Quant Library · The athlete · Athletics
Keller Optimal Pacing
The fastest way to spend a finite engine over a fixed distance.
Keller's two-equation runner, solved for the policy that minimises time: full force to a cruise, hold what the store can afford, arrive empty.
Inputs
3What it takes
| Name | Type | Units | Where it comes from |
|---|---|---|---|
| distance | float | m | - |
| F, tau, sigma, E0 optional | floats | m/s², s, W/kg, J/kg | the literature, or fitted to your athlete's own races |
| sigma_scale optional | float | - | altitude, heat, shoes and drafting all enter here |
Outputs
3What it gives back
| Name | Type | Units | Notes |
|---|---|---|---|
| time | float | s | |
| cruise speed | float | m/s | |
| velocity and energy profile | arrays | m/s, J/kg |
Method
How it works
- Integrate dv/dt = f − v/τ and dE/dt = σ − f v with f bounded by F and E bounded below by zero.
- 2 further steps in the licensed specification
Assumptions
- A single aerobic supply rate and a single anaerobic store, both constant through the race.
Limits
- It is a model of an even, optimal effort. It does not model tactics, packs, or the last-lap kick that decides championship races.
- 1 further limits in the licensed specification
In the package
- The full specification: inputs, method, assumptions and limits
- The validation panel as measured on the day of purchase
- A reference implementation extracted from the running source
- Perpetual commercial use for one entity
Reference modules:
- 2 module(s), named in the licensed specification and shipped in the package
Read this before you buy. The platform's own walk-forward tests say the closing line forecasts football better than these models do. What they give you is a coherent price for every market and match, before the market opens, with the band and the working attached. Any model here that has been beaten by something simpler says so in its limits, and the validation panel above is generated from the platform's result files rather than typed in.