⚡ electrical engineering

PM-EKF: A Physiological Model-Based Extended Kalman Filter for Daily-Life Physical Activity Energy Expenditure Estimation

This paper proposes a Physiological Model-based Extended Kalman Filter (PM-EKF) that integrates a simplified metabolic model with inertial and heart rate sensor data to provide personalized, interpretable, and accurate daily-life physical activity energy expenditure estimates, outperforming both linear regression and deep learning approaches on a validated dataset.

Original authors: Shuhao Que, Remco Poelarends, Valentina Breschi, Ying Wang

Published 2026-04-30

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Shuhao Que, Remco Poelarends, Valentina Breschi, Ying Wang

Original paper licensed under CC BY 4.0 (http://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

The Big Picture: Guessing How Much "Fuel" You Burn

Imagine your body is a car. When you drive, the engine burns fuel. In the human body, this "fuel" is energy, and the "exhaust" is the carbon dioxide you breathe out. Scientists want to know exactly how much fuel you burn every day (Physical Activity Energy Expenditure, or PAEE) to understand your health.

The "Gold Standard" for measuring this is a giant, clunky mask that measures every breath you take. It's accurate, but you can't wear it to the grocery store or while doing laundry. So, researchers use smartwatches and fitness trackers (wearable sensors) to guess the answer.

The Problem: Most current trackers are like black boxes. They look at your movement and heart rate, run it through a complex computer program (like a neural network), and spit out a number. They work okay, but they don't really understand why the number is what it is. If the data looks weird, the computer might guess wrong, or even give you a negative number for energy burned (which is impossible!).

The Solution: The authors of this paper built a new system called PM-EKF. Instead of a black box, they built a transparent, physics-based engine that mimics how your lungs, blood, and muscles actually work.

How It Works: The "Smart Pilot" Analogy

Think of estimating your energy burn as flying a plane from Point A to Point B.

The Map (The Physiological Model):
The authors created a simplified map of how your body works. It has three main stops:
- The Lungs: Where you take in oxygen.
- The Circulation: Your heart pumping blood.
- The Muscles: Where the energy is used.
  This map follows the laws of physics and biology. It knows that if you move your legs, your muscles need more oxygen, and your heart has to pump faster to deliver it.
The Sensors (The Inputs):
The system uses two types of data from your wearable devices:
- IMUs (Inertial Measurement Units): These are like accelerometers in your phone. They measure how fast and hard your body parts (pelvis and thighs) are moving. This tells the system the "mechanical work" you are doing.
- Heart Rate (HR): This is the pulse. It tells the system how hard your heart is working to deliver fuel.
The Pilot (The Extended Kalman Filter - EKF):
This is the "brain" of the system. Imagine a pilot flying a plane who has a map (the physiological model) but also has a GPS that sometimes glitches (the noisy sensor data).
- The Map predicts where the plane should be based on physics.
- The GPS tells the pilot where the plane actually is based on sensors.
- The Pilot (EKF) constantly compares the two. If the GPS is shaky, the pilot trusts the map more. If the map seems off, the pilot adjusts based on the GPS.
- This allows the system to smooth out the noise and give a steady, realistic estimate of your energy burn, even if the sensors are a little jumpy.

The Experiment: The "Living Lab" Test

The researchers tested this system on 9 people in a lab that simulated real life.

The Setup: The participants wore sensors on their pelvis and thighs, plus a heart rate monitor. They also wore a high-tech "gold standard" mask (COSMED K5) to measure their actual breath-by-breath gas exchange.
The Activities: They did a mix of things: sitting, reading, standing, mopping, stacking books, walking on a treadmill, climbing stairs, and cycling.
The Comparison: They pitted their new "Smart Pilot" (PM-EKF) against two other methods:
1. Linear Regression (LR): A simple, old-school math formula.
2. CNN-LSTM: A fancy, modern "black box" AI (Deep Learning).

The Results: Who Won?

