SSM-FastICANet: a hybrid state space and FastICA model for economic growth forecasting in energy-economy-environment systems

Data preprocessing

We standardize heterogeneous energy, emissions, and GDP for fair analysis to prevent variables such as non-renewable energy, which often significantly exceed GDP in Least Developed countries, from overshadowing the analysis.

• • Normalization: Variables (renewable and non-renewable energy, CO₂ emissions, GDP) are normalized: $$\widetilde{x_i}={\displaystyle \frac{x_i-{\mu }_i}{{\sigma }_i},\tilde{y}={\displaystyle \frac{y-{\mu }_y}{{\sigma }_y}}}$$

• • Integration: We build a structured input for FastICA to untangle mixed signals using normalized data X = $[\widetilde{x_i},\widetilde{x_2},\widetilde{x_3}]$.

FastICA for pattern recognition and feature selection

FastICA is engineered to attain comparable accuracy to ICA (Stone, 2004) nonetheless, with expedited convergence. It is typically superior for performance and complexity owing to its expedited convergence and reduced computing requirements. It is employed to identify statistically independent components within the data. These components proficiently discern the essential patterns or correlations among the variables. In this context, FastICA can reveal the independent dynamics of renewable energy, non-renewable energy, and CO₂ emissions, together with the correlations of these distinct patterns with variations in economic growth. For instance, one IC may demonstrate a correlation between increasing non-renewable energy and CO₂ emissions alongside economic growth, while another could indicate a tendency of expanding renewable energy in tandem with economic development.

FastICA is particularly well-suited for this analysis because its extracted ICs are:

• a.

  Non-Gaussian data: Energy and emissions datasets frequently exhibit non-Gaussian (irregular) data characteristics. FastICA is meticulously crafted to manage such data, rendering it exceptionally suitable for uncovering the intricate patterns in energy consumption and CO₂ emissions among various nations.

• b.

  Blind source separation: FastICA is capable of separating mixed signals into unique, independent sources. This ability is essential for distinguishing the roles of different energy sources (both renewable and non-renewable) and emissions, facilitating a more precise comprehension of their separate effects on economic growth.

• c.

  Interpretability: FastICA identifies independent components that relate to separate underlying processes. The components derived from this approach offer greater interpretability compared to alternatives, facilitating a clearer explanation of the connections among energy consumption, emissions, and economic growth.

State space model (SSM)

We have selected SSMs for its capability to adeptly model temporal dependencies in sequential data. SSMs are particularly beneficial for time-series analysis because of their intrinsic properties.

• a.

  Temporal dependencies: SSMs maintain an internal state that preserves information from previous time steps, making them naturally suitable for time-series data.

• b.

  Efficiency: SSMs demonstrate linear temporal complexity, making them computationally economical, especially for large sequences.

• c.

  Interpretability: SSMs provide a precise mathematical framework for understanding the temporal relationships among variables.

Model validation and forecasting

To guarantee generalizability and produce forecasts, we:

• a.

  Fine-tuning: Train the customized model on the complete dataset following convergence.

• b.

  Hold-out testing: Evaluate on unseen data using:

  • RMSE: $\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{1}{M}{\sum }_{i=1}^M{\left(\widehat{y_i}-y_i\right)}^2}}$

• c.

  GDP forecasting:

  • forecaste $\hat{y}\left(t\right)$ for future time steps utilizing energy/emission projections.

Insight generation and policy recommendations

We translate model outputs into actionable solutions by:

• a.

  Non-linear relationship analysis:

  • Compute sensitivity ${\displaystyle \frac{\mathrm{\partial }\hat{y}}{\mathrm{\partial }{x}_i}=C\cdot {\left(I-A\right)}^{-1}\cdot B_i}$ to quantify how changes in energy consumption (x₁, x₂, x₃) impact GDP ( $\hat{y}$).

• b.

  Scenario testing:

Simulate GDP under policies $\mathrm{\Delta }\hat{y}=f\left(\mathrm{\Delta }{x}_1,\mathrm{\Delta }{x}_2,\mathrm{\Delta }{x}_3\right)$ to derive actionable policy insights. The process commences by delineating hypothetical scenarios that model alterations in critical inputs, such as augmenting renewable energy utilization by 50% ( $\mathrm{\Delta }{x}_1$ = +50%) or diminishing CO₂ emissions by 20% ( $\mathrm{\Delta }{x}_3$ = −20). The GDP impact (Δ $\hat{y}$) is projected by aggregating the contributions of each input change, utilizing the model’s precomputed sensitivity values $\left(\frac{\mathrm{\partial }\hat{y}}{\mathrm{\partial }{x}_i}\right)$. For instance, increasing renewable energy ( $\mathrm{\Delta }{x}_1$ = +100%) with a sensitivity of 0.3 would elevate GDP by 0.3% (Δ $\hat{y}$ = 0.3 * 1.0). Subsequently, inputs are prioritized based on sensitivity (e.g., renewable energy’s $\frac{\mathrm{\partial }\hat{y}}{\mathrm{\partial }{x}_i}$ being the most significant) to guide interventions. Policies are then customized to enhance high-sensitivity drivers (e.g., incentives for renewable energy) or to alleviate detrimental ones (e.g., carbon taxes). Ultimately, proposed policies are assessed by simulations of their GDP and emission results, confirming their viability prior to execution. This method guarantees data-informed, focused tactics for enduring growth.

