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The model

wdm describes one person’s body weight as a five-state, continuous-time dynamical system driven by a daily food log. This page states each mechanism with its equation and source, then the observation model that connects the hidden state to a bathroom scale and an optional tape measure. Estimation covers discretization and inference, Control the diet planner, and Notation every symbol. The slide deck is a visual tour of the same material. Units: kilograms, kilocalories, days.

The five states

Symbol	Code name	Units	Meaning
$T$	`tissue_kg`	kg	Tissue mass: fat plus fat-free solids and their structural water. What a diet changes, and not what the scale reads.
$F$	`fat_kg`	kg	Fat mass, a component of $T$.
$G_c$	`gly_water_kg`	kg	Glycogen and its bound water, as a deviation from its level at habitual intake. Fast, carbohydrate-driven.
$G_n$	`na_water_kg`	kg	Sodium-associated extracellular water, as a deviation from its habitual level. Zero unless sodium is logged.
$A$	`adapt_kcal`	kcal/day	Adaptive thermogenesis. Negative during a deficit.

Lean mass $M = T - F$ is not a separate state, so the fat/lean split is consistent by construction. The scale weighs total mass $W = T + G_c + G_n$.

Energy balance

Stored energy $S$ changes by intake minus expenditure, $\mathrm{d}S/\mathrm{d}t = I - E$, with $I$ the metabolizable intake and $E$ the expenditure (kcal/day). Nobody disputes this; the hard parts are what form the stored energy takes, how $E$ depends on the body, and how well either is measured. Expenditure is not measured outside a metabolic ward, and self-reported intake is biased, often badly (Dhurandhar et al., 2015), so logged intake is treated as a noisy input. A constant bias is absorbed into an estimated intercept; a drifting one is the framework’s most serious weakness (Limitations). Before use, logged energy is rebuilt from Atwater factors (4, 4, 9, 7 kcal/g) when only macronutrients are given, and 2 kcal per gram of fiber is subtracted, since labels count fiber at the full carbohydrate factor.

Fat and lean: where the mass comes from

Adipose tissue stores $\rho_F = 9441$ kcal/kg (39.5 MJ/kg) and fat-free tissue $\rho_L = 1816$ kcal/kg (7.6 MJ/kg) (Hall, 2008); fat-free tissue is about 70% water.

The most important fact in the model

The same deficit moves the scale five times as far if the mass comes off as lean tissue rather than fat. A rule such as 3500 kcal per pound assumes this away, along with falling expenditure and adaptation, and so predicts unbounded linear loss where real trajectories flatten (Hall et al., 2011).

Forbes’ law. Forbes (1987) found lean mass approximately logarithmic in fat mass across a wide range of adiposity, $M = 10.4 \ln F + \text{const}$ (kg). Since $\mathrm{d}M/\mathrm{d}F = 10.4/F$ and $\mathrm{d}W = \mathrm{d}F + \mathrm{d}M$, the fat share of any mass change is

\[ p_{\mathrm{Forbes}} = \frac{\mathrm{d}F}{\mathrm{d}W} = \frac{F}{F + 10.4}, \]

with $F$ floored at 2 kg in code, where the relation is singular. The energy cost of a kilogram of tissue is then

\[ \rho_{\mathrm{eff}} = p\,\rho_F + (1 - p)\,\rho_L . \]

$F$ (kg)	50	30	20	15
$p_{\mathrm{Forbes}}$	0.83	0.74	0.66	0.59
$\rho_{\mathrm{eff}}$ (kcal/kg)	8128	7478	6832	6319

The fatter you are, the larger the fat share of each kilogram lost, so the last stretch of a diet is harder than the first. Forbes’ law is a population regression with substantial individual scatter, used as a structural prior that the scale barely constrains.

Protein. Higher protein intake during a deficit spares lean mass (Helms et al., 2014). With $P_g$ the protein eaten (g/day),

\[ p = \operatorname{clip}\!\left( \frac{F}{F + 10.4} + \kappa_P \left( \frac{P_g}{W} - 1.6 \right),\; 0.50,\; 0.98 \right), \]

applied only on days with a logged macronutrient split. The effect is real but its size is uncertain, and $\kappa_P$ is essentially not recoverable from scale data (identifiability). It stays because omitting it would assert that protein has no effect.

