Footprint-Aware Regression Overview

• Footprint-Aware Regression (FAR) is a framework that integrates carbon, energy, and spatial measurements into regression models for robust environmental predictions.
• It employs dual methodologies by combining machine learning for carbon footprint minimization with spatial attention to accurately estimate CO₂ flux at fine resolutions.
• Experimental benchmarks demonstrate FAR's ability to reduce energy consumption and enhance prediction accuracy compared to baseline models, especially in heterogeneous ecosystems.

Footprint-Aware Regression (FAR) encompasses a collection of methodologies designed to explicitly account for carbon, energy, and spatial measurement footprints in regression model training and environmental prediction. In machine learning for carbon footprint minimization, FAR constitutes a systematic workflow for quantifying, profiling, and reducing the carbon emissions inherent to model training. In ecosystem-scale CO₂ flux estimation, FAR addresses the spatial mismatch between ground sensors and remote sensing by jointly learning pixel-level outcomes and dynamic measurement footprints. The following outlines methodological foundations, experimental protocols, mathematical formulations, benchmarking results, and best-practice recommendations as synthesized from Antonopoulos and Antonopoulos (2024) and recent deep learning work in footprint-aware upscaling (Antonopoulos, 17 Sep 2024, Searcy et al., 1 Dec 2025).

1. Formal Definition and Objectives

Footprint-Aware Regression (FAR) in ML carbon footprint reduction is a structured protocol for designing, measuring, and minimizing the environmental impact of regression tasks. Its objective is to optimize model and system parameters to reduce power consumption (P̄), energy use (E), and carbon footprint (CF), operationalized by the following definitions:

• Power consumption at any time: $P\ [\mathrm{W}]$
• Total energy consumed over $t$ hours: $E = \frac{P̄}{1000} \times t\ [\mathrm{kWh}]$
• Carbon footprint: $CF = E \times CI\ [\mathrm{gCO}_{2}\mathrm{e}]$, where $CI$ is grid carbon intensity in $[\mathrm{gCO}_{2}\mathrm{e}/\mathrm{kWh}]$
• Example: For $CI = 123\ \mathrm{gCO}_{2}\mathrm{e}/\mathrm{kWh}$, $t \approx 1\ \mathrm{h}$, $CF = \bigl(\frac{P̄}{1000}\bigr) \times 123$

In satellite-based environmental modeling, FAR refers to a unified joint learning framework for:

• Predicting net $\mathrm{CO}_2$ flux ($\mathrm{FC}$) at fine spatial ($30\ \mathrm{m}$) resolution
• Simultaneously learning the effective measurement footprint of each tower—a spatial attention mask weighting each pixel’s contribution to tower-aggregated observations (Searcy et al., 1 Dec 2025)

2. Experimental Design and Data Protocols

ML Carbon Footprint Optimization

• Hardware: Custom PC (MSI Z690 DDR4, i5-12600K, 32GB RAM, NVIDIA RTX4060 Ti (Tensor Core), Kingston SSD, Windows 10 Pro, EVGA 1600W P2 PSU)
• Power/Carbon measurement: CodeCarbon (v3.35.3), Comet Emissions Tracker, HWINFO, Core Temp, MSI Afterburner, Corsair iCUE, Intel Power Gadget, wall wattmeter
• Component shares: GPU $\approx 70\%$, CPU $\approx 15\%$, RAM $\approx 10\%$, other $\approx 5\%$
• Dataset: Used-car sales (150,000 rows; numeric/categorical cleaning, one-hot-encoding, min-max normalization)
• DNN architecture:
  • Input layer: features
  • Two hidden Dense layers ($N\in\{1024, 2048\}$ neurons, ReLU)
  • Output: linear (price), mean squared error loss, Adam optimizer
  • Hyperparameters: Batch size $B\in\{256, 512, 1024\}$, epochs to target $t\approx 1\,h$, FP32 vs mixed precision (Antonopoulos, 17 Sep 2024)