This paradigm connects technical precision with practical policy formulation by tackling the energy-economy-environment nexus across many economies. It significantly surpasses traditional linear models by incorporating non-Gaussian dynamics (through FastICA) to elucidate asymmetric interactions—such as the unique GDP effects of renewable energy adoption in solar-centric economies and utilizing temporal dependencies (via SSMs) to characterize localized, quarterly GDP reactions to energy fluctuations. Furthermore, FastICA separates multi-scale patterns, distinguishing global trends such as emission reductions linked with the Paris Agreement from regional economic activities, while SSMs dynamically adjust through real-time policy modifications (e.g., post-COVID energy reforms). By incorporating ICA-guided feature extraction and adaptive block topologies, the methodology guarantees reproducibility and policy relevance, addressing significant deficiencies in conventional GDP forecasting, including inflexible linear assumptions and inadequate scalability to diverse contexts. Its organized yet adaptable framework provides a scalable instrument for sustainable development, allowing policymakers to assess trade-offs (e.g., emission reductions vs growth) and prioritize high-impact measures.

Dataset

The “Nexus between Carbon Emissions, Energy Consumption, and Economic Growth” dataset (Dissanayake et al., 2023) offers an in-depth analysis of the interconnections among carbon emissions, energy consumption, and economic growth across different countries and regions from 1990 to 2019. This dataset encompasses information from developed, developing, and least developed nations, facilitating a comparative examination of sustainability practices alongside economic performance. The dataset comprises essential variables such as carbon emissions quantified in metric tons, energy consumption represented as a percentage of total energy sources, and GDP. The dataset encompasses a broad geographical spectrum, featuring countries from North America, Europe, Asia, Africa, and the Caribbean, including the United States, United Kingdom, Denmark, Cyprus, Botswana, and the United Arab Emirates. The temporal coverage extends across three decades, offering valuable insights into the trends and shifts in energy policies, economic growth, and environmental impact throughout this period. This comprehensive dataset provides an essential resource for those seeking to comprehend the intricacies of energy consumption and its effects on carbon emissions and economic growth.

Materials

The studies were performed on a Windows 11 Pro (64-bit) PC, featuring an Intel Core i7-10700K CPU operating at 3.80GHz and 24 GB of DDR4 RAM. All model development and testing were conducted in VSCode Notebooks utilizing Python 3.8 within a Miniconda environment.

The solution utilized PyTorch 2.0.0, employing Adam optimization and symbolic gradient-based policy sensitivity for training purposes. Data preprocessing, statistical analysis, and scenario testing were performed utilizing standard Python modules such as pandas 2.0, NumPy 1.26.4, scikit-learn 1.3.2, and SciPy 1.13.0. The FastICA method and statistical measures were amalgamated to improve interpretability. Establishing fixed random seeds guaranteed reproducibility. The comprehensive, annotated implementation code is included as additional material.

Training and validation loss

Figure 2 illustrates the training and validation loss across epochs for each model, providing essential insights into their performance. All models reach low loss values, yet the rate of convergence and the ultimate loss levels differ considerably. BaseSSM and SSM-FastICANet demonstrate swift convergence in the initial epochs and exhibit low loss levels, with validation loss trends closely mirroring training loss, suggesting negligible overfitting. In comparison, S4 begins with a greater loss and exhibits a slower convergence rate, indicating a more pronounced disparity between training and validation loss, which implies possible generalization challenges.

SSM-FastICANet emerges as the top-performing model, achieving an ideal equilibrium between convergence speed and generalization. Although BaseSSM demonstrates strong performance, its elevated parameter count could suggest overparameterization, given that SSM-FastICANet attains similar or superior results with a reduced number of parameters. Conversely, S4 faces challenges with convergence and generalization, evident in its elevated test loss, which renders it less appropriate for this task, even with its lightweight design. In summary, SSM-FastICANet exhibits exceptional efficiency and performance, establishing it as the optimal selection for this analysis.

Figure 3 illustrates a temporal comparison of actual vs. expected GDP for five Arab nations from the test set: Saudi Arabia, United Arab Emirates, Algeria, Iraq, and Morocco, to assess the model’s predicting performance across time. These nations exemplify a varied spectrum of resource-dependent and transitional economies. The x-axis represents the years (2010–2019), and the y-axis illustrates normalized GDP.

The SSM-FastICANet model effectively delineates overall economic paths, particularly for nations exhibiting relatively stable energy and emissions trends. In instances such as Saudi Arabia and Iraq, the model demonstrates significant concordance with macroeconomic growth trends while mitigating sudden shocks or recessionary phases. This effect is probably attributable to the impact of block-averaged ICA features, which prioritize long-term structure over short-term fluctuations.

Figure 3 underscores the model’s efficacy in strategic forecasting across several nations, while also revealing opportunities for enhancement in addressing localized or sudden economic disturbances. These findings validate the model’s applicability in energy-growth planning contexts, especially for economies with moderate to high emissions.