Expenditure

Total expenditure is roughly 60 to 70% resting metabolism, 20 to 30% activity and 10% thermic effect of food. The model writes

\[ E = \underbrace{c_0 + c_1 W}_{\text{resting + baseline activity}} + \mathrm{TEF} + X + A , \]

with $X$ the logged exercise (kcal/day).

Resting expenditure. The Mifflin-St Jeor equation (Mifflin et al., 1990), $\mathrm{RMR} = 10\,W + 6.25 \times \text{height} - 5 \times \text{age} + s$ (kg, cm, years; $s = +5$ for men, $-161$ for women), only centers the priors. The code scales it by an activity level of 1.45: $c_1$ is centered at 14.5 kcal/day per kg and $c_0$ at $1.45\,(6.25 \times \text{height} - 5 \times \text{age} + s)$. Population equations miss an individual by several hundred kilocalories a day, so both are estimated. Never quote $c_0$ as physiology: it absorbs mass-independent expenditure, baseline activity and any constant bias in the food log (a subject who under-reports by 20% has that 20% absorbed into $c_0$), which is what makes a constant bias harmless.

Thermic effect of food. With carbohydrate $C$, fat $F_g$ and alcohol $A_g$ in g/day,

\[ \mathrm{TEF} = \alpha_P \cdot 4 P_g + \alpha_C \cdot 4 C + \alpha_F \cdot 9 F_g + \alpha_A \cdot 7 A_g , \]

with $\alpha_P = 0.25$, $\alpha_C = 0.075$, $\alpha_F = 0.02$ and $\alpha_A = 0.15$ (reported ranges 20 to 30%, 5 to 10%, 0 to 3% and 10 to 20%; Westerterp, 2004), or 10% of intake when no split is logged. Moving 100 g of carbohydrate to protein at constant energy raises expenditure by about $(0.25 - 0.075) \times 400 \approx 70$ kcal/day. The coefficients are held fixed because their effect is confounded with maintenance energy.

Adaptive thermogenesis. During restriction, expenditure falls by more than mass loss explains (Leibel et al., 1995; Rosenbaum and Leibel, 2010), and the effect can persist for years (Fothergill et al., 2016). Adaptation responds to the gap against expenditure without adaptation (using total expenditure would make the equation refer to itself):

\[ E_{\mathrm{base}} = c_0 + c_1 W + \mathrm{TEF} + X, \qquad D = E_{\mathrm{base}} - I, \qquad \dot A = \frac{-\beta_A D - A}{\tau_A}, \]

with $\beta_A \approx 0.15$ (reported 0.10 to 0.25; Hall, 2010) and $\tau_A \approx 14$ days, an order-of-magnitude figure.

The plateau

Under a sustained gap $D$, adaptation settles at $-\beta_A D$. A 500 kcal/day deficit at $\beta_A = 0.15$ eventually lowers expenditure by 75 kcal/day beyond what mass loss explains, shrinking the effective deficit to 425 kcal/day; loss slows toward a new equilibrium at constant intake. The plateau is a property of the system, not a failure of the person in it.

The intake that holds weight steady is not $c_0 + c_1 W$ but $(c_0 + c_1 W)/(1 - \phi)$ for a thermic fraction $\phi$, about 11% higher at $\phi = 0.10$; mid-deficit it overestimates, since it ignores adaptation.

Water compartments

Two first-order lags move the scale within days with no change in tissue.

Glycogen water. Glycogen is stored with roughly 3 g of water per gram (Olsson and Saltin, 1970), and swinging between carbohydrate restriction and repletion moves about 200 g of glycogen plus 600 g of water:

\[ \dot G_c = \frac{k_G\,\Delta C - G_c}{\tau_G}, \qquad \Delta C = C - \bar C . \]

A sustained +250 g/day swing gives about 0.8 kg on the scale, so $k_G \approx 0.8/250 = 0.0032$, rounded to a prior center of 0.0030 kg per g/day. This compartment explains the dramatic first week of a low-carbohydrate diet, the “relapse” after one large meal, and much of the daily scale noise; modeling it keeps that water from masquerading as fat.

