# Causal Model for Bike Riding Time

*Author: Danilo Lessa Bernardineli*

## Intro

## The 'magic' formula

The effective distance $\hat{D}$ is given by:

$$\hat{D} = x + \alpha u + \beta d$$

While the estimated duration $\hat{t}$ is given by:
$$\hat{t} = \frac{\hat{D}}{\hat{v_f}}$$

$\alpha$ and $\beta$ are unitless parameters to be found, while $\hat{v_f}$ is an estimate of the average speed when adopting a specific power strategy (non-corrected by the wind)

## Rules of thumb for finding the parameters

A priori, the model parameters can be found statistically by doing linear regression on a dataset of existing rides that show similiar profile, however, there are some rules of thumb which has show a good enough result when comparing empirically.

### $\hat{v}_f$

This parameter is associated with the mean speed on a flat profile. A good enough estimate is to use interactive calculators for computing the expected speed at 70% FTP (ref 1). CdA can be estimated by using frontal area photos with a relatively good accuracy (ref 2).

Testing multiple values of CdA, weights and power outputs can generate a range of possible flat speeds that can be used for providing uncertainty.

### $\alpha$

This is associated with the theoretical maximum VAM and some effiency measure in regards to using stateful energy for overcoming short climbs.

$$\alpha = \alpha_g (1 - \epsilon_g)$$

$$\alpha_g = \frac{v_f}{\gamma} \approx 40$$

$$\gamma = \frac{P}{mg} $$ (Unit: [P] / [kg] * [m * s^-2] -> [kg * m^2 * s^-3] * [kg^-1] * [m^-1 * s^2] -> [m * s^-1])

$$\epsilon_g \approx 0.5$$

### $\beta$

$\beta \approx \frac{-1}{120}$

## Correcting the formula for different conditions

### Constant wind speed

One empirical finding says that:

$\hat{v_f}^* \approx \hat{v_f} - \frac{2}{3} u$

### Different power strategy

Given the sensitivity of the estimated duration on the flat speed estimate and the non-linear dependences of marginal power into speed, one good estimate for the new speed $\hat{v_f}^*$ is given by:

$$\hat{v_f}^* = \hat{v_f} \sqrt{\frac{P_{new}}{P_{old}}}$$.

Supposing $P_{new} > P_{old}$, this correction tends to sub-estimate the estimate when the average speed is as slow such so that the dominating terms are the ones associated with linear dependencies (namely, rolling resistance and gravity). The opposite conclusion is made when the average speed is as fast such that the terms associated with aerodynamic drag dominates, although the scale of the error is less.

### Different terrain

It has been found empirically that a good descriptor of the effect of having mixed terrains is to change the mean power, therefore changing the estimate for $\hat{v}_f$, while also making the $\beta$ term more resistive.

## Update from statistical fit

v_f = 26.5 +- 1.37 -> (5.1% variance)
alpha = 0.0196 +- 0.004777 (positively skewed, 2.4% variance)
beta = -0.007348 +- 0.004495 -> (61.2% variance)
gamma = 0.52 +- 0.1205 (23.2% variance)

$\hat{D} \approx x + \frac{u}{50} - \frac{d}{140}$
$\hat{t} \approx \frac{\hat{D}}{26.5 - \frac{w}{2}}$

## References

ref 1: https://www.gribble.org/cycling/power_v_speed.html

ref 2: https://www.triradar.com/training-advice/how-to-calculate-your-drag/