Energy Attribution Model

• Energy attribution models are mathematical frameworks that decompose total energy consumption into contributions from individual components using statistical and physical methods.
• They employ empirical regression and analytical techniques to map resource counters, correct for shared resources, and achieve high-granularity attributions in heterogeneous systems.
• These models are crucial for enabling energy-aware optimizations in areas such as mobile devices, data centers, sensor networks, and climate modeling.

An energy attribution model is a formal mathematical and algorithmic construct that quantifies, apportions, and explains the energy usage of complex systems by tracing total measured consumption back to internal components, resources, algorithmic operations, or high-level tasks. Energy attribution is fundamental across domains including mobile devices, datacenters, sensor networks, deep learning, video encoding pipelines, building systems, and even climate modeling, enabling energy-aware optimization, accountability, and interpretability at granular levels.

1. Theoretical Foundations and Core Principles

Energy attribution models decompose total observed or metered energy into additive or factorable contributions from a set of sub-units (hardware components, code segments, resource counters, tasks, etc.), through explicit mapping, statistical modeling, or resource accounting mechanisms. The formalization typically satisfies

$E_{\text{total}} = \sum_{k} E_k$

where $E_k$ denotes the attributed energy share for component, resource, or task $k$. The attribution process must address the assignment of costs in presence of shared resources, measurement noise, indirect dependencies, and often under constraints of limited hardware support, multi-tenancy, or dynamic system state.

Two high-level modeling classes dominate:

• Empirical regression-based models: Fit coefficients that map observable resource counters (e.g., CPU cycles, memory access counts, I/O events) to measured energy, using linear or nonlinear regressions, decision trees, or neural networks (Dong et al., 2010, Povoa et al., 2017).
• Physical or analytical models: Leverage known instruction, hardware, or process-level energy costs, associating them with execution traces or static program structure (Grech et al., 2014, Weigell et al., 7 Dec 2025).

Robust attribution models incorporate normalization, handle idleness and contention, adjust for hardware heterogeneity, account for multi-level system topology (e.g., NUMA, SMT), and must reconcile measured aggregate data with per-entity accounting under partial observability.

2. Resource Counter-Based and Regression Models

A prominent energy attribution approach involves mapping readily available system counters (hardware PMC, OS-level stats) to energy, using regression:

$\hat{E} = \beta_0 + \sum_{j} \beta_j x_j$

where $x_j$ are resource-utilization features (such as CPU user time, memory accesses, disk I/O, network activity), and $\beta_j$ are learned coefficients (Povoa et al., 2017).

Key implementation details:

• Feature selection relies on statistical dependence (e.g., MIC), yielding compact, informative feature sets of $\sim$15 predictors from >40 raw metrics.
• Both parametric (MLR) and non-parametric (regression trees, multilayer perceptrons) regressors are used, with MLPs achieving $R^2 = 99.94\%$ and mean absolute errors $<2\,\text{W}$ on mixed commodity hardware (Povoa et al., 2017).
• Fast calculation: estimation at 10–100 ms granularity with negligible overhead.
• Layerwise variants allow deep learning energy attribution: accumulate predicted layer-wise energies for models defined within frameworks such as PyTorch, using static parsing and hand-trained regression per layer type (Getzner et al., 2023).
• Attribution is granular: energy is allocated to components (e.g., CPU, memory, disk) or even to code-layer entities (layers, submodules) by summing the contributions across the feature set weighted by their coefficients.

This approach is highly portable and enables accurate per-app, per-component, or per-task reporting in real time. Its main limitation is the dependence on the quality of underlying counters and the assumption of (piecewise) linearity between counters and true energy.

3. Thread-, Application-, and NUMA-Level Attribution

Contemporary computing environments require attribution at the finest operational granularity—at the thread or process level, across architectures characterized by simultaneous multi-threading (SMT), dynamic voltage and frequency scaling (DVFS), multi-socket (NUMA) memory hierarchies, and co-tenancy.