Sodium water. Extracellular fluid tracks sodium load over one to two days, often a larger and almost never logged source of scale noise:

\[ \dot G_n = \frac{k_{Na}\,\Delta\mathrm{Na} - G_n}{\tau_{Na}}, \qquad \Delta\mathrm{Na} = \mathrm{Na} - \overline{\mathrm{Na}} , \]

with $\mathrm{Na}$ in mg/day. Days without a sodium entry have $\Delta\mathrm{Na} = 0$.

The gain is the equilibrium water per unit of sustained sodium change. Extracellular fluid holds sodium at about 140 mmol/L (3.2 g per liter), and its volume moves until excretion matches intake: raising intake by about 4,400 mg/day adds on the order of a kilogram over several days, about $2.3 \times 10^{-4}$ kg per mg/day. The prior is centered at $2 \times 10^{-4}$ with a log-scale spread of 1.0 (a factor of about 2.7 either way), since some studies find sodium stored without water. Excretion is the lag’s job: a salty restaurant meal, a one-day pulse of 3,000 mg, raises the next morning’s weight by roughly 0.4 to 0.7 kg, gone within two or three days.

The habitual-intake reference

$\bar C$ and $\overline{\mathrm{Na}}$ are the subject’s habitual, pre-diet intakes: by default 260 g and 4000 mg a day for men, 200 g and 3000 mg for women (roughly US adult averages), overridden by habitual_carb_g and habitual_sodium_mg in subject.toml (Configuration). A diet that starts from ordinary eating starts with both compartments at their habitual fill, so $G_c(0) = G_n(0) = 0$, and the drop to diet intake drains them over the first days: the model predicts that drain instead of crediting it to tissue. For example, 80 g/day of carbohydrate against a habitual 260 g sends $G_c$ toward $k_G \Delta C \approx -0.54$ kg with a 1.5-day time constant. The earlier reference, each episode’s mean intake, hid that drop and changed as days were added; on the report’s logged record the habitual reference fits better. Called without a habitual level, the library still falls back to episode means.

The assembled model

Energy not spent goes into tissue, except energy diverted into glycogen, which is unavailable to fat. With state $x = (T, F, G_c, G_n, A)$ and daily input $u = (I, P_g, \Delta C, F_g, A_g, \Delta\mathrm{Na}, X)$:

\[ \begin{aligned} \rho_{\mathrm{eff}}\,\dot T &= I - E - \rho_G\,\dot G_c, & \dot F &= p\,\dot T, \\ \dot G_c &= (k_G\,\Delta C - G_c)/\tau_G, & \dot G_n &= (k_{Na}\,\Delta\mathrm{Na} - G_n)/\tau_{Na}, \\ \dot A &= \bigl(-\beta_A (E_{\mathrm{base}} - I) - A\bigr)/\tau_A . \end{aligned} \]

$\rho_G \approx 1025$ kcal/kg is the energy density of the glycogen-water complex: glycogen carries about 4.1 kcal/g but is stored with 3 g of water, so $4100/4 \approx 1025$. Using 4100 would overstate the cost of refeeding fourfold. Sodium water carries no energy term.

Parameters and priors

Nine parameters are estimated. Each prior is a transform of a standard-normal coordinate $\eta$ (reparameterization): “log, $s$” means $\theta = c\,e^{s\eta}$ ($s$ is the sd of $\ln\theta$), “logit, $s$” means $\theta = \sigma(\operatorname{logit} c + s\eta)$ with $\sigma$ the logistic function, and “normal, $s$” means $\theta = c + s\eta$.

