# How Vitality Tracker estimates your energy expenditure

Method note, version 1.0 · 8 October 2026 · TDEE Engine v2, the version in production · Csanad Novak

Vitality Tracker does not take the calories-burned figure from your watch, and it does not rely on a fixed formula. It estimates your total daily energy expenditure (TDEE) from the one relationship that depends on no device's guess: how your weight moved against what you logged eating. A physiological prior, computed from your lean mass and training load, anchors the estimate when the data is thin. The two are combined into a single figure with a credible interval, and the interval, not the point estimate, is the headline. The method is not ours. It is the energy-balance arithmetic of the research literature: Hall's group showed that changes in intake can be recovered from repeated body weights alone, checked against doubly labeled water and DXA over two years, and the same principle drives the adaptive coaching apps that re-estimate expenditure from weight and intake. What is ours is the way the pieces are put together for one person with sparse data, and the care taken to say how sure the number is.

## What it is for, and what it is not

The estimate sets the calorie targets in the Weight Loss Plan and underlies the energy-balance view. It is a description of a physiological quantity, estimated from your own records, for general wellness and for discussion with a health professional. It does not diagnose, treat or manage any disease, and it is not a measurement of metabolic rate; a clinical question about metabolism belongs with your doctor or dietitian.

## Inputs

- Daily logged energy intake, from the meal plan and the exceptions you record against it.
- Daily body weight, from any scale. The device rules that apply everywhere in VT apply here: each scale keeps its own series.
- Body composition when you have it: bioimpedance lean mass, corrected against a DEXA scan where one exists.
- Training load, from session-RPE.
- Optionally, overnight resting heart rate, heart-rate variability and temperature. These feed an observational arm (see below) and do not move the estimate.

Every input has a defined behavior when it is absent. Without body composition, the prior falls back to an anthropometric equation. Without a training-load record, the activity term sits at its floor. With sparse or noisy logging, the estimate leans toward the prior and the interval widens. Nothing is penalized for being missing: a person with a scale and a food log is a first-class user of this engine.

## Method

**1. The energy-balance measurement.** Over a window of n days, the measured expenditure is the average logged intake minus the energy content of the weight change, spread over the window:

$$
\mathrm{TDEE}_{\text{measured}} = \overline{EI} - \rho \cdot \frac{\Delta W}{n}
$$

Intake is the confounder-weighted mean of what you logged (step 6). The weight change is estimated by weighted regression on the raw daily weights, not by subtracting two readings. The energy density of weight change, ρ, is not fixed at the folklore 7,700 kcal per kilogram. Early weight loss is largely water, glycogen and lean tissue; later loss is mostly fat. VT adapts ρ to the composition of the tissue being gained or lost, using Forbes's partition of fat-free and fat mass as a function of body fatness and Hall's derivation of the energy deficit per unit of weight change from it, with the sign handled correctly in deficit and in surplus.

**2. The window is bounded by change points.** A measurement like this is valid only within one metabolic and behavioral regime, and pooling across a regime change biases it. VT runs Bayesian online change-point detection jointly on the raw daily intake and the raw daily weight-rate series, deliberately not on smoothed series, whose induced autocorrelation lags the detector and miscalibrates it. A change point is committed only when the run-length probability and a confirmation-day criterion are both met, and it is dated to its onset. The estimation window then starts at the most recent committed change point: a new training block, a diet started, an illness, each resets the clock rather than contaminating the estimate.

**3. The physiological prior.** The prior expectation is resting metabolic rate times an activity multiplier. When a trustworthy lean-mass value exists, operationally when bioimpedance lean mass has been corrected against a personal DEXA anchor, resting metabolic rate comes from a fat-free-mass equation: by default the one popularly called Katch–McArdle, which is Cunningham's 1991 equation, with his 1980 form as the alternative. Otherwise it comes from an anthropometric equation that needs only height, weight, age and sex (the Müller or Mifflin–St Jeor forms); the engine never assumes a body-composition device exists. The activity multiplier maps training load through an asymptotic curve with an explicit ceiling, following the constrained-expenditure model, in which total expenditure plateaus as activity rises rather than adding up linearly. The prior contains no intake term, so it is independent of how faithfully you log food, a property step 5 depends on.