The paper claims their new system (PM-EKF) was the winner, but with some interesting nuances:

Overall Accuracy: The PM-EKF was the most accurate overall. It matched the "gold standard" mask better than the AI or the simple math formula.
- Analogy: If the mask score is 100, PM-EKF scored 72, the AI scored 65, and the simple math scored 52.
The "Negative Energy" Problem: The AI and the simple math sometimes made a silly mistake: they predicted that people were burning negative calories while standing still (which is physically impossible). The PM-EKF never made this mistake because its "map" of the body physically prevents it from calculating impossible numbers.
The Low-Intensity Struggle: The AI and simple math were terrible at guessing energy burn during low-intensity tasks (like sitting or standing). They often guessed wildly high numbers. The PM-EKF stayed calm and accurate during these quiet moments.
The Cycling Glitch: The PM-EKF had one specific weakness: Cycling. It underestimated how much energy people burned while cycling.
- Why? The sensors were on the legs and pelvis. When you cycle, your legs move in a circle, but your body doesn't bounce up and down much. The sensors couldn't "see" the full effort of the bike's resistance, so the "mechanical work" calculation was too low. The AI, however, was better at guessing the cycling numbers because it just memorized patterns from the data without caring about the physics.
The Heart Rate Surprise: The researchers tested what happened if they removed the heart rate sensor from the equation.
- Result: It didn't matter much! The PM-EKF (and the other models) performed almost the same with or without heart rate data.
- Why? In the activities they tested (mostly walking, standing, and light chores), the movement of the body (captured by the leg sensors) was the biggest driver of energy burn. The heart rate didn't add much extra information.

The Bottom Line

The paper concludes that you don't need a "black box" AI to guess how much energy you burn. By building a system that understands the physics of the human body (how lungs, blood, and muscles interact), you can get a more reliable, realistic, and "sensible" estimate.

It's interpretable: You can look inside the "black box" and see the estimated oxygen levels in the blood and the air in the lungs.
It's robust: It doesn't give crazy answers (like negative calories).
It works without a heart rate monitor: For daily activities, just knowing how much your body is moving is often enough to get a good estimate.

The authors note that while the system is great, it needs more testing with more people and different types of activities to be perfect for everyone.

1. Problem Statement

Monitoring Physical Activity Energy Expenditure (PAEE) in daily life is critical for assessing metabolic health and disease risk. While indirect calorimetry (gas analysis) is the gold standard, it is impractical for continuous, free-living monitoring. Consequently, wearable sensors (Inertial Measurement Units - IMUs, and Heart Rate - HR) are widely used.

However, existing methods face significant limitations:

Lack of Physiological Interpretability: Most current approaches are purely data-driven (e.g., Linear Regression, Neural Networks). They map sensor signals to PAEE statistically without modeling the underlying physiological mechanisms (gas exchange, metabolism).
Black-Box Nature: These models lack intrinsic interpretability, making it difficult to explain why an estimate was made or to ensure physiological consistency (e.g., avoiding negative energy expenditure values).
Generalizability Issues: Data-driven models often struggle with distribution shifts (different individuals or activity types) and may produce physiologically implausible results.

2. Methodology: The PM-EKF Framework

The authors propose a Physiological Model-based Extended Kalman Filter (PM-EKF). This approach integrates a simplified mechanistic model of human physiology with an EKF to fuse sensor data. The framework operates on three levels:

A. Signal Level (Preprocessing & Proxies)

Inputs: Heart Rate (HR) from ECG and body acceleration from three IMUs (pelvis, left thigh, right thigh).
IMU-Derived Metabolic Proxy: The system calculates the mechanical kinetic energy ( $E(t)$ $E (t)$ ) required to move body segments based on velocity derived from IMU integration.
- This mechanical energy is converted into an estimated oxygen consumption rate ( $R_{MO2}$ ) and carbon dioxide production rate ( $R_{MCO2}$ ) using a lumped efficiency factor ( $\mu$ ) and a conversion factor (19.6 KJ/L $O_2$ ).
- Note: This proxy serves as the "observation" for the filter, representing movement-related metabolic demand.

B. Modeling Level (Physiological State-Space)

The core innovation is a three-compartment physiological model (Lungs, Circulation, Muscle Tissue) representing the respiratory control loop and gas exchange.