Advances typified by models such as METRION (Weigell et al., 7 Dec 2025) and EnergAt (Hè et al., 2023) employ:

• Measurement of active vs. idle energy per hardware component (CPU packages, DRAM domains), subtracting calibrated idle baselines.
• Apportionment of active energy for each unit based on detailed “work” measures—such as unhalted core cycles, frequency ratios (for DVFS), and, for memory, number of local/remote LLC-miss loads.
• Handling of NUMA: Each package/domain is attributed energy independently; remote memory accesses are up-weighted (by empirical factor, e.g., 9.67) to capture higher energy for cross-domain traffic.
• Nonlinearity in utilization-to-energy is corrected by raising resource fractions to the power of empirically fitted exponents (e.g., $\gamma, \sigma$ for CPU, DRAM).
• Formal equations for per-component, per-thread energy include terms like

$$E_{\text{CPU}}(a) = \sum_{s\in S} \Delta E^{\text{CPU}}(s) [f_s(a)]^\gamma$$

where $f_s(a)$ is thread $a$'s fraction of CPU time on socket $s$.

• Proven robustness against multi-tenancy effects: as co-resident (“noisy neighbor”) jobs arrive, the averaging denominator grows, yielding self-correcting per-thread attributions that avoid spurious spikes (Hè et al., 2023).
• Energy is attributed and logged per-thread at millisecond resolution, with system-wide sums matching hardware RAPL readings to within $4.5\%$ (Hè et al., 2023), and CPU energy MAPE <10% in controlled scenarios (Weigell et al., 7 Dec 2025).

Such frameworks are extensible to accommodate new hardware entities (GPUs, NICs), further performance events, or alternate attribution policies (fair-splitting, time-sharing, etc.).

4. Attribution for Application Workloads and Program Structure

In performance-critical or embedded domains, static and dynamic program analysis enables energy attribution at function, block, or even instruction granularity.

• Instruction-level energy models: Base energy values ($B_i$), inter-instruction overheads ($O_{i,j}$), and external effects are empirically measured or calibrated for specific ISAs and mapped to higher-level program representations (e.g., LLVM IR) (Grech et al., 2014).
• Static attribution methodology:
  • Map source or IR instructions to physical energy costs using either detailed ISA mapping or category-based models when mapping is incomplete.
  • Use symbolic execution and extraction of cost relations (recurrence equations) to generate closed-form or parametric energy functions for arbitrary inputs:

  $E_f(\vec{N}) = \sum_{i} n_i(\vec{N})\, c_i$

  where $n_i(\vec{N})$ is the symbolic count for instruction $i$ as a function of input parameters. - Recurrences are solved using automated solvers, yielding formulas such as $E_{\text{sort}}(P) = 25P^2 + 11P + 7.1$ nJ for insertion sort on Cortex-M3 (Grech et al., 2014).

• Dynamic code coverage is also possible with per-layer, per-block, or event-level metering in instrumented systems.

These techniques enable early, hardware-independent optimization of code for energy efficiency and precise attribution of consumption to logical units within software or algorithmic pipelines.

5. Domain-Specific Attribution: Sensor Networks, Buildings, Human and Climate Systems

Energy attribution models are adapted to specific application domains by tailoring attribution entities and linking them to domain-specific phenomena:

• Wireless sensor networks: Energy is partitioned among five “constituents”—individual, local, global, environment, sink—each defined by a formal packet-flow ($b_i$) and per-packet energy coefficients ($a_i$), estimated via least-squares regression from measured energy and activity traces. The model realizes:

$E_{\text{overall}} = \sum_{i=1}^{5} a_i b_i$

and facilitates both real-time energy budget tracking and optimization of network parameters (e.g., duty cycling, routing) to prolong system lifetime (Kamyabpour et al., 2012).

