gains and reliable probabilistic forecasts, while also highlighting AAD’s computational efficiency over alternatives. The results support RNN($p$) as an interpretable and effective tool for energy forecasting and decision-making in fintech applications, offering practical benefits in both accuracy and training efficiency. $RNN(p)$ thus provides a compelling balance between model simplicity, interpretability, and predictive power in seasonally rich time series.

Abstract

An elementary Recurrent Neural Network that operates on p time lags, called an RNN(p), is the natural generalisation of a linear autoregressive model ARX(p). It is a powerful forecasting tool for variables displaying inherent seasonal patterns across multiple time scales, as is often observed in energy, economic, and financial time series. The architecture of RNN(p) models, characterised by structured feedbacks across time lags, enables the design of efficient training strategies. We conduct a comparative study of learning algorithms for these models, providing a rigorous analysis of their computational complexity and training performance. We present two applications of RNN(p) models in power consumption forecasting, a key domain within the energy sector where accurate forecasts inform both operational and financial decisions. Experimental results show that RNN(p) models achieve excellent forecasting accuracy while maintaining a high degree of interpretability. These features make them well-suited for decision-making in energy markets and other fintech applications where reliable predictions play a significant economic role.

Figures (8)

• Figure 1: RNN($p$) models represent the simplest extension of ARX($p$) models, achieved by incorporating an additional nonlinear hidden layer $\textbf{h}^{(t)}$. The RNN($p$) architecture maintains high degree of interpretability and it is easy to implement and to train.
• Figure 2: Gradient backpropagation in an RNN($1$) (left) and in an RNN($2$) (right). In the RNN($1$) case, the unrolling procedure results in a linear chain, with each output $\hat{\textbf{y}}^{(t)}$ depending only on the previous output $\hat{\textbf{y}}^{(t-1)}$. In the RNN($2$) case, each output $\hat{\textbf{y}}^{(t)}$ depends on the two previous outputs, $\hat{\textbf{y}}^{(t-1)}$ and $\hat{\textbf{y}}^{(t-2)}$, resulting in a binary tree structure.
• Figure 3: The first two full weeks of training data for the New England dataset (above) and for the London dataset (below). Both time series display evident intra-day seasonal patterns, characterised by peaks in the morning and late afternoon. Additionally, a weekly seasonality is observed, with noticeable load variations between weekdays (solid blue line) and weekends (dashed orange line).
• Figure 4: Autocorrelation structure of the deseasonalised time series on the New England dataset (left) and on the London dataset (right). PACF diagrams are obtained by analysing the two time series on the training set (2007-2010 and 2014-2017, respectively). The plots highlight the presence of a distinctive serial correlation, involving the first 2 lags (hourly autocorrelation) and those around lag 24 (daily autocorrelation).
• Figure 5: Pipeline of the forecasting methodology.

Theorems & Definitions (4)

• Proposition 1
• proof
• Lemma C.1
• proof