State Vector ( $x$ ): The model tracks five latent physiological states:
1. Alveolar $O_2$ partial pressure ( $P_{A,O2}$ )
2. Alveolar $CO_2$ partial pressure ( $P_{A,CO2}$ )
3. Venous $O_2$ concentration ( $C_{v,O2}$ )
4. Venous $CO_2$ concentration ( $C_{v,CO2}$ )
5. Alveolar Ventilation ( $V_{TA}$ )
Process Model: A set of nonlinear differential equations describing mass balance for $O_2$ and $CO_2$ across the compartments. HR is used as an external input to drive cardiac output ( $Q$ ), which influences gas transport.
Measurement Model: The IMU-derived metabolic proxies ( $R_{MO2}, R_{MCO2}$ ) are treated as noisy observations of the actual gas-exchange fluxes predicted by the physiological model.

C. Inference Level (Extended Kalman Filter)

The EKF recursively estimates the latent physiological states.
Prediction Step: Propagates states forward based on the physiological dynamics driven by HR.
Correction Step: Updates the state estimates using the IMU-derived metabolic proxies to correct for model uncertainty and measurement noise.
Output: The inferred latent states are used to calculate the final PAEE using Weir's formula, which relates $O_2$ consumption and $CO_2$ production to energy expenditure.

3. Key Contributions

Mechanistic Interpretability: Unlike black-box AI models, PM-EKF explicitly models the causal chain from body movement $\to$ mechanical work $\to$ gas exchange $\to$ energy expenditure. The latent states (e.g., alveolar pressure) have direct physiological meaning.
Physiological Constraints: By embedding the model within an EKF, the estimates are constrained by biological laws (e.g., mass balance, ideal gas law), preventing physiologically impossible outputs (like negative energy expenditure).
Novel Sensor Fusion: The method treats IMU data not just as a direct predictor, but as an observation of metabolic demand that corrects a physiological process model driven by HR.
No Training Required: The model is based on first principles and does not require a large dataset for "training" in the machine learning sense, making it potentially more generalizable across individuals.

4. Experimental Results

The model was validated on a dataset of 9 healthy subjects performing various Activities of Daily Living (ADL), including sitting, walking, mopping, cycling, and stair climbing. Ground truth was provided by a COSMED K5 breath-by-breath respiratory analyzer.

Performance Metrics (Median $R^2$ ):
- PM-EKF (with HR): 0.72 (Range: 0.60–0.87)
- CNN-LSTM (Data-driven): 0.65
- Linear Regression (Data-driven): 0.52
- Statistical Significance: PM-EKF significantly outperformed both LR ( $p=0.012$ ) and CNN-LSTM ( $p=0.039$ ).
Low-Intensity Performance:
- PM-EKF excelled at low intensities (NRMSE = 0.19), whereas data-driven models performed poorly (NRMSE > 1.0, worse than a mean predictor) and occasionally predicted negative PAEE values.
- PM-EKF had a 0% violation rate for physiological bounds (PAEE $\ge$ 0), compared to 0.53% for CNN-LSTM.
HR Dependence:
- Surprisingly, removing HR input from the PM-EKF (fixing it to a constant) did not significantly degrade performance ( $R^2$ remained ~0.76).
- This suggests that for the ADLs tested, IMU-captured mechanical workload was the dominant driver of PAEE estimation, and HR provided redundant information in this specific context.
Limitations Observed:
- The model systematically underestimated PAEE during cycling. This is attributed to the IMU-based mechanical proxy failing to capture external resistance (ergometer load) and the fixed efficiency factor ( $\mu$ ) not adapting to cycling dynamics.

5. Significance and Conclusion

The PM-EKF framework represents a paradigm shift from purely data-driven PAEE estimation to physiology-guided estimation.

Clinical Relevance: It provides not just a number, but interpretable latent states (e.g., gas exchange rates) that could serve as biomarkers for metabolic health.
Robustness: The model maintains physiological plausibility even in low-intensity scenarios where data-driven models fail, making it suitable for continuous, real-world monitoring.
Future Outlook: While the current study shows IMUs are sufficient for ADL, future work aims to improve cycling estimation (by adapting the efficiency factor) and validate the model on larger, more diverse populations including those with metabolic disorders.

In summary, the paper demonstrates that embedding a simplified physiological model within a Kalman Filter yields more accurate, interpretable, and robust PAEE estimates than purely statistical methods, particularly for low-to-moderate intensity daily activities.