Test loss comparison

Figure 4 illustrates the test loss associated with three models: SSM-FastICANet, BaseSSM, and S4. SSM-FastICANet demonstrates the lowest test loss, reflecting exceptional performance and robust generalization abilities. BaseSSM is closely aligned, exhibiting a marginally elevated test loss; however, its performance does not correspondingly enhance in relation to its substantial parameter count, indicating a lack of efficiency. S4 exhibits a notably elevated test loss relative to the other two models, suggesting potential underfitting or an architectural mismatch for this particular task, even with its lightweight and efficient design. In summary, SSM-FastICANet emerges as the most effective model, achieving a commendable balance of performance, efficiency, and generalization. Conversely, S4 appears to necessitate additional optimization or architectural modifications to enhance its applicability for this specific use case.

Model complexity (number of parameters)

Figure 5 illustrates the comparison of the number of parameters among the three models: BaseSSM, SSM-FastICANet, and S4. BaseSSM possesses the largest number of parameters, considerably exceeding those of the other models. This suggests a more intricate architecture that probably demands increased memory and computational resources. In comparison, SSM-FastICANet possesses a reduced number of parameters relative to BaseSSM and is marginally lower than S4, achieving a harmonious balance between complexity and efficiency. S4, featuring a compact parameter set, showcases its lightweight characteristics, rendering it ideal for environments with limited resources.

We evaluate our model against state-of-the-art models as in selected for their representation of the latest achievements in time-series forecasting, each tackling distinct issues such as long-sequence dependencies, seasonal trends, and multivariate data. Informer (Ojeda, Guerra-Artal & Hernández-Tejera, 2021), engineered for long-sequence forecasting, employs ProbSparse Self-Attention to mitigate computing complexity but encounters difficulties with noisy input. Autoformer (Wu et al., 2021) is proficient at identifying trends and seasonality via a decomposition architecture, although it may exhibit suboptimal performance with non-periodic data. PatchTST (Nie et al., 2022) employs a patching methodology for multivariate forecasting, necessitating meticulous hyperparameter optimization.We intend to assess the SSM-FastICANet model’s efficacy in addressing comparable difficulties and its capacity to surpass current methodologies by juxtaposing it with these benchmarks. The results as in Table 2 evaluate the performance of the SSM-FastICANet model against three advanced models—Informer, Autoformer, and PatchTST—on a forecasting assignment, employing criteria including mean absolute error (MAE), root mean squared error (RMSE), and R² (coefficient of determination). The SSM-FastICANet model markedly surpasses the other models in MAE and RMSE, attaining values of 0.0171 and 0.0268, respectively, in contrast to the considerably elevated errors of the other models (MAE: 2.128–2.423, RMSE: 3.373–3.714). This signifies that the SSM-FastICANet model generates predictions that are significantly nearer to the actual values on average and exhibits reduced overall errors. The R² values for all models are notably low, with the SSM-FastICANet model attaining the greatest value of 0.0055, whereas the other models exhibit negative R² values, ranging from −0.010 to −0.225. This indicates that although the SSM-FastICANet model outperforms the others, none of the models are notably effective in elucidating the variance in the target variable (GDP). The diminished R² values may be ascribed to variables including substantial noise within the dataset, feeble correlations between the input features and GDP, or constraints in the models’ capacity to elucidate underlying patterns. To enhance performance, additional examination of feature engineering, model optimization, and data pretreatment may be required.

Generalization to unseen countries

To assess generalizability, we designated some nations as a holdout group that was not exposed during training. The model exhibited moderate generalization to developing economies with analogous emissions and energy patterns but encountered difficulties with outliers or structurally distinct nations, such as post-oil-shock Gulf states and small island economies. This indicates that SSM-FastICANet apprehends fundamental energy-economic frameworks rather than merely recalling country-specific IDs.

These findings underscore that the approach is more appropriate for regional or structurally uniform forecasting jobs. To enhance performance in more diverse environments, subsequent research could investigate:

• Integrating region-specific embeddings to represent macro-contextual elements (e.g., economic complexity, policy changes)

• Implementing meta-learning or transfer learning methodologies to adjust across various economic clusters or phases of development

In summary, SSM-FastICANet attained the minimal test loss, expedited convergence, and superior efficiency relative to baseline and transformer-based models. Nonetheless, its poor R² signifies that it cannot comprehensively account for GDP variance based just on energy-emissions data. Consequently, its efficacy is not in comprehensive macroeconomic forecasting, but in modeling nonlinear linkages between energy and growth, as well as simulating policy-driven impact scenarios across many countries and regions.

Future work

Future efforts will concentrate on augmenting the utility and explanatory capacity of the SSM-FastICANet model by addressing:

• Incorporating additional macroeconomic factors such as trade, inflation, and capital formation into the input features.

• Enhancing temporal resolution by transitioning from annual to quarterly data to more effectively capture short-term trends.

• Enhancing generalizability by the application of regional or economic-cluster embeddings.

• Utilizing meta-learning or transfer learning to modify the model for various national profiles.