Carbon Flux Prediction with FAR

• Data: AMERI-FAR25 (439 site-years, 209 towers, 45,124 Landsat patches, 14 ecosystem types)
• Inputs: Half-hourly tower variables ($$\mathrm{WD, WS, U}_{⋆}, \mathrm{TA, H, RH, SW}_{IN}, \mathrm{FC}$$), Landsat patch ($128\times128$ pixels, 11 bands, 4 angle channels), meteorological normals
• Data cleaning: >99.5th/<0.5th flux outliers dropped; negative radiation/nighttime uptake excluded
• Preprocessing: Zero-mean/unit-variance scaling; random image/WD rotations for augmentation
• Train/valid/test: By IGBP code, 40% site holdout for test/validation, 20% timepoints for global validation (Searcy et al., 1 Dec 2025)

3. Mathematical Formulation and Model Architectures

ML Carbon Analysis

Power and carbon computation adhere strictly to:

$$E = \frac{P̄}{1000} \times t, \qquad CF = E \times CI$$

Batch size, neuron count, and floating-point policy are systematically varied to empirically assess $P̄$ and $CF$.

FAR Spatial Regression (Environmental Prediction)

Let $L\times W$ be pixel spatial dimensions, $D$ input channels, $M$ footprint variables:

• $$x_\text{landsat,T}\in\mathbb{R}^{L \times W \times D}$$: Landsat image cube
• $x_\text{drivers,t}\in\mathbb{R}^{K}$: environmental drivers
• $x_\text{footprint,t}\in\mathbb{R}^{M}$: footprint physics features
• $FC_\text{pixel}\in\mathbb{R}^{L \times W}$: pixel-level flux predictions
• $FP\in\mathbb{R}^{L \times W}$: footprint attention mask, $\sum_{i,j} FP_{i,j}=1$
• $y_t\in\mathbb{R}$: tower-measured flux

Model equations:

Flux regression arm: $$FC_\text{pixel} = f_\text{flux}(x_\text{landsat,T}, x_\text{drivers,t}; \Theta_\text{flux})$$

Footprint prediction arm: $$FP = \text{softmax}_{2D}\, f_\text{fp}(x_\text{footprint,t}, \text{coords}; \Theta_\text{fp})$$

Tower flux prediction:

$$\hat{y}_t = \sum_{i=1}^L \sum_{j=1}^W [FC_{i,j}(x_\text{landsat,T},x_\text{drivers,t}) \times FP_{i,j}(x_\text{footprint,t})]$$

Loss function:

$$\mathcal{L}(\Theta) = E_t[(y_t - \hat{y}_t)^2] + \lambda\,E_t\left[\sum_{i,j} FP_{i,j} \ln FP_{i,j}\right]$$

$\lambda$ regularizes footprint entropy: $\lambda>0$ enforces compactness, $\lambda<0$ fosters spread, $\lambda=0$ neutral.

4. Quantitative Results and Benchmarking

ML Training Power and Carbon Metrics

Mean total power ($P̄$) and carbon footprint ($CF$) for a range of batch size, neuron configuration, and float policy are summarized:

Dataset	G₁ (FP32, B=256, N=1024)	G₂ (Mix, B=256, N=1024)	G₃ (Mix, B=512, N=1024)	G₄ (Mix, B=256, N=2048)
CSV $P̄$ (W)	126	125	115	127
CSV $CF$	15.50	15.38	14.15	15.62
Parquet $P̄$ (W)	126	125	119	128
Parquet $CF$	15.50	15.38	14.64	15.74

• Parquet format: Empirically faster I/O ($\sim$1.2–1.5$\times$ faster read), not measurably affecting GPU power draw.
• Lowest mean power: Mixed precision, $B=512$, $N=1024$ (–11 W vs benchmark in CSV, –7 W in Parquet)
• All comparisons lacked statistical significance (ANOVA/T-test $P$-values $>$ threshold); increased sample size recommended (Antonopoulos, 17 Sep 2024)

FAR Environmental Prediction Metrics

FAR model validation against XGBoost baseline with uniform footprint:

