Silhouette

The intensity-duration modelling toolkit for endurance sports. Scikit-learn compatible.

Try the interactive playground 🚀

Models

Parametric models

Model	Parameters
TwoParameterRegressor	CP, W'
ThreeParameterRegressor	CP, W', P_max
OmniDurationRegressor	CP, W', P_max, a, tcp_max

Data-driven models

Model	Parameters
FPCARegressor	FPC1 (gain), FPC2 (sprint/endurance bias), FPC3 (mid-duration)

Installation

uv add silhouette

Or with pip:

pip install silhouette

Quick start

Parametric models

import numpy as np
from silhouette import OmniDurationRegressor

durations = np.array([5, 10, 30, 60, 120, 300, 600, 1200, 1800, 3600])
power = np.array([1050, 850, 600, 480, 400, 340, 310, 290, 275, 255])

reg = OmniDurationRegressor()
reg.fit(durations.reshape(-1, 1), power)

reg.cp_       # critical power (W)
reg.p_max_    # peak power (W)
reg.w_prime_  # anaerobic work capacity (J)

reg.predict(np.array([[300]]))  # predicted power at 5 minutes

All parametric models share the same interface. Swap OmniDurationRegressor for TwoParameterRegressor or ThreeParameterRegressor and the code works the same way.

FPCA model

from silhouette import FPCARegressor

reg = FPCARegressor.from_model()
reg.fit(durations.reshape(-1, 1), power)

reg.fpc1_     # overall power level
reg.fpc2_     # sprint vs endurance bias
reg.fpc3_     # mid-duration specialization

reg.predict(np.array([[300]]))
reg.percentiles()  # {"fpc1": 72.3, "fpc2": 34.1, "fpc3": 55.8}
reg.z_scores()     # {"fpc1": 0.87, "fpc2": -0.41, "fpc3": 0.14}

Known parameters

When parameters are already known, use curve directly without fitting:

from silhouette import TwoParameterRegressor, FPCARegressor

t = np.arange(1, 3601)
power = TwoParameterRegressor.curve(t, cp=250, w_prime=20_000)
power = FPCARegressor.curve(t, fpc1=0.5, fpc2=-0.1, fpc3=0.0)

Custom bounds

reg = OmniDurationRegressor(
    bounds={"cp": (200, 400), "p_max": (800, 1500)},
    initial_params={"cp": 280},
)

Fitting methods

TwoParameterRegressor supports an alternative fitting method that minimizes error in work space instead of power space:

reg = TwoParameterRegressor(fitting="work_duration")
reg.fit(X, power)

This linearizes the model to W = W' + CP·t and fits via OLS, giving more weight to longer durations. The default (fitting="nonlinear") minimizes error in power space.

Time to exhaustion

The inverse of the power-duration curve: given a power, how long can it be sustained?

# On a fitted model
tte = reg.predict_inverse(np.array([250, 300, 350]))

# With known parameters
tte = TwoParameterRegressor.curve_inverse(350, cp=250, w_prime=20_000)

Plotting

Install with plotting support:

uv add silhouette[plotting]

Plot data with fitted models (sklearn Display pattern):

from silhouette.plotting import PowerDurationDisplay

# Single model
display = PowerDurationDisplay.from_estimator(reg, durations.reshape(-1, 1), power)

# Compare models
display = PowerDurationDisplay.from_estimators(
    [reg_2p, reg_omni], durations.reshape(-1, 1), power,
)

FPCA mode of variance:

from silhouette.plotting import ModeOfVarianceDisplay

display = ModeOfVarianceDisplay.from_estimator(fpca_reg)

References

• Monod, H., & Scherrer, J. (1965). The work capacity of a synergic muscular group. Ergonomics, 8(3), 329-338.
• Morton, R. H. (1996). A 3-parameter critical power model. Ergonomics, 39(4), 611-619.
• Puchowicz, M. J., Baker, J., & Clarke, D. C. (2020). Development and field validation of an omni-domain power-duration model. Journal of Sports Sciences, 38(7), 801-813.
• Puchowicz, M. J., & Skiba, P. F. (2025). Functional Data Analysis of the Power-Duration Relationship in Cyclists. International Journal of Sports Physiology and Performance, 1(aop), 1-10.