Code name	Symbol	Meaning	Prior center $c$	Spread	Units	Source
`rmr_intercept`	$c_0$	Mass-independent expenditure, plus constant log bias	$1.45\,(6.25 \times \text{height} - 5 \times \text{age} + s)$	log, 0.15	kcal/day	Mifflin et al. (1990)
`rmr_slope`	$c_1$	Mass dependence of expenditure	14.5	log, 0.25	kcal/day per kg	Mifflin et al. (1990)
`at_gain`	$\beta_A$	Fraction of a sustained deficit offset by adaptation	0.15	logit, 0.70	dimensionless	Hall (2010)
`at_tau`	$\tau_A$	Adaptation time constant	14	log, 0.50	days	Order of magnitude
`gly_gain`	$k_G$	Glycogen-water gain	0.0030	log, 0.50	kg per g/day	Olsson and Saltin (1970)
`gly_tau`	$\tau_G$	Glycogen-water time constant	1.5	log, 0.40	days	Assumed fast lag
`na_gain`	$k_{Na}$	Sodium-water gain	$2 \times 10^{-4}$	log, 1.0	kg per mg/day	Extracellular fluid at 140 mmol/L
`na_tau`	$\tau_{Na}$	Sodium-water time constant	1.0	log, 0.40	days	Standard physiology
`partition_kappa`	$\kappa_P$	Fat-share shift per g/kg/day of protein above 1.6	0.05	normal, 0.05	per g/kg/day	Helms et al. (2014)

The thermic-effect coefficients, tissue energy densities and Forbes coefficient are fixed; identifiability explains why. The tape adds two estimated constants, $a_w$ and $b_w$ (below).

Unlogged days and timescales

On a day with no food log, $I$ is set to current total expenditure (no net imbalance), carbohydrate and sodium to habitual levels, and the process noise is inflated (process noise). People forget to log a party more often than a salad; the inflation widens the uncertainty but does not correct that bias.

Component	Timescale	Consequence
Water compartments	1 to 2 days	Fast scale swings with no tissue change
Adaptation	about 14 days	The plateau; separable from mass effects only because it lags
Tissue	months	The actual objective

If adaptation were instantaneous it would be perfectly confounded with an error in $c_0$. Because it lags, a change in intake produces a brief transient in which the two differ, and that transient is the whole basis for estimating $\beta_A$.

The observation model

The scale reads total mass plus noise on the morning of day $k$:

\[ y_k = H x_k + v_k, \qquad H = \begin{pmatrix} 1 & 0 & 1 & 1 & 0 \end{pmatrix}, \qquad v_k \sim \mathcal{N}(0, R). \]

The map is linear, so the scale update is exact. $R$ is not instrument error (tens of grams for a decent scale) but fluid and gut-content variation the water compartments miss; for consistent morning fasted weights its standard deviation is near 0.4 to 0.6 kg (prior 0.45 kg), and phase 1 estimates it from your history. Consistent conditions therefore matter more than accuracy: a constant scale offset is absorbed into the initial condition, while weighing at random times adds variance no filter can remove. The noise is Gaussian on evidence, with a gate against typing errors (noise model).

The waist tape

The scale sees fat only through $\rho_{\mathrm{eff}}$, which is weak evidence; a tape measure responds to fat directly. The optional waist_in or waist_cm column (Data) is modeled, in centimeters, as

\[ y^w_k = a_w + b_w\,F_k + v^w_k, \qquad v^w_k \sim \mathcal{N}(0, \sigma_w^2), \qquad \sigma_w = 1.5\ \text{cm}, \]

the agreement of repeated self-measurements. The offset $a_w$ and slope $b_w$ are personal constants, estimated like the metabolic parameters and added to the filter only when some episode has a waist reading.

The priors come from Lean, Han and Deurenberg (1996), who predict $\%\mathrm{BF} = 0.567\,y^w + 0.101\,\text{age} - 31.8$ for men and $0.439\,y^w + 0.221\,\text{age} - 9.4$ for women, with standard errors of 4.1 and 4.9 percentage points. With $k_w$ the waist coefficient and $\%\mathrm{BF} = 100F/W$, differentiating at fixed lean mass gives the prior slope

\[ b_{w,0} = \frac{100\,M}{k_w\,W^2}, \]

at the starting composition of the latest episode, with a log-scale prior sd of 0.2 (about $\pm 20\%$). A man of 100 kg with 30 kg of fat gets about 1.2 cm per kilogram of fat, consistent with the common observation of about a centimeter per kilogram lost. The prior offset puts the line through the Lean prediction at the starting fat mass, with sd equal to the standard error in centimeters, $4.1/0.567 \approx 7.2$ cm for men ($4.9/0.439 \approx 11.2$ cm for women). The Lean equation’s implied slope steepens as fat falls (about 10% per 5 kg lost), so the prior slope is a starting value that the estimated slope must track. The tape constrains only the product of $b_w$ and the fat share, so it anchors the fat level without pinning down the share (what the tape identifies).