Temporal scale	FAR $R^2$	XGBoost $R^2$	FAR RMSE	XGBoost RMSE
Half-hour	0.65	0.59	11.51 Mg C ha⁻¹ yr⁻¹	12.41
Monthly	0.78	0.66	3.14 Mg C ha⁻¹ mo⁻¹	3.14
Yearly	0.81	0.69	1.39 Mg C ha⁻¹ yr⁻¹	1.39

• FAR outperforms baseline, particularly in sites with spatial heterogeneity (shrublands, wetlands, grasslands)
• Footprint regularization $\lambda = 1 \times 10^{-3}$ optimal ($R^2=0.78$); replacement by a physics-based footprint yields similar but slightly lower $R^2=0.77$
• SHAP analyses indicate strongest features: $\mathrm{SW}_{IN}$, red, blue, NIR, SWIR1, SWIR2 (aligned with NDVI, NDWI, NBR) (Searcy et al., 1 Dec 2025)

5. Statistical Analysis Protocols

• Normality: Skewness $|\text{skew}| < 2$, Kurtosis $|\text{kurtosis}| < 7$
• Homoscedasticity: Pairwise $F$-test, variance ratio $< 1.5$
• Independence: Ensured by experimental design
• ANOVA ($\alpha = 0.05$, $F_{\text{critical}} = 3.49$): No significant difference in mean power draw; fail to reject $H_0$
• Bonferroni-corrected one-sided $t$-test ($\alpha \approx 0.0083$ for 6 comparisons): All $P$-values $> \alpha$; no significant pairwise differences at sample size (Antonopoulos, 17 Sep 2024)

A plausible implication is that more extensive experiments on multi-GPU clusters may yield statistical significance in future studies.

6. Limitations, Sensitivity Analyses, and Extensions

• ML Carbon FAR: Single-node sample size underpowered for statistical inference; Parquet format unlikely to affect model carbon footprint when preprocessing is similar.
• FAR (Ecosystem): AMERI-FAR25 restricted to North America (PRISM meteorology), limiting global generalization; additional driver variables (land-use, irrigation, historical land-cover change) may be needed for full spatial and temporal integrity.
• Sensitivity: Footprint regularization parameter $\lambda$ influences prediction accuracy and footprint compactness; learned footprints behave consistently with atmospheric tracer theory but tend toward higher circularity, reflecting integration over dynamic conditions.
• Extensibility: FAR framework is compatible with alternative atmospheric species (e.g., $\mathrm{CH}_4$), higher-resolution imagery, and multisensor fusion (Searcy et al., 1 Dec 2025).

7. Methodological Recommendations and Best Practices

• Systematically embed wattage/carbon tracking (CodeCarbon+, Comet) in every regression pipeline
• Exploit mixed precision (“mixed_float16” policy) on Tensor Core-equipped NVIDIA GPUs for optimal carbon reduction
• Hyperparameter tuning for carbon minimization requires explicit attention to batch size and layer width; avoid excessively large DNN layers
• Cross-validate power readings using independent sensors; compute component-level (GPU/CPU/RAM) and complete system measurements for robust mean estimates
• When experimental differences are marginal, increase sample size and/or server cluster scale to enable inferential tests
• Dataset format (CSV vs Parquet) yields insignificant differences for GPU power if preprocessing regimes are matched; Parquet accelerates I/O but not total footprint
• Always report $P̄$, $CF$, effect sizes, and $P$-values; statistically non-significant results still contribute to best practice and motivate scalability for robust statistical evaluation (Antonopoulos, 17 Sep 2024)
• FAR model regularization ($\lambda$ tuning) and SHAP feature analysis enhance interpretability and site-specific deployment in environmental contexts (Searcy et al., 1 Dec 2025)

By following the above protocols—rigorous measurement, precision optimization, controlled hyperparameter sweeps, and comprehensive statistical testing—Footprint-Aware Regression enables researchers to generate data-driven reductions in the carbon footprint of regression modeling and improve the representativeness and accuracy of environmental predictions.