**4. Combining the two.** The prior and the measurement are treated as two Gaussian estimates and combined by precision weighting:

$$
\mu_{\text{post}} = \frac{\mu_{\text{prior}}/\sigma_{\text{prior}}^{2} + \mathrm{TDEE}_{\text{measured}}/\sigma_{\text{meas}}^{2}}{1/\sigma_{\text{prior}}^{2} + 1/\sigma_{\text{meas}}^{2}}
$$

The measurement's uncertainty scales with the effective number of clean days in the window and with the variance of the intake and weight series. A sparse or confounded window therefore leans toward the prior; a dense, clean window leans toward the measurement. The output is a mean and a credible interval, and the interval is what the app shows first.

**5. The equilibrium check and the logging-consistency signal.** When your weight is statistically stationary, the energy-balance identity collapses to TDEE equals mean intake, and the engine takes a fast path: expenditure is your mean reported intake, in reported-calorie units. The ratio of that figure to the prior is a self-calibration signal. Because the prior does not depend on logging, a ratio well below one suggests under-reporting, and a ratio above one suggests the prior under-reads your expenditure. The ratio cannot tell those two apart on its own; that is an intrinsic limit, not a bug. VT therefore shows it as a consistency diagnostic, never as a verdict on a single cause.

**6. Confounded days.** Days on which physiology is perturbed are handled by mechanism rather than by a generic outlier rule. Acute one-off confounders (illness, alcohol, a poor night, a late meal) down-weight the day. Cyclic confounders (menstrual water retention) are a recurring level shift, so they are offset-subtracted rather than down-weighted. Correlated confounders are clustered and combined by their strongest single effect, so one physiological event is not penalized several times, and a severe single confounder is allowed to dominate. The per-day weights yield an effective sample size that propagates into the measurement's uncertainty, and the alignment respects that an end-of-day rating bears on the next morning's weight.

**7. Metabolic adaptation, held observational.** Under a sustained deficit, expenditure can fall below what lost mass predicts; the effect is well documented, and in a small group of extreme weight losers it persisted for six years. VT computes a candidate adaptation signal from the overnight autonomic triad (resting heart rate, heart-rate variability, temperature), read against your own baseline on your own device, gated to detected deficit periods, and conditioned on lost mass and training load so that a training-induced drop in heart rate is not mistaken for adaptation. This arm ships observational: it computes and logs its signal but does not move the estimate. On eighteen months of the founder's own data, resting heart rate moved in the expected direction during a deficit, heart-rate variability was flat, and the heart-rate movement was confounded with a concurrent change in training load; the signal was not separable from the energy-balance measurement. The energy-balance measurement already captures adaptation implicitly, because a falling expenditure shows up as a slower weight change. For the same reason the engine excludes wearable calories-burned figures entirely: in an independent laboratory evaluation of seven wrist-worn devices, median energy-expenditure errors ran from 27 to 93 percent by device.

## What you see

- The estimate as a range first: a credible interval around a central figure, not a single number with false precision.
- A confidence band built from named, inspectable dimensions: data sufficiency, logging consistency, confounder load, estimate stability and prior basis. There is no single opaque score, which would manufacture precision and double-count uncertainty already in the interval.
- The logging-consistency diagnostic, when the equilibrium path is active.
- When the data is not good enough, a wider interval or no estimate, rather than a confident wrong one.
- The Weight Loss Plan sets its deficit from this estimate; diet choices and training plans are separate decisions layered on top.

