## Lap Time - DR Correlations

Beta version

#### Objective

1. Compare car performance with reduced effects of driver
2. Estimate standard lap times by DR

#### Summary

A new "Convergent Learning Curve" is a suitable model for representing the correlation between lap time and driver rating points (Fig.1). With this model, we can estimate approximate curves for each car and lap times for each DR points. If we compare the lap times of some cars on the same DR points, we can reduce the effects of driver skill on car performance. Also, the approximate curve is considered the standard lap time for each DR, so it is a statistically calculated optimal target time for players.

Fig.1 Correlations btween Lap Times and DR
 Scatter plots : Fastest lap times of players Line plots : Approximate curves for each car
###### Data Conditions of Plots
 Event : GTWS 2022 Manufacturers Cup Season 2 Rd.8 Category : Gr.3 Track : Interlagos Session : Free Practice Time Trial Conditions : Afternoon, Sunny, No wind
 BoP : On (Mid-speed) Settings : Specified Tyre : Racing: Hard Others : No Tyre Wear, No Fuel Consumption App Ver. : 1.21 and 1.23

At Manufacturers Rd.8 Interlagos, MR cars like NSX, 911, 4C and 650S were fast in the high DR band. On the other hand, the correlation curves of FR cars are close to the one of the whole cars, and there were fewer differences between each car.
It's worth noting that 911 outperforms other models in the low and middle DR bands. 911 is more stable and has a more understeer trend than other MR cars. So, it can be inferred that a stable car tends to be easier for beginners and middle-level players to set times.

#### Approximate Model

I introduced a new modified learning curve with convergency (Eq.1) as the approximate model function for curve fitting.

\begin{eqnarray} t &=& a e^{-bx} x^{c} + d \\ \end{eqnarray}

where:

 $t$ [s] : Lap time $x$ [-] : DR points $a, b, c, d$ : Prameters

When using a basic learning curve $t = ax^c + d \ \ (c<0)$ as the model function, there were some errors in the high DR points area. The basic learning curve is a power function and decremental but generally not convergent. On the other hand, limitations of car and tyre performance restrict lap time and converge its improvement. I thought the presence or absence of convergence was a cause of the errors. And introducing the exponential decay term $e^{-bx}$ as in Eq.1 reduced the errors (Fig.2). As a result, this new model improves the accuracy of lap time predictions.

Fig.2 Curve fitting improvement with new approximate model
(e.g., NSX)