I wasn't going to write more posts about this subject but I've got some results to share. If you remember my previous post about yaw, I calculated the relation between the standard deviation of the Maxwellian distribution and the average of the Rayleigh distribution for wind speeds. One of the main problems of the Rayleigh distribution is that there is only one tuning parameter. That means that average value and variability can't be chosen indepently or, alternatively, that a high average wind speed means high variability.
For that reason, I have tried to modify Dan Connelly's yaw model for a more evolved wind speed model. The Weibull distribution is a two-parameter distribution that is used to model wind speed in the energy industry but its complexity means that no analytical expression can be obtained for the yaw probability model for the fixed bike speed case. If we consider that bike speed is variable, it becomes too much computing intensive and impractical.
As I wanted to take into account variable bike speeds, the Rayleigh model was the only option for wind speeds. Assuming a Weibull distribution of bike speeds, the yaw probability function for variable bike and wind speed and bike and wind direction is:
|κ, shape factor of the Weibull distribution. λ, scale parameter of the Weibull distribution. va, average wind speed|
The main problem of this distribution is that it isn't normalized and, as no analytical solution exists, the normalization constant can't be computed. The normalization constant could be calculated numerically but this would be too computing intensive so I have used an alternative method. If you calculate the probability for a given yaw, the real probability would be that value divided by the normalization constant. So, if you calculate the probability for different yaws and you fit that data to a null-average gaussian distribution (as Mavic's data) multiplied by that unkown constant, the standard deviation and the normalization constant can be determined.
An example. Weibull distribution for bike speeds:
More soon. Thanks for reading!