## Where it comes from

1. Hall KD, Sacks G, Chandramohan D, et al. Quantification of the effect of energy imbalance on bodyweight. *Lancet* 2011;378(9793):826–837.
2. Hall KD. Predicting metabolic adaptation, body weight change, and energy intake in humans. *American Journal of Physiology: Endocrinology and Metabolism* 2010;298(3):E449–E466.
3. Hall KD. What is the required energy deficit per unit weight loss? *International Journal of Obesity* 2008;32(3):573–576.
4. Sanghvi A, Redman LM, Martin CK, Ravussin E, Hall KD. Validation of an inexpensive and accurate mathematical method to measure long-term changes in free-living energy intake. *American Journal of Clinical Nutrition* 2015;102(2):353–358.
5. Thomas DM, Martin CK, Lettieri S, et al. Can a weight loss of one pound a week be achieved with a 3500-kcal deficit? Commentary on a commonly accepted rule. *International Journal of Obesity* 2013;37(12):1611–1613.
6. Forbes GB. Lean body mass–body fat interrelationships in humans. *Nutrition Reviews* 1987;45(8):225–231.
7. Mifflin MD, St Jeor ST, Hill LA, Scott BJ, Daugherty SA, Koh YO. A new predictive equation for resting energy expenditure in healthy individuals. *American Journal of Clinical Nutrition* 1990;51(2):241–247.
8. Cunningham JJ. A reanalysis of the factors influencing basal metabolic rate in normal adults. *American Journal of Clinical Nutrition* 1980;33(11):2372–2374.
9. Cunningham JJ. Body composition as a determinant of energy expenditure: a synthetic review and a proposed general prediction equation. *American Journal of Clinical Nutrition* 1991;54(6):963–969. (The equation popularly called Katch–McArdle.)
10. McArdle WD, Katch FI, Katch VL. *Exercise Physiology: Nutrition, Energy, and Human Performance.* 8th ed. Philadelphia: Wolters Kluwer/Lippincott Williams & Wilkins; 2015. (The textbook that gave the Katch–McArdle equation its name.)
11. Müller MJ, Bosy-Westphal A, Klaus S, et al. World Health Organization equations have shortcomings for predicting resting energy expenditure in persons from a modern, affluent population: generation of a new reference standard from a retrospective analysis of a German database of resting energy expenditure. *American Journal of Clinical Nutrition* 2004;80(5):1379–1390.
12. Pontzer H, Durazo-Arvizu R, Dugas LR, et al. Constrained total energy expenditure and metabolic adaptation to physical activity in adult humans. *Current Biology* 2016;26(3):410–417.
13. Leibel RL, Rosenbaum M, Hirsch J. Changes in energy expenditure resulting from altered body weight. *New England Journal of Medicine* 1995;332(10):621–628.
14. Fothergill E, Guo J, Howard L, et al. Persistent metabolic adaptation 6 years after "The Biggest Loser" competition. *Obesity* 2016;24(8):1612–1619.
15. Adams RP, MacKay DJC. Bayesian online changepoint detection. arXiv:0710.3742, 2007.
16. Shcherbina A, Mattsson CM, Waggott D, et al. Accuracy in wrist-worn, sensor-based measurements of heart rate and energy expenditure in a diverse cohort. *Journal of Personalized Medicine* 2017;7(2):3.

## Limitations

- It is an estimate with an interval, not a measurement. The dominant source of error in practice is self-reported intake, which is why the consistency signal exists and why the interval is shown first.
- The engine has so far been run on one person's data, eighteen months of the founder's own records, with no reference method such as doubly labeled water. That establishes feasibility and predictive coherence, not generalizable accuracy. A simulation study with known ground truth, built to test whether the stated intervals are calibrated, is in preparation, and a cohort validation will follow once there is a user base to run it on.
- The logging-consistency ratio cannot separate under-reporting from an inaccurate prior without an external anchor; it is a diagnostic.
- The metabolic-adaptation arm is observational and, on present evidence, marginal.
- Several parameters were set by hand on a single subject and may not transfer. They will be re-estimated as data accrues, under the versioning rule below.

## Version

TDEE Engine v2, specified April 2026; this note follows the methodology draft of June 2026. Version 2 replaced the fixed energy density of weight change with the composition-adaptive form, added the change-point-bounded window and the confounder layer, moved the prior to a fat-free-mass basis when a DEXA anchor exists, and made the credible interval the headline output. Any change to a formula, a parameter or an input rule bumps the version, is logged here with its reason, and recomputes history with both values kept. The version shown in the app matches this note. This note: version 1.0, 8 October 2026, deposited on Zenodo as [doi:10.5281/zenodo.23224326](https://doi.org/10.5281/zenodo.23224326); all versions [doi:10.5281/zenodo.23224325](https://doi.org/10.5281/zenodo.23224325), which always resolves to the latest.

## Cite and license

Cite as: Novak C. How Vitality Tracker estimates your energy expenditure. Vitality Tracker Methods, method note version 1.0 for TDEE Engine v2, 8 October 2026. vitalitytracker.app/science/methods/energy-expenditure. [doi:10.5281/zenodo.23224326](https://doi.org/10.5281/zenodo.23224326) (all versions: [doi:10.5281/zenodo.23224325](https://doi.org/10.5281/zenodo.23224325)).

Text © Novak Consulting Ltd, licensed under CC BY 4.0: reuse with attribution. The engine's code is not covered by this license.

Full specification: [doi:10.5281/zenodo.23224326](https://doi.org/10.5281/zenodo.23224326). TDEE methodology paper, draft v2 (June 2026). The paper stays a paper: the simulation study with known ground truth is in preparation, then preprint. Method note deposited on Zenodo.

Source: https://vitalitytracker.app/science/methods/energy-expenditure