# Trendline estimation

Let's say you had a scatter plot with lap times on the y axis and tyre pressure on the x-axis from an experiment where your dependent variable is lap time and your independent variable is tyre pressure. If you fitted a quadratic trendline to that data with an R squared value of between 0.6 - 0.7, could you find the minimum of the quadratic to estimate the optimum tyre pressure that leads to the fastest lap?

• Use derivative calculus on the fitted equation of the trendline.
– r13
Mar 21, 2022 at 19:50

The value $$R^2$$ is a regression metric that is best used to compare how well different data sets fit to (or scatter about) a fixed (linear) regression model. You are better to obtain the uncertainty factors for your regression coefficients and use them to determine the uncertainty in your regression extrapolation.
By example, supposed you state that lap time $$t$$ (s) versus pressure $$p$$ (Pa) should fit $$t = k(p - po)^2 + to$$. Use a regression method that does not linearize the function but instead fits it directly to return $$k \pm \Delta k$$, $$po + \Delta po$$, and $$to \pm \Delta to$$. You will have your predicted best time $$to$$ at the proposed optimal pressure $$po$$.
Is there some physical basis for the assumption that the system actually, truly follows a quadratic behavior? If not, then your quadratic trendline may be completely bogus. What if the behavior is actually $$x^{2.2}$$ or $$x^{1.8}$$ or some other function entirely?