• Building energy benchmarking: Models use multi-order linear regression (MLRi) and gradient-boosted trees (GBT), often enhanced with Shapley (SHAP) value analysis to decompose aggregate energy performance metrics (e.g., EUI) into attributed contributions from factors (area, occupancy, equipment, usage patterns) and their interactions. SHAP force plots visualize individual-building attributions, directly supporting management decisions (Arjunan et al., 2019).
• Athlete energy systems: Systems biology-inspired hydraulic models represent energy stores as interconnected tanks (aerobic, anaerobic slow, anaerobic fast), with energy attribution at each moment apportioned to modeled physiological pathways via the flow of power between tanks. Pareto-optimized, parameter-fitted models enable detailed, interpretable attribution by time or exertion profile (Weigend et al., 2021).
• Climate models: Attribution frameworks leverage feature-importance and counterfactual analysis (e.g., LSTM-VAE attribution for ERA5 and GEMB snowmelt outputs) to identify physical drivers (e.g., radiative, thermal, meteorological variables) of anomalous energy fluxes, supporting both anomaly detection and physical interpretability in multivariate geophysical systems (Ale et al., 11 Feb 2025).

6. Evaluations, Limitations, and Practical Recommendations

Models are routinely benchmarked against high-frequency metered ground truth, competing attribution frameworks, and domain-specific accuracy standards:

• High granularity (10–100 ms) model accuracy against RAPL or power meter: 4–18% MAPE (CPU/DRAM), depending on workload (Weigell et al., 7 Dec 2025, Hè et al., 2023).
• Attribution accuracy is validated by summing per-component or per-thread attributions to total system consumption and by synthetic workload “mixtures” testing isolation from co-tenant impact.
• Regression- and statistical-model-based methods are deployable on commodity OSs, requiring minimal overhead and no specialized hardware (Povoa et al., 2017, Dong et al., 2010).
• Domain-centric extensions require ongoing model calibration as hardware (e.g., hidden domain energy, cache effects) or resource metrics (e.g., new perf counters, energy domain exposure) evolve.

Core limitations include:

• Attribution precision is bounded by the accuracy of resource counter reporting, static model completeness, and calibration fidelity.
• Hardware components that are not exposed or measurable via OS/firmware counters remain unmodeled.
• Fine-grained attribution (sub-10 ms, high concurrency) is subject to increased measurement and polling overhead, which must be calibrated out for <1% perturbation (Dong et al., 2010).
• Energy attribution for highly dynamic, interactive, or virtualized environments (e.g., heavy migration, rapid context switches) entails increased model complexity and systemic uncertainty.

Best practices include frequent model retraining for new hardware/software configurations, low-frequency error monitoring to identify drift, selective polling for overhead minimization, and offloading numerically intensive tasks (e.g., PCA, SVD) during idle periods or to remote resources (Dong et al., 2010, Weigell et al., 7 Dec 2025).

7. Impact and Future Directions

Energy attribution models have transformed the observability, optimization, and accountability of complex technological and scientific systems. Their evolution has enabled:

• Self-modeling devices that dynamically adapt to workload and hardware variability (e.g., battery-powered mobile systems (Dong et al., 2010)).
• Real-time, fine-grained, and multi-tenant energy accounting for cloud platforms, with demonstrated resilience to noisy neighbors and dynamic workload environments (Hè et al., 2023, Weigell et al., 7 Dec 2025).
• Early-stage, design-time energy budgeting during algorithm or system development—prior to deployment.
• Transparent, physically interpretable attribution in societal infrastructure (buildings, climate) and biological or human-in-the-loop systems.

Ongoing research is extending the scope of attribution to encompass additional hardware entities (GPUs, accelerators), richer runtime features (cache, memory patterns), coupled or nested attribution hierarchies (from operator to VM to physical host), and integration with broader sustainability optimization frameworks.

Energy attribution models thus provide a rigorous foundation for multidimensional energy-aware computing, optimization, and scientific discovery across scales, domains, and disciplines.