How firm each piece is

Relationship	Source	Firmness
Energy balance	First law	Certain
Tissue energy densities	Hall (2008)	Strong
Forbes partitioning	Forbes (1987)	Moderate
Protein lean-sparing	Helms et al. (2014)	Weak
RMR from body measurements	Mifflin et al. (1990)	Prior only
Thermic-effect ordering; magnitudes	Westerterp (2004)	Strong; moderate, held fixed
Adaptive thermogenesis exists	Leibel et al. (1995)	Strong
Adaptation parameters	Hall (2010)	Weak
Glycogen-water coupling	Olsson and Saltin (1970)	Mechanism certain, ratio moderate
Sodium-water coupling	Standard physiology	Moderate
Habitual intake levels	US adult averages	Default only; set your own
Waist against fat mass	Lean et al. (1996)	Prior only; slope estimated

A wide posterior on a weakly founded parameter is the estimator correctly reporting that neither the literature nor your data has much to say.

References

Dhurandhar, N. V., Schoeller, D., Brown, A. W., Heymsfield, S. B., Thomas, D., Sørensen, T. I. A., Speakman, J. R., Jeansonne, M., and Allison, D. B. (2015). Energy balance measurement: when something is not better than nothing. International Journal of Obesity.
Forbes, G. B. (1987). Lean body mass-body fat interrelationships in humans. Nutrition Reviews.
Fothergill, E., Guo, J., Howard, L., Kerns, J. C., Knuth, N. D., Brychta, R., Chen, K. Y., Skarulis, M. C., Walter, M., Walter, P. J., and Hall, K. D. (2016). Persistent metabolic adaptation 6 years after “The Biggest Loser” competition. Obesity.
Hall, K. D. (2008). What is the required energy deficit per unit weight loss? International Journal of Obesity.
Hall, K. D. (2010). Predicting metabolic adaptation, body weight change, and energy intake in humans. American Journal of Physiology: Endocrinology and Metabolism.
Hall, K. D., Sacks, G., Chandramohan, D., Chow, C. C., Wang, Y. C., Gortmaker, S. L., and Swinburn, B. A. (2011). Quantification of the effect of energy imbalance on bodyweight. The Lancet.
Helms, E. R., Zinn, C., Rowlands, D. S., and Brown, S. R. (2014). A systematic review of dietary protein during caloric restriction in resistance trained lean athletes: a case for higher intakes. International Journal of Sport Nutrition and Exercise Metabolism.
Lean, M. E. J., Han, T. S., and Deurenberg, P. (1996). Predicting body composition by densitometry from simple anthropometric measurements. The American Journal of Clinical Nutrition, 63(1), 4-14.
Leibel, R. L., Rosenbaum, M., and Hirsch, J. (1995). Changes in energy expenditure resulting from altered body weight. New England Journal of Medicine, 332, 621-628.
Mifflin, M. D., St Jeor, S. T., Hill, L. A., Scott, B. J., Daugherty, S. A., and Koh, Y. O. (1990). A new predictive equation for resting energy expenditure in healthy individuals. American Journal of Clinical Nutrition, 51, 241-247.
Olsson, K.-E., and Saltin, B. (1970). Variation in total body water with muscle glycogen changes in man. Acta Physiologica Scandinavica.
Rosenbaum, M., and Leibel, R. L. (2010). Adaptive thermogenesis in humans. International Journal of Obesity.
Westerterp, K. R. (2004). Diet induced thermogenesis. Nutrition & Metabolism.